WEBVTT

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Any<00:00:00.400><c> shape</c><00:00:00.800><c> you</c><00:00:01.040><c> can</c><00:00:01.280><c> draw</c><00:00:01.760><c> with</c><00:00:02.000><c> a</c><00:00:02.320><c> pen</c><00:00:02.560><c> on</c><00:00:02.800><c> a</c>

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Any shape you can draw with a pen on a
 

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Any shape you can draw with a pen on a
paper<00:00:03.919><c> can</c><00:00:04.400><c> be</c><00:00:04.640><c> built</c><00:00:05.279><c> by</c><00:00:05.600><c> adding</c><00:00:05.920><c> up</c><00:00:06.080><c> a</c><00:00:06.400><c> lot</c><00:00:06.480><c> of</c>

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paper can be built by adding up a lot of
 

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paper can be built by adding up a lot of
sine<00:00:06.960><c> waves.</c><00:00:07.680><c> Literally</c><00:00:08.559><c> any</c><00:00:08.960><c> shape</c><00:00:09.440><c> at</c><00:00:09.679><c> all.</c>

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sine waves. Literally any shape at all.
 

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sine waves. Literally any shape at all.
This<00:00:10.639><c> is</c><00:00:10.800><c> the</c><00:00:11.040><c> idea</c><00:00:11.360><c> behind</c><00:00:11.920><c> Foryear</c><00:00:12.559><c> series</c>

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This is the idea behind Foryear series
 

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This is the idea behind Foryear series
and<00:00:13.920><c> Forier</c><00:00:14.559><c> transforms</c><00:00:15.519><c> named</c><00:00:15.920><c> after</c><00:00:16.480><c> Joseph</c>

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and Forier transforms named after Joseph
 

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and Forier transforms named after Joseph
Forier.<00:00:17.920><c> And</c><00:00:18.160><c> without</c><00:00:18.480><c> it,</c><00:00:18.800><c> you</c><00:00:19.039><c> won't</c><00:00:19.199><c> be</c>

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Forier. And without it, you won't be
 

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Forier. And without it, you won't be
able<00:00:19.439><c> to</c><00:00:19.600><c> watch</c><00:00:19.840><c> this</c><00:00:20.160><c> video</c><00:00:20.560><c> or</c><00:00:20.880><c> send</c><00:00:21.119><c> a</c><00:00:21.359><c> voice</c>

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able to watch this video or send a voice
 

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able to watch this video or send a voice
message<00:00:22.160><c> or</c><00:00:22.480><c> even</c><00:00:22.720><c> take</c><00:00:23.039><c> pictures</c><00:00:23.439><c> on</c><00:00:23.680><c> your</c>

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message or even take pictures on your
 

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message or even take pictures on your
phone.<00:00:24.640><c> Not</c><00:00:24.880><c> only</c><00:00:25.039><c> does</c><00:00:25.279><c> this</c><00:00:25.439><c> idea</c><00:00:25.840><c> power</c>

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phone. Not only does this idea power
 

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phone. Not only does this idea power
Wi-Fi,<00:00:27.039><c> radar,</c><00:00:27.760><c> MRI,</c><00:00:28.720><c> you</c><00:00:28.960><c> name</c><00:00:29.119><c> it,</c><00:00:29.599><c> but</c>

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Wi-Fi, radar, MRI, you name it, but
 

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Wi-Fi, radar, MRI, you name it, but
about<00:00:30.000><c> a</c><00:00:30.240><c> century</c><00:00:30.560><c> ago,</c><00:00:30.960><c> it</c><00:00:31.199><c> helped</c><00:00:31.599><c> power</c><00:00:32.079><c> the</c>

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about a century ago, it helped power the
 

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about a century ago, it helped power the
British<00:00:32.719><c> Empire's</c><00:00:33.440><c> global</c><00:00:33.920><c> sea</c><00:00:34.239><c> trade.</c><00:00:34.800><c> And</c>

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British Empire's global sea trade. And
 

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British Empire's global sea trade. And
decades<00:00:35.440><c> later,</c><00:00:35.920><c> it</c><00:00:36.079><c> decided</c><00:00:36.480><c> the</c><00:00:36.719><c> date</c><00:00:37.040><c> of</c>

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decades later, it decided the date of
 

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decades later, it decided the date of
the<00:00:37.520><c> largest</c><00:00:37.920><c> seabbone</c><00:00:38.399><c> invasion</c><00:00:39.200><c> in</c><00:00:39.520><c> human</c>

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the largest seabbone invasion in human
 

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the largest seabbone invasion in human
history.<00:00:41.520><c> But</c><00:00:41.760><c> why</c><00:00:41.920><c> is</c><00:00:42.160><c> this</c><00:00:42.399><c> true</c><00:00:42.559><c> in</c><00:00:42.800><c> the</c>

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history. But why is this true in the
 

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history. But why is this true in the
first<00:00:43.120><c> place?</c><00:00:43.680><c> And</c><00:00:43.920><c> how</c><00:00:44.160><c> does</c><00:00:44.399><c> that</c><00:00:44.879><c> help</c><00:00:45.200><c> my</c>

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first place? And how does that help my
 

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first place? And how does that help my
phone<00:00:45.760><c> take</c><00:00:46.079><c> pictures?</c><00:00:47.039><c> And</c><00:00:47.360><c> where</c><00:00:47.680><c> did</c><00:00:47.840><c> it</c>

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phone take pictures? And where did it
 

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phone take pictures? And where did it
all<00:00:48.239><c> come</c><00:00:48.399><c> from?</c><00:00:49.039><c> I</c><00:00:49.360><c> never</c><00:00:49.600><c> thought</c><00:00:49.840><c> I'll</c><00:00:50.079><c> get</c>

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all come from? I never thought I'll get
 

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all come from? I never thought I'll get
any<00:00:50.399><c> intuition</c><00:00:50.879><c> behind</c><00:00:51.200><c> this</c><00:00:51.680><c> until</c><00:00:52.239><c> now.</c>

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any intuition behind this until now.
 

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any intuition behind this until now.
Forier<00:00:53.520><c> says</c><00:00:53.920><c> all</c><00:00:54.239><c> of</c><00:00:54.399><c> this</c><00:00:54.960><c> every</c><00:00:55.360><c> single</c>

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Forier says all of this every single
 

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Forier says all of this every single
aspect<00:00:56.000><c> of</c><00:00:56.160><c> it</c><00:00:56.480><c> including</c><00:00:56.960><c> all</c><00:00:57.199><c> the</c><00:00:57.360><c> math</c><00:00:58.160><c> can</c>

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aspect of it including all the math can
 

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aspect of it including all the math can
be<00:00:58.640><c> rediscovered</c><00:00:59.280><c> from</c><00:00:59.600><c> just</c><00:00:59.920><c> one</c><00:01:00.399><c> fact</c><00:01:01.120><c> that</c>

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be rediscovered from just one fact that
 

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be rediscovered from just one fact that
heat<00:01:01.840><c> flows</c><00:01:02.640><c> from</c><00:01:03.039><c> hot</c><00:01:03.359><c> region</c><00:01:03.920><c> to</c><00:01:04.159><c> a</c><00:01:04.400><c> cold</c>

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heat flows from hot region to a cold
 

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heat flows from hot region to a cold
region.<00:01:05.439><c> That's</c><00:01:05.680><c> it.</c><00:01:06.400><c> Nothing</c><00:01:06.799><c> else.</c>

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region. That's it. Nothing else.
 

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region. That's it. Nothing else.
Wait,<00:01:08.960><c> what?</c><00:01:09.680><c> What?</c><00:01:10.400><c> How</c><00:01:10.640><c> does</c><00:01:11.040><c> this</c><00:01:11.680><c> lead</c><00:01:12.000><c> to</c>

01:12.950 --> 01:12.960 align:start position:0%
Wait, what? What? How does this lead to
 

01:12.960 --> 01:16.630 align:start position:0%
Wait, what? What? How does this lead to
what?<00:01:14.240><c> For</c><00:01:14.720><c> says,</c><00:01:15.119><c> Mahesh,</c><00:01:15.920><c> here's</c><00:01:16.240><c> a</c><00:01:16.400><c> hot</c>

01:16.630 --> 01:16.640 align:start position:0%
what? For says, Mahesh, here's a hot
 

01:16.640 --> 01:18.710 align:start position:0%
what? For says, Mahesh, here's a hot
piece<00:01:16.880><c> of</c><00:01:17.040><c> metal.</c><00:01:17.439><c> It's</c><00:01:17.759><c> touching</c><00:01:18.000><c> a</c><00:01:18.240><c> colder</c>

01:18.710 --> 01:18.720 align:start position:0%
piece of metal. It's touching a colder
 

01:18.720 --> 01:21.910 align:start position:0%
piece of metal. It's touching a colder
one.<00:01:19.439><c> What</c><00:01:19.680><c> happens</c><00:01:20.080><c> to</c><00:01:20.320><c> its</c><00:01:20.720><c> temperature?</c>

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one. What happens to its temperature?
 

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one. What happens to its temperature?
Well,<00:01:22.320><c> heat</c><00:01:22.640><c> flows</c><00:01:22.960><c> from</c><00:01:23.200><c> hot</c><00:01:23.520><c> to</c><00:01:23.759><c> cold,</c><00:01:24.240><c> so</c>

01:24.390 --> 01:24.400 align:start position:0%
Well, heat flows from hot to cold, so
 

01:24.400 --> 01:26.469 align:start position:0%
Well, heat flows from hot to cold, so
its<00:01:24.720><c> temperature</c><00:01:25.040><c> is</c><00:01:25.280><c> going</c><00:01:25.360><c> to</c><00:01:25.600><c> reduce.</c>

01:26.469 --> 01:26.479 align:start position:0%
its temperature is going to reduce.
 

01:26.479 --> 01:30.870 align:start position:0%
its temperature is going to reduce.
Obviously,<00:01:27.759><c> for</c><00:01:28.159><c> says</c><00:01:28.560><c> wrong.</c><00:01:29.600><c> Why?</c><00:01:30.560><c> Because</c>

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Obviously, for says wrong. Why? Because
 

01:30.880 --> 01:33.109 align:start position:0%
Obviously, for says wrong. Why? Because
it's<00:01:31.280><c> also</c><00:01:31.520><c> touching</c><00:01:31.920><c> a</c><00:01:32.159><c> hotter</c><00:01:32.560><c> piece</c><00:01:32.799><c> on</c><00:01:32.960><c> the</c>

01:33.109 --> 01:33.119 align:start position:0%
it's also touching a hotter piece on the
 

01:33.119 --> 01:34.950 align:start position:0%
it's also touching a hotter piece on the
right<00:01:33.280><c> with</c><00:01:33.520><c> the</c><00:01:33.759><c> exact</c><00:01:34.159><c> same</c><00:01:34.479><c> temperature</c>

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right with the exact same temperature
 

01:34.960 --> 01:36.870 align:start position:0%
right with the exact same temperature
difference.<00:01:35.600><c> So,</c><00:01:35.759><c> the</c><00:01:36.000><c> same</c><00:01:36.159><c> amount</c><00:01:36.400><c> of</c><00:01:36.560><c> heat</c>

01:36.870 --> 01:36.880 align:start position:0%
difference. So, the same amount of heat
 

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difference. So, the same amount of heat
flows<00:01:37.200><c> back</c><00:01:37.520><c> in,</c><00:01:37.920><c> making</c><00:01:38.159><c> the</c><00:01:38.400><c> net</c><00:01:38.799><c> heat</c><00:01:39.119><c> flow</c>

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flows back in, making the net heat flow
 

01:39.520 --> 01:41.749 align:start position:0%
flows back in, making the net heat flow
zero.<00:01:40.240><c> So,</c><00:01:40.400><c> our</c><00:01:40.720><c> temperature</c><00:01:41.200><c> remains</c>

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zero. So, our temperature remains
 

01:41.759 --> 01:44.710 align:start position:0%
zero. So, our temperature remains
unchanged.<00:01:42.799><c> For</c><00:01:43.520><c> you</c><00:01:43.840><c> tricked</c><00:01:44.159><c> me.</c><00:01:44.479><c> What's</c>

01:44.710 --> 01:44.720 align:start position:0%
unchanged. For you tricked me. What's
 

01:44.720 --> 01:47.910 align:start position:0%
unchanged. For you tricked me. What's
your<00:01:45.040><c> point?</c><00:01:45.840><c> The</c><00:01:46.079><c> point</c><00:01:46.320><c> is</c><00:01:47.040><c> if</c><00:01:47.280><c> the</c><00:01:47.680><c> left</c>

01:47.910 --> 01:47.920 align:start position:0%
your point? The point is if the left
 

01:47.920 --> 01:51.270 align:start position:0%
your point? The point is if the left
piece<00:01:48.320><c> was</c><00:01:48.560><c> only</c><00:01:49.119><c> slightly</c><00:01:49.759><c> colder</c><00:01:50.640><c> then</c><00:01:50.960><c> look</c>

01:51.270 --> 01:51.280 align:start position:0%
piece was only slightly colder then look
 

01:51.280 --> 01:53.109 align:start position:0%
piece was only slightly colder then look
the<00:01:51.520><c> amount</c><00:01:51.680><c> of</c><00:01:51.840><c> heat</c><00:01:52.079><c> flowing</c><00:01:52.479><c> out</c><00:01:52.720><c> would</c><00:01:52.960><c> be</c>

01:53.109 --> 01:53.119 align:start position:0%
the amount of heat flowing out would be
 

01:53.119 --> 01:55.830 align:start position:0%
the amount of heat flowing out would be
lesser<00:01:53.920><c> and</c><00:01:54.240><c> now</c><00:01:54.560><c> the</c><00:01:54.880><c> temperature</c><00:01:55.280><c> of</c><00:01:55.600><c> our</c>

01:55.830 --> 01:55.840 align:start position:0%
lesser and now the temperature of our
 

01:55.840 --> 01:59.749 align:start position:0%
lesser and now the temperature of our
piece<00:01:56.240><c> will</c><00:01:56.880><c> increase.</c><00:01:57.920><c> Makes</c><00:01:58.240><c> sense.</c><00:01:59.200><c> And</c><00:01:59.439><c> if</c>

01:59.749 --> 01:59.759 align:start position:0%
piece will increase. Makes sense. And if
 

01:59.759 --> 02:03.030 align:start position:0%
piece will increase. Makes sense. And if
this<00:02:00.079><c> hot</c><00:02:00.320><c> piece</c><00:02:00.640><c> was</c><00:02:00.960><c> even</c><00:02:01.280><c> hotter</c><00:02:02.159><c> then</c><00:02:02.719><c> the</c>

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this hot piece was even hotter then the
 

02:03.040 --> 02:06.389 align:start position:0%
this hot piece was even hotter then the
temperature<00:02:03.600><c> is</c><00:02:03.920><c> going</c><00:02:04.079><c> to</c><00:02:04.479><c> increase</c><00:02:05.200><c> faster.</c>

02:06.389 --> 02:06.399 align:start position:0%
temperature is going to increase faster.
 

02:06.399 --> 02:08.790 align:start position:0%
temperature is going to increase faster.
Yeah,<00:02:06.640><c> that</c><00:02:06.960><c> also</c><00:02:07.360><c> makes</c><00:02:07.600><c> sense.</c><00:02:08.319><c> So</c><00:02:08.479><c> the</c>

02:08.790 --> 02:08.800 align:start position:0%
Yeah, that also makes sense. So the
 

02:08.800 --> 02:12.550 align:start position:0%
Yeah, that also makes sense. So the
question<00:02:09.039><c> is</c><00:02:09.759><c> what</c><00:02:10.160><c> decides</c><00:02:11.280><c> how</c><00:02:11.680><c> quickly</c>

02:12.550 --> 02:12.560 align:start position:0%
question is what decides how quickly
 

02:12.560 --> 02:14.309 align:start position:0%
question is what decides how quickly
does<00:02:12.800><c> the</c><00:02:13.040><c> temperature</c><00:02:13.440><c> of</c><00:02:13.760><c> our</c><00:02:13.920><c> piece</c>

02:14.309 --> 02:14.319 align:start position:0%
does the temperature of our piece
 

02:14.319 --> 02:16.470 align:start position:0%
does the temperature of our piece
change?<00:02:14.800><c> What</c><00:02:15.120><c> decides</c><00:02:15.599><c> the</c><00:02:15.840><c> rate</c><00:02:16.080><c> of</c><00:02:16.319><c> change</c>

02:16.470 --> 02:16.480 align:start position:0%
change? What decides the rate of change
 

02:16.480 --> 02:20.390 align:start position:0%
change? What decides the rate of change
of<00:02:16.800><c> temperature?</c>

02:20.390 --> 02:20.400 align:start position:0%
 
 

02:20.400 --> 02:23.350 align:start position:0%
 
Well,<00:02:21.280><c> I</c><00:02:22.000><c> I</c><00:02:22.400><c> know</c><00:02:22.560><c> that</c><00:02:22.640><c> it's</c><00:02:22.879><c> got</c><00:02:23.120><c> something</c>

02:23.350 --> 02:23.360 align:start position:0%
Well, I I know that it's got something
 

02:23.360 --> 02:24.790 align:start position:0%
Well, I I know that it's got something
to<00:02:23.440><c> do</c><00:02:23.599><c> with</c><00:02:23.680><c> the</c><00:02:24.239><c> difference</c><00:02:24.560><c> in</c>

02:24.790 --> 02:24.800 align:start position:0%
to do with the difference in
 

02:24.800 --> 02:26.550 align:start position:0%
to do with the difference in
temperature,<00:02:25.200><c> but</c><00:02:25.360><c> I</c><00:02:25.520><c> can't</c><00:02:25.680><c> put</c><00:02:25.840><c> my</c><00:02:26.239><c> finger</c>

02:26.550 --> 02:26.560 align:start position:0%
temperature, but I can't put my finger
 

02:26.560 --> 02:29.750 align:start position:0%
temperature, but I can't put my finger
on<00:02:26.720><c> it</c><00:02:27.040><c> exactly.</c><00:02:27.920><c> For</c><00:02:28.400><c> says,</c><00:02:28.720><c> no</c><00:02:28.959><c> worries.</c>

02:29.750 --> 02:29.760 align:start position:0%
on it exactly. For says, no worries.
 

02:29.760 --> 02:33.030 align:start position:0%
on it exactly. For says, no worries.
Think<00:02:30.480><c> geometrically.</c><00:02:31.840><c> So,</c><00:02:32.400><c> let's</c><00:02:32.720><c> go</c><00:02:32.879><c> back</c>

02:33.030 --> 02:33.040 align:start position:0%
Think geometrically. So, let's go back
 

02:33.040 --> 02:34.869 align:start position:0%
Think geometrically. So, let's go back
to<00:02:33.200><c> our</c><00:02:33.440><c> first</c><00:02:33.599><c> case</c><00:02:33.920><c> and</c><00:02:34.239><c> draw</c><00:02:34.480><c> the</c>

02:34.869 --> 02:34.879 align:start position:0%
to our first case and draw the
 

02:34.879 --> 02:36.550 align:start position:0%
to our first case and draw the
temperature<00:02:35.280><c> position</c><00:02:35.760><c> graph.</c><00:02:36.239><c> Remember,</c>

02:36.550 --> 02:36.560 align:start position:0%
temperature position graph. Remember,
 

02:36.560 --> 02:38.869 align:start position:0%
temperature position graph. Remember,
the<00:02:36.800><c> temperature</c><00:02:37.200><c> was</c><00:02:37.440><c> unchanged</c><00:02:38.000><c> over</c><00:02:38.239><c> here.</c>

02:38.869 --> 02:38.879 align:start position:0%
the temperature was unchanged over here.
 

02:38.879 --> 02:40.869 align:start position:0%
the temperature was unchanged over here.
What<00:02:39.120><c> does</c><00:02:39.280><c> the</c><00:02:39.599><c> graph</c><00:02:39.920><c> look</c><00:02:40.160><c> like?</c><00:02:40.640><c> Well,</c>

02:40.869 --> 02:40.879 align:start position:0%
What does the graph look like? Well,
 

02:40.879 --> 02:42.309 align:start position:0%
What does the graph look like? Well,
since<00:02:41.120><c> the</c><00:02:41.280><c> temperature</c><00:02:41.599><c> difference</c><00:02:41.920><c> is</c><00:02:42.160><c> the</c>

02:42.309 --> 02:42.319 align:start position:0%
since the temperature difference is the
 

02:42.319 --> 02:43.990 align:start position:0%
since the temperature difference is the
same,<00:02:42.720><c> the</c><00:02:42.959><c> graph</c><00:02:43.200><c> is</c><00:02:43.440><c> going</c><00:02:43.519><c> to</c><00:02:43.680><c> be</c><00:02:43.760><c> a</c>

02:43.990 --> 02:44.000 align:start position:0%
same, the graph is going to be a
 

02:44.000 --> 02:46.949 align:start position:0%
same, the graph is going to be a
straight<00:02:44.319><c> line.</c><00:02:45.120><c> In</c><00:02:45.360><c> other</c><00:02:45.519><c> words,</c><00:02:46.080><c> the</c><00:02:46.560><c> slope</c>

02:46.949 --> 02:46.959 align:start position:0%
straight line. In other words, the slope
 

02:46.959 --> 02:50.150 align:start position:0%
straight line. In other words, the slope
of<00:02:47.200><c> that</c><00:02:47.519><c> graph</c><00:02:48.319><c> is</c><00:02:48.720><c> unchanged</c><00:02:49.440><c> as</c><00:02:49.680><c> I</c><00:02:49.840><c> go</c><00:02:50.000><c> from</c>

02:50.150 --> 02:50.160 align:start position:0%
of that graph is unchanged as I go from
 

02:50.160 --> 02:51.990 align:start position:0%
of that graph is unchanged as I go from
here<00:02:50.319><c> to</c><00:02:50.560><c> here.</c><00:02:51.120><c> And</c><00:02:51.280><c> when</c><00:02:51.440><c> the</c><00:02:51.599><c> slope</c><00:02:51.840><c> is</c>

02:51.990 --> 02:52.000 align:start position:0%
here to here. And when the slope is
 

02:52.000 --> 02:54.550 align:start position:0%
here to here. And when the slope is
unchanged,<00:02:52.720><c> the</c><00:02:53.120><c> temperature</c><00:02:53.599><c> is</c><00:02:53.840><c> unchanged.</c>

02:54.550 --> 02:54.560 align:start position:0%
unchanged, the temperature is unchanged.
 

02:54.560 --> 02:57.509 align:start position:0%
unchanged, the temperature is unchanged.
Ooh,<00:02:55.200><c> is</c><00:02:55.440><c> that</c><00:02:55.760><c> is</c><00:02:56.000><c> that</c><00:02:56.160><c> a</c><00:02:56.319><c> coincidence?</c>

02:57.509 --> 02:57.519 align:start position:0%
Ooh, is that is that a coincidence?
 

02:57.519 --> 02:59.589 align:start position:0%
Ooh, is that is that a coincidence?
Let's<00:02:57.840><c> see.</c><00:02:58.480><c> In</c><00:02:58.640><c> the</c><00:02:58.800><c> second</c><00:02:59.040><c> case,</c><00:02:59.440><c> this</c>

02:59.589 --> 02:59.599 align:start position:0%
Let's see. In the second case, this
 

02:59.599 --> 03:02.390 align:start position:0%
Let's see. In the second case, this
piece<00:02:59.920><c> was</c><00:03:00.160><c> only</c><00:03:00.560><c> slightly</c><00:03:01.120><c> colder.</c><00:03:02.000><c> And</c><00:03:02.159><c> so</c>

03:02.390 --> 03:02.400 align:start position:0%
piece was only slightly colder. And so
 

03:02.400 --> 03:04.390 align:start position:0%
piece was only slightly colder. And so
now,<00:03:02.640><c> as</c><00:03:02.800><c> I</c><00:03:03.040><c> move</c><00:03:03.200><c> from</c><00:03:03.360><c> here</c><00:03:03.599><c> to</c><00:03:03.760><c> here,</c><00:03:04.080><c> the</c>

03:04.390 --> 03:04.400 align:start position:0%
now, as I move from here to here, the
 

03:04.400 --> 03:07.350 align:start position:0%
now, as I move from here to here, the
the<00:03:04.720><c> slope</c><00:03:05.200><c> increases</c><00:03:06.239><c> and</c><00:03:06.560><c> the</c><00:03:06.879><c> temperature</c>

03:07.350 --> 03:07.360 align:start position:0%
the slope increases and the temperature
 

03:07.360 --> 03:09.910 align:start position:0%
the slope increases and the temperature
also<00:03:07.760><c> increases.</c><00:03:08.800><c> And</c><00:03:09.040><c> in</c><00:03:09.200><c> the</c><00:03:09.360><c> third</c><00:03:09.599><c> case,</c>

03:09.910 --> 03:09.920 align:start position:0%
also increases. And in the third case,
 

03:09.920 --> 03:12.710 align:start position:0%
also increases. And in the third case,
this<00:03:10.159><c> piece</c><00:03:10.480><c> was</c><00:03:10.879><c> even</c><00:03:11.360><c> harder,</c><00:03:12.159><c> which</c><00:03:12.480><c> means</c>

03:12.710 --> 03:12.720 align:start position:0%
this piece was even harder, which means
 

03:12.720 --> 03:15.750 align:start position:0%
this piece was even harder, which means
the<00:03:13.200><c> slope</c><00:03:13.599><c> increases</c><00:03:14.239><c> faster</c><00:03:15.040><c> as</c><00:03:15.280><c> I</c><00:03:15.440><c> go</c><00:03:15.599><c> from</c>

03:15.750 --> 03:15.760 align:start position:0%
the slope increases faster as I go from
 

03:15.760 --> 03:17.509 align:start position:0%
the slope increases faster as I go from
here<00:03:16.000><c> to</c><00:03:16.159><c> here.</c><00:03:16.400><c> And</c><00:03:16.560><c> the</c><00:03:16.800><c> temperature</c><00:03:17.200><c> also</c>

03:17.509 --> 03:17.519 align:start position:0%
here to here. And the temperature also
 

03:17.519 --> 03:19.750 align:start position:0%
here to here. And the temperature also
increases<00:03:18.159><c> faster.</c><00:03:18.640><c> So</c><00:03:18.800><c> there</c><00:03:19.040><c> is</c><00:03:19.200><c> a</c><00:03:19.440><c> nice</c>

03:19.750 --> 03:19.760 align:start position:0%
increases faster. So there is a nice
 

03:19.760 --> 03:22.630 align:start position:0%
increases faster. So there is a nice
correlation<00:03:20.400><c> between</c><00:03:20.720><c> this.</c><00:03:21.440><c> We</c><00:03:21.680><c> see</c><00:03:22.319><c> the</c>

03:22.630 --> 03:22.640 align:start position:0%
correlation between this. We see the
 

03:22.640 --> 03:24.550 align:start position:0%
correlation between this. We see the
rate<00:03:22.879><c> of</c><00:03:23.040><c> change</c><00:03:23.200><c> of</c><00:03:23.440><c> temperature</c><00:03:24.080><c> is</c>

03:24.550 --> 03:24.560 align:start position:0%
rate of change of temperature is
 

03:24.560 --> 03:27.270 align:start position:0%
rate of change of temperature is
directly<00:03:25.120><c> related</c><00:03:25.599><c> to</c><00:03:25.840><c> how</c><00:03:26.239><c> quickly</c><00:03:26.800><c> the</c>

03:27.270 --> 03:27.280 align:start position:0%
directly related to how quickly the
 

03:27.280 --> 03:31.110 align:start position:0%
directly related to how quickly the
slope<00:03:28.239><c> changes.</c><00:03:29.440><c> The</c><00:03:29.760><c> rate</c><00:03:30.159><c> of</c><00:03:30.480><c> change</c><00:03:30.720><c> of</c>

03:31.110 --> 03:31.120 align:start position:0%
slope changes. The rate of change of
 

03:31.120 --> 03:34.789 align:start position:0%
slope changes. The rate of change of
slope<00:03:31.680><c> itself.</c><00:03:32.959><c> But</c><00:03:33.200><c> slope</c><00:03:33.680><c> itself</c><00:03:34.239><c> is</c><00:03:34.560><c> rate</c>

03:34.789 --> 03:34.799 align:start position:0%
slope itself. But slope itself is rate
 

03:34.799 --> 03:37.509 align:start position:0%
slope itself. But slope itself is rate
of<00:03:34.959><c> change</c><00:03:35.200><c> of</c><00:03:35.440><c> temperature.</c><00:03:36.400><c> So</c><00:03:36.640><c> we</c><00:03:36.879><c> get</c><00:03:37.120><c> a</c>

03:37.509 --> 03:37.519 align:start position:0%
of change of temperature. So we get a
 

03:37.519 --> 03:40.710 align:start position:0%
of change of temperature. So we get a
second<00:03:38.319><c> derivative.</c><00:03:39.440><c> And</c><00:03:39.599><c> there</c><00:03:39.920><c> we</c><00:03:40.159><c> have</c><00:03:40.319><c> it,</c>

03:40.710 --> 03:40.720 align:start position:0%
second derivative. And there we have it,
 

03:40.720 --> 03:44.949 align:start position:0%
second derivative. And there we have it,
the<00:03:41.200><c> legendary</c><00:03:42.319><c> heat</c><00:03:42.720><c> equation.</c>

03:44.949 --> 03:44.959 align:start position:0%
the legendary heat equation.
 

03:44.959 --> 03:47.190 align:start position:0%
the legendary heat equation.
Okay.<00:03:45.599><c> But</c><00:03:46.000><c> what</c><00:03:46.159><c> is</c><00:03:46.319><c> this</c><00:03:46.560><c> equation</c><00:03:46.959><c> telling</c>

03:47.190 --> 03:47.200 align:start position:0%
Okay. But what is this equation telling
 

03:47.200 --> 03:50.630 align:start position:0%
Okay. But what is this equation telling
us?<00:03:47.440><c> And</c><00:03:47.920><c> why</c><00:03:48.159><c> is</c><00:03:48.319><c> it</c><00:03:48.560><c> so</c><00:03:48.720><c> legendary?</c><00:03:50.080><c> Well,</c>

03:50.630 --> 03:50.640 align:start position:0%
us? And why is it so legendary? Well,
 

03:50.640 --> 03:53.830 align:start position:0%
us? And why is it so legendary? Well,
the<00:03:50.879><c> second</c><00:03:51.280><c> derivative</c><00:03:52.319><c> means</c><00:03:52.799><c> a</c><00:03:53.040><c> curvature.</c>

03:53.830 --> 03:53.840 align:start position:0%
the second derivative means a curvature.
 

03:53.840 --> 03:56.149 align:start position:0%
the second derivative means a curvature.
So<00:03:54.560><c> how</c><00:03:54.799><c> quickly</c><00:03:55.120><c> the</c><00:03:55.360><c> temperature</c><00:03:55.760><c> changes</c>

03:56.149 --> 03:56.159 align:start position:0%
So how quickly the temperature changes
 

03:56.159 --> 03:58.710 align:start position:0%
So how quickly the temperature changes
at<00:03:56.400><c> a</c><00:03:56.720><c> spot</c><00:03:57.200><c> depends</c><00:03:57.680><c> on</c><00:03:57.920><c> the</c><00:03:58.159><c> temperature</c>

03:58.710 --> 03:58.720 align:start position:0%
at a spot depends on the temperature
 

03:58.720 --> 04:01.030 align:start position:0%
at a spot depends on the temperature
curvature<00:03:59.280><c> at</c><00:03:59.519><c> that</c><00:03:59.760><c> spot.</c><00:04:00.400><c> For</c><00:04:00.480><c> example,</c><00:04:00.879><c> if</c>

04:01.030 --> 04:01.040 align:start position:0%
curvature at that spot. For example, if
 

04:01.040 --> 04:02.949 align:start position:0%
curvature at that spot. For example, if
there<00:04:01.200><c> is</c><00:04:01.360><c> no</c><00:04:01.519><c> curvature,</c><00:04:02.400><c> the</c><00:04:02.640><c> temperature</c>

04:02.949 --> 04:02.959 align:start position:0%
there is no curvature, the temperature
 

04:02.959 --> 04:05.589 align:start position:0%
there is no curvature, the temperature
is<00:04:03.120><c> unchanged.</c><00:04:04.159><c> If</c><00:04:04.400><c> there</c><00:04:04.640><c> is</c><00:04:04.799><c> a</c><00:04:05.040><c> positive</c>

04:05.589 --> 04:05.599 align:start position:0%
is unchanged. If there is a positive
 

04:05.599 --> 04:07.670 align:start position:0%
is unchanged. If there is a positive
upward<00:04:06.000><c> curvature,</c><00:04:06.720><c> look,</c><00:04:06.959><c> the</c><00:04:07.280><c> temperature</c>

04:07.670 --> 04:07.680 align:start position:0%
upward curvature, look, the temperature
 

04:07.680 --> 04:09.429 align:start position:0%
upward curvature, look, the temperature
increases<00:04:08.159><c> at</c><00:04:08.400><c> that</c><00:04:08.640><c> point.</c><00:04:09.040><c> If</c><00:04:09.200><c> that</c>

04:09.429 --> 04:09.439 align:start position:0%
increases at that point. If that
 

04:09.439 --> 04:11.350 align:start position:0%
increases at that point. If that
curvature<00:04:09.920><c> is</c><00:04:10.080><c> stronger,</c><00:04:10.720><c> the</c><00:04:10.959><c> temperature</c>

04:11.350 --> 04:11.360 align:start position:0%
curvature is stronger, the temperature
 

04:11.360 --> 04:13.509 align:start position:0%
curvature is stronger, the temperature
increases<00:04:11.920><c> faster.</c><00:04:12.879><c> And</c><00:04:13.040><c> of</c><00:04:13.200><c> course,</c><00:04:13.360><c> if</c>

04:13.509 --> 04:13.519 align:start position:0%
increases faster. And of course, if
 

04:13.519 --> 04:15.030 align:start position:0%
increases faster. And of course, if
there's<00:04:13.760><c> a</c><00:04:13.920><c> downward</c><00:04:14.239><c> curvature,</c><00:04:14.799><c> the</c>

04:15.030 --> 04:15.040 align:start position:0%
there's a downward curvature, the
 

04:15.040 --> 04:17.990 align:start position:0%
there's a downward curvature, the
temperature<00:04:15.439><c> would</c><00:04:15.840><c> reduce.</c><00:04:17.040><c> Now,</c><00:04:17.440><c> let's</c><00:04:17.759><c> put</c>

04:17.990 --> 04:18.000 align:start position:0%
temperature would reduce. Now, let's put
 

04:18.000 --> 04:21.270 align:start position:0%
temperature would reduce. Now, let's put
this<00:04:18.239><c> into</c><00:04:18.639><c> action.</c><00:04:19.280><c> Suppose</c><00:04:19.600><c> we</c><00:04:19.919><c> have</c><00:04:20.239><c> a</c><00:04:20.639><c> rod</c>

04:21.270 --> 04:21.280 align:start position:0%
this into action. Suppose we have a rod
 

04:21.280 --> 04:23.909 align:start position:0%
this into action. Suppose we have a rod
sitting<00:04:21.680><c> here</c><00:04:21.919><c> at</c><00:04:22.240><c> room</c><00:04:22.639><c> temperature.</c><00:04:23.600><c> We</c>

04:23.909 --> 04:23.919 align:start position:0%
sitting here at room temperature. We
 

04:23.919 --> 04:26.150 align:start position:0%
sitting here at room temperature. We
heat<00:04:24.160><c> up</c><00:04:24.400><c> one</c><00:04:24.800><c> specific</c><00:04:25.360><c> spot.</c><00:04:25.840><c> So,</c><00:04:25.919><c> the</c>

04:26.150 --> 04:26.160 align:start position:0%
heat up one specific spot. So, the
 

04:26.160 --> 04:28.310 align:start position:0%
heat up one specific spot. So, the
temperature<00:04:26.639><c> spikes</c><00:04:27.120><c> over</c><00:04:27.360><c> there.</c><00:04:28.080><c> The</c>

04:28.310 --> 04:28.320 align:start position:0%
temperature spikes over there. The
 

04:28.320 --> 04:30.230 align:start position:0%
temperature spikes over there. The
question<00:04:28.479><c> is,</c><00:04:28.800><c> what</c><00:04:29.040><c> happens</c><00:04:29.280><c> a</c><00:04:29.520><c> little</c><00:04:29.680><c> while</c>

04:30.230 --> 04:30.240 align:start position:0%
question is, what happens a little while
 

04:30.240 --> 04:33.189 align:start position:0%
question is, what happens a little while
later?<00:04:30.960><c> How</c><00:04:31.120><c> do</c><00:04:31.280><c> we</c><00:04:31.440><c> figure</c><00:04:31.680><c> that</c><00:04:31.840><c> out?</c><00:04:32.639><c> Well,</c>

04:33.189 --> 04:33.199 align:start position:0%
later? How do we figure that out? Well,
 

04:33.199 --> 04:35.590 align:start position:0%
later? How do we figure that out? Well,
just<00:04:33.680><c> look</c><00:04:33.840><c> at</c><00:04:34.000><c> the</c><00:04:34.240><c> curvature</c><00:04:34.720><c> everywhere.</c>

04:35.590 --> 04:35.600 align:start position:0%
just look at the curvature everywhere.
 

04:35.600 --> 04:37.830 align:start position:0%
just look at the curvature everywhere.
At<00:04:35.840><c> the</c><00:04:36.240><c> peak,</c><00:04:36.639><c> the</c><00:04:36.800><c> curvature</c><00:04:37.360><c> is</c><00:04:37.600><c> the</c>

04:37.830 --> 04:37.840 align:start position:0%
At the peak, the curvature is the
 

04:37.840 --> 04:39.749 align:start position:0%
At the peak, the curvature is the
strongest.<00:04:38.479><c> So,</c><00:04:38.639><c> that</c><00:04:38.880><c> is</c><00:04:38.960><c> the</c><00:04:39.199><c> point</c><00:04:39.440><c> where</c>

04:39.749 --> 04:39.759 align:start position:0%
strongest. So, that is the point where
 

04:39.759 --> 04:42.950 align:start position:0%
strongest. So, that is the point where
it'll<00:04:40.160><c> cool</c><00:04:40.479><c> down</c><00:04:41.040><c> the</c><00:04:41.360><c> fastest.</c><00:04:42.240><c> As</c><00:04:42.479><c> we</c><00:04:42.720><c> go</c>

04:42.950 --> 04:42.960 align:start position:0%
it'll cool down the fastest. As we go
 

04:42.960 --> 04:46.150 align:start position:0%
it'll cool down the fastest. As we go
farther<00:04:43.360><c> away,</c><00:04:44.080><c> the</c><00:04:44.320><c> curvature</c><00:04:44.880><c> reduces</c><00:04:46.000><c> and</c>

04:46.150 --> 04:46.160 align:start position:0%
farther away, the curvature reduces and
 

04:46.160 --> 04:48.150 align:start position:0%
farther away, the curvature reduces and
then<00:04:46.400><c> the</c><00:04:46.560><c> curvature</c><00:04:47.040><c> goes</c><00:04:47.280><c> to</c><00:04:47.440><c> zero</c><00:04:47.759><c> and</c><00:04:48.000><c> then</c>

04:48.150 --> 04:48.160 align:start position:0%
then the curvature goes to zero and then
 

04:48.160 --> 04:50.550 align:start position:0%
then the curvature goes to zero and then
the<00:04:48.320><c> curvature</c><00:04:48.880><c> becomes</c><00:04:49.440><c> upward</c><00:04:50.080><c> and</c><00:04:50.320><c> so</c>

04:50.550 --> 04:50.560 align:start position:0%
the curvature becomes upward and so
 

04:50.560 --> 04:54.710 align:start position:0%
the curvature becomes upward and so
these<00:04:50.880><c> spots</c><00:04:51.680><c> actually</c><00:04:52.479><c> warm</c><00:04:52.880><c> up.</c>

04:54.710 --> 04:54.720 align:start position:0%
these spots actually warm up.
 

04:54.720 --> 04:57.030 align:start position:0%
these spots actually warm up.
So,<00:04:55.120><c> a</c><00:04:55.360><c> little</c><00:04:55.600><c> later</c><00:04:56.080><c> our</c><00:04:56.320><c> profile</c><00:04:56.720><c> is</c><00:04:56.960><c> going</c>

04:57.030 --> 04:57.040 align:start position:0%
So, a little later our profile is going
 

04:57.040 --> 05:00.710 align:start position:0%
So, a little later our profile is going
to<00:04:57.199><c> look</c><00:04:57.680><c> like</c><00:04:58.320><c> this.</c><00:04:59.440><c> This</c><00:04:59.759><c> equation</c><00:05:00.320><c> allows</c>

05:00.710 --> 05:00.720 align:start position:0%
to look like this. This equation allows
 

05:00.720 --> 05:03.990 align:start position:0%
to look like this. This equation allows
us<00:05:00.880><c> to</c><00:05:01.199><c> predict</c><00:05:02.000><c> what</c><00:05:02.560><c> happens</c><00:05:03.440><c> to</c><00:05:03.759><c> the</c>

05:03.990 --> 05:04.000 align:start position:0%
us to predict what happens to the
 

05:04.000 --> 05:06.070 align:start position:0%
us to predict what happens to the
temperature<00:05:04.560><c> profile.</c><00:05:05.440><c> It</c><00:05:05.680><c> helps</c><00:05:05.840><c> us</c>

05:06.070 --> 05:06.080 align:start position:0%
temperature profile. It helps us
 

05:06.080 --> 05:10.070 align:start position:0%
temperature profile. It helps us
understand<00:05:06.560><c> how</c><00:05:06.880><c> the</c><00:05:07.120><c> profile</c><00:05:08.080><c> evolves</c><00:05:09.199><c> over</c>

05:10.070 --> 05:10.080 align:start position:0%
understand how the profile evolves over
 

05:10.080 --> 05:14.469 align:start position:0%
understand how the profile evolves over
time.<00:05:11.520><c> That's</c><00:05:11.840><c> awesome.</c><00:05:13.280><c> But</c><00:05:13.759><c> I</c><00:05:14.000><c> still</c><00:05:14.240><c> don't</c>

05:14.469 --> 05:14.479 align:start position:0%
time. That's awesome. But I still don't
 

05:14.479 --> 05:18.310 align:start position:0%
time. That's awesome. But I still don't
get<00:05:14.880><c> why</c><00:05:15.600><c> this</c><00:05:15.919><c> is</c><00:05:16.240><c> so</c><00:05:16.800><c> legendary.</c>

05:18.310 --> 05:18.320 align:start position:0%
get why this is so legendary.
 

05:18.320 --> 05:20.390 align:start position:0%
get why this is so legendary.
Just<00:05:18.639><c> like</c><00:05:18.800><c> how</c><00:05:19.120><c> heat</c><00:05:19.520><c> moves</c><00:05:19.840><c> from</c><00:05:20.080><c> high</c>

05:20.390 --> 05:20.400 align:start position:0%
Just like how heat moves from high
 

05:20.400 --> 05:22.629 align:start position:0%
Just like how heat moves from high
temperature<00:05:20.800><c> to</c><00:05:21.039><c> low</c><00:05:21.280><c> temperature,</c><00:05:22.080><c> charges</c>

05:22.629 --> 05:22.639 align:start position:0%
temperature to low temperature, charges
 

05:22.639 --> 05:25.830 align:start position:0%
temperature to low temperature, charges
inside<00:05:23.280><c> a</c><00:05:23.600><c> neuron,</c><00:05:24.240><c> for</c><00:05:24.240><c> example,</c><00:05:25.199><c> move</c><00:05:25.520><c> from</c>

05:25.830 --> 05:25.840 align:start position:0%
inside a neuron, for example, move from
 

05:25.840 --> 05:28.950 align:start position:0%
inside a neuron, for example, move from
high<00:05:26.080><c> voltage</c><00:05:26.800><c> to</c><00:05:27.120><c> low</c><00:05:27.440><c> voltage.</c><00:05:28.479><c> Which</c><00:05:28.720><c> means</c>

05:28.950 --> 05:28.960 align:start position:0%
high voltage to low voltage. Which means
 

05:28.960 --> 05:30.790 align:start position:0%
high voltage to low voltage. Which means
the<00:05:29.199><c> exact</c><00:05:29.600><c> same</c><00:05:29.840><c> physics</c><00:05:30.240><c> can</c><00:05:30.400><c> be</c><00:05:30.479><c> used</c><00:05:30.639><c> to</c>

05:30.790 --> 05:30.800 align:start position:0%
the exact same physics can be used to
 

05:30.800 --> 05:32.950 align:start position:0%
the exact same physics can be used to
model<00:05:31.120><c> how</c><00:05:31.440><c> electricity</c><00:05:32.080><c> electricity</c><00:05:32.639><c> flows</c>

05:32.950 --> 05:32.960 align:start position:0%
model how electricity electricity flows
 

05:32.960 --> 05:35.029 align:start position:0%
model how electricity electricity flows
through<00:05:33.120><c> a</c><00:05:33.360><c> neuron.</c><00:05:34.000><c> All</c><00:05:34.240><c> we</c><00:05:34.400><c> need</c><00:05:34.479><c> to</c><00:05:34.639><c> do</c><00:05:34.800><c> is</c>

05:35.029 --> 05:35.039 align:start position:0%
through a neuron. All we need to do is
 

05:35.039 --> 05:38.629 align:start position:0%
through a neuron. All we need to do is
replace<00:05:35.440><c> temperature</c><00:05:36.320><c> with</c><00:05:37.039><c> voltage.</c><00:05:38.320><c> That's</c>

05:38.629 --> 05:38.639 align:start position:0%
replace temperature with voltage. That's
 

05:38.639 --> 05:40.710 align:start position:0%
replace temperature with voltage. That's
awesome,<00:05:39.120><c> right?</c><00:05:39.600><c> And</c><00:05:39.759><c> the</c><00:05:40.080><c> exact</c><00:05:40.400><c> same</c>

05:40.710 --> 05:40.720 align:start position:0%
awesome, right? And the exact same
 

05:40.720 --> 05:43.270 align:start position:0%
awesome, right? And the exact same
equation<00:05:41.120><c> governs</c><00:05:41.680><c> how</c><00:05:42.080><c> mutations</c><00:05:42.880><c> spread</c>

05:43.270 --> 05:43.280 align:start position:0%
equation governs how mutations spread
 

05:43.280 --> 05:46.070 align:start position:0%
equation governs how mutations spread
through<00:05:43.520><c> a</c><00:05:43.759><c> population</c><00:05:44.479><c> or</c><00:05:44.800><c> how</c><00:05:45.120><c> uncertainty</c>

05:46.070 --> 05:46.080 align:start position:0%
through a population or how uncertainty
 

05:46.080 --> 05:48.950 align:start position:0%
through a population or how uncertainty
spreads<00:05:46.479><c> through</c><00:05:46.800><c> financial</c><00:05:47.360><c> markets.</c><00:05:48.320><c> Oh</c><00:05:48.560><c> my</c>

05:48.950 --> 05:48.960 align:start position:0%
spreads through financial markets. Oh my
 

05:48.960 --> 05:52.150 align:start position:0%
spreads through financial markets. Oh my
god,<00:05:50.160><c> why</c><00:05:50.479><c> does</c><00:05:50.720><c> the</c><00:05:50.880><c> single</c><00:05:51.199><c> equation</c><00:05:51.680><c> work</c>

05:52.150 --> 05:52.160 align:start position:0%
god, why does the single equation work
 

05:52.160 --> 05:54.550 align:start position:0%
god, why does the single equation work
everywhere?

05:54.550 --> 05:54.560 align:start position:0%
everywhere?
 

05:54.560 --> 05:56.629 align:start position:0%
everywhere?
Because<00:05:54.880><c> at</c><00:05:55.120><c> the</c><00:05:55.280><c> heart</c><00:05:55.440><c> of</c><00:05:55.600><c> it,</c><00:05:55.919><c> the</c><00:05:56.240><c> central</c>

05:56.629 --> 05:56.639 align:start position:0%
Because at the heart of it, the central
 

05:56.639 --> 05:59.990 align:start position:0%
Because at the heart of it, the central
phenomena<00:05:57.440><c> is</c><00:05:57.919><c> diffusion.</c><00:05:58.960><c> So</c><00:05:59.199><c> this</c><00:05:59.440><c> equation</c>

05:59.990 --> 06:00.000 align:start position:0%
phenomena is diffusion. So this equation
 

06:00.000 --> 06:02.550 align:start position:0%
phenomena is diffusion. So this equation
is<00:06:00.560><c> fundamentally</c><00:06:01.280><c> a</c><00:06:01.600><c> diffusion</c><00:06:02.080><c> equation.</c>

06:02.550 --> 06:02.560 align:start position:0%
is fundamentally a diffusion equation.
 

06:02.560 --> 06:03.909 align:start position:0%
is fundamentally a diffusion equation.
It's<00:06:02.720><c> one</c><00:06:02.880><c> of</c><00:06:02.960><c> the</c><00:06:03.120><c> most</c><00:06:03.280><c> common</c><00:06:03.600><c> things</c><00:06:03.759><c> that</c>

06:03.909 --> 06:03.919 align:start position:0%
It's one of the most common things that
 

06:03.919 --> 06:05.909 align:start position:0%
It's one of the most common things that
happen<00:06:04.160><c> in</c><00:06:04.319><c> our</c><00:06:04.560><c> universe.</c><00:06:05.199><c> That's</c><00:06:05.520><c> why</c><00:06:05.680><c> this</c>

06:05.909 --> 06:05.919 align:start position:0%
happen in our universe. That's why this
 

06:05.919 --> 06:08.469 align:start position:0%
happen in our universe. That's why this
equation<00:06:06.319><c> is</c><00:06:06.639><c> legendary.</c><00:06:07.600><c> And</c><00:06:07.840><c> we</c><00:06:08.000><c> derived</c><00:06:08.319><c> it</c>

06:08.469 --> 06:08.479 align:start position:0%
equation is legendary. And we derived it
 

06:08.479 --> 06:10.390 align:start position:0%
equation is legendary. And we derived it
from<00:06:08.720><c> scratch</c><00:06:09.440><c> just</c><00:06:09.680><c> from</c><00:06:09.840><c> the</c><00:06:10.000><c> fact</c><00:06:10.160><c> that</c>

06:10.390 --> 06:10.400 align:start position:0%
from scratch just from the fact that
 

06:10.400 --> 06:12.550 align:start position:0%
from scratch just from the fact that
heat<00:06:10.639><c> moves</c><00:06:10.960><c> from</c><00:06:11.199><c> hot</c><00:06:11.520><c> to</c><00:06:11.759><c> cold.</c><00:06:12.160><c> That's</c>

06:12.550 --> 06:12.560 align:start position:0%
heat moves from hot to cold. That's
 

06:12.560 --> 06:15.430 align:start position:0%
heat moves from hot to cold. That's
awesome,<00:06:13.039><c> right?</c><00:06:14.319><c> Okay.</c><00:06:14.639><c> But</c><00:06:14.800><c> what</c><00:06:15.039><c> does</c><00:06:15.199><c> this</c>

06:15.430 --> 06:15.440 align:start position:0%
awesome, right? Okay. But what does this
 

06:15.440 --> 06:18.710 align:start position:0%
awesome, right? Okay. But what does this
have<00:06:15.680><c> anything</c><00:06:16.080><c> to</c><00:06:16.319><c> do</c><00:06:16.479><c> with</c><00:06:16.960><c> 4year</c><00:06:17.520><c> series?</c>

06:18.710 --> 06:18.720 align:start position:0%
have anything to do with 4year series?
 

06:18.720 --> 06:20.469 align:start position:0%
have anything to do with 4year series?
Everything.<00:06:19.360><c> Because</c><00:06:19.600><c> now</c><00:06:19.759><c> we</c><00:06:20.000><c> have</c><00:06:20.080><c> a</c><00:06:20.319><c> big</c>

06:20.469 --> 06:20.479 align:start position:0%
Everything. Because now we have a big
 

06:20.479 --> 06:23.029 align:start position:0%
Everything. Because now we have a big
challenge<00:06:20.880><c> in</c><00:06:21.120><c> front</c><00:06:21.280><c> of</c><00:06:21.440><c> us.</c><00:06:22.240><c> How</c><00:06:22.560><c> do</c><00:06:22.720><c> we</c>

06:23.029 --> 06:23.039 align:start position:0%
challenge in front of us. How do we
 

06:23.039 --> 06:26.309 align:start position:0%
challenge in front of us. How do we
actually<00:06:23.759><c> solve</c><00:06:24.560><c> this</c><00:06:24.960><c> equation?</c><00:06:25.680><c> I</c><00:06:25.919><c> mean,</c><00:06:26.080><c> so</c>

06:26.309 --> 06:26.319 align:start position:0%
actually solve this equation? I mean, so
 

06:26.319 --> 06:29.029 align:start position:0%
actually solve this equation? I mean, so
far<00:06:26.479><c> we</c><00:06:26.960><c> solved</c><00:06:27.360><c> it</c><00:06:27.840><c> qualitatively</c><00:06:28.880><c> looking</c>

06:29.029 --> 06:29.039 align:start position:0%
far we solved it qualitatively looking
 

06:29.039 --> 06:32.150 align:start position:0%
far we solved it qualitatively looking
at<00:06:29.199><c> it</c><00:06:29.520><c> step</c><00:06:29.919><c> by</c><00:06:30.240><c> step.</c><00:06:31.120><c> But</c><00:06:31.360><c> how</c><00:06:31.520><c> do</c><00:06:31.680><c> we</c><00:06:31.840><c> get</c><00:06:31.919><c> a</c>

06:32.150 --> 06:32.160 align:start position:0%
at it step by step. But how do we get a
 

06:32.160 --> 06:35.270 align:start position:0%
at it step by step. But how do we get a
mathematical<00:06:32.960><c> expression</c><00:06:33.840><c> for</c><00:06:34.160><c> it?</c><00:06:34.960><c> U</c>

06:35.270 --> 06:35.280 align:start position:0%
mathematical expression for it? U
 

06:35.280 --> 06:37.749 align:start position:0%
mathematical expression for it? U
solving<00:06:35.680><c> differential</c><00:06:36.160><c> equations</c><00:06:36.800><c> are</c><00:06:37.120><c> not</c>

06:37.749 --> 06:37.759 align:start position:0%
solving differential equations are not
 

06:37.759 --> 06:41.430 align:start position:0%
solving differential equations are not
easy.<00:06:38.479><c> So</c><00:06:39.120><c> what</c><00:06:39.520><c> do</c><00:06:39.759><c> we</c><00:06:40.000><c> do?</c><00:06:40.639><c> Forier</c><00:06:41.039><c> uses</c><00:06:41.360><c> one</c>

06:41.430 --> 06:41.440 align:start position:0%
easy. So what do we do? Forier uses one
 

06:41.440 --> 06:42.950 align:start position:0%
easy. So what do we do? Forier uses one
of<00:06:41.520><c> the</c><00:06:41.680><c> best</c><00:06:41.840><c> problems</c><00:06:42.240><c> solving</c><00:06:42.560><c> techniques.</c>

06:42.950 --> 06:42.960 align:start position:0%
of the best problems solving techniques.
 

06:42.960 --> 06:45.110 align:start position:0%
of the best problems solving techniques.
If<00:06:43.199><c> you</c><00:06:43.360><c> can't</c><00:06:43.600><c> solve</c><00:06:43.840><c> a</c><00:06:44.080><c> hard</c><00:06:44.319><c> problem,</c><00:06:44.960><c> see</c>

06:45.110 --> 06:45.120 align:start position:0%
If you can't solve a hard problem, see
 

06:45.120 --> 06:47.430 align:start position:0%
If you can't solve a hard problem, see
if<00:06:45.280><c> you</c><00:06:45.440><c> can</c><00:06:45.600><c> tackle</c><00:06:46.080><c> an</c><00:06:46.479><c> easier</c><00:06:46.960><c> version</c><00:06:47.280><c> of</c>

06:47.430 --> 06:47.440 align:start position:0%
if you can tackle an easier version of
 

06:47.440 --> 06:49.830 align:start position:0%
if you can tackle an easier version of
the<00:06:47.680><c> problem.</c><00:06:48.479><c> Okay,</c><00:06:48.960><c> what's</c><00:06:49.199><c> the</c><00:06:49.440><c> easier</c>

06:49.830 --> 06:49.840 align:start position:0%
the problem. Okay, what's the easier
 

06:49.840 --> 06:52.950 align:start position:0%
the problem. Okay, what's the easier
version<00:06:50.080><c> of</c><00:06:50.160><c> the</c><00:06:50.400><c> problem</c><00:06:50.639><c> here?</c>

06:52.950 --> 06:52.960 align:start position:0%
version of the problem here?
 

06:52.960 --> 06:54.710 align:start position:0%
version of the problem here?
Well,<00:06:53.360><c> here</c><00:06:53.600><c> if</c><00:06:53.840><c> you</c><00:06:53.919><c> look</c><00:06:54.000><c> at</c><00:06:54.160><c> the</c><00:06:54.319><c> initial</c>

06:54.710 --> 06:54.720 align:start position:0%
Well, here if you look at the initial
 

06:54.720 --> 06:57.189 align:start position:0%
Well, here if you look at the initial
profile,<00:06:55.360><c> the</c><00:06:55.600><c> curvature</c><00:06:56.240><c> is</c><00:06:56.560><c> all</c><00:06:56.800><c> over</c><00:06:56.960><c> the</c>

06:57.189 --> 06:57.199 align:start position:0%
profile, the curvature is all over the
 

06:57.199 --> 06:59.430 align:start position:0%
profile, the curvature is all over the
place<00:06:57.520><c> and</c><00:06:57.919><c> that's</c><00:06:58.240><c> why</c><00:06:58.800><c> things</c><00:06:59.120><c> look</c>

06:59.430 --> 06:59.440 align:start position:0%
place and that's why things look
 

06:59.440 --> 07:01.670 align:start position:0%
place and that's why things look
complicated.<00:07:00.319><c> So,</c><00:07:00.639><c> can</c><00:07:00.880><c> we</c><00:07:01.039><c> start</c><00:07:01.280><c> with</c><00:07:01.440><c> a</c>

07:01.670 --> 07:01.680 align:start position:0%
complicated. So, can we start with a
 

07:01.680 --> 07:06.230 align:start position:0%
complicated. So, can we start with a
profile<00:07:02.080><c> whose</c><00:07:02.400><c> curvature</c><00:07:02.880><c> is</c><00:07:03.440><c> nice?</c><00:07:04.880><c> Yes.</c>

07:06.230 --> 07:06.240 align:start position:0%
profile whose curvature is nice? Yes.
 

07:06.240 --> 07:10.070 align:start position:0%
profile whose curvature is nice? Yes.
What<00:07:06.479><c> about</c><00:07:07.520><c> a</c><00:07:08.000><c> sine</c><00:07:08.319><c> wave?</c><00:07:09.039><c> How</c><00:07:09.280><c> on</c><00:07:09.599><c> earth</c>

07:10.070 --> 07:10.080 align:start position:0%
What about a sine wave? How on earth
 

07:10.080 --> 07:13.029 align:start position:0%
What about a sine wave? How on earth
does<00:07:10.479><c> sign</c><00:07:10.880><c> make</c><00:07:11.360><c> things</c><00:07:11.919><c> easier?</c><00:07:12.880><c> Well,</c>

07:13.029 --> 07:13.039 align:start position:0%
does sign make things easier? Well,
 

07:13.039 --> 07:15.350 align:start position:0%
does sign make things easier? Well,
Forier<00:07:13.599><c> says</c><00:07:13.919><c> sign</c><00:07:14.240><c> has</c><00:07:14.400><c> a</c><00:07:14.800><c> beautiful</c>

07:15.350 --> 07:15.360 align:start position:0%
Forier says sign has a beautiful
 

07:15.360 --> 07:18.870 align:start position:0%
Forier says sign has a beautiful
property.<00:07:16.479><c> Its</c><00:07:16.800><c> curvature</c><00:07:17.680><c> is</c><00:07:18.080><c> proportional</c>

07:18.870 --> 07:18.880 align:start position:0%
property. Its curvature is proportional
 

07:18.880 --> 07:21.589 align:start position:0%
property. Its curvature is proportional
to<00:07:19.199><c> the</c><00:07:19.440><c> height.</c><00:07:20.400><c> What</c><00:07:20.639><c> does</c><00:07:20.800><c> that</c><00:07:21.039><c> mean?</c>

07:21.589 --> 07:21.599 align:start position:0%
to the height. What does that mean?
 

07:21.599 --> 07:23.749 align:start position:0%
to the height. What does that mean?
Well,<00:07:22.000><c> if</c><00:07:22.240><c> you</c><00:07:22.400><c> look</c><00:07:22.479><c> at</c><00:07:22.720><c> this</c><00:07:22.960><c> point,</c><00:07:23.520><c> the</c>

07:23.749 --> 07:23.759 align:start position:0%
Well, if you look at this point, the
 

07:23.759 --> 07:26.150 align:start position:0%
Well, if you look at this point, the
height<00:07:24.080><c> is</c><00:07:24.319><c> zero,</c><00:07:24.880><c> right?</c><00:07:25.440><c> Guess</c><00:07:25.680><c> what?</c><00:07:26.000><c> The</c>

07:26.150 --> 07:26.160 align:start position:0%
height is zero, right? Guess what? The
 

07:26.160 --> 07:28.629 align:start position:0%
height is zero, right? Guess what? The
curvature<00:07:26.639><c> is</c><00:07:26.800><c> also</c><00:07:27.120><c> zero.</c><00:07:28.160><c> This</c><00:07:28.319><c> is</c><00:07:28.479><c> the</c>

07:28.629 --> 07:28.639 align:start position:0%
curvature is also zero. This is the
 

07:28.639 --> 07:30.790 align:start position:0%
curvature is also zero. This is the
point<00:07:28.800><c> where</c><00:07:29.039><c> the</c><00:07:29.199><c> height</c><00:07:29.360><c> is</c><00:07:29.599><c> maximum</c><00:07:30.400><c> and</c>

07:30.790 --> 07:30.800 align:start position:0%
point where the height is maximum and
 

07:30.800 --> 07:33.350 align:start position:0%
point where the height is maximum and
the<00:07:31.120><c> curvature</c><00:07:31.520><c> is</c><00:07:31.680><c> also</c><00:07:32.000><c> maximum.</c><00:07:33.039><c> If</c><00:07:33.280><c> you</c>

07:33.350 --> 07:33.360 align:start position:0%
the curvature is also maximum. If you
 

07:33.360 --> 07:35.029 align:start position:0%
the curvature is also maximum. If you
consider<00:07:33.680><c> a</c><00:07:33.840><c> point</c><00:07:34.000><c> where</c><00:07:34.160><c> the</c><00:07:34.319><c> height</c><00:07:34.639><c> is</c>

07:35.029 --> 07:35.039 align:start position:0%
consider a point where the height is
 

07:35.039 --> 07:37.189 align:start position:0%
consider a point where the height is
half<00:07:35.360><c> the</c><00:07:35.599><c> maximum</c><00:07:36.080><c> height,</c><00:07:36.560><c> the</c><00:07:36.800><c> curvature</c>

07:37.189 --> 07:37.199 align:start position:0%
half the maximum height, the curvature
 

07:37.199 --> 07:39.510 align:start position:0%
half the maximum height, the curvature
at<00:07:37.360><c> that</c><00:07:37.520><c> point</c><00:07:37.759><c> is</c><00:07:37.919><c> also</c><00:07:38.560><c> half</c><00:07:38.880><c> the</c><00:07:39.039><c> maximum</c>

07:39.510 --> 07:39.520 align:start position:0%
at that point is also half the maximum
 

07:39.520 --> 07:41.990 align:start position:0%
at that point is also half the maximum
curvature.<00:07:40.400><c> See,</c><00:07:40.880><c> curvature</c><00:07:41.680><c> is</c>

07:41.990 --> 07:42.000 align:start position:0%
curvature. See, curvature is
 

07:42.000 --> 07:45.350 align:start position:0%
curvature. See, curvature is
proportional<00:07:42.880><c> to</c><00:07:43.120><c> the</c><00:07:43.360><c> height.</c><00:07:44.319><c> Okay?</c><00:07:45.039><c> So</c>

07:45.350 --> 07:45.360 align:start position:0%
proportional to the height. Okay? So
 

07:45.360 --> 07:47.589 align:start position:0%
proportional to the height. Okay? So
this<00:07:45.680><c> means</c><00:07:46.000><c> if</c><00:07:46.319><c> the</c><00:07:46.479><c> starting</c><00:07:46.800><c> profile</c><00:07:47.199><c> was</c><00:07:47.360><c> a</c>

07:47.589 --> 07:47.599 align:start position:0%
this means if the starting profile was a
 

07:47.599 --> 07:49.830 align:start position:0%
this means if the starting profile was a
sign<00:07:48.000><c> curve</c><00:07:48.639><c> then</c><00:07:48.880><c> the</c><00:07:49.120><c> rate</c><00:07:49.280><c> of</c><00:07:49.440><c> change</c><00:07:49.599><c> of</c>

07:49.830 --> 07:49.840 align:start position:0%
sign curve then the rate of change of
 

07:49.840 --> 07:51.749 align:start position:0%
sign curve then the rate of change of
temperature<00:07:50.400><c> is</c><00:07:50.639><c> going</c><00:07:50.800><c> to</c><00:07:50.880><c> be</c><00:07:51.120><c> proportional</c>

07:51.749 --> 07:51.759 align:start position:0%
temperature is going to be proportional
 

07:51.759 --> 07:54.629 align:start position:0%
temperature is going to be proportional
to<00:07:52.080><c> the</c><00:07:52.479><c> height</c>

07:54.629 --> 07:54.639 align:start position:0%
to the height
 

07:54.639 --> 07:56.629 align:start position:0%
to the height
which<00:07:54.879><c> means</c><00:07:55.120><c> the</c><00:07:55.360><c> temperature</c><00:07:55.759><c> at</c><00:07:56.160><c> every</c>

07:56.629 --> 07:56.639 align:start position:0%
which means the temperature at every
 

07:56.639 --> 07:59.270 align:start position:0%
which means the temperature at every
point<00:07:57.039><c> changes</c><00:07:57.599><c> in</c><00:07:58.080><c> proportion</c><00:07:58.720><c> to</c><00:07:58.960><c> its</c>

07:59.270 --> 07:59.280 align:start position:0%
point changes in proportion to its
 

07:59.280 --> 08:02.230 align:start position:0%
point changes in proportion to its
height.<00:07:59.919><c> And</c><00:08:00.080><c> so</c><00:08:00.319><c> a</c><00:08:00.560><c> little</c><00:08:00.639><c> while</c><00:08:00.879><c> later</c><00:08:01.840><c> the</c>

08:02.230 --> 08:02.240 align:start position:0%
height. And so a little while later the
 

08:02.240 --> 08:06.710 align:start position:0%
height. And so a little while later the
shape<00:08:02.800><c> looks</c><00:08:03.360><c> exactly</c><00:08:04.240><c> the</c><00:08:04.560><c> same.</c><00:08:05.120><c> Sign</c><00:08:06.000><c> just</c>

08:06.710 --> 08:06.720 align:start position:0%
shape looks exactly the same. Sign just
 

08:06.720 --> 08:09.670 align:start position:0%
shape looks exactly the same. Sign just
shrunk<00:08:07.520><c> down.</c>

08:09.670 --> 08:09.680 align:start position:0%
shrunk down.
 

08:09.680 --> 08:13.990 align:start position:0%
shrunk down.
That<00:08:09.919><c> is</c><00:08:10.240><c> cool</c><00:08:11.120><c> right?</c><00:08:12.000><c> And</c><00:08:12.240><c> that</c><00:08:12.879><c> continues</c>

08:13.990 --> 08:14.000 align:start position:0%
That is cool right? And that continues
 

08:14.000 --> 08:18.070 align:start position:0%
That is cool right? And that continues
forever<00:08:14.560><c> and</c><00:08:14.960><c> ever.</c><00:08:16.240><c> And</c><00:08:16.479><c> the</c><00:08:16.720><c> whole</c><00:08:16.960><c> point</c><00:08:17.199><c> is</c>

08:18.070 --> 08:18.080 align:start position:0%
forever and ever. And the whole point is
 

08:18.080 --> 08:22.550 align:start position:0%
forever and ever. And the whole point is
that<00:08:18.720><c> is</c><00:08:19.120><c> super</c><00:08:19.840><c> easy</c><00:08:20.240><c> to</c><00:08:20.639><c> write</c><00:08:21.199><c> down.</c><00:08:22.240><c> The</c>

08:22.550 --> 08:22.560 align:start position:0%
that is super easy to write down. The
 

08:22.560 --> 08:24.150 align:start position:0%
that is super easy to write down. The
whole<00:08:22.879><c> graph</c><00:08:23.199><c> is</c><00:08:23.440><c> just</c><00:08:23.680><c> shrinking</c>

08:24.150 --> 08:24.160 align:start position:0%
whole graph is just shrinking
 

08:24.160 --> 08:25.589 align:start position:0%
whole graph is just shrinking
exponentially.<00:08:25.039><c> So</c><00:08:25.199><c> how</c><00:08:25.360><c> can</c><00:08:25.520><c> we</c>

08:25.589 --> 08:25.599 align:start position:0%
exponentially. So how can we
 

08:25.599 --> 08:27.670 align:start position:0%
exponentially. So how can we
mathematically<00:08:26.160><c> write</c><00:08:26.400><c> down?</c><00:08:26.879><c> Well,</c><00:08:27.280><c> we</c><00:08:27.520><c> can</c>

08:27.670 --> 08:27.680 align:start position:0%
mathematically write down? Well, we can
 

08:27.680 --> 08:30.150 align:start position:0%
mathematically write down? Well, we can
say<00:08:27.919><c> the</c><00:08:28.240><c> temperature</c><00:08:28.639><c> profile</c><00:08:29.199><c> is</c><00:08:29.599><c> basically</c>

08:30.150 --> 08:30.160 align:start position:0%
say the temperature profile is basically
 

08:30.160 --> 08:34.630 align:start position:0%
say the temperature profile is basically
be<00:08:30.800><c> sign</c><00:08:31.280><c> graph</c><00:08:32.240><c> whose</c><00:08:32.719><c> amplitude</c><00:08:33.680><c> shrinks</c>

08:34.630 --> 08:34.640 align:start position:0%
be sign graph whose amplitude shrinks
 

08:34.640 --> 08:36.149 align:start position:0%
be sign graph whose amplitude shrinks
exponentially.

08:36.149 --> 08:36.159 align:start position:0%
exponentially.
 

08:36.159 --> 08:38.870 align:start position:0%
exponentially.
So<00:08:36.560><c> this</c><00:08:36.880><c> part</c><00:08:37.200><c> stays</c><00:08:37.599><c> the</c><00:08:37.839><c> same</c><00:08:38.000><c> because</c><00:08:38.399><c> the</c>

08:38.870 --> 08:38.880 align:start position:0%
So this part stays the same because the
 

08:38.880 --> 08:41.190 align:start position:0%
So this part stays the same because the
graph<00:08:39.599><c> stays</c><00:08:40.000><c> the</c><00:08:40.159><c> same,</c><00:08:40.399><c> the</c><00:08:40.640><c> shape</c><00:08:40.880><c> stays</c>

08:41.190 --> 08:41.200 align:start position:0%
graph stays the same, the shape stays
 

08:41.200 --> 08:43.829 align:start position:0%
graph stays the same, the shape stays
the<00:08:41.440><c> same.</c><00:08:42.000><c> And</c><00:08:42.240><c> this</c><00:08:42.479><c> part</c><00:08:42.719><c> represents</c><00:08:43.440><c> the</c>

08:43.829 --> 08:43.839 align:start position:0%
the same. And this part represents the
 

08:43.839 --> 08:46.710 align:start position:0%
the same. And this part represents the
whole<00:08:44.159><c> graph</c><00:08:44.640><c> shrinking</c><00:08:45.279><c> exponentially.</c><00:08:46.560><c> So</c>

08:46.710 --> 08:46.720 align:start position:0%
whole graph shrinking exponentially. So
 

08:46.720 --> 08:48.870 align:start position:0%
whole graph shrinking exponentially. So
if<00:08:46.959><c> the</c><00:08:47.120><c> initial</c><00:08:47.519><c> graph</c><00:08:47.760><c> is</c><00:08:47.920><c> a</c><00:08:48.080><c> sign</c><00:08:48.399><c> curve,</c>

08:48.870 --> 08:48.880 align:start position:0%
if the initial graph is a sign curve,
 

08:48.880 --> 08:52.710 align:start position:0%
if the initial graph is a sign curve,
look,<00:08:49.680><c> this</c><00:08:50.160><c> is</c><00:08:50.399><c> the</c><00:08:50.880><c> exact</c><00:08:51.519><c> equation</c><00:08:52.320><c> that</c>

08:52.710 --> 08:52.720 align:start position:0%
look, this is the exact equation that
 

08:52.720 --> 08:55.110 align:start position:0%
look, this is the exact equation that
represents<00:08:53.600><c> this</c><00:08:54.080><c> evolution.</c><00:08:54.880><c> That's</c>

08:55.110 --> 08:55.120 align:start position:0%
represents this evolution. That's
 

08:55.120 --> 08:58.070 align:start position:0%
represents this evolution. That's
awesome.<00:08:55.600><c> Okay,</c><00:08:56.480><c> now</c><00:08:56.800><c> here's</c><00:08:57.040><c> a</c><00:08:57.200><c> question.</c>

08:58.070 --> 08:58.080 align:start position:0%
awesome. Okay, now here's a question.
 

08:58.080 --> 09:01.750 align:start position:0%
awesome. Okay, now here's a question.
What<00:08:58.320><c> if</c><00:08:59.200><c> the</c><00:08:59.600><c> initial</c><00:09:00.080><c> graph</c><00:09:00.399><c> had</c><00:09:00.800><c> two</c><00:09:01.440><c> sign</c>

09:01.750 --> 09:01.760 align:start position:0%
What if the initial graph had two sign
 

09:01.760 --> 09:04.150 align:start position:0%
What if the initial graph had two sign
curves?<00:09:02.399><c> This</c><00:09:02.640><c> also</c><00:09:03.040><c> behaves</c><00:09:03.360><c> the</c><00:09:03.600><c> same</c><00:09:03.839><c> way.</c>

09:04.150 --> 09:04.160 align:start position:0%
curves? This also behaves the same way.
 

09:04.160 --> 09:06.870 align:start position:0%
curves? This also behaves the same way.
But<00:09:04.320><c> the</c><00:09:04.560><c> question</c><00:09:04.880><c> is,</c><00:09:05.760><c> will</c><00:09:06.000><c> it</c><00:09:06.240><c> shrink</c><00:09:06.640><c> at</c>

09:06.870 --> 09:06.880 align:start position:0%
But the question is, will it shrink at
 

09:06.880 --> 09:08.870 align:start position:0%
But the question is, will it shrink at
the<00:09:07.040><c> same</c><00:09:07.279><c> rate</c><00:09:07.600><c> as</c><00:09:07.839><c> the</c><00:09:08.080><c> previous</c><00:09:08.399><c> one?</c><00:09:08.720><c> Will</c>

09:08.870 --> 09:08.880 align:start position:0%
the same rate as the previous one? Will
 

09:08.880 --> 09:11.750 align:start position:0%
the same rate as the previous one? Will
it<00:09:09.040><c> shrink</c><00:09:09.440><c> faster</c><00:09:09.920><c> or</c><00:09:10.320><c> slower?</c><00:09:11.440><c> Why</c><00:09:11.600><c> don't</c>

09:11.750 --> 09:11.760 align:start position:0%
it shrink faster or slower? Why don't
 

09:11.760 --> 09:15.030 align:start position:0%
it shrink faster or slower? Why don't
you<00:09:11.920><c> pause</c><00:09:12.160><c> the</c><00:09:12.320><c> video</c><00:09:12.480><c> and</c><00:09:12.720><c> think</c><00:09:12.880><c> about</c><00:09:13.120><c> it?</c>

09:15.030 --> 09:15.040 align:start position:0%
you pause the video and think about it?
 

09:15.040 --> 09:17.430 align:start position:0%
you pause the video and think about it?
Well,<00:09:15.360><c> now</c><00:09:15.519><c> we</c><00:09:15.760><c> have</c><00:09:15.920><c> squeezed</c><00:09:16.560><c> two</c><00:09:17.120><c> sine</c>

09:17.430 --> 09:17.440 align:start position:0%
Well, now we have squeezed two sine
 

09:17.440 --> 09:19.590 align:start position:0%
Well, now we have squeezed two sine
waves<00:09:17.760><c> in</c><00:09:17.920><c> that</c><00:09:18.160><c> same</c><00:09:18.320><c> place</c><00:09:18.560><c> as</c><00:09:18.800><c> before.</c><00:09:19.440><c> And</c>

09:19.590 --> 09:19.600 align:start position:0%
waves in that same place as before. And
 

09:19.600 --> 09:21.430 align:start position:0%
waves in that same place as before. And
so<00:09:19.839><c> now</c><00:09:19.920><c> the</c><00:09:20.160><c> curvature</c><00:09:20.560><c> everywhere</c><00:09:20.959><c> is</c><00:09:21.200><c> much</c>

09:21.430 --> 09:21.440 align:start position:0%
so now the curvature everywhere is much
 

09:21.440 --> 09:23.509 align:start position:0%
so now the curvature everywhere is much
higher,<00:09:21.920><c> right?</c><00:09:22.160><c> We</c><00:09:22.320><c> can</c><00:09:22.480><c> visually</c><00:09:22.880><c> see</c><00:09:23.040><c> it.</c>

09:23.509 --> 09:23.519 align:start position:0%
higher, right? We can visually see it.
 

09:23.519 --> 09:25.110 align:start position:0%
higher, right? We can visually see it.
So<00:09:23.760><c> the</c><00:09:24.000><c> temperature</c><00:09:24.399><c> change</c><00:09:24.640><c> is</c><00:09:24.880><c> going</c><00:09:24.959><c> to</c><00:09:25.040><c> be</c>

09:25.110 --> 09:25.120 align:start position:0%
So the temperature change is going to be
 

09:25.120 --> 09:27.269 align:start position:0%
So the temperature change is going to be
much<00:09:25.360><c> quicker.</c><00:09:26.160><c> The</c><00:09:26.399><c> whole</c><00:09:26.640><c> graph</c><00:09:26.959><c> is</c><00:09:27.200><c> going</c>

09:27.269 --> 09:27.279 align:start position:0%
much quicker. The whole graph is going
 

09:27.279 --> 09:30.790 align:start position:0%
much quicker. The whole graph is going
to<00:09:27.440><c> shrink</c><00:09:27.920><c> much</c><00:09:28.320><c> faster</c><00:09:28.880><c> than</c><00:09:29.200><c> before.</c>

09:30.790 --> 09:30.800 align:start position:0%
to shrink much faster than before.
 

09:30.800 --> 09:34.230 align:start position:0%
to shrink much faster than before.
Higher<00:09:31.200><c> the</c><00:09:31.440><c> frequency,</c><00:09:32.800><c> the</c><00:09:33.120><c> faster</c><00:09:33.760><c> it</c>

09:34.230 --> 09:34.240 align:start position:0%
Higher the frequency, the faster it
 

09:34.240 --> 09:38.070 align:start position:0%
Higher the frequency, the faster it
shrinks.<00:09:35.120><c> Makes</c><00:09:35.519><c> sense,</c><00:09:36.160><c> right?</c><00:09:37.519><c> Okay,</c><00:09:37.839><c> this</c>

09:38.070 --> 09:38.080 align:start position:0%
shrinks. Makes sense, right? Okay, this
 

09:38.080 --> 09:40.949 align:start position:0%
shrinks. Makes sense, right? Okay, this
is<00:09:38.240><c> great.</c><00:09:38.560><c> But</c><00:09:38.720><c> for</c><00:09:39.519><c> what's</c><00:09:39.839><c> the</c><00:09:40.080><c> point?</c><00:09:40.720><c> I</c>

09:40.949 --> 09:40.959 align:start position:0%
is great. But for what's the point? I
 

09:40.959 --> 09:42.870 align:start position:0%
is great. But for what's the point? I
mean,<00:09:41.279><c> this</c><00:09:41.600><c> would</c><00:09:41.760><c> only</c><00:09:42.080><c> work</c><00:09:42.320><c> if</c><00:09:42.640><c> the</c>

09:42.870 --> 09:42.880 align:start position:0%
mean, this would only work if the
 

09:42.880 --> 09:44.870 align:start position:0%
mean, this would only work if the
initial<00:09:43.279><c> profile</c><00:09:43.760><c> just</c><00:09:44.080><c> happens</c><00:09:44.399><c> to</c><00:09:44.560><c> be</c><00:09:44.640><c> a</c>

09:44.870 --> 09:44.880 align:start position:0%
initial profile just happens to be a
 

09:44.880 --> 09:47.269 align:start position:0%
initial profile just happens to be a
sign<00:09:45.120><c> curve.</c><00:09:46.000><c> Real</c><00:09:46.320><c> world</c><00:09:46.480><c> profiles</c><00:09:46.959><c> are</c><00:09:47.120><c> not</c>

09:47.269 --> 09:47.279 align:start position:0%
sign curve. Real world profiles are not
 

09:47.279 --> 09:49.269 align:start position:0%
sign curve. Real world profiles are not
sign<00:09:47.600><c> curves.</c><00:09:48.080><c> So,</c><00:09:48.399><c> what</c><00:09:48.640><c> can</c><00:09:48.800><c> we</c><00:09:48.959><c> do</c><00:09:49.120><c> with</c>

09:49.269 --> 09:49.279 align:start position:0%
sign curves. So, what can we do with
 

09:49.279 --> 09:51.990 align:start position:0%
sign curves. So, what can we do with
that?<00:09:50.000><c> Well,</c><00:09:50.240><c> Forier</c><00:09:50.720><c> asks,</c><00:09:51.200><c> what</c><00:09:51.440><c> if</c><00:09:51.600><c> we</c><00:09:51.839><c> had</c>

09:51.990 --> 09:52.000 align:start position:0%
that? Well, Forier asks, what if we had
 

09:52.000 --> 09:55.750 align:start position:0%
that? Well, Forier asks, what if we had
the<00:09:52.240><c> initial</c><00:09:52.640><c> profile</c><00:09:53.120><c> which</c><00:09:53.360><c> was</c><00:09:54.000><c> sum</c><00:09:54.800><c> of</c><00:09:55.200><c> two</c>

09:55.750 --> 09:55.760 align:start position:0%
the initial profile which was sum of two
 

09:55.760 --> 09:57.670 align:start position:0%
the initial profile which was sum of two
sign<00:09:56.080><c> curves</c><00:09:56.480><c> of</c><00:09:56.640><c> different</c><00:09:56.880><c> frequencies?</c>

09:57.670 --> 09:57.680 align:start position:0%
sign curves of different frequencies?
 

09:57.680 --> 09:59.590 align:start position:0%
sign curves of different frequencies?
The<00:09:58.000><c> result</c><00:09:58.240><c> will</c><00:09:58.480><c> no</c><00:09:58.720><c> longer</c><00:09:58.959><c> be</c><00:09:59.120><c> a</c><00:09:59.279><c> sign</c>

09:59.590 --> 09:59.600 align:start position:0%
The result will no longer be a sign
 

09:59.600 --> 10:01.509 align:start position:0%
The result will no longer be a sign
curve.<00:10:00.080><c> Its</c><00:10:00.480><c> shape</c><00:10:00.720><c> will</c><00:10:00.959><c> look</c><00:10:01.200><c> pretty</c>

10:01.509 --> 10:01.519 align:start position:0%
curve. Its shape will look pretty
 

10:01.519 --> 10:03.430 align:start position:0%
curve. Its shape will look pretty
complicated<00:10:02.080><c> right</c><00:10:02.399><c> now.</c><00:10:02.560><c> If</c><00:10:02.720><c> you</c><00:10:02.880><c> try</c><00:10:03.120><c> to</c>

10:03.430 --> 10:03.440 align:start position:0%
complicated right now. If you try to
 

10:03.440 --> 10:05.910 align:start position:0%
complicated right now. If you try to
solve<00:10:03.760><c> how</c><00:10:04.000><c> it</c><00:10:04.240><c> will</c><00:10:04.399><c> evolve</c><00:10:04.959><c> directly,</c><00:10:05.680><c> that</c>

10:05.910 --> 10:05.920 align:start position:0%
solve how it will evolve directly, that
 

10:05.920 --> 10:07.990 align:start position:0%
solve how it will evolve directly, that
would<00:10:06.080><c> be</c><00:10:06.160><c> a</c><00:10:06.320><c> nightmare.</c><00:10:07.120><c> But</c><00:10:07.279><c> we</c><00:10:07.519><c> don't</c><00:10:07.839><c> have</c>

10:07.990 --> 10:08.000 align:start position:0%
would be a nightmare. But we don't have
 

10:08.000 --> 10:10.790 align:start position:0%
would be a nightmare. But we don't have
to.<00:10:08.640><c> We</c><00:10:08.880><c> can</c><00:10:09.120><c> split</c><00:10:09.440><c> it</c><00:10:09.680><c> back</c><00:10:10.000><c> into</c><00:10:10.399><c> its</c>

10:10.790 --> 10:10.800 align:start position:0%
to. We can split it back into its
 

10:10.800 --> 10:13.269 align:start position:0%
to. We can split it back into its
component<00:10:11.519><c> sign</c><00:10:11.839><c> curves.</c><00:10:12.800><c> And</c><00:10:12.959><c> the</c><00:10:13.120><c> whole</c>

10:13.269 --> 10:13.279 align:start position:0%
component sign curves. And the whole
 

10:13.279 --> 10:16.389 align:start position:0%
component sign curves. And the whole
point<00:10:13.440><c> is</c><00:10:13.680><c> we</c><00:10:14.000><c> already</c><00:10:14.399><c> know</c><00:10:15.040><c> exactly</c><00:10:16.000><c> how</c>

10:16.389 --> 10:16.399 align:start position:0%
point is we already know exactly how
 

10:16.399 --> 10:19.350 align:start position:0%
point is we already know exactly how
each<00:10:16.720><c> one</c><00:10:17.200><c> evolves.</c><00:10:18.399><c> The</c><00:10:18.640><c> high</c><00:10:18.959><c> frequency</c>

10:19.350 --> 10:19.360 align:start position:0%
each one evolves. The high frequency
 

10:19.360 --> 10:23.269 align:start position:0%
each one evolves. The high frequency
ones<00:10:19.920><c> die</c><00:10:20.160><c> out</c><00:10:20.480><c> much</c><00:10:20.720><c> faster,</c><00:10:21.760><c> right?</c><00:10:22.880><c> And</c><00:10:23.120><c> so</c>

10:23.269 --> 10:23.279 align:start position:0%
ones die out much faster, right? And so
 

10:23.279 --> 10:25.750 align:start position:0%
ones die out much faster, right? And so
then<00:10:23.519><c> what</c><00:10:23.680><c> we</c><00:10:23.920><c> do</c><00:10:24.079><c> is</c><00:10:24.399><c> we</c><00:10:24.640><c> simply</c><00:10:25.120><c> add</c><00:10:25.519><c> them</c>

10:25.750 --> 10:25.760 align:start position:0%
then what we do is we simply add them
 

10:25.760 --> 10:29.670 align:start position:0%
then what we do is we simply add them
back<00:10:26.560><c> together</c><00:10:27.120><c> to</c><00:10:27.440><c> get</c><00:10:27.600><c> our</c><00:10:27.920><c> final</c><00:10:28.399><c> profile.</c>

10:29.670 --> 10:29.680 align:start position:0%
back together to get our final profile.
 

10:29.680 --> 10:32.150 align:start position:0%
back together to get our final profile.
Oh<00:10:30.079><c> my</c><00:10:30.560><c> god,</c>

10:32.150 --> 10:32.160 align:start position:0%
Oh my god,
 

10:32.160 --> 10:35.190 align:start position:0%
Oh my god,
that<00:10:32.399><c> is</c><00:10:32.640><c> amazing.</c><00:10:33.440><c> So</c><00:10:33.680><c> the</c><00:10:33.920><c> trick</c><00:10:34.160><c> is</c><00:10:34.800><c> we</c>

10:35.190 --> 10:35.200 align:start position:0%
that is amazing. So the trick is we
 

10:35.200 --> 10:37.350 align:start position:0%
that is amazing. So the trick is we
never<00:10:35.680><c> solve</c><00:10:36.000><c> a</c><00:10:36.320><c> complicated</c><00:10:36.959><c> graph</c>

10:37.350 --> 10:37.360 align:start position:0%
never solve a complicated graph
 

10:37.360 --> 10:39.670 align:start position:0%
never solve a complicated graph
directly.<00:10:38.160><c> Instead,</c><00:10:38.480><c> we</c><00:10:38.800><c> just</c><00:10:38.959><c> break</c><00:10:39.200><c> it</c><00:10:39.440><c> down</c>

10:39.670 --> 10:39.680 align:start position:0%
directly. Instead, we just break it down
 

10:39.680 --> 10:42.310 align:start position:0%
directly. Instead, we just break it down
into<00:10:40.079><c> its</c><00:10:40.320><c> component</c><00:10:40.800><c> sine</c><00:10:41.040><c> waves.</c><00:10:41.760><c> Let</c><00:10:42.079><c> each</c>

10:42.310 --> 10:42.320 align:start position:0%
into its component sine waves. Let each
 

10:42.320 --> 10:44.949 align:start position:0%
into its component sine waves. Let each
one<00:10:42.720><c> evolve</c><00:10:43.279><c> over</c><00:10:43.600><c> time</c><00:10:44.160><c> because</c><00:10:44.399><c> we</c><00:10:44.640><c> know</c><00:10:44.720><c> the</c>

10:44.949 --> 10:44.959 align:start position:0%
one evolve over time because we know the
 

10:44.959 --> 10:47.350 align:start position:0%
one evolve over time because we know the
exact<00:10:45.200><c> math</c><00:10:45.440><c> for</c><00:10:45.680><c> that</c><00:10:46.240><c> and</c><00:10:46.480><c> then</c><00:10:46.720><c> stitch</c><00:10:47.120><c> them</c>

10:47.350 --> 10:47.360 align:start position:0%
exact math for that and then stitch them
 

10:47.360 --> 10:51.829 align:start position:0%
exact math for that and then stitch them
back<00:10:47.680><c> together.</c><00:10:48.240><c> That</c><00:10:48.480><c> is</c><00:10:48.640><c> so</c><00:10:49.360><c> ingenious.</c>

10:51.829 --> 10:51.839 align:start position:0%
back together. That is so ingenious.
 

10:51.839 --> 10:53.829 align:start position:0%
back together. That is so ingenious.
This<00:10:52.079><c> works</c><00:10:52.399><c> because</c><00:10:52.800><c> derivatives</c><00:10:53.440><c> are</c>

10:53.829 --> 10:53.839 align:start position:0%
This works because derivatives are
 

10:53.839 --> 10:56.389 align:start position:0%
This works because derivatives are
linear<00:10:54.320><c> operators.</c><00:10:55.200><c> Meaning</c><00:10:55.680><c> derivative</c><00:10:56.160><c> of</c>

10:56.389 --> 10:56.399 align:start position:0%
linear operators. Meaning derivative of
 

10:56.399 --> 10:59.910 align:start position:0%
linear operators. Meaning derivative of
a<00:10:56.640><c> +</c><00:10:56.959><c> b</c><00:10:57.600><c> equals</c><00:10:58.079><c> derivative</c><00:10:58.560><c> of</c><00:10:58.800><c> a</c><00:10:59.279><c> plus</c>

10:59.910 --> 10:59.920 align:start position:0%
a + b equals derivative of a plus
 

10:59.920 --> 11:03.590 align:start position:0%
a + b equals derivative of a plus
derivative<00:11:00.480><c> of</c><00:11:01.040><c> b.</c><00:11:01.680><c> But</c><00:11:01.920><c> again</c><00:11:02.399><c> for</c><00:11:02.959><c> this</c><00:11:03.200><c> only</c>

11:03.590 --> 11:03.600 align:start position:0%
derivative of b. But again for this only
 

11:03.600 --> 11:07.590 align:start position:0%
derivative of b. But again for this only
works<00:11:04.480><c> provided</c><00:11:05.600><c> the</c><00:11:05.920><c> initial</c><00:11:06.399><c> profile</c><00:11:07.200><c> just</c>

11:07.590 --> 11:07.600 align:start position:0%
works provided the initial profile just
 

11:07.600 --> 11:09.910 align:start position:0%
works provided the initial profile just
happens<00:11:08.000><c> to</c><00:11:08.240><c> be</c><00:11:08.480><c> a</c><00:11:08.720><c> sum</c><00:11:08.959><c> of</c><00:11:09.120><c> two</c><00:11:09.279><c> or</c><00:11:09.519><c> more</c><00:11:09.680><c> sine</c>

11:09.910 --> 11:09.920 align:start position:0%
happens to be a sum of two or more sine
 

11:09.920 --> 11:14.550 align:start position:0%
happens to be a sum of two or more sine
waves.<00:11:10.399><c> Right?</c><00:11:11.200><c> What</c><00:11:11.440><c> if</c><00:11:11.600><c> it's</c><00:11:12.000><c> not?</c><00:11:13.200><c> And</c><00:11:13.600><c> for</c>

11:14.550 --> 11:14.560 align:start position:0%
waves. Right? What if it's not? And for
 

11:14.560 --> 11:17.590 align:start position:0%
waves. Right? What if it's not? And for
makes<00:11:15.120><c> an</c><00:11:15.519><c> incredible</c><00:11:16.240><c> leap</c><00:11:16.640><c> at</c><00:11:16.800><c> this</c><00:11:17.040><c> point.</c>

11:17.590 --> 11:17.600 align:start position:0%
makes an incredible leap at this point.
 

11:17.600 --> 11:21.350 align:start position:0%
makes an incredible leap at this point.
He<00:11:17.839><c> asks</c><00:11:18.880><c> what</c><00:11:19.120><c> if</c><00:11:19.519><c> this</c><00:11:19.760><c> is</c><00:11:20.240><c> always</c><00:11:20.640><c> true?</c>

11:21.350 --> 11:21.360 align:start position:0%
He asks what if this is always true?
 

11:21.360 --> 11:24.949 align:start position:0%
He asks what if this is always true?
What<00:11:21.600><c> if</c><00:11:22.240><c> any</c><00:11:22.959><c> graph</c><00:11:23.440><c> or</c><00:11:23.839><c> any</c><00:11:24.320><c> shape</c><00:11:24.640><c> that</c><00:11:24.800><c> you</c>

11:24.949 --> 11:24.959 align:start position:0%
What if any graph or any shape that you
 

11:24.959 --> 11:27.750 align:start position:0%
What if any graph or any shape that you
can<00:11:25.200><c> draw</c><00:11:25.440><c> with</c><00:11:25.680><c> a</c><00:11:25.839><c> pen</c><00:11:26.000><c> and</c><00:11:26.320><c> paper</c><00:11:27.200><c> can</c><00:11:27.519><c> be</c>

11:27.750 --> 11:27.760 align:start position:0%
can draw with a pen and paper can be
 

11:27.760 --> 11:30.630 align:start position:0%
can draw with a pen and paper can be
broken<00:11:28.160><c> down</c><00:11:28.480><c> or</c><00:11:28.800><c> can</c><00:11:29.040><c> be</c><00:11:29.200><c> built</c><00:11:29.920><c> by</c><00:11:30.240><c> adding</c>

11:30.630 --> 11:30.640 align:start position:0%
broken down or can be built by adding
 

11:30.640 --> 11:33.430 align:start position:0%
broken down or can be built by adding
lots<00:11:30.959><c> and</c><00:11:31.200><c> lots</c><00:11:31.440><c> of</c><00:11:31.600><c> sine</c><00:11:31.920><c> waves.</c><00:11:32.800><c> Then</c><00:11:33.120><c> the</c>

11:33.430 --> 11:33.440 align:start position:0%
lots and lots of sine waves. Then the
 

11:33.440 --> 11:36.710 align:start position:0%
lots and lots of sine waves. Then the
heat<00:11:33.680><c> equation</c><00:11:34.160><c> would</c><00:11:34.480><c> be</c><00:11:34.720><c> officially</c><00:11:35.920><c> solved</c>

11:36.710 --> 11:36.720 align:start position:0%
heat equation would be officially solved
 

11:36.720 --> 11:38.470 align:start position:0%
heat equation would be officially solved
because<00:11:36.959><c> you</c><00:11:37.200><c> can</c><00:11:37.279><c> use</c><00:11:37.440><c> the</c><00:11:37.680><c> same</c><00:11:37.920><c> technique</c>

11:38.470 --> 11:38.480 align:start position:0%
because you can use the same technique
 

11:38.480 --> 11:43.350 align:start position:0%
because you can use the same technique
to<00:11:38.800><c> solve</c><00:11:39.279><c> any</c><00:11:39.680><c> shape.</c><00:11:41.040><c> This</c><00:11:41.440><c> is</c><00:11:41.680><c> the</c><00:11:42.079><c> birth</c><00:11:42.720><c> of</c>

11:43.350 --> 11:43.360 align:start position:0%
to solve any shape. This is the birth of
 

11:43.360 --> 11:49.030 align:start position:0%
to solve any shape. This is the birth of
Forier<00:11:44.480><c> series.</c><00:11:45.680><c> Here</c><00:11:46.320><c> it</c><00:11:46.640><c> is.</c>

11:49.030 --> 11:49.040 align:start position:0%
Forier series. Here it is.
 

11:49.040 --> 11:51.350 align:start position:0%
Forier series. Here it is.
Forier's<00:11:49.680><c> intuition</c><00:11:50.399><c> is</c><00:11:50.720><c> kind</c><00:11:50.880><c> of</c><00:11:51.040><c> like</c><00:11:51.200><c> a</c>

11:51.350 --> 11:51.360 align:start position:0%
Forier's intuition is kind of like a
 

11:51.360 --> 11:53.430 align:start position:0%
Forier's intuition is kind of like a
mathematical<00:11:52.000><c> prism.</c><00:11:52.640><c> Just</c><00:11:52.880><c> like</c><00:11:53.040><c> how</c><00:11:53.200><c> a</c>

11:53.430 --> 11:53.440 align:start position:0%
mathematical prism. Just like how a
 

11:53.440 --> 11:56.069 align:start position:0%
mathematical prism. Just like how a
prism<00:11:54.240><c> splits</c><00:11:54.720><c> light</c><00:11:54.959><c> into</c><00:11:55.200><c> its</c><00:11:55.440><c> constituent</c>

11:56.069 --> 11:56.079 align:start position:0%
prism splits light into its constituent
 

11:56.079 --> 11:59.509 align:start position:0%
prism splits light into its constituent
colors.<00:11:56.959><c> For</c><00:11:57.519><c> asserts</c><00:11:58.079><c> that</c><00:11:58.560><c> any</c><00:11:58.959><c> shape</c><00:11:59.360><c> can</c>

11:59.509 --> 11:59.519 align:start position:0%
colors. For asserts that any shape can
 

11:59.519 --> 12:02.630 align:start position:0%
colors. For asserts that any shape can
be<00:11:59.920><c> split</c><00:12:00.320><c> into</c><00:12:00.560><c> its</c><00:12:00.959><c> constituent</c><00:12:02.079><c> sign</c>

12:02.630 --> 12:02.640 align:start position:0%
be split into its constituent sign
 

12:02.640 --> 12:05.990 align:start position:0%
be split into its constituent sign
curves.<00:12:04.000><c> This</c><00:12:04.160><c> means</c><00:12:04.399><c> we</c><00:12:04.640><c> can</c><00:12:04.880><c> transform</c><00:12:05.600><c> any</c>

12:05.990 --> 12:06.000 align:start position:0%
curves. This means we can transform any
 

12:06.000 --> 12:08.710 align:start position:0%
curves. This means we can transform any
shape<00:12:06.399><c> into</c><00:12:06.720><c> a</c><00:12:06.959><c> graph</c><00:12:07.760><c> showing</c><00:12:08.079><c> the</c><00:12:08.399><c> strength</c>

12:08.710 --> 12:08.720 align:start position:0%
shape into a graph showing the strength
 

12:08.720 --> 12:11.829 align:start position:0%
shape into a graph showing the strength
of<00:12:09.040><c> those</c><00:12:09.519><c> constituent</c><00:12:10.560><c> frequencies.</c><00:12:11.600><c> Today</c>

12:11.829 --> 12:11.839 align:start position:0%
of those constituent frequencies. Today
 

12:11.839 --> 12:14.629 align:start position:0%
of those constituent frequencies. Today
we<00:12:12.000><c> call</c><00:12:12.160><c> this</c><00:12:12.480><c> graph</c><00:12:13.120><c> the</c><00:12:13.440><c> forier</c><00:12:14.000><c> transform</c>

12:14.629 --> 12:14.639 align:start position:0%
we call this graph the forier transform
 

12:14.639 --> 12:16.629 align:start position:0%
we call this graph the forier transform
because<00:12:14.959><c> we</c><00:12:15.200><c> are</c><00:12:15.519><c> literally</c><00:12:16.079><c> transforming</c>

12:16.629 --> 12:16.639 align:start position:0%
because we are literally transforming
 

12:16.639 --> 12:19.750 align:start position:0%
because we are literally transforming
this<00:12:16.880><c> graph</c><00:12:17.279><c> into</c><00:12:18.000><c> the</c><00:12:18.320><c> frequency</c><00:12:18.880><c> domain.</c>

12:19.750 --> 12:19.760 align:start position:0%
this graph into the frequency domain.
 

12:19.760 --> 12:22.150 align:start position:0%
this graph into the frequency domain.
But<00:12:20.000><c> Forier</c><00:12:20.639><c> right</c><00:12:20.880><c> now</c><00:12:21.279><c> this</c><00:12:21.519><c> is</c><00:12:21.680><c> just</c><00:12:21.839><c> a</c>

12:22.150 --> 12:22.160 align:start position:0%
But Forier right now this is just a
 

12:22.160 --> 12:24.550 align:start position:0%
But Forier right now this is just a
fantastic<00:12:22.880><c> idea.</c><00:12:23.440><c> I</c><00:12:23.680><c> know</c><00:12:23.760><c> we</c><00:12:24.000><c> want</c><00:12:24.079><c> it</c><00:12:24.240><c> to</c><00:12:24.399><c> be</c>

12:24.550 --> 12:24.560 align:start position:0%
fantastic idea. I know we want it to be
 

12:24.560 --> 12:26.069 align:start position:0%
fantastic idea. I know we want it to be
true<00:12:24.720><c> but</c><00:12:24.959><c> how</c><00:12:25.120><c> do</c><00:12:25.200><c> we</c><00:12:25.360><c> know</c><00:12:25.519><c> that</c><00:12:25.680><c> this</c><00:12:25.920><c> is</c>

12:26.069 --> 12:26.079 align:start position:0%
true but how do we know that this is
 

12:26.079 --> 12:28.230 align:start position:0%
true but how do we know that this is
indeed<00:12:26.639><c> true</c><00:12:26.880><c> and</c><00:12:27.200><c> not</c><00:12:27.440><c> just</c><00:12:27.680><c> wishful</c>

12:28.230 --> 12:28.240 align:start position:0%
indeed true and not just wishful
 

12:28.240 --> 12:31.509 align:start position:0%
indeed true and not just wishful
thinking?<00:12:29.279><c> For</c><00:12:29.680><c> says</c><00:12:29.920><c> m</c><00:12:30.320><c> the</c><00:12:30.480><c> only</c><00:12:30.720><c> way</c><00:12:31.040><c> to</c>

12:31.509 --> 12:31.519 align:start position:0%
thinking? For says m the only way to
 

12:31.519 --> 12:34.310 align:start position:0%
thinking? For says m the only way to
trust<00:12:31.920><c> ideas</c><00:12:32.320><c> as</c><00:12:32.560><c> wild</c><00:12:32.880><c> as</c><00:12:33.040><c> these</c><00:12:33.760><c> is</c><00:12:33.920><c> to</c><00:12:34.160><c> try</c>

12:34.310 --> 12:34.320 align:start position:0%
trust ideas as wild as these is to try
 

12:34.320 --> 12:36.870 align:start position:0%
trust ideas as wild as these is to try
and<00:12:34.639><c> build</c><00:12:34.880><c> it</c><00:12:35.120><c> ourselves</c><00:12:35.519><c> from</c><00:12:35.839><c> scratch</c><00:12:36.720><c> just</c>

12:34.320 --> 12:37.429 align:start position:0%
the annual subscription. So, happy
learning.<00:14:03.360><c> Back</c><00:14:03.519><c> to</c><00:14:03.680><c> the</c><00:14:03.920><c> video.</c>

12:36.706 --> 12:36.716 align:start position:0%
learning. Back to the video.
 

12:36.716 --> 12:38.706 align:start position:0%
learning. Back to the video.
So,<00:14:06.079><c> for</c><00:14:06.560><c> how</c><00:14:06.639><c> do</c><00:14:06.800><c> we</c><00:14:06.959><c> actually</c><00:14:07.360><c> prove</c><00:14:07.600><c> that</c>

12:38.706 --> 12:38.716 align:start position:0%
So, for how do we actually prove that
 

12:38.716 --> 12:41.987 align:start position:0%
So, for how do we actually prove that
any<00:14:08.160><c> shape</c><00:14:08.560><c> can</c><00:14:08.800><c> indeed</c><00:14:09.360><c> be</c><00:14:09.519><c> built</c><00:14:10.160><c> using</c><00:14:10.800><c> sign</c>

12:41.987 --> 12:41.997 align:start position:0%
any shape can indeed be built using sign
 

12:41.997 --> 12:45.107 align:start position:0%
any shape can indeed be built using sign
curves?<00:14:12.079><c> Well,</c><00:14:12.720><c> for</c><00:14:13.040><c> any</c><00:14:13.360><c> given</c><00:14:13.680><c> shape,</c><00:14:14.079><c> we</c>

12:45.107 --> 12:45.117 align:start position:0%
curves? Well, for any given shape, we
 

12:45.117 --> 12:47.187 align:start position:0%
curves? Well, for any given shape, we
just<00:14:14.399><c> need</c><00:14:14.560><c> to</c><00:14:14.720><c> know</c><00:14:15.040><c> what</c><00:14:15.440><c> frequencies</c><00:14:16.160><c> make</c>

12:47.187 --> 12:47.197 align:start position:0%
just need to know what frequencies make
 

12:47.197 --> 12:50.627 align:start position:0%
just need to know what frequencies make
it<00:14:16.560><c> up</c><00:14:17.040><c> and</c><00:14:17.519><c> what</c><00:14:17.839><c> the</c><00:14:18.240><c> height</c><00:14:18.959><c> of</c><00:14:19.279><c> those</c>

12:50.627 --> 12:50.637 align:start position:0%
it up and what the height of those
 

12:50.637 --> 12:53.587 align:start position:0%
it up and what the height of those
frequencies<00:14:20.639><c> are.</c><00:14:21.440><c> Then</c><00:14:21.680><c> we</c><00:14:21.839><c> just</c><00:14:22.160><c> add</c><00:14:22.480><c> them</c>

12:53.587 --> 12:53.597 align:start position:0%
frequencies are. Then we just add them
 

12:53.597 --> 12:55.907 align:start position:0%
frequencies are. Then we just add them
up<00:14:23.040><c> and</c><00:14:23.360><c> check</c><00:14:23.600><c> if</c><00:14:23.839><c> it</c><00:14:24.000><c> gives</c><00:14:24.240><c> us</c><00:14:24.480><c> the</c><00:14:24.720><c> right</c>

12:55.907 --> 12:55.917 align:start position:0%
up and check if it gives us the right
 

12:55.917 --> 12:57.907 align:start position:0%
up and check if it gives us the right
shape.<00:14:25.680><c> We</c><00:14:25.839><c> can</c><00:14:25.920><c> add</c><00:14:26.160><c> them</c><00:14:26.320><c> up</c><00:14:26.480><c> just</c><00:14:26.639><c> by</c><00:14:26.800><c> using</c>

12:57.907 --> 12:57.917 align:start position:0%
shape. We can add them up just by using
 

12:57.917 --> 13:00.227 align:start position:0%
shape. We can add them up just by using
pen<00:14:27.279><c> and</c><00:14:27.519><c> paper</c><00:14:27.839><c> or</c><00:14:28.079><c> we</c><00:14:28.320><c> could</c><00:14:28.480><c> use</c><00:14:28.880><c> machines</c>

13:00.227 --> 13:00.237 align:start position:0%
pen and paper or we could use machines
 

13:00.237 --> 13:01.907 align:start position:0%
pen and paper or we could use machines
to<00:14:29.519><c> do</c><00:14:29.680><c> it</c><00:14:29.839><c> or</c><00:14:30.079><c> we</c><00:14:30.160><c> can</c><00:14:30.320><c> just</c><00:14:30.480><c> put</c><00:14:30.560><c> it</c><00:14:30.720><c> in</c><00:14:30.800><c> a</c>

13:01.907 --> 13:01.917 align:start position:0%
to do it or we can just put it in a
 

13:01.917 --> 13:04.547 align:start position:0%
to do it or we can just put it in a
simulator.<00:14:31.920><c> But</c><00:14:32.320><c> how</c><00:14:32.560><c> do</c><00:14:32.720><c> we</c><00:14:32.959><c> figure</c><00:14:33.199><c> out</c>

13:04.547 --> 13:04.557 align:start position:0%
simulator. But how do we figure out
 

13:04.557 --> 13:06.867 align:start position:0%
simulator. But how do we figure out
their<00:14:34.079><c> constituent</c><00:14:34.639><c> frequencies</c><00:14:35.360><c> and</c><00:14:35.680><c> the</c>

13:06.867 --> 13:06.877 align:start position:0%
their constituent frequencies and the
 

13:06.877 --> 13:08.946 align:start position:0%
their constituent frequencies and the
heights<00:14:36.399><c> that</c><00:14:36.639><c> make</c><00:14:36.800><c> them</c><00:14:37.040><c> up?</c><00:14:37.519><c> I</c><00:14:37.760><c> don't</c><00:14:37.920><c> even</c>

13:08.946 --> 13:08.956 align:start position:0%
heights that make them up? I don't even
 

13:08.956 --> 13:11.506 align:start position:0%
heights that make them up? I don't even
know<00:14:38.240><c> where</c><00:14:38.399><c> to</c><00:14:38.560><c> begin.</c><00:14:39.199><c> For</c><00:14:39.600><c> says,</c><00:14:40.160><c> let's</c>

13:11.506 --> 13:11.516 align:start position:0%
know where to begin. For says, let's
 

13:11.516 --> 13:15.826 align:start position:0%
know where to begin. For says, let's
assume<00:14:41.519><c> that</c><00:14:42.160><c> our</c><00:14:42.480><c> signal</c><00:14:43.680><c> repeats</c><00:14:44.240><c> forever</c>

13:15.826 --> 13:15.836 align:start position:0%
assume that our signal repeats forever
 

13:15.836 --> 13:19.266 align:start position:0%
assume that our signal repeats forever
with<00:14:45.199><c> a</c><00:14:45.440><c> period</c><00:14:46.160><c> t.</c><00:14:47.360><c> Then</c><00:14:47.839><c> finding</c><00:14:48.160><c> the</c>

13:19.266 --> 13:19.276 align:start position:0%
with a period t. Then finding the
 

13:19.276 --> 13:21.347 align:start position:0%
with a period t. Then finding the
frequencies<00:14:48.880><c> is</c><00:14:49.199><c> very</c><00:14:49.440><c> easy.</c><00:14:49.760><c> He</c><00:14:50.000><c> says,</c>

13:21.347 --> 13:21.357 align:start position:0%
frequencies is very easy. He says,
 

13:21.357 --> 13:24.066 align:start position:0%
frequencies is very easy. He says,
"Wait,<00:14:50.880><c> wait,</c><00:14:51.120><c> wait.</c><00:14:51.920><c> Why</c><00:14:52.160><c> are</c><00:14:52.399><c> you</c><00:14:52.639><c> assuming</c>

13:24.066 --> 13:24.076 align:start position:0%
"Wait, wait, wait. Why are you assuming
 

13:24.076 --> 13:26.467 align:start position:0%
"Wait, wait, wait. Why are you assuming
that<00:14:53.519><c> our</c><00:14:53.920><c> signal</c><00:14:54.399><c> is</c><00:14:54.639><c> periodic</c><00:14:55.040><c> and</c><00:14:55.279><c> repeats</c>

13:26.467 --> 13:26.477 align:start position:0%
that our signal is periodic and repeats
 

13:26.477 --> 13:29.826 align:start position:0%
that our signal is periodic and repeats
forever?<00:14:56.160><c> It's</c><00:14:56.480><c> not.</c><00:14:57.199><c> And</c><00:14:57.440><c> if</c><00:14:57.600><c> it</c><00:14:57.839><c> did,</c><00:14:58.720><c> why</c>

13:29.826 --> 13:29.836 align:start position:0%
forever? It's not. And if it did, why
 

13:29.836 --> 13:31.506 align:start position:0%
forever? It's not. And if it did, why
would<00:14:59.199><c> figuring</c><00:14:59.600><c> out</c><00:14:59.839><c> frequencies</c><00:15:00.320><c> would</c><00:15:00.560><c> be</c>

13:31.506 --> 13:31.516 align:start position:0%
would figuring out frequencies would be
 

13:31.516 --> 13:34.307 align:start position:0%
would figuring out frequencies would be
easy?<00:15:00.959><c> I</c><00:15:01.199><c> don't</c><00:15:01.279><c> get</c><00:15:01.440><c> it."</c><00:15:02.880><c> For</c><00:15:03.199><c> says,</c>

13:34.307 --> 13:34.317 align:start position:0%
easy? I don't get it." For says,
 

13:34.317 --> 13:36.066 align:start position:0%
easy? I don't get it." For says,
"Remember<00:15:03.920><c> Mahes,</c><00:15:04.399><c> how</c><00:15:04.639><c> when</c><00:15:04.800><c> you're</c><00:15:04.959><c> stuck</c>

13:36.066 --> 13:36.076 align:start position:0%
"Remember Mahes, how when you're stuck
 

13:36.076 --> 13:37.987 align:start position:0%
"Remember Mahes, how when you're stuck
with<00:15:05.360><c> a</c><00:15:05.519><c> hard</c><00:15:05.760><c> problem,</c><00:15:06.160><c> you</c><00:15:06.399><c> need</c><00:15:06.480><c> to</c><00:15:06.720><c> first</c>

13:37.987 --> 13:37.997 align:start position:0%
with a hard problem, you need to first
 

13:37.997 --> 13:39.747 align:start position:0%
with a hard problem, you need to first
see<00:15:07.199><c> if</c><00:15:07.360><c> you</c><00:15:07.519><c> can</c><00:15:07.680><c> solve</c><00:15:08.000><c> an</c><00:15:08.240><c> easier</c><00:15:08.639><c> version</c>

13:39.747 --> 13:39.757 align:start position:0%
see if you can solve an easier version
 

13:39.757 --> 13:41.826 align:start position:0%
see if you can solve an easier version
of<00:15:09.040><c> that</c><00:15:09.279><c> problem.</c><00:15:10.000><c> Periodic</c><00:15:10.480><c> signals</c><00:15:10.800><c> are</c>

13:41.826 --> 13:41.836 align:start position:0%
of that problem. Periodic signals are
 

13:41.836 --> 13:43.587 align:start position:0%
of that problem. Periodic signals are
much<00:15:11.199><c> easier</c><00:15:11.440><c> to</c><00:15:11.680><c> solve.</c><00:15:12.079><c> So</c><00:15:12.240><c> that's</c><00:15:12.399><c> why</c>

13:43.587 --> 13:43.597 align:start position:0%
much easier to solve. So that's why
 

13:43.597 --> 13:46.547 align:start position:0%
much easier to solve. So that's why
let's<00:15:13.120><c> start</c><00:15:13.360><c> with</c><00:15:13.600><c> that."</c><00:15:14.560><c> Okay.</c><00:15:15.279><c> But</c><00:15:15.519><c> how</c>

13:46.547 --> 13:46.557 align:start position:0%
let's start with that." Okay. But how
 

13:46.557 --> 13:49.187 align:start position:0%
let's start with that." Okay. But how
does<00:15:16.079><c> making</c><00:15:16.399><c> it</c><00:15:16.560><c> periodic</c><00:15:17.199><c> make</c><00:15:17.360><c> it</c><00:15:17.600><c> easy?</c>

13:49.187 --> 13:49.197 align:start position:0%
does making it periodic make it easy?
 

13:49.197 --> 13:52.066 align:start position:0%
does making it periodic make it easy?
Here's<00:15:18.639><c> how.</c><00:15:19.360><c> If</c><00:15:19.600><c> a</c><00:15:19.760><c> signal</c><00:15:20.160><c> keeps</c><00:15:20.560><c> repeating</c>

13:52.066 --> 13:52.076 align:start position:0%
Here's how. If a signal keeps repeating
 

13:52.076 --> 13:54.867 align:start position:0%
Here's how. If a signal keeps repeating
after<00:15:21.760><c> a</c><00:15:22.000><c> period</c><00:15:22.399><c> t</c><00:15:22.720><c> for</c><00:15:22.800><c> example,</c><00:15:23.440><c> then</c><00:15:23.680><c> any</c>

13:54.867 --> 13:54.877 align:start position:0%
after a period t for example, then any
 

13:54.877 --> 13:57.266 align:start position:0%
after a period t for example, then any
sine<00:15:24.240><c> wave</c><00:15:24.399><c> that</c><00:15:24.639><c> contributes</c><00:15:25.120><c> must</c><00:15:25.600><c> also</c>

13:57.266 --> 13:57.276 align:start position:0%
sine wave that contributes must also
 

13:57.276 --> 14:00.146 align:start position:0%
sine wave that contributes must also
repeat<00:15:26.800><c> perfectly</c><00:15:27.519><c> after</c><00:15:27.920><c> period</c><00:15:28.320><c> t.</c><00:15:29.040><c> This</c>

14:00.146 --> 14:00.156 align:start position:0%
repeat perfectly after period t. This
 

14:00.156 --> 14:02.146 align:start position:0%
repeat perfectly after period t. This
means<00:15:29.519><c> we</c><00:15:29.839><c> can</c><00:15:30.000><c> have</c><00:15:30.240><c> a</c><00:15:30.399><c> sine</c><00:15:30.720><c> wave</c><00:15:31.040><c> that</c>

14:02.146 --> 14:02.156 align:start position:0%
means we can have a sine wave that
 

14:02.156 --> 14:05.027 align:start position:0%
means we can have a sine wave that
completes<00:15:31.760><c> exactly</c><00:15:32.560><c> one</c><00:15:32.880><c> cycle</c><00:15:33.600><c> then</c><00:15:33.839><c> it'll</c>

14:05.027 --> 14:05.037 align:start position:0%
completes exactly one cycle then it'll
 

14:05.037 --> 14:07.826 align:start position:0%
completes exactly one cycle then it'll
perfectly<00:15:34.560><c> repeat</c><00:15:35.199><c> or</c><00:15:35.600><c> two</c><00:15:35.839><c> cycles</c><00:15:36.639><c> again</c>

14:07.826 --> 14:07.836 align:start position:0%
perfectly repeat or two cycles again
 

14:07.836 --> 14:10.227 align:start position:0%
perfectly repeat or two cycles again
it'll<00:15:37.279><c> perfectly</c><00:15:37.760><c> repeat</c><00:15:38.160><c> or</c><00:15:38.480><c> three</c><00:15:38.880><c> and</c><00:15:39.120><c> so</c>

14:10.227 --> 14:10.237 align:start position:0%
it'll perfectly repeat or three and so
 

14:10.237 --> 14:12.706 align:start position:0%
it'll perfectly repeat or three and so
on<00:15:39.519><c> and</c><00:15:39.760><c> so</c><00:15:39.920><c> forth.</c><00:15:40.560><c> But</c><00:15:40.720><c> we</c><00:15:40.959><c> cannot</c><00:15:41.360><c> have</c>

14:12.706 --> 14:12.716 align:start position:0%
on and so forth. But we cannot have
 

14:12.716 --> 14:15.187 align:start position:0%
on and so forth. But we cannot have
anything<00:15:42.480><c> in</c><00:15:42.800><c> between</c><00:15:43.199><c> like</c><00:15:43.440><c> these</c><00:15:43.839><c> because</c>

14:15.187 --> 14:15.197 align:start position:0%
anything in between like these because
 

14:15.197 --> 14:18.307 align:start position:0%
anything in between like these because
they<00:15:44.880><c> won't</c><00:15:45.279><c> perfectly</c><00:15:46.079><c> repeat.</c><00:15:46.959><c> So</c><00:15:47.199><c> they</c>

14:18.307 --> 14:18.317 align:start position:0%
they won't perfectly repeat. So they
 

14:18.317 --> 14:21.427 align:start position:0%
they won't perfectly repeat. So they
won't<00:15:47.920><c> perfectly</c><00:15:48.720><c> line</c><00:15:49.040><c> up.</c><00:15:49.920><c> Okay,</c><00:15:50.240><c> that's</c>

14:21.427 --> 14:21.437 align:start position:0%
won't perfectly line up. Okay, that's
 

14:21.437 --> 14:23.427 align:start position:0%
won't perfectly line up. Okay, that's
pretty<00:15:50.959><c> ingenious.</c><00:15:51.920><c> And</c><00:15:52.079><c> by</c><00:15:52.240><c> the</c><00:15:52.399><c> way,</c>

14:23.427 --> 14:23.437 align:start position:0%
pretty ingenious. And by the way,
 

14:23.437 --> 14:26.386 align:start position:0%
pretty ingenious. And by the way,
there's<00:15:52.959><c> also</c><00:15:53.360><c> a</c><00:15:53.759><c> zero</c><00:15:54.240><c> frequency</c><00:15:54.800><c> term,</c><00:15:55.279><c> a</c>

14:26.386 --> 14:26.396 align:start position:0%
there's also a zero frequency term, a
 

14:26.396 --> 14:29.427 align:start position:0%
there's also a zero frequency term, a
constant<00:15:55.920><c> function.</c><00:15:56.639><c> Wait,</c><00:15:57.040><c> why?</c><00:15:58.000><c> It</c><00:15:58.240><c> simply</c>

14:29.427 --> 14:29.437 align:start position:0%
constant function. Wait, why? It simply
 

14:29.437 --> 14:31.747 align:start position:0%
constant function. Wait, why? It simply
tells<00:15:58.800><c> us</c><00:15:59.040><c> how</c><00:15:59.279><c> much</c><00:15:59.440><c> the</c><00:15:59.759><c> whole</c><00:16:00.079><c> signal</c><00:16:00.639><c> is</c>

14:31.747 --> 14:31.757 align:start position:0%
tells us how much the whole signal is
 

14:31.757 --> 14:36.227 align:start position:0%
tells us how much the whole signal is
shifted<00:16:01.360><c> above</c><00:16:02.160><c> or</c><00:16:02.800><c> below</c><00:16:03.680><c> the</c><00:16:04.000><c> zero.</c><00:16:05.040><c> So</c>

14:36.227 --> 14:36.237 align:start position:0%
shifted above or below the zero. So
 

14:36.237 --> 14:39.347 align:start position:0%
shifted above or below the zero. So
look,<00:16:05.839><c> by</c><00:16:06.079><c> assuming</c><00:16:06.639><c> periodicity,</c><00:16:07.759><c> we</c><00:16:08.079><c> have</c>

14:39.347 --> 14:39.357 align:start position:0%
look, by assuming periodicity, we have
 

14:39.357 --> 14:42.146 align:start position:0%
look, by assuming periodicity, we have
easily<00:16:08.959><c> figured</c><00:16:09.360><c> out</c><00:16:09.519><c> the</c><00:16:09.920><c> exact</c><00:16:10.720><c> frequency</c>

14:42.146 --> 14:42.156 align:start position:0%
easily figured out the exact frequency
 

14:42.156 --> 14:44.386 align:start position:0%
easily figured out the exact frequency
constituents.<00:16:12.480><c> But</c><00:16:12.639><c> now</c><00:16:12.880><c> comes</c><00:16:13.120><c> the</c><00:16:13.279><c> main</c>

14:44.386 --> 14:44.396 align:start position:0%
constituents. But now comes the main
 

14:44.396 --> 14:47.587 align:start position:0%
constituents. But now comes the main
question.<00:16:14.160><c> How</c><00:16:14.399><c> do</c><00:16:14.560><c> we</c><00:16:14.720><c> figure</c><00:16:14.959><c> out</c><00:16:15.519><c> the</c><00:16:15.839><c> exact</c>

14:47.587 --> 14:47.597 align:start position:0%
question. How do we figure out the exact
 

14:47.597 --> 14:50.706 align:start position:0%
question. How do we figure out the exact
heights<00:16:17.120><c> of</c><00:16:17.360><c> those</c><00:16:17.600><c> constituents?</c>

14:50.706 --> 14:50.716 align:start position:0%
heights of those constituents?
 

14:50.716 --> 14:54.227 align:start position:0%
heights of those constituents?
For<00:16:20.240><c> says</c><00:16:20.800><c> easy</c><00:16:21.759><c> algebra.</c>

14:54.227 --> 14:54.237 align:start position:0%
For says easy algebra.
 

14:54.237 --> 14:58.386 align:start position:0%
For says easy algebra.
Okay.<00:16:23.839><c> What?</c><00:16:25.120><c> Let's</c><00:16:25.440><c> call</c><00:16:25.680><c> our</c><00:16:26.000><c> shape</c><00:16:26.480><c> f</c><00:16:26.720><c> of</c><00:16:26.959><c> t.</c>

14:58.386 --> 14:58.396 align:start position:0%
Okay. What? Let's call our shape f of t.
 

14:58.396 --> 15:01.667 align:start position:0%
Okay. What? Let's call our shape f of t.
Then<00:16:27.759><c> we</c><00:16:28.000><c> can</c><00:16:28.160><c> write</c><00:16:28.639><c> this</c><00:16:29.120><c> series</c><00:16:30.000><c> as</c><00:16:30.399><c> a</c>

15:01.667 --> 15:01.677 align:start position:0%
Then we can write this series as a
 

15:01.677 --> 15:04.946 align:start position:0%
Then we can write this series as a
summation.<00:16:32.160><c> Now,</c><00:16:32.399><c> if</c><00:16:32.560><c> you</c><00:16:32.720><c> have</c><00:16:32.959><c> one</c><00:16:33.199><c> unknown,</c>

15:04.946 --> 15:04.956 align:start position:0%
summation. Now, if you have one unknown,
 

15:04.956 --> 15:06.867 align:start position:0%
summation. Now, if you have one unknown,
you<00:16:34.240><c> need</c><00:16:34.399><c> one</c><00:16:34.639><c> equation</c><00:16:35.040><c> to</c><00:16:35.199><c> solve</c><00:16:35.440><c> it.</c><00:16:35.839><c> If</c>

15:06.867 --> 15:06.877 align:start position:0%
you need one equation to solve it. If
 

15:06.877 --> 15:08.706 align:start position:0%
you need one equation to solve it. If
you<00:16:36.160><c> have</c><00:16:36.240><c> two</c><00:16:36.480><c> unknowns,</c><00:16:37.120><c> you</c><00:16:37.360><c> need</c><00:16:37.600><c> two</c>

15:08.706 --> 15:08.716 align:start position:0%
you have two unknowns, you need two
 

15:08.716 --> 15:11.027 align:start position:0%
you have two unknowns, you need two
equations<00:16:38.320><c> to</c><00:16:38.480><c> solve</c><00:16:38.720><c> it.</c><00:16:39.199><c> Here,</c><00:16:39.600><c> we</c><00:16:39.920><c> have</c>

15:11.027 --> 15:11.037 align:start position:0%
equations to solve it. Here, we have
 

15:11.037 --> 15:13.107 align:start position:0%
equations to solve it. Here, we have
infinite<00:16:40.639><c> unknowns.</c><00:16:41.440><c> So</c><00:16:41.680><c> we</c><00:16:41.839><c> just</c><00:16:42.000><c> need</c>

15:13.107 --> 15:13.117 align:start position:0%
infinite unknowns. So we just need
 

15:13.117 --> 15:15.107 align:start position:0%
infinite unknowns. So we just need
infinite<00:16:42.720><c> equations</c><00:16:43.360><c> to</c><00:16:43.600><c> solve</c><00:16:43.839><c> it.</c><00:16:44.079><c> That's</c>

15:15.107 --> 15:15.117 align:start position:0%
infinite equations to solve it. That's
 

15:15.117 --> 15:18.307 align:start position:0%
infinite equations to solve it. That's
it.<00:16:44.880><c> Oh,</c><00:16:45.279><c> okay.</c><00:16:45.759><c> That's</c><00:16:46.000><c> it.</c><00:16:46.480><c> For</c><00:16:47.040><c> we</c><00:16:47.199><c> just</c>

15:18.307 --> 15:18.317 align:start position:0%
it. Oh, okay. That's it. For we just
 

15:18.317 --> 15:21.427 align:start position:0%
it. Oh, okay. That's it. For we just
have<00:16:47.680><c> one</c><00:16:48.000><c> equation.</c><00:16:48.880><c> For</c><00:16:49.279><c> says</c><00:16:49.519><c> no,</c><00:16:50.000><c> I</c><00:16:50.320><c> see</c>

15:21.427 --> 15:21.437 align:start position:0%
have one equation. For says no, I see
 

15:21.437 --> 15:24.066 align:start position:0%
have one equation. For says no, I see
infinite<00:16:51.120><c> equations.</c>

15:24.066 --> 15:24.076 align:start position:0%
infinite equations.
 

15:24.076 --> 15:27.667 align:start position:0%
infinite equations.
How?<00:16:54.480><c> Differentiate.</c><00:16:55.759><c> If</c><00:16:56.000><c> you</c><00:16:56.240><c> differentiate</c>

15:27.667 --> 15:27.677 align:start position:0%
How? Differentiate. If you differentiate
 

15:27.677 --> 15:30.706 align:start position:0%
How? Differentiate. If you differentiate
this<00:16:57.040><c> equation,</c><00:16:58.000><c> you</c><00:16:58.240><c> will</c><00:16:58.480><c> get</c><00:16:59.040><c> another</c>

15:30.706 --> 15:30.716 align:start position:0%
this equation, you will get another
 

15:30.716 --> 15:34.146 align:start position:0%
this equation, you will get another
equation.<00:17:00.800><c> And</c><00:17:00.959><c> the</c><00:17:01.120><c> thing</c><00:17:01.279><c> is</c><00:17:01.839><c> s</c><00:17:02.240><c> and</c><00:17:02.480><c> cos</c><00:17:02.959><c> are</c>

15:34.146 --> 15:34.156 align:start position:0%
equation. And the thing is s and cos are
 

15:34.156 --> 15:36.227 align:start position:0%
equation. And the thing is s and cos are
infinitely<00:17:03.839><c> differentiable.</c><00:17:04.720><c> So</c><00:17:04.799><c> I</c><00:17:05.039><c> can</c><00:17:05.120><c> just</c>

15:36.227 --> 15:36.237 align:start position:0%
infinitely differentiable. So I can just
 

15:36.237 --> 15:37.826 align:start position:0%
infinitely differentiable. So I can just
keep<00:17:05.520><c> differentiating</c><00:17:06.079><c> this</c><00:17:06.319><c> over</c><00:17:06.559><c> and</c><00:17:06.799><c> over</c>

15:37.826 --> 15:37.836 align:start position:0%
keep differentiating this over and over
 

15:37.836 --> 15:40.066 align:start position:0%
keep differentiating this over and over
again.<00:17:07.439><c> I'll</c><00:17:07.760><c> get</c><00:17:08.000><c> infinite</c><00:17:08.400><c> equations.</c><00:17:09.039><c> Now</c>

15:40.066 --> 15:40.076 align:start position:0%
again. I'll get infinite equations. Now
 

15:40.076 --> 15:41.507 align:start position:0%
again. I'll get infinite equations. Now
I<00:17:09.439><c> just</c><00:17:09.520><c> have</c><00:17:09.679><c> to</c><00:17:09.839><c> solve</c><00:17:10.000><c> these</c><00:17:10.240><c> infinite</c>

15:41.507 --> 15:41.517 align:start position:0%
I just have to solve these infinite
 

15:41.517 --> 15:43.426 align:start position:0%
I just have to solve these infinite
equations<00:17:11.039><c> for</c><00:17:11.280><c> my</c><00:17:11.439><c> infinite</c><00:17:11.919><c> variables</c>

15:43.426 --> 15:43.436 align:start position:0%
equations for my infinite variables
 

15:43.436 --> 15:45.107 align:start position:0%
equations for my infinite variables
simultaneously.

15:45.107 --> 15:45.117 align:start position:0%
simultaneously.
 

15:45.117 --> 15:48.066 align:start position:0%
simultaneously.
Madman,<00:17:15.199><c> you're</c><00:17:15.439><c> a</c><00:17:15.679><c> madman.</c><00:17:16.640><c> No</c><00:17:16.880><c> man,</c>

15:48.066 --> 15:48.076 align:start position:0%
Madman, you're a madman. No man,
 

15:48.076 --> 15:50.147 align:start position:0%
Madman, you're a madman. No man,
physicist<00:17:17.679><c> or</c><00:17:17.919><c> mathematician</c><00:17:18.559><c> threatens</c><00:17:18.959><c> a</c>

15:50.147 --> 15:50.157 align:start position:0%
physicist or mathematician threatens a
 

15:50.157 --> 15:52.946 align:start position:0%
physicist or mathematician threatens a
learner<00:17:19.760><c> with</c><00:17:20.240><c> infinite</c><00:17:20.799><c> equations.</c><00:17:21.839><c> I'm</c>

15:52.946 --> 15:52.956 align:start position:0%
learner with infinite equations. I'm
 

15:52.956 --> 15:57.987 align:start position:0%
learner with infinite equations. I'm
done.<00:17:22.400><c> I</c><00:17:22.640><c> I</c><00:17:22.720><c> I</c><00:17:23.280><c> quit.</c>

15:57.987 --> 15:57.997 align:start position:0%
 
 

15:57.997 --> 16:00.387 align:start position:0%
 
For<00:17:27.439><c> agrees,</c><00:17:28.000><c> this</c><00:17:28.240><c> is</c><00:17:28.799><c> absolutely</c>

16:00.387 --> 16:00.397 align:start position:0%
For agrees, this is absolutely
 

16:00.397 --> 16:03.027 align:start position:0%
For agrees, this is absolutely
ridiculous.<00:17:30.720><c> We</c><00:17:30.960><c> need</c><00:17:31.120><c> to</c><00:17:31.280><c> find</c><00:17:31.520><c> a</c><00:17:31.760><c> better</c><00:17:32.000><c> way</c>

16:03.027 --> 16:03.037 align:start position:0%
ridiculous. We need to find a better way
 

16:03.037 --> 16:04.947 align:start position:0%
ridiculous. We need to find a better way
to<00:17:32.320><c> go</c><00:17:32.480><c> about</c><00:17:32.720><c> it.</c><00:17:33.280><c> So</c><00:17:33.520><c> if</c><00:17:33.679><c> we</c><00:17:33.919><c> can't</c>

16:04.947 --> 16:04.957 align:start position:0%
to go about it. So if we can't
 

16:04.957 --> 16:07.187 align:start position:0%
to go about it. So if we can't
differentiate,<00:17:35.440><c> maybe</c><00:17:35.679><c> we</c><00:17:35.840><c> can</c><00:17:36.000><c> try</c>

16:07.187 --> 16:07.197 align:start position:0%
differentiate, maybe we can try
 

16:07.197 --> 16:08.947 align:start position:0%
differentiate, maybe we can try
integrating.

16:08.947 --> 16:08.957 align:start position:0%
integrating.
 

16:08.957 --> 16:11.906 align:start position:0%
integrating.
That's<00:17:38.320><c> it.</c><00:17:39.120><c> Integration</c><00:17:40.080><c> that</c><00:17:40.320><c> is</c><00:17:40.480><c> the</c><00:17:40.720><c> key.</c>

16:11.906 --> 16:11.916 align:start position:0%
That's it. Integration that is the key.
 

16:11.916 --> 16:14.947 align:start position:0%
That's it. Integration that is the key.
And<00:17:41.200><c> I</c><00:17:41.440><c> say</c><00:17:41.679><c> stop</c><00:17:42.160><c> for</c><00:17:42.720><c> stop</c><00:17:43.120><c> doing</c><00:17:43.440><c> that</c><00:17:43.760><c> thing</c>

16:14.947 --> 16:14.957 align:start position:0%
And I say stop for stop doing that thing
 

16:14.957 --> 16:17.587 align:start position:0%
And I say stop for stop doing that thing
where<00:17:44.400><c> you</c><00:17:44.720><c> get</c><00:17:45.039><c> things</c><00:17:45.360><c> instantly</c><00:17:46.080><c> and</c><00:17:46.400><c> I</c>

16:17.587 --> 16:17.597 align:start position:0%
where you get things instantly and I
 

16:17.597 --> 16:19.426 align:start position:0%
where you get things instantly and I
don't<00:17:46.960><c> and</c><00:17:47.360><c> you</c><00:17:47.600><c> have</c><00:17:47.679><c> to</c><00:17:47.840><c> spell</c><00:17:48.000><c> it</c><00:17:48.160><c> out</c><00:17:48.320><c> for</c>

16:19.426 --> 16:19.436 align:start position:0%
don't and you have to spell it out for
 

16:19.436 --> 16:22.307 align:start position:0%
don't and you have to spell it out for
me.<00:17:49.440><c> For</c><00:17:49.760><c> your</c><00:17:49.919><c> asks,</c><00:17:50.480><c> what</c><00:17:50.720><c> exactly</c><00:17:51.039><c> is</c><00:17:51.280><c> an</c>

16:22.307 --> 16:22.317 align:start position:0%
me. For your asks, what exactly is an
 

16:22.317 --> 16:24.546 align:start position:0%
me. For your asks, what exactly is an
integral?<00:17:52.160><c> Well,</c><00:17:52.400><c> I</c><00:17:52.640><c> know</c><00:17:52.880><c> integrals</c><00:17:53.520><c> are</c>

16:24.546 --> 16:24.556 align:start position:0%
integral? Well, I know integrals are
 

16:24.556 --> 16:26.627 align:start position:0%
integral? Well, I know integrals are
basically<00:17:54.160><c> calculating</c><00:17:54.799><c> area</c><00:17:55.200><c> under</c><00:17:55.600><c> a</c>

16:26.627 --> 16:26.637 align:start position:0%
basically calculating area under a
 

16:26.637 --> 16:29.666 align:start position:0%
basically calculating area under a
curve,<00:17:56.880><c> right?</c><00:17:57.600><c> And</c><00:17:57.840><c> integrals</c><00:17:58.320><c> are</c><00:17:58.480><c> also</c>

16:29.666 --> 16:29.676 align:start position:0%
curve, right? And integrals are also
 

16:29.676 --> 16:32.147 align:start position:0%
curve, right? And integrals are also
linear.<00:17:59.440><c> That</c><00:17:59.600><c> means</c><00:18:00.160><c> the</c><00:18:00.480><c> area</c><00:18:00.720><c> under</c><00:18:01.039><c> this</c>

16:32.147 --> 16:32.157 align:start position:0%
linear. That means the area under this
 

16:32.157 --> 16:34.467 align:start position:0%
linear. That means the area under this
curve<00:18:01.679><c> equals</c><00:18:02.160><c> area</c><00:18:02.480><c> under</c><00:18:02.799><c> this</c><00:18:03.039><c> plus</c><00:18:03.280><c> area</c>

16:34.467 --> 16:34.477 align:start position:0%
curve equals area under this plus area
 

16:34.477 --> 16:36.147 align:start position:0%
curve equals area under this plus area
under<00:18:03.840><c> this</c><00:18:04.080><c> plus</c><00:18:04.320><c> area</c><00:18:04.559><c> under</c><00:18:04.799><c> this</c><00:18:04.960><c> and</c><00:18:05.200><c> so</c>

16:36.147 --> 16:36.157 align:start position:0%
under this plus area under this and so
 

16:36.157 --> 16:39.267 align:start position:0%
under this plus area under this and so
on<00:18:05.440><c> and</c><00:18:05.679><c> so</c><00:18:05.840><c> forth,</c><00:18:06.720><c> right?</c><00:18:07.360><c> Yeah,</c><00:18:07.919><c> but</c><00:18:08.240><c> look</c>

16:39.267 --> 16:39.277 align:start position:0%
on and so forth, right? Yeah, but look
 

16:39.277 --> 16:42.147 align:start position:0%
on and so forth, right? Yeah, but look
at<00:18:08.559><c> the</c><00:18:08.799><c> sign</c><00:18:09.120><c> curves.</c><00:18:09.679><c> Over</c><00:18:10.000><c> a</c><00:18:10.160><c> full</c><00:18:10.400><c> period,</c>

16:42.147 --> 16:42.157 align:start position:0%
at the sign curves. Over a full period,
 

16:42.157 --> 16:44.067 align:start position:0%
at the sign curves. Over a full period,
their<00:18:11.600><c> positive</c><00:18:11.919><c> and</c><00:18:12.160><c> negative</c><00:18:12.559><c> areas</c>

16:44.067 --> 16:44.077 align:start position:0%
their positive and negative areas
 

16:44.077 --> 16:45.827 align:start position:0%
their positive and negative areas
perfectly<00:18:13.760><c> cancel</c><00:18:14.080><c> out.</c><00:18:14.400><c> So</c><00:18:14.559><c> they</c><00:18:14.720><c> all</c>

16:45.827 --> 16:45.837 align:start position:0%
perfectly cancel out. So they all
 

16:45.837 --> 16:47.987 align:start position:0%
perfectly cancel out. So they all
disappear.<00:18:15.760><c> That</c><00:18:16.000><c> means</c><00:18:16.160><c> the</c><00:18:16.400><c> only</c><00:18:16.640><c> survival</c>

16:47.987 --> 16:47.997 align:start position:0%
disappear. That means the only survival
 

16:47.997 --> 16:50.467 align:start position:0%
disappear. That means the only survival
is<00:18:17.280><c> this</c><00:18:17.520><c> one.</c><00:18:17.919><c> So</c><00:18:18.080><c> the</c><00:18:18.320><c> area</c><00:18:18.640><c> under</c><00:18:18.960><c> our</c><00:18:19.280><c> curve</c>

16:50.467 --> 16:50.477 align:start position:0%
is this one. So the area under our curve
 

16:50.477 --> 16:52.467 align:start position:0%
is this one. So the area under our curve
is<00:18:19.840><c> exactly</c><00:18:20.320><c> equal</c><00:18:20.640><c> to</c><00:18:20.799><c> the</c><00:18:20.960><c> area</c><00:18:21.200><c> of</c><00:18:21.360><c> this</c>

16:52.467 --> 16:52.477 align:start position:0%
is exactly equal to the area of this
 

16:52.477 --> 16:54.867 align:start position:0%
is exactly equal to the area of this
rectangle,<00:18:22.160><c> which</c><00:18:22.400><c> is</c><00:18:22.559><c> basically</c><00:18:22.960><c> the</c><00:18:23.200><c> height</c>

16:54.867 --> 16:54.877 align:start position:0%
rectangle, which is basically the height
 

16:54.877 --> 16:57.747 align:start position:0%
rectangle, which is basically the height
times<00:18:24.320><c> the</c><00:18:24.559><c> width</c><00:18:25.039><c> t.</c><00:18:25.840><c> And</c><00:18:26.000><c> so</c><00:18:26.240><c> we</c><00:18:26.480><c> can</c><00:18:26.640><c> now</c>

16:57.747 --> 16:57.757 align:start position:0%
times the width t. And so we can now
 

16:57.757 --> 17:00.227 align:start position:0%
times the width t. And so we can now
write<00:18:27.360><c> hnot</c><00:18:28.080><c> *</c><00:18:28.240><c> t.</c><00:18:28.640><c> The</c><00:18:28.799><c> area</c><00:18:29.039><c> of</c><00:18:29.120><c> this</c>

17:00.227 --> 17:00.237 align:start position:0%
write hnot * t. The area of this
 

17:00.237 --> 17:03.747 align:start position:0%
write hnot * t. The area of this
rectangle<00:18:30.240><c> equals</c><00:18:30.880><c> the</c><00:18:31.280><c> area</c><00:18:31.840><c> under</c><00:18:32.320><c> this</c>

17:03.747 --> 17:03.757 align:start position:0%
rectangle equals the area under this
 

17:03.757 --> 17:06.867 align:start position:0%
rectangle equals the area under this
curve.<00:18:33.760><c> And</c><00:18:34.000><c> from</c><00:18:34.240><c> this</c><00:18:34.640><c> we</c><00:18:35.039><c> can</c><00:18:35.360><c> isolate</c>

17:06.867 --> 17:06.877 align:start position:0%
curve. And from this we can isolate
 

17:06.877 --> 17:10.867 align:start position:0%
curve. And from this we can isolate
hnot.<00:18:37.120><c> We</c><00:18:37.360><c> have</c><00:18:37.760><c> found</c><00:18:38.559><c> the</c><00:18:38.880><c> value</c><00:18:39.200><c> of</c><00:18:39.360><c> hnot.</c>

17:10.867 --> 17:10.877 align:start position:0%
hnot. We have found the value of hnot.
 

17:10.877 --> 17:13.747 align:start position:0%
hnot. We have found the value of hnot.
It's<00:18:40.160><c> the</c><00:18:40.320><c> area</c><00:18:40.640><c> under</c><00:18:40.880><c> the</c><00:18:41.039><c> curve</c><00:18:41.760><c> divided</c><00:18:42.400><c> by</c>

17:13.747 --> 17:13.757 align:start position:0%
It's the area under the curve divided by
 

17:13.757 --> 17:16.786 align:start position:0%
It's the area under the curve divided by
t.<00:18:44.080><c> That's</c><00:18:44.480><c> pretty</c><00:18:44.720><c> cool.</c><00:18:45.039><c> That's</c><00:18:45.280><c> how</c>

17:16.786 --> 17:16.796 align:start position:0%
t. That's pretty cool. That's how
 

17:16.796 --> 17:20.067 align:start position:0%
t. That's pretty cool. That's how
integrals<00:18:46.559><c> can</c><00:18:46.880><c> help</c><00:18:47.039><c> us.</c><00:18:48.000><c> Okay,</c><00:18:48.640><c> I</c><00:18:48.880><c> admit</c>

17:20.067 --> 17:20.077 align:start position:0%
integrals can help us. Okay, I admit
 

17:20.077 --> 17:21.987 align:start position:0%
integrals can help us. Okay, I admit
that's<00:18:49.520><c> cool.</c><00:18:50.080><c> But</c><00:18:50.320><c> how</c><00:18:50.559><c> are</c><00:18:50.720><c> we</c><00:18:50.880><c> going</c><00:18:50.960><c> to</c>

17:21.987 --> 17:21.997 align:start position:0%
that's cool. But how are we going to
 

17:21.997 --> 17:24.867 align:start position:0%
that's cool. But how are we going to
find<00:18:51.360><c> the</c><00:18:51.679><c> values</c><00:18:52.000><c> of</c><00:18:52.160><c> h1,</c><00:18:52.720><c> h2,</c><00:18:53.200><c> h3,</c><00:18:53.679><c> and</c><00:18:53.840><c> so</c>

17:24.867 --> 17:24.877 align:start position:0%
find the values of h1, h2, h3, and so
 

17:24.877 --> 17:27.027 align:start position:0%
find the values of h1, h2, h3, and so
forth?<00:18:54.400><c> Now,</c>

17:27.027 --> 17:27.037 align:start position:0%
forth? Now,
 

17:27.037 --> 17:28.947 align:start position:0%
forth? Now,
so<00:18:56.400><c> this</c><00:18:56.640><c> time</c><00:18:57.120><c> we</c><00:18:57.360><c> want</c><00:18:57.440><c> to</c><00:18:57.600><c> do</c><00:18:57.760><c> something</c>

17:28.947 --> 17:28.957 align:start position:0%
so this time we want to do something
 

17:28.957 --> 17:30.387 align:start position:0%
so this time we want to do something
similar,<00:18:58.559><c> but</c><00:18:58.799><c> this</c><00:18:58.960><c> time</c><00:18:59.120><c> we</c><00:18:59.280><c> want</c><00:18:59.440><c> to</c>

17:30.387 --> 17:30.397 align:start position:0%
similar, but this time we want to
 

17:30.397 --> 17:33.827 align:start position:0%
similar, but this time we want to
isolate<00:19:00.080><c> h1.</c><00:19:01.280><c> So</c><00:19:01.440><c> the</c><00:19:01.679><c> question</c><00:19:01.840><c> is</c><00:19:02.320><c> can</c><00:19:02.559><c> we</c><00:19:02.720><c> do</c>

17:33.827 --> 17:33.837 align:start position:0%
isolate h1. So the question is can we do
 

17:33.837 --> 17:35.827 align:start position:0%
isolate h1. So the question is can we do
some<00:19:03.200><c> operation</c><00:19:03.600><c> on</c><00:19:03.840><c> this</c><00:19:04.080><c> entire</c><00:19:04.480><c> equation</c>

17:35.827 --> 17:35.837 align:start position:0%
some operation on this entire equation
 

17:35.837 --> 17:38.067 align:start position:0%
some operation on this entire equation
such<00:19:05.200><c> that</c><00:19:05.440><c> when</c><00:19:05.679><c> we</c><00:19:05.840><c> calculate</c><00:19:06.240><c> the</c><00:19:06.480><c> areas</c>

17:38.067 --> 17:38.077 align:start position:0%
such that when we calculate the areas
 

17:38.077 --> 17:40.227 align:start position:0%
such that when we calculate the areas
only<00:19:07.520><c> H1</c><00:19:08.080><c> survives</c><00:19:08.480><c> and</c><00:19:08.799><c> everything</c><00:19:09.039><c> else</c>

17:40.227 --> 17:40.237 align:start position:0%
only H1 survives and everything else
 

17:40.237 --> 17:43.507 align:start position:0%
only H1 survives and everything else
cancels<00:19:09.840><c> out.</c><00:19:10.960><c> Okay.</c><00:19:11.280><c> So</c><00:19:11.440><c> the</c><00:19:11.679><c> problem</c><00:19:11.840><c> is</c><00:19:12.240><c> for</c>

17:43.507 --> 17:43.517 align:start position:0%
cancels out. Okay. So the problem is for
 

17:43.517 --> 17:45.906 align:start position:0%
cancels out. Okay. So the problem is for
H1<00:19:13.200><c> the</c><00:19:13.520><c> areas</c><00:19:14.000><c> are</c><00:19:14.240><c> positive</c><00:19:14.559><c> and</c><00:19:14.720><c> negative</c>

17:45.906 --> 17:45.916 align:start position:0%
H1 the areas are positive and negative
 

17:45.916 --> 17:48.467 align:start position:0%
H1 the areas are positive and negative
that<00:19:15.200><c> cancels</c><00:19:15.600><c> out.</c><00:19:16.000><c> We</c><00:19:16.240><c> don't</c><00:19:16.400><c> want</c><00:19:16.480><c> that.</c><00:19:17.200><c> We</c>

17:48.467 --> 17:48.477 align:start position:0%
that cancels out. We don't want that. We
 

17:48.477 --> 17:50.627 align:start position:0%
that cancels out. We don't want that. We
want<00:19:18.000><c> the</c><00:19:18.320><c> area</c><00:19:18.559><c> to</c><00:19:18.799><c> be</c><00:19:18.960><c> just</c><00:19:19.120><c> positive.</c><00:19:19.679><c> For</c>

17:50.627 --> 17:50.637 align:start position:0%
want the area to be just positive. For
 

17:50.637 --> 17:52.546 align:start position:0%
want the area to be just positive. For
example,<00:19:20.080><c> we</c><00:19:20.320><c> want</c><00:19:20.480><c> this</c><00:19:20.720><c> to</c><00:19:20.880><c> flip</c><00:19:21.200><c> over</c><00:19:21.440><c> and</c>

17:52.546 --> 17:52.556 align:start position:0%
example, we want this to flip over and
 

17:52.556 --> 17:55.107 align:start position:0%
example, we want this to flip over and
become<00:19:22.000><c> positive.</c><00:19:22.640><c> How</c><00:19:22.799><c> do</c><00:19:22.880><c> we</c><00:19:23.039><c> do</c><00:19:23.520><c> I</c><00:19:23.760><c> got</c><00:19:23.919><c> it?</c>

17:55.107 --> 17:55.117 align:start position:0%
become positive. How do we do I got it?
 

17:55.117 --> 17:58.067 align:start position:0%
become positive. How do we do I got it?
I<00:19:24.480><c> got</c><00:19:24.640><c> it.</c><00:19:25.440><c> Square.</c><00:19:26.559><c> Because</c><00:19:26.799><c> when</c><00:19:27.039><c> you</c>

17:58.067 --> 17:58.077 align:start position:0%
I got it. Square. Because when you
 

17:58.077 --> 17:59.827 align:start position:0%
I got it. Square. Because when you
square<00:19:27.520><c> something</c><00:19:27.840><c> the</c><00:19:28.240><c> you</c><00:19:28.400><c> know</c><00:19:28.480><c> the</c><00:19:28.720><c> thing</c>

17:59.827 --> 17:59.837 align:start position:0%
square something the you know the thing
 

17:59.837 --> 18:01.426 align:start position:0%
square something the you know the thing
becomes<00:19:29.280><c> positive.</c><00:19:29.679><c> And</c><00:19:29.919><c> so</c><00:19:30.080><c> we'll</c><00:19:30.320><c> end</c><00:19:30.400><c> up</c>

18:01.426 --> 18:01.436 align:start position:0%
becomes positive. And so we'll end up
 

18:01.436 --> 18:03.747 align:start position:0%
becomes positive. And so we'll end up
with<00:19:31.280><c> positive</c><00:19:31.760><c> area.</c><00:19:32.160><c> So</c><00:19:32.240><c> we</c><00:19:32.480><c> have</c><00:19:32.559><c> to</c><00:19:32.640><c> square</c>

18:03.747 --> 18:03.757 align:start position:0%
with positive area. So we have to square
 

18:03.757 --> 18:05.666 align:start position:0%
with positive area. So we have to square
it<00:19:33.120><c> for</c><00:19:33.280><c> year,</c><00:19:33.600><c> right?</c><00:19:33.919><c> And</c><00:19:34.160><c> for</c><00:19:34.400><c> year</c><00:19:34.640><c> says</c>

18:05.666 --> 18:05.676 align:start position:0%
it for year, right? And for year says
 

18:05.676 --> 18:08.307 align:start position:0%
it for year, right? And for year says
you're<00:19:35.039><c> on</c><00:19:35.200><c> the</c><00:19:35.280><c> right</c><00:19:35.440><c> track,</c><00:19:35.679><c> Mahesh.</c>

18:08.307 --> 18:08.317 align:start position:0%
you're on the right track, Mahesh.
 

18:08.317 --> 18:11.187 align:start position:0%
you're on the right track, Mahesh.
But<00:19:37.679><c> the</c><00:19:37.919><c> problem</c><00:19:38.240><c> is</c><00:19:38.559><c> when</c><00:19:38.799><c> you</c><00:19:39.120><c> square</c>

18:11.187 --> 18:11.197 align:start position:0%
But the problem is when you square
 

18:11.197 --> 18:13.587 align:start position:0%
But the problem is when you square
everything<00:19:40.799><c> gets</c><00:19:41.280><c> squared</c><00:19:42.160><c> and</c><00:19:42.400><c> so</c>

18:13.587 --> 18:13.597 align:start position:0%
everything gets squared and so
 

18:13.597 --> 18:15.267 align:start position:0%
everything gets squared and so
everything<00:19:43.120><c> will</c><00:19:43.360><c> become</c><00:19:43.600><c> positive.</c><00:19:44.160><c> Nothing</c>

18:15.267 --> 18:15.277 align:start position:0%
everything will become positive. Nothing
 

18:15.277 --> 18:17.507 align:start position:0%
everything will become positive. Nothing
will<00:19:44.640><c> cancel</c><00:19:44.960><c> out.</c><00:19:45.520><c> Also</c><00:19:45.919><c> squares</c><00:19:46.240><c> are</c><00:19:46.400><c> not</c>

18:17.507 --> 18:17.517 align:start position:0%
will cancel out. Also squares are not
 

18:17.517 --> 18:20.627 align:start position:0%
will cancel out. Also squares are not
linear.<00:19:47.120><c> So</c><00:19:47.280><c> that's</c><00:19:47.679><c> an</c><00:19:48.000><c> additional</c><00:19:48.480><c> problem.</c>

18:20.627 --> 18:20.637 align:start position:0%
linear. So that's an additional problem.
 

18:20.637 --> 18:24.227 align:start position:0%
linear. So that's an additional problem.
Okay.<00:19:50.240><c> So</c><00:19:51.280><c> what</c><00:19:51.600><c> we</c><00:19:51.840><c> need</c><00:19:52.240><c> is</c><00:19:52.480><c> some</c><00:19:52.799><c> operation</c>

18:24.227 --> 18:24.237 align:start position:0%
Okay. So what we need is some operation
 

18:24.237 --> 18:27.987 align:start position:0%
Okay. So what we need is some operation
that<00:19:53.760><c> only</c><00:19:54.400><c> squares</c><00:19:55.200><c> this</c><00:19:55.520><c> one</c><00:19:56.320><c> but</c><00:19:56.640><c> doesn't</c>

18:27.987 --> 18:27.997 align:start position:0%
that only squares this one but doesn't
 

18:27.997 --> 18:29.906 align:start position:0%
that only squares this one but doesn't
square<00:19:57.520><c> the</c><00:19:57.760><c> rest.</c><00:19:58.400><c> And</c><00:19:58.559><c> it</c><00:19:58.799><c> should</c><00:19:58.960><c> be</c>

18:29.906 --> 18:29.916 align:start position:0%
square the rest. And it should be
 

18:29.916 --> 18:32.546 align:start position:0%
square the rest. And it should be
linear.<00:20:00.240><c> What</c><00:20:00.480><c> kind</c><00:20:00.720><c> of</c><00:20:00.799><c> an</c><00:20:01.039><c> operation</c><00:20:01.440><c> would</c>

18:32.546 --> 18:32.556 align:start position:0%
linear. What kind of an operation would
 

18:32.556 --> 18:35.107 align:start position:0%
linear. What kind of an operation would
that<00:20:01.919><c> be?</c>

18:35.107 --> 18:35.117 align:start position:0%
that be?
 

18:35.117 --> 18:38.546 align:start position:0%
that be?
For<00:20:04.720><c> says</c><00:20:05.120><c> simple</c><00:20:05.919><c> just</c><00:20:06.400><c> multiply</c><00:20:06.960><c> the</c><00:20:07.280><c> whole</c>

18:38.546 --> 18:38.556 align:start position:0%
For says simple just multiply the whole
 

18:38.556 --> 18:42.947 align:start position:0%
For says simple just multiply the whole
equation<00:20:09.039><c> with</c><00:20:09.760><c> sign.</c><00:20:11.200><c> When</c><00:20:11.440><c> you</c><00:20:11.600><c> do</c><00:20:11.760><c> that,</c>

18:42.947 --> 18:42.957 align:start position:0%
equation with sign. When you do that,
 

18:42.957 --> 18:47.027 align:start position:0%
equation with sign. When you do that,
look<00:20:12.720><c> this</c><00:20:13.120><c> gets</c><00:20:13.679><c> squared</c><00:20:15.039><c> but</c><00:20:15.360><c> the</c><00:20:15.600><c> rest</c>

18:47.027 --> 18:47.037 align:start position:0%
look this gets squared but the rest
 

18:47.037 --> 18:51.267 align:start position:0%
look this gets squared but the rest
don't.<00:20:17.600><c> And</c><00:20:18.960><c> sign</c><00:20:19.600><c> multiplication</c><00:20:20.240><c> is</c>

18:51.267 --> 18:51.277 align:start position:0%
don't. And sign multiplication is
 

18:51.277 --> 18:53.987 align:start position:0%
don't. And sign multiplication is
linear.<00:20:21.360><c> So</c><00:20:21.520><c> because</c><00:20:21.760><c> this</c><00:20:22.080><c> got</c><00:20:22.320><c> squared,</c>

18:53.987 --> 18:53.997 align:start position:0%
linear. So because this got squared,
 

18:53.997 --> 18:56.227 align:start position:0%
linear. So because this got squared,
this<00:20:23.360><c> will</c><00:20:23.600><c> now</c><00:20:23.919><c> end</c><00:20:24.080><c> up</c><00:20:24.320><c> with</c><00:20:24.559><c> a</c><00:20:24.799><c> positive</c>

18:56.227 --> 18:56.237 align:start position:0%
this will now end up with a positive
 

18:56.237 --> 18:58.707 align:start position:0%
this will now end up with a positive
area.<00:20:26.480><c> Okay.</c><00:20:26.799><c> But</c><00:20:26.960><c> what</c><00:20:27.200><c> about</c><00:20:27.360><c> the</c><00:20:27.600><c> rest?</c>

18:58.707 --> 18:58.717 align:start position:0%
area. Okay. But what about the rest?
 

18:58.717 --> 19:03.187 align:start position:0%
area. Okay. But what about the rest?
What<00:20:28.080><c> happens</c><00:20:28.320><c> to</c><00:20:28.480><c> them?</c><00:20:29.039><c> Will</c><00:20:29.360><c> these</c><00:20:29.679><c> cancel?</c>

19:03.187 --> 19:03.197 align:start position:0%
What happens to them? Will these cancel?
 

19:03.197 --> 19:04.867 align:start position:0%
What happens to them? Will these cancel?
To<00:20:32.559><c> think</c><00:20:32.720><c> about</c><00:20:32.880><c> the</c><00:20:33.120><c> areas,</c><00:20:33.679><c> think</c><00:20:33.840><c> about</c>

19:04.867 --> 19:04.877 align:start position:0%
To think about the areas, think about
 

19:04.877 --> 19:07.347 align:start position:0%
To think about the areas, think about
the<00:20:34.240><c> product.</c><00:20:34.960><c> If</c><00:20:35.200><c> they</c><00:20:35.360><c> both</c><00:20:35.679><c> have</c><00:20:35.919><c> the</c><00:20:36.240><c> same</c>

19:07.347 --> 19:07.357 align:start position:0%
the product. If they both have the same
 

19:07.357 --> 19:09.906 align:start position:0%
the product. If they both have the same
signs,<00:20:37.520><c> then</c><00:20:37.840><c> the</c><00:20:38.080><c> product</c><00:20:38.400><c> is</c><00:20:38.640><c> going</c><00:20:38.720><c> to</c><00:20:38.880><c> be</c>

19:09.906 --> 19:09.916 align:start position:0%
signs, then the product is going to be
 

19:09.916 --> 19:11.906 align:start position:0%
signs, then the product is going to be
positive<00:20:39.440><c> and</c><00:20:39.679><c> so</c><00:20:39.840><c> the</c><00:20:40.000><c> area</c><00:20:40.320><c> will</c><00:20:40.559><c> also</c><00:20:40.720><c> be</c><00:20:40.880><c> a</c>

19:11.906 --> 19:11.916 align:start position:0%
positive and so the area will also be a
 

19:11.916 --> 19:14.707 align:start position:0%
positive and so the area will also be a
positive.<00:20:42.159><c> But</c><00:20:42.400><c> if</c><00:20:42.640><c> they</c><00:20:42.960><c> have</c><00:20:43.280><c> opposite</c>

19:14.707 --> 19:14.717 align:start position:0%
positive. But if they have opposite
 

19:14.717 --> 19:17.187 align:start position:0%
positive. But if they have opposite
signs,<00:20:44.799><c> then</c><00:20:45.039><c> the</c><00:20:45.280><c> product</c><00:20:45.600><c> will</c><00:20:45.760><c> be</c><00:20:45.919><c> negative</c>

19:17.187 --> 19:17.197 align:start position:0%
signs, then the product will be negative
 

19:17.197 --> 19:18.947 align:start position:0%
signs, then the product will be negative
and<00:20:46.480><c> the</c><00:20:46.640><c> area</c><00:20:46.880><c> under</c><00:20:47.200><c> the</c><00:20:47.360><c> curve</c><00:20:47.679><c> will</c><00:20:47.840><c> also</c>

19:18.947 --> 19:18.957 align:start position:0%
and the area under the curve will also
 

19:18.957 --> 19:21.747 align:start position:0%
and the area under the curve will also
end<00:20:48.240><c> up</c><00:20:48.400><c> becoming</c><00:20:49.360><c> negative.</c><00:20:50.320><c> Okay.</c><00:20:50.640><c> What</c>

19:21.747 --> 19:21.757 align:start position:0%
end up becoming negative. Okay. What
 

19:21.757 --> 19:23.747 align:start position:0%
end up becoming negative. Okay. What
happens<00:20:51.200><c> here?</c><00:20:51.760><c> Over</c><00:20:52.000><c> a</c><00:20:52.240><c> positive</c><00:20:52.640><c> half</c>

19:23.747 --> 19:23.757 align:start position:0%
happens here? Over a positive half
 

19:23.757 --> 19:26.147 align:start position:0%
happens here? Over a positive half
cycle,<00:20:53.600><c> you</c><00:20:53.840><c> have</c><00:20:54.000><c> the</c><00:20:54.240><c> same</c><00:20:54.400><c> sign.</c><00:20:54.799><c> Both</c><00:20:55.039><c> are</c>

19:26.147 --> 19:26.157 align:start position:0%
cycle, you have the same sign. Both are
 

19:26.157 --> 19:28.707 align:start position:0%
cycle, you have the same sign. Both are
positive.<00:20:55.760><c> So</c><00:20:56.080><c> you</c><00:20:56.400><c> get</c><00:20:56.480><c> a</c><00:20:56.720><c> positive</c><00:20:57.200><c> area.</c>

19:28.707 --> 19:28.717 align:start position:0%
positive. So you get a positive area.
 

19:28.717 --> 19:30.387 align:start position:0%
positive. So you get a positive area.
But<00:20:58.000><c> over</c><00:20:58.159><c> the</c><00:20:58.320><c> negative</c><00:20:58.720><c> half</c><00:20:58.960><c> cycle</c><00:20:59.360><c> you</c>

19:30.387 --> 19:30.397 align:start position:0%
But over the negative half cycle you
 

19:30.397 --> 19:32.227 align:start position:0%
But over the negative half cycle you
have<00:20:59.600><c> a</c><00:20:59.840><c> negative</c><00:21:00.320><c> *</c><00:21:00.559><c> positive.</c><00:21:01.039><c> So</c><00:21:01.200><c> the</c>

19:32.227 --> 19:32.237 align:start position:0%
have a negative * positive. So the
 

19:32.237 --> 19:34.147 align:start position:0%
have a negative * positive. So the
product<00:21:01.600><c> becomes</c><00:21:02.000><c> negative.</c><00:21:02.640><c> So</c><00:21:02.720><c> you</c><00:21:02.880><c> get</c><00:21:03.039><c> an</c>

19:34.147 --> 19:34.157 align:start position:0%
product becomes negative. So you get an
 

19:34.157 --> 19:36.947 align:start position:0%
product becomes negative. So you get an
equal<00:21:03.679><c> negative</c><00:21:04.080><c> area.</c><00:21:04.559><c> So</c><00:21:04.799><c> this</c><00:21:05.039><c> time</c><00:21:05.679><c> this</c>

19:36.947 --> 19:36.957 align:start position:0%
equal negative area. So this time this
 

19:36.957 --> 19:40.147 align:start position:0%
equal negative area. So this time this
cancels<00:21:06.559><c> out.</c><00:21:07.760><c> Okay.</c><00:21:08.080><c> What</c><00:21:08.320><c> happens</c><00:21:08.640><c> here?</c>

19:40.147 --> 19:40.157 align:start position:0%
cancels out. Okay. What happens here?
 

19:40.157 --> 19:42.947 align:start position:0%
cancels out. Okay. What happens here?
Notice<00:21:09.760><c> over</c><00:21:10.000><c> a</c><00:21:10.400><c> positive</c><00:21:10.799><c> cycle</c><00:21:11.360><c> you</c><00:21:11.679><c> have</c><00:21:11.840><c> a</c>

19:42.947 --> 19:42.957 align:start position:0%
Notice over a positive cycle you have a
 

19:42.957 --> 19:45.987 align:start position:0%
Notice over a positive cycle you have a
positive<00:21:12.799><c> and</c><00:21:13.039><c> a</c><00:21:13.280><c> negative</c><00:21:13.600><c> area.</c><00:21:14.400><c> But</c><00:21:14.559><c> over</c><00:21:14.799><c> a</c>

19:45.987 --> 19:45.997 align:start position:0%
positive and a negative area. But over a
 

19:45.997 --> 19:48.147 align:start position:0%
positive and a negative area. But over a
negative<00:21:15.440><c> cycle</c><00:21:15.760><c> the</c><00:21:16.000><c> whole</c><00:21:16.159><c> thing</c><00:21:16.400><c> reverses.</c>

19:48.147 --> 19:48.157 align:start position:0%
negative cycle the whole thing reverses.
 

19:48.157 --> 19:50.627 align:start position:0%
negative cycle the whole thing reverses.
And<00:21:17.520><c> so</c><00:21:17.760><c> again</c><00:21:18.240><c> it</c><00:21:18.559><c> perfectly</c><00:21:19.120><c> cancels</c><00:21:19.520><c> out.</c>

19:50.627 --> 19:50.637 align:start position:0%
And so again it perfectly cancels out.
 

19:50.637 --> 19:52.307 align:start position:0%
And so again it perfectly cancels out.
And<00:21:19.919><c> the</c><00:21:20.159><c> same</c><00:21:20.240><c> thing</c><00:21:20.400><c> applies</c><00:21:20.799><c> everywhere.</c>

19:52.307 --> 19:52.317 align:start position:0%
And the same thing applies everywhere.
 

19:52.317 --> 19:56.546 align:start position:0%
And the same thing applies everywhere.
So<00:21:21.919><c> indeed</c><00:21:23.200><c> these</c><00:21:23.600><c> all</c><00:21:24.000><c> cancel</c><00:21:24.480><c> out</c><00:21:25.039><c> and</c><00:21:25.360><c> we</c>

19:56.546 --> 19:56.556 align:start position:0%
So indeed these all cancel out and we
 

19:56.556 --> 20:00.627 align:start position:0%
So indeed these all cancel out and we
are<00:21:26.400><c> left</c><00:21:26.799><c> with</c><00:21:27.120><c> H1.</c><00:21:28.000><c> Oh</c><00:21:28.320><c> my</c><00:21:28.640><c> god,</c><00:21:29.120><c> this</c><00:21:29.360><c> is</c>

20:00.627 --> 20:00.637 align:start position:0%
are left with H1. Oh my god, this is
 

20:00.637 --> 20:02.707 align:start position:0%
are left with H1. Oh my god, this is
working.

20:02.707 --> 20:02.717 align:start position:0%
working.
 

20:02.717 --> 20:06.387 align:start position:0%
working.
But<00:21:32.159><c> for</c><00:21:33.280><c> there's</c><00:21:33.520><c> a</c><00:21:33.679><c> small</c><00:21:34.000><c> problem.</c><00:21:35.200><c> We</c><00:21:35.440><c> need</c>

20:06.387 --> 20:06.397 align:start position:0%
But for there's a small problem. We need
 

20:06.397 --> 20:08.707 align:start position:0%
But for there's a small problem. We need
to<00:21:35.600><c> now</c><00:21:35.840><c> find</c><00:21:36.000><c> the</c><00:21:36.159><c> area</c><00:21:36.400><c> under</c><00:21:36.640><c> this</c><00:21:36.880><c> curve.</c>

20:08.707 --> 20:08.717 align:start position:0%
to now find the area under this curve.
 

20:08.717 --> 20:10.627 align:start position:0%
to now find the area under this curve.
That's<00:21:38.159><c> going</c><00:21:38.240><c> to</c><00:21:38.400><c> be</c><00:21:38.480><c> hard</c><00:21:38.640><c> to</c><00:21:38.799><c> do,</c><00:21:39.120><c> right?</c>

20:10.627 --> 20:10.637 align:start position:0%
That's going to be hard to do, right?
 

20:10.637 --> 20:12.546 align:start position:0%
That's going to be hard to do, right?
And<00:21:39.919><c> Foria</c><00:21:40.400><c> says,</c><00:21:40.640><c> actually</c><00:21:40.960><c> Mahesh,</c><00:21:41.440><c> it's</c>

20:12.546 --> 20:12.556 align:start position:0%
And Foria says, actually Mahesh, it's
 

20:12.556 --> 20:14.227 align:start position:0%
And Foria says, actually Mahesh, it's
going<00:21:41.760><c> to</c><00:21:41.840><c> be</c><00:21:42.000><c> super</c><00:21:42.320><c> easy.</c><00:21:42.720><c> Barely</c><00:21:43.120><c> an</c>

20:14.227 --> 20:14.237 align:start position:0%
going to be super easy. Barely an
 

20:14.237 --> 20:23.666 align:start position:0%
going to be super easy. Barely an
inconvenience.<00:21:44.559><c> What?</c><00:21:45.200><c> How?</c>

20:23.666 --> 20:23.676 align:start position:0%
 
 

20:23.676 --> 20:25.507 align:start position:0%
 
I'm<00:21:53.120><c> embarrassed.</c><00:21:53.760><c> I'm</c><00:21:54.000><c> sorry.</c><00:21:54.320><c> I'm</c>

20:25.507 --> 20:25.517 align:start position:0%
I'm embarrassed. I'm sorry. I'm
 

20:25.517 --> 20:28.467 align:start position:0%
I'm embarrassed. I'm sorry. I'm
embarrassed.<00:21:55.360><c> Okay.</c><00:21:56.080><c> Anyways,</c><00:21:56.880><c> so</c><00:21:57.200><c> we</c><00:21:57.360><c> now</c>

20:28.467 --> 20:28.477 align:start position:0%
embarrassed. Okay. Anyways, so we now
 

20:28.477 --> 20:30.147 align:start position:0%
embarrassed. Okay. Anyways, so we now
need<00:21:57.760><c> to</c><00:21:57.919><c> find</c><00:21:58.080><c> the</c><00:21:58.240><c> area</c><00:21:58.480><c> of</c><00:21:58.640><c> this</c><00:21:58.799><c> rectangle,</c>

20:30.147 --> 20:30.157 align:start position:0%
need to find the area of this rectangle,
 

20:30.157 --> 20:33.026 align:start position:0%
need to find the area of this rectangle,
which<00:21:59.520><c> is</c><00:21:59.600><c> just</c><00:21:59.760><c> the</c><00:22:00.000><c> height</c><00:22:00.720><c> times</c><00:22:01.760><c> half</c><00:22:02.000><c> of</c>

20:33.026 --> 20:33.036 align:start position:0%
which is just the height times half of
 

20:33.036 --> 20:37.426 align:start position:0%
which is just the height times half of
the<00:22:02.320><c> period.</c><00:22:02.880><c> So</c><00:22:03.120><c> now</c><00:22:03.280><c> we</c><00:22:03.520><c> can</c><00:22:03.679><c> say</c><00:22:04.080><c> h1</c><00:22:04.799><c> *</c><00:22:05.360><c> t</c><00:22:05.600><c> /2</c>

20:37.426 --> 20:37.436 align:start position:0%
the period. So now we can say h1 * t /2
 

20:37.436 --> 20:40.307 align:start position:0%
the period. So now we can say h1 * t /2
should<00:22:06.960><c> equal</c><00:22:07.679><c> the</c><00:22:08.000><c> area</c><00:22:08.400><c> not</c><00:22:08.640><c> under</c><00:22:08.880><c> f_t</c><00:22:09.280><c> but</c>

20:40.307 --> 20:40.317 align:start position:0%
should equal the area not under f_t but
 

20:40.317 --> 20:44.867 align:start position:0%
should equal the area not under f_t but
f_t*s<00:22:10.559><c> the</c><00:22:10.880><c> sign.</c><00:22:11.919><c> This</c><00:22:12.480><c> is</c><00:22:12.720><c> how</c><00:22:13.039><c> we</c><00:22:13.360><c> isolate</c>

20:44.867 --> 20:44.877 align:start position:0%
f_t*s the sign. This is how we isolate
 

20:44.877 --> 20:48.546 align:start position:0%
f_t*s the sign. This is how we isolate
h1.<00:22:15.440><c> And</c><00:22:15.679><c> now</c><00:22:15.919><c> we</c><00:22:16.080><c> can</c><00:22:16.240><c> do</c><00:22:16.400><c> the</c><00:22:16.640><c> same</c><00:22:16.880><c> thing</c><00:22:17.280><c> for</c>

20:48.546 --> 20:48.556 align:start position:0%
h1. And now we can do the same thing for
 

20:48.556 --> 20:51.026 align:start position:0%
h1. And now we can do the same thing for
H2.<00:22:18.320><c> Multiply</c><00:22:18.720><c> the</c><00:22:18.960><c> whole</c><00:22:19.120><c> thing</c><00:22:19.520><c> with</c><00:22:19.840><c> that</c>

20:51.026 --> 20:51.036 align:start position:0%
H2. Multiply the whole thing with that
 

20:51.036 --> 20:53.587 align:start position:0%
H2. Multiply the whole thing with that
particular<00:22:20.640><c> sign</c><00:22:20.960><c> curve</c><00:22:21.760><c> and</c><00:22:22.000><c> then</c><00:22:22.320><c> calculate</c>

20:53.587 --> 20:53.597 align:start position:0%
particular sign curve and then calculate
 

20:53.597 --> 20:56.067 align:start position:0%
particular sign curve and then calculate
the<00:22:22.960><c> area</c><00:22:23.280><c> here</c><00:22:24.000><c> and</c><00:22:24.240><c> we'll</c><00:22:24.559><c> find</c><00:22:24.720><c> that</c><00:22:24.960><c> this</c>

20:56.067 --> 20:56.077 align:start position:0%
the area here and we'll find that this
 

20:56.077 --> 20:57.827 align:start position:0%
the area here and we'll find that this
area<00:22:25.600><c> remains</c><00:22:25.919><c> positive</c><00:22:26.320><c> where</c><00:22:26.640><c> everything</c>

20:57.827 --> 20:57.837 align:start position:0%
area remains positive where everything
 

20:57.837 --> 21:00.467 align:start position:0%
area remains positive where everything
else<00:22:27.280><c> cancels</c><00:22:27.679><c> out.</c><00:22:28.480><c> And</c><00:22:28.640><c> then</c><00:22:28.960><c> to</c><00:22:29.280><c> find</c><00:22:29.440><c> the</c>

21:00.467 --> 21:00.477 align:start position:0%
else cancels out. And then to find the
 

21:00.477 --> 21:02.467 align:start position:0%
else cancels out. And then to find the
area<00:22:29.919><c> under</c><00:22:30.159><c> this</c><00:22:30.400><c> curve,</c><00:22:30.720><c> we</c><00:22:30.960><c> do</c><00:22:31.120><c> the</c><00:22:31.280><c> same</c>

21:02.467 --> 21:02.477 align:start position:0%
area under this curve, we do the same
 

21:02.477 --> 21:05.347 align:start position:0%
area under this curve, we do the same
trick<00:22:32.000><c> that</c><00:22:32.400><c> we</c><00:22:32.640><c> did</c><00:22:32.880><c> before.</c>

21:05.347 --> 21:05.357 align:start position:0%
trick that we did before.
 

21:05.357 --> 21:08.947 align:start position:0%
trick that we did before.
And<00:22:34.720><c> notice</c><00:22:35.280><c> H2</c><00:22:35.919><c> *</c><00:22:36.159><c> T2</c><00:22:36.640><c> now</c><00:22:36.880><c> equals</c><00:22:37.360><c> area</c><00:22:37.679><c> under</c>

21:08.947 --> 21:08.957 align:start position:0%
And notice H2 * T2 now equals area under
 

21:08.957 --> 21:12.387 align:start position:0%
And notice H2 * T2 now equals area under
this<00:22:38.400><c> curve</c><00:22:38.880><c> times</c><00:22:39.600><c> the</c><00:22:39.919><c> sign.</c>

21:12.387 --> 21:12.397 align:start position:0%
this curve times the sign.
 

21:12.397 --> 21:16.786 align:start position:0%
this curve times the sign.
This<00:22:42.640><c> is</c><00:22:42.960><c> it.</c>

21:16.786 --> 21:16.796 align:start position:0%
This is it.
 

21:16.796 --> 21:19.666 align:start position:0%
This is it.
Okay,<00:22:46.320><c> why</c><00:22:46.720><c> did</c><00:22:46.960><c> this</c><00:22:47.280><c> work?</c><00:22:48.159><c> The</c><00:22:48.480><c> general</c>

21:19.666 --> 21:19.676 align:start position:0%
Okay, why did this work? The general
 

21:19.676 --> 21:21.747 align:start position:0%
Okay, why did this work? The general
idea<00:22:49.039><c> is</c><00:22:49.360><c> if</c><00:22:49.600><c> you</c><00:22:49.760><c> when</c><00:22:50.000><c> you</c><00:22:50.159><c> multiply</c><00:22:50.640><c> two</c>

21:21.747 --> 21:21.757 align:start position:0%
idea is if you when you multiply two
 

21:21.757 --> 21:24.067 align:start position:0%
idea is if you when you multiply two
sign<00:22:51.200><c> curves,</c><00:22:51.919><c> if</c><00:22:52.080><c> they</c><00:22:52.320><c> perfectly</c><00:22:52.799><c> line</c><00:22:53.039><c> up</c>

21:24.067 --> 21:24.077 align:start position:0%
sign curves, if they perfectly line up
 

21:24.077 --> 21:26.147 align:start position:0%
sign curves, if they perfectly line up
with<00:22:53.440><c> matching</c><00:22:53.840><c> frequencies,</c><00:22:54.720><c> then</c><00:22:54.960><c> you'll</c>

21:26.147 --> 21:26.157 align:start position:0%
with matching frequencies, then you'll
 

21:26.157 --> 21:28.147 align:start position:0%
with matching frequencies, then you'll
end<00:22:55.440><c> up</c><00:22:55.679><c> with</c><00:22:55.919><c> a</c><00:22:56.240><c> positive</c><00:22:56.559><c> area</c><00:22:56.799><c> under</c><00:22:57.120><c> the</c>

21:28.147 --> 21:28.157 align:start position:0%
end up with a positive area under the
 

21:28.157 --> 21:30.707 align:start position:0%
end up with a positive area under the
curve.<00:22:57.840><c> But</c><00:22:58.000><c> if</c><00:22:58.320><c> their</c><00:22:58.559><c> frequencies</c><00:22:59.360><c> don't</c>

21:30.707 --> 21:30.717 align:start position:0%
curve. But if their frequencies don't
 

21:30.717 --> 21:32.867 align:start position:0%
curve. But if their frequencies don't
match,<00:23:00.400><c> they</c><00:23:00.640><c> will</c><00:23:00.880><c> not</c><00:23:01.120><c> line</c><00:23:01.360><c> up</c><00:23:01.600><c> perfectly.</c>

21:32.867 --> 21:32.877 align:start position:0%
match, they will not line up perfectly.
 

21:32.877 --> 21:34.867 align:start position:0%
match, they will not line up perfectly.
They<00:23:02.159><c> will</c><00:23:02.320><c> go</c><00:23:02.400><c> in</c><00:23:02.559><c> and</c><00:23:02.799><c> out</c><00:23:02.960><c> of</c><00:23:03.120><c> alignment.</c>

21:34.867 --> 21:34.877 align:start position:0%
They will go in and out of alignment.
 

21:34.877 --> 21:37.267 align:start position:0%
They will go in and out of alignment.
And<00:23:04.159><c> when</c><00:23:04.400><c> that</c><00:23:04.640><c> happens,</c><00:23:05.200><c> the</c><00:23:05.520><c> area</c><00:23:05.840><c> under</c><00:23:06.159><c> a</c>

21:37.267 --> 21:37.277 align:start position:0%
And when that happens, the area under a
 

21:37.277 --> 21:41.187 align:start position:0%
And when that happens, the area under a
curve<00:23:07.039><c> over</c><00:23:07.280><c> a</c><00:23:07.440><c> full</c><00:23:07.679><c> period</c><00:23:08.880><c> will</c><00:23:09.120><c> be</c><00:23:09.280><c> zero.</c>

21:41.187 --> 21:41.197 align:start position:0%
curve over a full period will be zero.
 

21:41.197 --> 21:44.227 align:start position:0%
curve over a full period will be zero.
So<00:23:10.559><c> that's</c><00:23:10.960><c> how</c><00:23:11.360><c> you</c><00:23:11.600><c> can</c><00:23:11.840><c> use</c><00:23:12.480><c> each</c><00:23:12.880><c> sign</c>

21:44.227 --> 21:44.237 align:start position:0%
So that's how you can use each sign
 

21:44.237 --> 21:47.347 align:start position:0%
So that's how you can use each sign
frequency<00:23:13.840><c> as</c><00:23:14.080><c> a</c><00:23:14.240><c> probe</c><00:23:14.799><c> to</c><00:23:15.120><c> calculate</c><00:23:15.919><c> its</c>

21:47.347 --> 21:47.357 align:start position:0%
frequency as a probe to calculate its
 

21:47.357 --> 21:49.426 align:start position:0%
frequency as a probe to calculate its
strength.<00:23:17.120><c> It's</c><00:23:17.360><c> kind</c><00:23:17.520><c> of</c><00:23:17.600><c> like</c><00:23:17.840><c> using</c><00:23:18.159><c> a</c>

21:49.426 --> 21:49.436 align:start position:0%
strength. It's kind of like using a
 

21:49.436 --> 21:51.747 align:start position:0%
strength. It's kind of like using a
filter<00:23:19.440><c> to</c><00:23:19.679><c> find</c><00:23:19.919><c> the</c><00:23:20.080><c> strength</c><00:23:20.400><c> of</c><00:23:20.559><c> a</c>

21:51.747 --> 21:51.757 align:start position:0%
filter to find the strength of a
 

21:51.757 --> 21:53.747 align:start position:0%
filter to find the strength of a
specific<00:23:21.520><c> color</c><00:23:21.840><c> that</c><00:23:22.080><c> you're</c><00:23:22.320><c> looking</c><00:23:22.480><c> for.</c>

21:53.747 --> 21:53.757 align:start position:0%
specific color that you're looking for.
 

21:53.757 --> 21:57.507 align:start position:0%
specific color that you're looking for.
That<00:23:23.120><c> is</c><00:23:23.280><c> so</c><00:23:23.600><c> cool.</c><00:23:24.240><c> So</c><00:23:24.880><c> this</c><00:23:25.120><c> is</c><00:23:25.280><c> it.</c><00:23:25.840><c> We</c><00:23:26.159><c> now</c>

21:57.507 --> 21:57.517 align:start position:0%
That is so cool. So this is it. We now
 

21:57.517 --> 21:59.906 align:start position:0%
That is so cool. So this is it. We now
know<00:23:27.039><c> which</c><00:23:27.360><c> frequencies</c><00:23:28.159><c> and</c><00:23:28.559><c> what</c><00:23:28.799><c> the</c>

21:59.906 --> 21:59.916 align:start position:0%
know which frequencies and what the
 

21:59.916 --> 22:01.747 align:start position:0%
know which frequencies and what the
height<00:23:29.280><c> of</c><00:23:29.440><c> those</c><00:23:29.679><c> frequencies</c><00:23:30.240><c> are.</c><00:23:30.559><c> We</c><00:23:30.799><c> can</c>

22:01.747 --> 22:01.757 align:start position:0%
height of those frequencies are. We can
 

22:01.757 --> 22:03.906 align:start position:0%
height of those frequencies are. We can
calculate<00:23:31.360><c> it</c><00:23:31.840><c> and</c><00:23:32.080><c> we</c><00:23:32.320><c> can</c><00:23:32.480><c> put</c><00:23:32.640><c> it</c><00:23:32.799><c> back</c>

22:03.906 --> 22:03.916 align:start position:0%
calculate it and we can put it back
 

22:03.916 --> 22:05.666 align:start position:0%
calculate it and we can put it back
together<00:23:33.440><c> and</c><00:23:33.679><c> we</c><00:23:33.840><c> can</c><00:23:34.000><c> check</c><00:23:34.159><c> if</c><00:23:34.400><c> this</c><00:23:34.559><c> is</c>

22:05.666 --> 22:05.676 align:start position:0%
together and we can check if this is
 

22:05.676 --> 22:08.627 align:start position:0%
together and we can check if this is
indeed<00:23:35.280><c> true.</c><00:23:35.919><c> So</c><00:23:36.159><c> for</c><00:23:36.559><c> says</c><00:23:36.960><c> we</c><00:23:37.120><c> are</c><00:23:37.360><c> almost</c>

22:08.627 --> 22:08.637 align:start position:0%
indeed true. So for says we are almost
 

22:08.637 --> 22:11.267 align:start position:0%
indeed true. So for says we are almost
done<00:23:38.000><c> now.</c><00:23:38.880><c> And</c><00:23:39.039><c> I'm</c><00:23:39.280><c> like</c><00:23:39.440><c> wait</c><00:23:39.760><c> what?</c><00:23:40.080><c> Why</c><00:23:40.240><c> do</c>

22:11.267 --> 22:11.277 align:start position:0%
done now. And I'm like wait what? Why do
 

22:11.277 --> 22:13.987 align:start position:0%
done now. And I'm like wait what? Why do
you<00:23:40.559><c> say</c><00:23:40.880><c> almost?</c><00:23:41.520><c> We</c><00:23:41.760><c> are</c><00:23:41.919><c> done,</c><00:23:42.400><c> right?</c>

22:13.987 --> 22:13.997 align:start position:0%
you say almost? We are done, right?
 

22:13.997 --> 22:17.987 align:start position:0%
you say almost? We are done, right?
Well,<00:23:43.360><c> there</c><00:23:43.520><c> is</c><00:23:43.840><c> one</c><00:23:44.320><c> small</c><00:23:44.960><c> subtlety.</c>

22:17.987 --> 22:17.997 align:start position:0%
Well, there is one small subtlety.
 

22:17.997 --> 22:20.546 align:start position:0%
Well, there is one small subtlety.
Well,<00:23:47.440><c> so</c><00:23:47.679><c> far</c><00:23:47.919><c> we</c><00:23:48.240><c> assumed</c><00:23:48.960><c> that</c><00:23:49.360><c> every</c>

22:20.546 --> 22:20.556 align:start position:0%
Well, so far we assumed that every
 

22:20.556 --> 22:23.026 align:start position:0%
Well, so far we assumed that every
single<00:23:50.000><c> component</c><00:23:50.559><c> starts</c><00:23:50.960><c> with</c><00:23:51.200><c> this</c><00:23:51.600><c> exact</c>

22:23.026 --> 22:23.036 align:start position:0%
single component starts with this exact
 

22:23.036 --> 22:25.747 align:start position:0%
single component starts with this exact
phase.<00:23:53.120><c> But</c><00:23:53.360><c> what</c><00:23:53.600><c> if</c><00:23:53.760><c> it</c><00:23:54.000><c> doesn't</c><00:23:54.320><c> start</c><00:23:54.640><c> this</c>

22:25.747 --> 22:25.757 align:start position:0%
phase. But what if it doesn't start this
 

22:25.757 --> 22:28.307 align:start position:0%
phase. But what if it doesn't start this
way?<00:23:55.200><c> What</c><00:23:55.280><c> if</c><00:23:55.440><c> the</c><00:23:55.679><c> graph</c><00:23:56.080><c> is</c><00:23:56.640><c> shifted</c><00:23:57.200><c> like</c>

22:28.307 --> 22:28.317 align:start position:0%
way? What if the graph is shifted like
 

22:28.317 --> 22:31.507 align:start position:0%
way? What if the graph is shifted like
this<00:23:58.000><c> say</c><00:23:58.240><c> by</c><00:23:58.480><c> a</c><00:23:58.720><c> quarter</c><00:23:58.960><c> of</c><00:23:59.120><c> a</c><00:23:59.360><c> cycle?</c><00:24:00.159><c> Now</c><00:24:00.480><c> if</c>

22:31.507 --> 22:31.517 align:start position:0%
this say by a quarter of a cycle? Now if
 

22:31.517 --> 22:34.707 align:start position:0%
this say by a quarter of a cycle? Now if
we<00:24:00.799><c> were</c><00:24:00.960><c> to</c><00:24:01.200><c> multiply</c><00:24:01.840><c> by</c><00:24:02.400><c> sign</c><00:24:03.280><c> then</c><00:24:03.520><c> look</c><00:24:03.679><c> at</c>

22:34.707 --> 22:34.717 align:start position:0%
we were to multiply by sign then look at
 

22:34.717 --> 22:38.546 align:start position:0%
we were to multiply by sign then look at
what<00:24:04.080><c> happens.</c><00:24:04.720><c> They</c><00:24:05.120><c> will</c><00:24:05.360><c> not</c><00:24:06.159><c> be</c><00:24:06.480><c> aligned.</c>

22:38.546 --> 22:38.556 align:start position:0%
what happens. They will not be aligned.
 

22:38.556 --> 22:41.107 align:start position:0%
what happens. They will not be aligned.
Now<00:24:08.000><c> you</c><00:24:08.240><c> will</c><00:24:08.400><c> get</c><00:24:08.720><c> positives</c><00:24:09.440><c> and</c><00:24:09.760><c> negative</c>

22:41.107 --> 22:41.117 align:start position:0%
Now you will get positives and negative
 

22:41.117 --> 22:44.307 align:start position:0%
Now you will get positives and negative
areas<00:24:11.120><c> and</c><00:24:11.440><c> they</c><00:24:11.679><c> will</c><00:24:11.919><c> perfectly</c><00:24:12.480><c> cancel</c><00:24:13.120><c> to</c>

22:44.307 --> 22:44.317 align:start position:0%
areas and they will perfectly cancel to
 

22:44.317 --> 22:47.026 align:start position:0%
areas and they will perfectly cancel to
give<00:24:13.600><c> the</c><00:24:13.840><c> area</c><00:24:14.080><c> under</c><00:24:14.400><c> it</c><00:24:14.559><c> to</c><00:24:14.799><c> be</c><00:24:14.960><c> zero.</c><00:24:15.760><c> So</c><00:24:15.919><c> if</c>

22:47.026 --> 22:47.036 align:start position:0%
give the area under it to be zero. So if
 

22:47.036 --> 22:50.307 align:start position:0%
give the area under it to be zero. So if
you<00:24:16.320><c> did</c><00:24:16.480><c> this</c><00:24:16.880><c> we</c><00:24:17.200><c> might</c><00:24:17.919><c> wrongly</c><00:24:18.559><c> think</c><00:24:19.120><c> that</c>

22:50.307 --> 22:50.317 align:start position:0%
you did this we might wrongly think that
 

22:50.317 --> 22:53.347 align:start position:0%
you did this we might wrongly think that
the<00:24:19.679><c> height</c><00:24:20.000><c> of</c><00:24:20.159><c> that</c><00:24:20.400><c> frequency</c><00:24:20.960><c> is</c><00:24:21.520><c> zero.</c>

22:53.347 --> 22:53.357 align:start position:0%
the height of that frequency is zero.
 

22:53.357 --> 22:55.747 align:start position:0%
the height of that frequency is zero.
But<00:24:22.640><c> that's</c><00:24:22.960><c> wrong.</c><00:24:23.360><c> It</c><00:24:23.600><c> exists.</c><00:24:24.480><c> It's</c><00:24:24.640><c> just</c>

22:55.747 --> 22:55.757 align:start position:0%
But that's wrong. It exists. It's just
 

22:55.757 --> 22:58.546 align:start position:0%
But that's wrong. It exists. It's just
that<00:24:25.120><c> it</c><00:24:25.360><c> is</c><00:24:25.600><c> phase</c><00:24:26.000><c> shifted</c><00:24:26.799><c> by</c><00:24:27.039><c> a</c><00:24:27.279><c> quarter</c><00:24:27.600><c> of</c>

22:58.546 --> 22:58.556 align:start position:0%
that it is phase shifted by a quarter of
 

22:58.556 --> 23:03.666 align:start position:0%
that it is phase shifted by a quarter of
a<00:24:27.919><c> cycle.</c><00:24:28.799><c> So</c><00:24:29.520><c> how</c><00:24:29.760><c> do</c><00:24:29.919><c> we</c><00:24:30.960><c> take</c><00:24:31.200><c> care</c><00:24:31.440><c> of</c><00:24:31.600><c> this?</c>

23:03.666 --> 23:03.676 align:start position:0%
a cycle. So how do we take care of this?
 

23:03.676 --> 23:06.387 align:start position:0%
a cycle. So how do we take care of this?
Foria<00:24:33.279><c> says</c><00:24:33.760><c> simple</c><00:24:34.400><c> a</c><00:24:34.640><c> quarter</c><00:24:34.960><c> of</c><00:24:35.039><c> a</c><00:24:35.200><c> cycle</c>

23:06.387 --> 23:06.397 align:start position:0%
Foria says simple a quarter of a cycle
 

23:06.397 --> 23:08.867 align:start position:0%
Foria says simple a quarter of a cycle
phase<00:24:35.760><c> shift</c><00:24:36.080><c> is</c><00:24:36.320><c> basically</c><00:24:36.799><c> cost.</c><00:24:37.520><c> So</c><00:24:37.760><c> this</c>

23:08.867 --> 23:08.877 align:start position:0%
phase shift is basically cost. So this
 

23:08.877 --> 23:11.187 align:start position:0%
phase shift is basically cost. So this
time<00:24:38.559><c> we</c><00:24:38.880><c> should</c><00:24:39.039><c> have</c><00:24:39.279><c> actually</c><00:24:39.679><c> multiplied</c>

23:11.187 --> 23:11.197 align:start position:0%
time we should have actually multiplied
 

23:11.197 --> 23:14.227 align:start position:0%
time we should have actually multiplied
by<00:24:40.720><c> cost</c><00:24:41.600><c> and</c><00:24:41.919><c> calculated</c><00:24:42.400><c> the</c><00:24:42.640><c> area</c><00:24:43.120><c> that</c>

23:14.227 --> 23:14.237 align:start position:0%
by cost and calculated the area that
 

23:14.237 --> 23:16.867 align:start position:0%
by cost and calculated the area that
would<00:24:43.520><c> have</c><00:24:43.760><c> given</c><00:24:43.919><c> you</c><00:24:44.080><c> the</c><00:24:44.320><c> correct</c><00:24:44.880><c> answer.</c>

23:16.867 --> 23:16.877 align:start position:0%
would have given you the correct answer.
 

23:16.877 --> 23:20.227 align:start position:0%
would have given you the correct answer.
But<00:24:46.559><c> for</c><00:24:47.200><c> how</c><00:24:47.360><c> do</c><00:24:47.520><c> I</c><00:24:47.679><c> know</c><00:24:48.080><c> beforehand</c><00:24:49.039><c> whether</c>

23:20.227 --> 23:20.237 align:start position:0%
But for how do I know beforehand whether
 

23:20.237 --> 23:22.227 align:start position:0%
But for how do I know beforehand whether
I<00:24:49.520><c> should</c><00:24:49.679><c> be</c><00:24:49.840><c> multiplying</c><00:24:50.240><c> with</c><00:24:50.480><c> sign</c><00:24:50.880><c> or</c><00:24:51.200><c> a</c>

23:22.227 --> 23:22.237 align:start position:0%
I should be multiplying with sign or a
 

23:22.237 --> 23:27.267 align:start position:0%
I should be multiplying with sign or a
cost?<00:24:52.480><c> For</c><00:24:52.960><c> says</c><00:24:53.200><c> good</c><00:24:53.440><c> point</c><00:24:54.480><c> do</c><00:24:54.720><c> both.</c><00:24:55.679><c> What?</c>

23:27.267 --> 23:27.277 align:start position:0%
cost? For says good point do both. What?
 

23:27.277 --> 23:30.546 align:start position:0%
cost? For says good point do both. What?
Yeah.<00:24:57.279><c> If</c><00:24:57.520><c> you</c><00:24:57.919><c> multiply</c><00:24:58.400><c> by</c><00:24:58.720><c> sign</c><00:24:59.200><c> and</c><00:24:59.440><c> you</c>

23:30.546 --> 23:30.556 align:start position:0%
Yeah. If you multiply by sign and you
 

23:30.556 --> 23:33.267 align:start position:0%
Yeah. If you multiply by sign and you
get<00:25:00.159><c> say</c><00:25:00.480><c> the</c><00:25:00.640><c> height</c><00:25:00.880><c> to</c><00:25:01.120><c> be</c><00:25:01.279><c> zero</c><00:25:01.919><c> then</c><00:25:02.159><c> you</c>

23:33.267 --> 23:33.277 align:start position:0%
get say the height to be zero then you
 

23:33.277 --> 23:34.947 align:start position:0%
get say the height to be zero then you
know<00:25:02.640><c> the</c><00:25:02.880><c> height</c><00:25:03.120><c> from</c><00:25:03.279><c> the</c><00:25:03.440><c> cost</c><00:25:03.760><c> is</c><00:25:03.919><c> the</c>

23:34.947 --> 23:34.957 align:start position:0%
know the height from the cost is the
 

23:34.957 --> 23:37.026 align:start position:0%
know the height from the cost is the
real<00:25:04.320><c> height.</c><00:25:05.120><c> In</c><00:25:05.360><c> contrast</c><00:25:05.679><c> if</c><00:25:05.840><c> the</c><00:25:06.000><c> height</c>

23:37.026 --> 23:37.036 align:start position:0%
real height. In contrast if the height
 

23:37.036 --> 23:38.947 align:start position:0%
real height. In contrast if the height
from<00:25:06.400><c> the</c><00:25:06.559><c> cause</c><00:25:06.880><c> gives</c><00:25:07.120><c> you</c><00:25:07.360><c> zero</c><00:25:07.679><c> answer</c>

23:38.947 --> 23:38.957 align:start position:0%
from the cause gives you zero answer
 

23:38.957 --> 23:40.067 align:start position:0%
from the cause gives you zero answer
then<00:25:08.320><c> you</c><00:25:08.400><c> know</c><00:25:08.559><c> the</c><00:25:08.720><c> height</c><00:25:08.880><c> that</c><00:25:09.039><c> you</c>

23:40.067 --> 23:40.077 align:start position:0%
then you know the height that you
 

23:40.077 --> 23:41.987 align:start position:0%
then you know the height that you
calculated<00:25:09.600><c> from</c><00:25:09.760><c> the</c><00:25:09.919><c> sign</c><00:25:10.320><c> is</c><00:25:10.640><c> the</c><00:25:10.799><c> correct</c>

23:41.987 --> 23:41.997 align:start position:0%
calculated from the sign is the correct
 

23:41.997 --> 23:44.867 align:start position:0%
calculated from the sign is the correct
answer.<00:25:12.000><c> But</c><00:25:12.240><c> what</c><00:25:12.400><c> if</c><00:25:12.640><c> it's</c><00:25:13.279><c> not</c><00:25:13.520><c> shifted</c>

23:44.867 --> 23:44.877 align:start position:0%
answer. But what if it's not shifted
 

23:44.877 --> 23:46.786 align:start position:0%
answer. But what if it's not shifted
exactly<00:25:14.400><c> 90°?</c><00:25:15.039><c> What</c><00:25:15.200><c> if</c><00:25:15.279><c> it's</c><00:25:15.520><c> shifted</c>

23:46.786 --> 23:46.796 align:start position:0%
exactly 90°? What if it's shifted
 

23:46.796 --> 23:49.587 align:start position:0%
exactly 90°? What if it's shifted
somewhere<00:25:16.240><c> in</c><00:25:16.480><c> between?</c><00:25:17.200><c> Then</c><00:25:17.440><c> both</c><00:25:17.840><c> s</c><00:25:18.400><c> and</c>

23:49.587 --> 23:49.597 align:start position:0%
somewhere in between? Then both s and
 

23:49.597 --> 23:51.827 align:start position:0%
somewhere in between? Then both s and
cos<00:25:19.120><c> will</c><00:25:19.279><c> give</c><00:25:19.440><c> you</c><00:25:19.600><c> some</c><00:25:19.760><c> nonzero</c><00:25:20.400><c> answers.</c>

23:51.827 --> 23:51.837 align:start position:0%
cos will give you some nonzero answers.
 

23:51.837 --> 23:54.387 align:start position:0%
cos will give you some nonzero answers.
Now<00:25:21.279><c> do</c><00:25:21.600><c> now</c><00:25:21.840><c> what</c><00:25:22.000><c> do</c><00:25:22.159><c> we</c><00:25:22.320><c> do?</c><00:25:22.799><c> Well,</c><00:25:23.120><c> now</c><00:25:23.279><c> we</c>

23:54.387 --> 23:54.397 align:start position:0%
Now do now what do we do? Well, now we
 

23:54.397 --> 23:56.707 align:start position:0%
Now do now what do we do? Well, now we
just<00:25:23.679><c> use</c><00:25:23.919><c> the</c><00:25:24.080><c> Pythogus</c><00:25:24.799><c> theorem</c><00:25:25.120><c> to</c><00:25:25.360><c> get</c><00:25:25.600><c> the</c>

23:56.707 --> 23:56.717 align:start position:0%
just use the Pythogus theorem to get the
 

23:56.717 --> 24:01.267 align:start position:0%
just use the Pythogus theorem to get the
real<00:25:26.159><c> heights.</c><00:25:27.200><c> Why</c><00:25:27.520><c> Pythagoras?</c>

24:01.267 --> 24:01.277 align:start position:0%
real heights. Why Pythagoras?
 

24:01.277 --> 24:03.347 align:start position:0%
real heights. Why Pythagoras?
Well,<00:25:30.640><c> because</c><00:25:30.960><c> when</c><00:25:31.200><c> you</c><00:25:31.440><c> multiply</c><00:25:31.840><c> by</c><00:25:32.080><c> s</c><00:25:32.320><c> and</c>

24:03.347 --> 24:03.357 align:start position:0%
Well, because when you multiply by s and
 

24:03.357 --> 24:05.187 align:start position:0%
Well, because when you multiply by s and
cost,<00:25:32.880><c> what</c><00:25:33.120><c> you're</c><00:25:33.360><c> really</c><00:25:33.679><c> doing</c><00:25:34.000><c> is</c>

24:05.187 --> 24:05.197 align:start position:0%
cost, what you're really doing is
 

24:05.197 --> 24:07.026 align:start position:0%
cost, what you're really doing is
calculating<00:25:34.799><c> the</c><00:25:35.039><c> two</c><00:25:35.360><c> perpendicular</c>

24:07.026 --> 24:07.036 align:start position:0%
calculating the two perpendicular
 

24:07.036 --> 24:09.187 align:start position:0%
calculating the two perpendicular
components.<00:25:36.880><c> So,</c><00:25:37.120><c> you</c><00:25:37.279><c> use</c><00:25:37.440><c> the</c><00:25:37.679><c> Pythagoras</c>

24:09.187 --> 24:09.197 align:start position:0%
components. So, you use the Pythagoras
 

24:09.197 --> 24:12.467 align:start position:0%
components. So, you use the Pythagoras
to<00:25:38.480><c> calculate</c><00:25:38.880><c> the</c><00:25:39.200><c> full</c><00:25:39.679><c> height.</c><00:25:41.039><c> And</c><00:25:41.279><c> if</c><00:25:41.440><c> you</c>

24:12.467 --> 24:12.477 align:start position:0%
to calculate the full height. And if you
 

24:12.477 --> 24:15.267 align:start position:0%
to calculate the full height. And if you
take<00:25:41.840><c> their</c><00:25:42.159><c> ratios,</c><00:25:43.120><c> you</c><00:25:43.360><c> can</c><00:25:43.520><c> now</c><00:25:43.840><c> calculate</c>

24:15.267 --> 24:15.277 align:start position:0%
take their ratios, you can now calculate
 

24:15.277 --> 24:17.987 align:start position:0%
take their ratios, you can now calculate
the<00:25:44.720><c> phase</c><00:25:45.440><c> angle.</c><00:25:46.320><c> And</c><00:25:46.480><c> that</c><00:25:46.640><c> will</c><00:25:46.799><c> tell</c><00:25:46.880><c> you</c>

24:17.987 --> 24:17.997 align:start position:0%
the phase angle. And that will tell you
 

24:17.997 --> 24:20.707 align:start position:0%
the phase angle. And that will tell you
exactly<00:25:47.919><c> how</c><00:25:48.480><c> that</c><00:25:48.720><c> graph</c><00:25:48.960><c> is</c><00:25:49.200><c> shifted.</c><00:25:49.679><c> So</c>

24:20.707 --> 24:20.717 align:start position:0%
exactly how that graph is shifted. So
 

24:20.717 --> 24:22.867 align:start position:0%
exactly how that graph is shifted. So
for<00:25:50.000><c> example,</c><00:25:50.320><c> if</c><00:25:50.559><c> that</c><00:25:50.880><c> phase</c><00:25:51.200><c> angle</c><00:25:51.600><c> is</c>

24:22.867 --> 24:22.877 align:start position:0%
for example, if that phase angle is
 

24:22.877 --> 24:25.187 align:start position:0%
for example, if that phase angle is
zero,<00:25:52.640><c> then</c><00:25:52.960><c> you</c><00:25:53.120><c> know</c><00:25:53.279><c> that</c><00:25:53.520><c> it's</c><00:25:53.840><c> shifted</c><00:25:54.159><c> by</c>

24:25.187 --> 24:25.197 align:start position:0%
zero, then you know that it's shifted by
 

24:25.197 --> 24:27.426 align:start position:0%
zero, then you know that it's shifted by
0<00:25:54.640><c> degrees.</c><00:25:54.960><c> It's</c><00:25:55.039><c> a</c><00:25:55.279><c> perfect</c><00:25:55.520><c> sign.</c><00:25:56.159><c> If</c><00:25:56.320><c> that</c>

24:27.426 --> 24:27.436 align:start position:0%
0 degrees. It's a perfect sign. If that
 

24:27.436 --> 24:29.267 align:start position:0%
0 degrees. It's a perfect sign. If that
phase<00:25:56.799><c> angle</c><00:25:57.120><c> is</c><00:25:57.279><c> 90,</c><00:25:57.679><c> then</c><00:25:57.840><c> I</c><00:25:58.000><c> know</c><00:25:58.159><c> it's</c>

24:29.267 --> 24:29.277 align:start position:0%
phase angle is 90, then I know it's
 

24:29.277 --> 24:31.187 align:start position:0%
phase angle is 90, then I know it's
shifted<00:25:58.720><c> by</c><00:25:58.880><c> a</c><00:25:59.039><c> quarter</c><00:25:59.279><c> of</c><00:25:59.440><c> a</c><00:25:59.600><c> cycle</c><00:25:59.919><c> and</c><00:26:00.159><c> so</c>

24:31.187 --> 24:31.197 align:start position:0%
shifted by a quarter of a cycle and so
 

24:31.197 --> 24:35.267 align:start position:0%
shifted by a quarter of a cycle and so
on<00:26:00.480><c> and</c><00:26:00.720><c> so</c><00:26:00.880><c> forth.</c><00:26:01.679><c> So</c><00:26:02.240><c> now</c><00:26:03.120><c> by</c><00:26:03.679><c> calculating</c>

24:35.267 --> 24:35.277 align:start position:0%
on and so forth. So now by calculating
 

24:35.277 --> 24:38.707 align:start position:0%
on and so forth. So now by calculating
both<00:26:04.640><c> the</c><00:26:04.880><c> sign</c><00:26:05.120><c> and</c><00:26:05.440><c> cause,</c><00:26:06.480><c> we</c><00:26:07.200><c> have</c><00:26:07.440><c> the</c>

24:38.707 --> 24:38.717 align:start position:0%
both the sign and cause, we have the
 

24:38.717 --> 24:41.347 align:start position:0%
both the sign and cause, we have the
full<00:26:08.559><c> picture.</c><00:26:09.120><c> We</c><00:26:09.360><c> now</c><00:26:09.600><c> know</c><00:26:09.840><c> both</c><00:26:10.159><c> the</c>

24:41.347 --> 24:41.357 align:start position:0%
full picture. We now know both the
 

24:41.357 --> 24:45.187 align:start position:0%
full picture. We now know both the
heights<00:26:11.760><c> and</c><00:26:12.240><c> the</c><00:26:12.559><c> exact</c><00:26:13.039><c> phase</c><00:26:13.440><c> shift</c><00:26:13.840><c> of</c><00:26:14.159><c> the</c>

24:45.187 --> 24:45.197 align:start position:0%
heights and the exact phase shift of the
 

24:45.197 --> 24:47.587 align:start position:0%
heights and the exact phase shift of the
constituent<00:26:14.960><c> graphs.</c><00:26:15.760><c> This</c><00:26:16.080><c> tells</c><00:26:16.320><c> us</c><00:26:16.480><c> the</c>

24:47.587 --> 24:47.597 align:start position:0%
constituent graphs. This tells us the
 

24:47.597 --> 24:49.906 align:start position:0%
constituent graphs. This tells us the
exact<00:26:17.120><c> heights</c><00:26:17.840><c> and</c><00:26:18.159><c> this</c><00:26:18.400><c> tells</c><00:26:18.640><c> us</c><00:26:18.799><c> the</c>

24:49.906 --> 24:49.916 align:start position:0%
exact heights and this tells us the
 

24:49.916 --> 24:51.987 align:start position:0%
exact heights and this tells us the
exact<00:26:19.360><c> phase</c><00:26:19.679><c> shift.</c><00:26:20.159><c> And</c><00:26:20.400><c> now</c><00:26:20.640><c> we</c><00:26:20.880><c> are</c>

24:51.987 --> 24:51.997 align:start position:0%
exact phase shift. And now we are
 

24:51.997 --> 24:55.026 align:start position:0%
exact phase shift. And now we are
perfectly<00:26:21.919><c> done.</c>

24:55.026 --> 24:55.036 align:start position:0%
perfectly done.
 

24:55.036 --> 24:57.426 align:start position:0%
perfectly done.
And<00:26:24.320><c> so</c><00:26:24.559><c> now</c><00:26:24.799><c> if</c><00:26:25.120><c> we</c><00:26:25.440><c> account</c><00:26:25.760><c> for</c><00:26:26.000><c> the</c><00:26:26.240><c> phase</c>

24:57.426 --> 24:57.436 align:start position:0%
And so now if we account for the phase
 

24:57.436 --> 25:00.307 align:start position:0%
And so now if we account for the phase
shift,<00:26:26.960><c> we</c><00:26:27.200><c> are</c><00:26:27.360><c> done.</c><00:26:28.000><c> So</c><00:26:28.240><c> in</c><00:26:28.480><c> its</c><00:26:28.880><c> full</c>

25:00.307 --> 25:00.317 align:start position:0%
shift, we are done. So in its full
 

25:00.317 --> 25:03.507 align:start position:0%
shift, we are done. So in its full
glory,<00:26:30.559><c> the</c><00:26:30.960><c> four</c><00:26:31.520><c> C.</c><00:26:31.840><c> Whoa,</c><00:26:32.000><c> whoa,</c><00:26:32.240><c> whoa,</c>

25:03.507 --> 25:03.517 align:start position:0%
glory, the four C. Whoa, whoa, whoa,
 

25:03.517 --> 25:07.347 align:start position:0%
glory, the four C. Whoa, whoa, whoa,
waits<00:26:33.600><c> for</c><00:26:34.480><c> let's</c><00:26:34.799><c> first</c><00:26:35.200><c> clean</c><00:26:35.600><c> this</c><00:26:35.919><c> up.</c>

25:07.347 --> 25:07.357 align:start position:0%
waits for let's first clean this up.
 

25:07.357 --> 25:09.267 align:start position:0%
waits for let's first clean this up.
First<00:26:36.799><c> let's</c><00:26:37.039><c> write</c><00:26:37.279><c> this</c><00:26:37.520><c> down</c><00:26:37.840><c> as</c><00:26:38.159><c> a</c>

25:09.267 --> 25:09.277 align:start position:0%
First let's write this down as a
 

25:09.277 --> 25:12.707 align:start position:0%
First let's write this down as a
summation<00:26:39.200><c> where</c><00:26:39.520><c> n</c><00:26:39.840><c> is</c><00:26:40.080><c> an</c><00:26:40.240><c> integer.</c><00:26:41.279><c> Then</c><00:26:41.600><c> we</c>

25:12.707 --> 25:12.717 align:start position:0%
summation where n is an integer. Then we
 

25:12.717 --> 25:15.426 align:start position:0%
summation where n is an integer. Then we
turn<00:26:42.080><c> the</c><00:26:42.320><c> areas</c><00:26:42.960><c> into</c><00:26:43.440><c> integrals</c><00:26:44.080><c> over</c><00:26:44.320><c> a</c>

25:15.426 --> 25:15.436 align:start position:0%
turn the areas into integrals over a
 

25:15.436 --> 25:19.187 align:start position:0%
turn the areas into integrals over a
period.<00:26:45.520><c> Isolate</c><00:26:46.240><c> for</c><00:26:46.720><c> the</c><00:26:47.039><c> coefficients</c><00:26:48.080><c> and</c>

25:19.187 --> 25:19.197 align:start position:0%
period. Isolate for the coefficients and
 

25:19.197 --> 25:22.067 align:start position:0%
period. Isolate for the coefficients and
combine<00:26:48.720><c> them</c><00:26:48.960><c> in</c><00:26:49.120><c> terms</c><00:26:49.360><c> of</c><00:26:49.520><c> n</c><00:26:49.840><c> as</c><00:26:50.159><c> well.</c><00:26:50.960><c> And</c>

25:22.067 --> 25:22.077 align:start position:0%
combine them in terms of n as well. And
 

25:22.077 --> 25:24.387 align:start position:0%
combine them in terms of n as well. And
there<00:26:51.440><c> we</c><00:26:51.679><c> have</c><00:26:51.840><c> it.</c><00:26:52.320><c> Are</c><00:26:52.559><c> you</c><00:26:52.640><c> done</c><00:26:52.880><c> for</c><00:26:53.120><c> year?</c>

25:24.387 --> 25:24.397 align:start position:0%
there we have it. Are you done for year?
 

25:24.397 --> 25:27.987 align:start position:0%
there we have it. Are you done for year?
We're<00:26:53.840><c> done.</c><00:26:54.320><c> So</c><00:26:54.559><c> in</c><00:26:54.799><c> its</c><00:26:55.120><c> full</c><00:26:55.600><c> glory,</c><00:26:56.720><c> the</c>

25:27.987 --> 25:27.997 align:start position:0%
We're done. So in its full glory, the
 

25:27.997 --> 25:31.827 align:start position:0%
We're done. So in its full glory, the
forier<00:26:58.240><c> series.</c><00:26:58.960><c> I</c><00:26:59.279><c> mean</c><00:26:59.600><c> right</c><00:26:59.919><c> now</c><00:27:00.240><c> it</c><00:27:00.559><c> looks</c>

25:31.827 --> 25:31.837 align:start position:0%
forier series. I mean right now it looks
 

25:31.837 --> 25:34.387 align:start position:0%
forier series. I mean right now it looks
so<00:27:01.679><c> incredibly</c><00:27:02.559><c> mathematically</c>

25:34.387 --> 25:34.397 align:start position:0%
so incredibly mathematically
 

25:34.397 --> 25:37.426 align:start position:0%
so incredibly mathematically
complicated,<00:27:04.320><c> but</c><00:27:04.799><c> we</c><00:27:05.200><c> derived</c><00:27:05.840><c> all</c><00:27:06.159><c> of</c><00:27:06.320><c> it</c>

25:37.426 --> 25:37.436 align:start position:0%
complicated, but we derived all of it
 

25:37.436 --> 25:39.987 align:start position:0%
complicated, but we derived all of it
ourselves.<00:27:07.760><c> And</c><00:27:08.000><c> remember</c><00:27:08.400><c> where</c><00:27:08.640><c> we</c><00:27:08.799><c> got</c><00:27:08.960><c> it</c>

25:39.987 --> 25:39.997 align:start position:0%
ourselves. And remember where we got it
 

25:39.997 --> 25:43.107 align:start position:0%
ourselves. And remember where we got it
from?<00:27:10.159><c> from</c><00:27:10.559><c> heat</c><00:27:11.039><c> flowing</c><00:27:11.520><c> from</c><00:27:11.840><c> hot</c><00:27:12.080><c> to</c>

25:43.107 --> 25:43.117 align:start position:0%
from? from heat flowing from hot to
 

25:43.117 --> 25:45.666 align:start position:0%
from? from heat flowing from hot to
cold.<00:27:12.640><c> That</c><00:27:12.880><c> is</c><00:27:13.120><c> just</c><00:27:13.440><c> crazy</c><00:27:14.000><c> how</c><00:27:14.320><c> far</c><00:27:14.559><c> we</c>

25:45.666 --> 25:45.676 align:start position:0%
cold. That is just crazy how far we
 

25:45.676 --> 25:47.507 align:start position:0%
cold. That is just crazy how far we
reached<00:27:15.120><c> with</c><00:27:15.279><c> just</c><00:27:15.520><c> that</c><00:27:15.760><c> and</c><00:27:16.000><c> some</c><00:27:16.240><c> insane</c>

25:47.507 --> 25:47.517 align:start position:0%
reached with just that and some insane
 

25:47.517 --> 25:51.507 align:start position:0%
reached with just that and some insane
logic.<00:27:17.120><c> Oh</c><00:27:17.440><c> wow.</c><00:27:17.840><c> That</c><00:27:18.080><c> is</c><00:27:18.240><c> just</c><00:27:18.480><c> wow.</c>

25:51.507 --> 25:51.517 align:start position:0%
logic. Oh wow. That is just wow.
 

25:51.517 --> 25:53.507 align:start position:0%
logic. Oh wow. That is just wow.
So<00:27:20.880><c> now</c><00:27:21.039><c> that</c><00:27:21.200><c> we</c><00:27:21.440><c> know</c><00:27:21.679><c> exactly</c><00:27:22.240><c> how</c><00:27:22.480><c> to</c>

25:53.507 --> 25:53.517 align:start position:0%
So now that we know exactly how to
 

25:53.517 --> 25:54.947 align:start position:0%
So now that we know exactly how to
figure<00:27:22.880><c> out</c><00:27:23.039><c> the</c><00:27:23.279><c> frequencies</c><00:27:23.679><c> and</c><00:27:23.919><c> the</c>

25:54.947 --> 25:54.957 align:start position:0%
figure out the frequencies and the
 

25:54.957 --> 25:56.627 align:start position:0%
figure out the frequencies and the
heights,<00:27:24.480><c> we</c><00:27:24.640><c> can</c><00:27:24.880><c> test</c><00:27:25.120><c> it</c><00:27:25.360><c> in</c><00:27:25.520><c> our</c>

25:56.627 --> 25:56.637 align:start position:0%
heights, we can test it in our
 

25:56.637 --> 25:59.267 align:start position:0%
heights, we can test it in our
simulation<00:27:26.799><c> and</c><00:27:27.120><c> the</c><00:27:27.360><c> result</c><00:27:27.679><c> is</c><00:27:27.919><c> right</c><00:27:28.159><c> in</c>

25:59.267 --> 25:59.277 align:start position:0%
simulation and the result is right in
 

25:59.277 --> 26:02.307 align:start position:0%
simulation and the result is right in
front<00:27:28.559><c> of</c><00:27:28.640><c> us.</c><00:27:29.360><c> Indeed,</c><00:27:30.159><c> any</c><00:27:30.559><c> random</c><00:27:31.039><c> signal</c>

26:02.307 --> 26:02.317 align:start position:0%
front of us. Indeed, any random signal
 

26:02.317 --> 26:05.267 align:start position:0%
front of us. Indeed, any random signal
can<00:27:31.760><c> be</c><00:27:31.919><c> built</c><00:27:32.320><c> using</c><00:27:32.720><c> sine</c><00:27:32.960><c> waves,</c><00:27:33.760><c> even</c><00:27:34.080><c> ones</c>

26:05.267 --> 26:05.277 align:start position:0%
can be built using sine waves, even ones
 

26:05.277 --> 26:07.507 align:start position:0%
can be built using sine waves, even ones
with<00:27:34.559><c> the</c><00:27:34.720><c> sharp</c><00:27:35.120><c> corners.</c><00:27:35.840><c> Ideally,</c><00:27:36.320><c> we</c><00:27:36.480><c> need</c>

26:07.507 --> 26:07.517 align:start position:0%
with the sharp corners. Ideally, we need
 

26:07.517 --> 26:09.426 align:start position:0%
with the sharp corners. Ideally, we need
infinite<00:27:37.120><c> components.</c><00:27:37.760><c> But</c><00:27:37.919><c> the</c><00:27:38.080><c> more</c><00:27:38.320><c> we</c>

26:09.426 --> 26:09.436 align:start position:0%
infinite components. But the more we
 

26:09.436 --> 26:13.666 align:start position:0%
infinite components. But the more we
add,<00:27:39.120><c> the</c><00:27:39.440><c> closer</c><00:27:40.159><c> we</c><00:27:40.480><c> get</c><00:27:40.720><c> to</c><00:27:40.960><c> it.</c><00:27:42.080><c> Gosh,</c><00:27:42.559><c> this</c>

26:13.666 --> 26:13.676 align:start position:0%
add, the closer we get to it. Gosh, this
 

26:13.676 --> 26:16.067 align:start position:0%
add, the closer we get to it. Gosh, this
is<00:27:42.960><c> awesome.</c><00:27:43.679><c> There's</c><00:27:44.000><c> one</c><00:27:44.320><c> loose</c><00:27:44.640><c> end</c><00:27:44.880><c> left,</c>

26:16.067 --> 26:16.077 align:start position:0%
is awesome. There's one loose end left,
 

26:16.077 --> 26:18.387 align:start position:0%
is awesome. There's one loose end left,
though.<00:27:45.760><c> Remember</c><00:27:46.000><c> how</c><00:27:46.240><c> we</c><00:27:46.480><c> assume</c><00:27:46.799><c> that</c><00:27:47.120><c> our</c>

26:18.387 --> 26:18.397 align:start position:0%
though. Remember how we assume that our
 

26:18.397 --> 26:21.507 align:start position:0%
though. Remember how we assume that our
signal<00:27:48.080><c> repeats</c><00:27:48.559><c> forever,</c><00:27:49.760><c> but</c><00:27:50.159><c> real</c><00:27:50.400><c> world</c>

26:21.507 --> 26:21.517 align:start position:0%
signal repeats forever, but real world
 

26:21.517 --> 26:24.146 align:start position:0%
signal repeats forever, but real world
signals<00:27:51.279><c> don't.</c><00:27:51.919><c> So,</c><00:27:52.320><c> again,</c><00:27:52.640><c> for</c><00:27:53.120><c> I</c><00:27:53.200><c> think</c>

26:24.146 --> 26:24.156 align:start position:0%
signals don't. So, again, for I think
 

26:24.156 --> 26:25.587 align:start position:0%
signals don't. So, again, for I think
it's<00:27:53.440><c> going</c><00:27:53.520><c> to</c><00:27:53.600><c> be</c><00:27:53.760><c> hard</c><00:27:53.919><c> to</c><00:27:54.080><c> translate</c>

26:25.587 --> 26:25.597 align:start position:0%
it's going to be hard to translate
 

26:25.597 --> 26:27.507 align:start position:0%
it's going to be hard to translate
everything<00:27:54.960><c> that</c><00:27:55.200><c> we</c><00:27:55.360><c> derived</c><00:27:55.919><c> to</c><00:27:56.240><c> the</c><00:27:56.399><c> real</c>

26:27.507 --> 26:27.517 align:start position:0%
everything that we derived to the real
 

26:27.517 --> 26:29.827 align:start position:0%
everything that we derived to the real
world.<00:27:57.200><c> And</c><00:27:57.440><c> for</c><00:27:57.760><c> says,</c><00:27:57.919><c> actually</c><00:27:58.240><c> m</c><00:27:58.720><c> super</c>

26:29.827 --> 26:29.837 align:start position:0%
world. And for says, actually m super
 

26:29.837 --> 26:33.587 align:start position:0%
world. And for says, actually m super
easy,<00:27:59.440><c> barely</c><00:27:59.840><c> an</c><00:28:00.000><c> inconvenience.</c><00:28:01.039><c> What?</c><00:28:01.760><c> How</c>

26:33.587 --> 26:33.597 align:start position:0%
easy, barely an inconvenience. What? How
 

26:33.597 --> 26:36.467 align:start position:0%
easy, barely an inconvenience. What? How
non-periodic<00:28:03.600><c> signals</c><00:28:04.240><c> can</c><00:28:04.480><c> be</c><00:28:04.720><c> thought</c><00:28:05.200><c> as</c>

26:36.467 --> 26:36.477 align:start position:0%
non-periodic signals can be thought as
 

26:36.477 --> 26:39.426 align:start position:0%
non-periodic signals can be thought as
periodic<00:28:06.080><c> signals</c><00:28:06.640><c> with</c><00:28:07.039><c> period</c>

26:39.426 --> 26:39.436 align:start position:0%
periodic signals with period
 

26:39.436 --> 26:42.387 align:start position:0%
periodic signals with period
infinity?

26:42.387 --> 26:42.397 align:start position:0%
infinity?
 

26:42.397 --> 26:45.347 align:start position:0%
infinity?
That<00:28:11.840><c> is</c><00:28:12.000><c> so</c><00:28:12.399><c> brilliant.</c><00:28:13.360><c> What</c><00:28:13.600><c> a</c><00:28:13.840><c> brilliant</c>

26:45.347 --> 26:45.357 align:start position:0%
That is so brilliant. What a brilliant
 

26:45.357 --> 26:48.707 align:start position:0%
That is so brilliant. What a brilliant
logic.<00:28:15.600><c> Non-periodic</c><00:28:16.320><c> signals</c><00:28:16.799><c> are</c><00:28:17.039><c> periodic</c>

26:48.707 --> 26:48.717 align:start position:0%
logic. Non-periodic signals are periodic
 

26:48.717 --> 26:50.947 align:start position:0%
logic. Non-periodic signals are periodic
with<00:28:18.159><c> infinite</c><00:28:18.720><c> period.</c><00:28:19.520><c> That</c><00:28:19.760><c> is</c><00:28:19.919><c> so</c>

26:50.947 --> 26:50.957 align:start position:0%
with infinite period. That is so
 

26:50.957 --> 26:52.067 align:start position:0%
with infinite period. That is so
brilliant.<00:28:20.480><c> You</c><00:28:20.720><c> got</c><00:28:20.799><c> to</c><00:28:20.960><c> love</c>

26:52.067 --> 26:52.077 align:start position:0%
brilliant. You got to love
 

26:52.077 --> 26:54.067 align:start position:0%
brilliant. You got to love
mathematicians,<00:28:21.919><c> man.</c><00:28:22.240><c> Come</c><00:28:22.399><c> on.</c><00:28:22.720><c> That</c><00:28:22.880><c> is</c>

26:54.067 --> 26:54.077 align:start position:0%
mathematicians, man. Come on. That is
 

26:54.077 --> 26:56.067 align:start position:0%
mathematicians, man. Come on. That is
that<00:28:23.440><c> is</c><00:28:23.679><c> cool.</c>

26:56.067 --> 26:56.077 align:start position:0%
that is cool.
 

26:56.077 --> 26:58.467 align:start position:0%
that is cool.
Okay,<00:28:25.600><c> let's</c><00:28:25.919><c> calm</c><00:28:26.240><c> down</c><00:28:26.960><c> because</c><00:28:27.279><c> when</c><00:28:27.520><c> you</c>

26:58.467 --> 26:58.477 align:start position:0%
Okay, let's calm down because when you
 

26:58.477 --> 27:00.867 align:start position:0%
Okay, let's calm down because when you
do,<00:28:28.080><c> you</c><00:28:28.320><c> realize</c><00:28:28.640><c> that's</c><00:28:28.880><c> a</c><00:28:29.039><c> big</c><00:28:29.279><c> problem.</c>

27:00.867 --> 27:00.877 align:start position:0%
do, you realize that's a big problem.
 

27:00.877 --> 27:04.467 align:start position:0%
do, you realize that's a big problem.
See,<00:28:30.240><c> if</c><00:28:30.399><c> the</c><00:28:30.640><c> period</c><00:28:31.279><c> goes</c><00:28:31.600><c> to</c><00:28:32.159><c> infinity,</c><00:28:33.360><c> the</c>

27:04.467 --> 27:04.477 align:start position:0%
See, if the period goes to infinity, the
 

27:04.477 --> 27:06.627 align:start position:0%
See, if the period goes to infinity, the
heights<00:28:34.080><c> all</c><00:28:34.399><c> go</c><00:28:34.559><c> to</c><00:28:34.720><c> zero.</c><00:28:35.200><c> Everything</c><00:28:35.520><c> goes</c>

27:06.627 --> 27:06.637 align:start position:0%
heights all go to zero. Everything goes
 

27:06.637 --> 27:09.347 align:start position:0%
heights all go to zero. Everything goes
to<00:28:35.919><c> zero.</c><00:28:36.399><c> The</c><00:28:36.640><c> whole</c><00:28:36.799><c> thing</c><00:28:37.200><c> breaks.</c><00:28:38.080><c> Forier</c>

27:09.347 --> 27:09.357 align:start position:0%
to zero. The whole thing breaks. Forier
 

27:09.357 --> 27:12.067 align:start position:0%
to zero. The whole thing breaks. Forier
says,<00:28:38.720><c> "Mah,</c><00:28:39.600><c> that's</c><00:28:40.000><c> not</c><00:28:40.240><c> a</c><00:28:40.480><c> bug.</c><00:28:40.880><c> That's</c>

27:12.067 --> 27:12.077 align:start position:0%
says, "Mah, that's not a bug. That's
 

27:12.077 --> 27:14.867 align:start position:0%
says, "Mah, that's not a bug. That's
actually<00:28:41.440><c> a</c><00:28:41.760><c> feature.</c><00:28:42.480><c> Let's</c><00:28:42.799><c> go</c><00:28:42.960><c> to</c><00:28:43.120><c> infinity</c>

27:14.867 --> 27:14.877 align:start position:0%
actually a feature. Let's go to infinity
 

27:14.877 --> 27:16.707 align:start position:0%
actually a feature. Let's go to infinity
step<00:28:44.320><c> by</c><00:28:44.559><c> step.</c><00:28:45.120><c> Let's</c><00:28:45.360><c> start</c><00:28:45.520><c> with</c><00:28:45.679><c> the</c>

27:16.707 --> 27:16.717 align:start position:0%
step by step. Let's start with the
 

27:16.717 --> 27:19.666 align:start position:0%
step by step. Let's start with the
period<00:28:46.240><c> t.</c><00:28:46.880><c> If</c><00:28:47.120><c> we</c><00:28:47.279><c> double</c><00:28:47.600><c> the</c><00:28:47.760><c> period,</c><00:28:48.559><c> the</c>

27:19.666 --> 27:19.676 align:start position:0%
period t. If we double the period, the
 

27:19.676 --> 27:21.827 align:start position:0%
period t. If we double the period, the
heights<00:28:49.200><c> all</c><00:28:49.440><c> get</c><00:28:49.679><c> halfed,</c><00:28:50.480><c> right?</c><00:28:50.799><c> Because</c>

27:21.827 --> 27:21.837 align:start position:0%
heights all get halfed, right? Because
 

27:21.837 --> 27:24.707 align:start position:0%
heights all get halfed, right? Because
there's<00:28:51.200><c> a</c><00:28:51.440><c> divided</c><00:28:51.840><c> by</c><00:28:52.000><c> t.</c><00:28:52.799><c> But</c><00:28:52.960><c> now</c><00:28:53.520><c> more</c>

27:24.707 --> 27:24.717 align:start position:0%
there's a divided by t. But now more
 

27:24.717 --> 27:27.107 align:start position:0%
there's a divided by t. But now more
frequencies<00:28:54.480><c> are</c><00:28:54.720><c> allowed</c><00:28:55.039><c> to</c><00:28:55.440><c> squeeze</c><00:28:55.840><c> in</c><00:28:56.000><c> in</c>

27:27.107 --> 27:27.117 align:start position:0%
frequencies are allowed to squeeze in in
 

27:27.117 --> 27:28.947 align:start position:0%
frequencies are allowed to squeeze in in
between.<00:28:56.799><c> So</c><00:28:57.039><c> although</c><00:28:57.360><c> the</c><00:28:57.520><c> heights</c><00:28:57.840><c> got</c>

27:28.947 --> 27:28.957 align:start position:0%
between. So although the heights got
 

27:28.957 --> 27:31.666 align:start position:0%
between. So although the heights got
half<00:28:58.559><c> the</c><00:28:58.880><c> number</c><00:28:59.120><c> of</c><00:28:59.440><c> components</c><00:29:00.320><c> doubled</c>

27:31.666 --> 27:31.676 align:start position:0%
half the number of components doubled
 

27:31.676 --> 27:33.906 align:start position:0%
half the number of components doubled
and<00:29:01.039><c> that</c><00:29:01.279><c> compensates</c><00:29:02.159><c> double</c><00:29:02.480><c> the</c><00:29:02.640><c> period</c>

27:33.906 --> 27:33.916 align:start position:0%
and that compensates double the period
 

27:33.916 --> 27:36.947 align:start position:0%
and that compensates double the period
again<00:29:03.679><c> the</c><00:29:04.000><c> same</c><00:29:04.320><c> thing</c><00:29:04.559><c> happens</c><00:29:05.039><c> again</c><00:29:05.840><c> as</c>

27:36.947 --> 27:36.957 align:start position:0%
again the same thing happens again as
 

27:36.957 --> 27:39.026 align:start position:0%
again the same thing happens again as
the<00:29:06.240><c> periods</c><00:29:06.720><c> get</c><00:29:06.880><c> bigger</c><00:29:07.200><c> and</c><00:29:07.440><c> bigger</c><00:29:07.919><c> each</c>

27:39.026 --> 27:39.036 align:start position:0%
the periods get bigger and bigger each
 

27:39.036 --> 27:41.187 align:start position:0%
the periods get bigger and bigger each
individual<00:29:08.799><c> components</c><00:29:09.360><c> get</c><00:29:09.520><c> shorter</c><00:29:10.000><c> but</c>

27:41.187 --> 27:41.197 align:start position:0%
individual components get shorter but
 

27:41.197 --> 27:42.947 align:start position:0%
individual components get shorter but
more<00:29:10.559><c> and</c><00:29:10.799><c> more</c><00:29:10.960><c> frequencies</c><00:29:11.520><c> get</c><00:29:11.760><c> squeezed</c>

27:42.947 --> 27:42.957 align:start position:0%
more and more frequencies get squeezed
 

27:42.957 --> 27:45.266 align:start position:0%
more and more frequencies get squeezed
in<00:29:12.720><c> and</c><00:29:12.880><c> in</c><00:29:13.120><c> the</c><00:29:13.360><c> limit</c><00:29:13.600><c> where</c><00:29:13.840><c> the</c><00:29:14.000><c> period</c>

27:45.266 --> 27:45.276 align:start position:0%
in and in the limit where the period
 

27:45.276 --> 27:47.347 align:start position:0%
in and in the limit where the period
goes<00:29:14.640><c> to</c><00:29:14.799><c> infinity</c><00:29:15.679><c> the</c><00:29:15.840><c> heights</c><00:29:16.159><c> of</c>

27:47.347 --> 27:47.357 align:start position:0%
goes to infinity the heights of
 

27:47.357 --> 27:49.827 align:start position:0%
goes to infinity the heights of
individual<00:29:17.039><c> ones</c><00:29:17.360><c> shrink</c><00:29:17.760><c> to</c><00:29:18.080><c> zero</c><00:29:18.480><c> but</c><00:29:18.720><c> the</c>

27:49.827 --> 27:49.837 align:start position:0%
individual ones shrink to zero but the
 

27:49.837 --> 27:52.627 align:start position:0%
individual ones shrink to zero but the
frequencies<00:29:19.440><c> themselves</c><00:29:19.919><c> becomes</c><00:29:20.559><c> a</c><00:29:20.880><c> smooth</c>

27:52.627 --> 27:52.637 align:start position:0%
frequencies themselves becomes a smooth
 

27:52.637 --> 27:56.627 align:start position:0%
frequencies themselves becomes a smooth
continuous<00:29:23.039><c> spectrum.</c><00:29:24.080><c> So</c><00:29:24.320><c> now</c><00:29:24.799><c> we</c><00:29:25.120><c> have</c><00:29:25.360><c> the</c>

27:56.627 --> 27:56.637 align:start position:0%
continuous spectrum. So now we have the
 

27:56.637 --> 28:01.587 align:start position:0%
continuous spectrum. So now we have the
full<00:29:26.320><c> picture</c><00:29:27.440><c> for</c><00:29:27.919><c> a</c><00:29:28.480><c> periodic</c><00:29:29.520><c> signal</c><00:29:30.480><c> we</c>

28:01.587 --> 28:01.597 align:start position:0%
full picture for a periodic signal we
 

28:01.597 --> 28:05.026 align:start position:0%
full picture for a periodic signal we
get<00:29:31.120><c> discrete</c><00:29:31.840><c> set</c><00:29:32.159><c> of</c><00:29:32.640><c> frequencies</c><00:29:33.679><c> and</c><00:29:33.919><c> we</c>

28:05.026 --> 28:05.036 align:start position:0%
get discrete set of frequencies and we
 

28:05.036 --> 28:09.187 align:start position:0%
get discrete set of frequencies and we
can<00:29:34.320><c> add</c><00:29:34.559><c> them</c><00:29:34.720><c> up</c><00:29:35.039><c> as</c><00:29:35.600><c> a</c><00:29:35.919><c> forier</c><00:29:36.799><c> series.</c><00:29:37.919><c> But</c>

28:09.187 --> 28:09.197 align:start position:0%
can add them up as a forier series. But
 

28:09.197 --> 28:12.947 align:start position:0%
can add them up as a forier series. But
when<00:29:38.640><c> we</c><00:29:38.880><c> have</c><00:29:39.039><c> a</c><00:29:39.360><c> non-periodic</c><00:29:40.320><c> signal</c><00:29:41.200><c> we</c><00:29:41.679><c> no</c>

28:12.947 --> 28:12.957 align:start position:0%
when we have a non-periodic signal we no
 

28:12.957 --> 28:15.987 align:start position:0%
when we have a non-periodic signal we no
longer<00:29:42.640><c> have</c><00:29:43.520><c> separate</c><00:29:44.399><c> countable</c>

28:15.987 --> 28:15.997 align:start position:0%
longer have separate countable
 

28:15.997 --> 28:19.587 align:start position:0%
longer have separate countable
components.<00:29:45.760><c> We</c><00:29:46.000><c> have</c><00:29:46.159><c> a</c><00:29:46.399><c> whole</c><00:29:46.960><c> continuum</c><00:29:48.000><c> of</c>

28:19.587 --> 28:19.597 align:start position:0%
components. We have a whole continuum of
 

28:19.597 --> 28:22.786 align:start position:0%
components. We have a whole continuum of
frequencies.<00:29:49.919><c> So</c><00:29:50.159><c> this</c><00:29:50.399><c> goes</c><00:29:50.640><c> away</c><00:29:51.440><c> and</c><00:29:51.679><c> the</c>

28:22.786 --> 28:22.796 align:start position:0%
frequencies. So this goes away and the
 

28:22.796 --> 28:24.786 align:start position:0%
frequencies. So this goes away and the
heights<00:29:52.240><c> and</c><00:29:52.480><c> the</c><00:29:52.640><c> phase</c><00:29:52.880><c> shifts</c><00:29:53.360><c> become</c>

28:24.786 --> 28:24.796 align:start position:0%
heights and the phase shifts become
 

28:24.796 --> 28:27.426 align:start position:0%
heights and the phase shifts become
functions<00:29:54.559><c> of</c><00:29:55.039><c> frequency.</c>

28:27.426 --> 28:27.436 align:start position:0%
functions of frequency.
 

28:27.436 --> 28:30.467 align:start position:0%
functions of frequency.
This<00:29:57.279><c> is</c><00:29:57.600><c> called</c><00:29:58.000><c> a</c><00:29:58.159><c> forier</c><00:29:58.720><c> transform</c>

28:30.467 --> 28:30.477 align:start position:0%
This is called a forier transform
 

28:30.477 --> 28:32.627 align:start position:0%
This is called a forier transform
because<00:30:00.080><c> we</c><00:30:00.320><c> have</c><00:30:00.559><c> transformed</c><00:30:01.120><c> the</c><00:30:01.360><c> signal</c>

28:32.627 --> 28:32.637 align:start position:0%
because we have transformed the signal
 

28:32.637 --> 28:36.067 align:start position:0%
because we have transformed the signal
into<00:30:02.559><c> a</c><00:30:02.880><c> frequency</c><00:30:03.760><c> domain.</c><00:30:04.640><c> And</c><00:30:04.880><c> this</c>

28:36.067 --> 28:36.077 align:start position:0%
into a frequency domain. And this
 

28:36.077 --> 28:38.707 align:start position:0%
into a frequency domain. And this
reconstruction<00:30:06.000><c> is</c><00:30:06.320><c> no</c><00:30:06.480><c> longer</c><00:30:06.720><c> a</c><00:30:06.960><c> summation</c>

28:38.707 --> 28:38.717 align:start position:0%
reconstruction is no longer a summation
 

28:38.717 --> 28:41.666 align:start position:0%
reconstruction is no longer a summation
but<00:30:08.080><c> it's</c><00:30:08.399><c> an</c><00:30:08.799><c> integral.</c><00:30:09.919><c> And</c><00:30:10.159><c> that</c><00:30:10.399><c> is</c><00:30:10.640><c> what</c>

28:41.666 --> 28:41.676 align:start position:0%
but it's an integral. And that is what
 

28:41.676 --> 28:43.587 align:start position:0%
but it's an integral. And that is what
we<00:30:10.960><c> call</c><00:30:11.200><c> an</c><00:30:11.440><c> inverse</c><00:30:11.919><c> transform</c><00:30:12.480><c> because</c>

28:43.587 --> 28:43.597 align:start position:0%
we call an inverse transform because
 

28:43.597 --> 28:46.146 align:start position:0%
we call an inverse transform because
we're<00:30:12.960><c> going</c><00:30:13.279><c> back.</c>

28:46.146 --> 28:46.156 align:start position:0%
we're going back.
 

28:46.156 --> 28:48.067 align:start position:0%
we're going back.
And<00:30:15.520><c> yes,</c><00:30:15.840><c> this</c><00:30:16.080><c> does</c><00:30:16.320><c> look</c><00:30:16.559><c> complicated</c><00:30:16.960><c> and</c>

28:48.067 --> 28:48.077 align:start position:0%
And yes, this does look complicated and
 

28:48.077 --> 28:49.666 align:start position:0%
And yes, this does look complicated and
I<00:30:17.360><c> may</c><00:30:17.520><c> have</c><00:30:17.679><c> glossed</c><00:30:18.080><c> over</c><00:30:18.320><c> some</c><00:30:18.480><c> of</c><00:30:18.559><c> the</c>

28:49.666 --> 28:49.676 align:start position:0%
I may have glossed over some of the
 

28:49.676 --> 28:51.906 align:start position:0%
I may have glossed over some of the
details,<00:30:19.440><c> but</c><00:30:19.679><c> the</c><00:30:19.840><c> central</c><00:30:20.240><c> idea</c><00:30:20.559><c> is</c><00:30:20.720><c> the</c>

28:51.906 --> 28:51.916 align:start position:0%
details, but the central idea is the
 

28:51.916 --> 28:55.187 align:start position:0%
details, but the central idea is the
exact<00:30:21.440><c> same.</c><00:30:22.399><c> 4year</c><00:30:22.880><c> transform</c><00:30:23.520><c> breaks</c><00:30:23.919><c> any</c>

28:55.187 --> 28:55.197 align:start position:0%
exact same. 4year transform breaks any
 

28:55.197 --> 28:57.347 align:start position:0%
exact same. 4year transform breaks any
signal<00:30:24.720><c> into</c><00:30:25.039><c> its</c><00:30:25.360><c> constituent</c><00:30:26.000><c> frequency</c>

28:57.347 --> 28:57.357 align:start position:0%
signal into its constituent frequency
 

28:57.357 --> 28:59.426 align:start position:0%
signal into its constituent frequency
components<00:30:27.200><c> and</c><00:30:27.360><c> the</c><00:30:27.520><c> inverse</c><00:30:27.919><c> transform</c>

28:59.426 --> 28:59.436 align:start position:0%
components and the inverse transform
 

28:59.436 --> 29:02.146 align:start position:0%
components and the inverse transform
just<00:30:28.880><c> adds</c><00:30:29.120><c> them</c><00:30:29.360><c> back</c><00:30:29.520><c> together.</c><00:30:30.880><c> At</c><00:30:31.120><c> this</c>

29:02.146 --> 29:02.156 align:start position:0%
just adds them back together. At this
 

29:02.156 --> 29:04.387 align:start position:0%
just adds them back together. At this
point,<00:30:31.520><c> I</c><00:30:31.760><c> thought</c><00:30:32.080><c> that</c><00:30:32.399><c> this</c><00:30:32.720><c> 4</c><00:30:32.880><c> years</c><00:30:33.200><c> idea</c>

29:04.387 --> 29:04.397 align:start position:0%
point, I thought that this 4 years idea
 

29:04.397 --> 29:06.707 align:start position:0%
point, I thought that this 4 years idea
with<00:30:33.760><c> all</c><00:30:33.919><c> this</c><00:30:34.159><c> math</c><00:30:34.880><c> wouldn't</c><00:30:35.279><c> have</c><00:30:35.440><c> any</c>

29:06.707 --> 29:06.717 align:start position:0%
with all this math wouldn't have any
 

29:06.717 --> 29:10.307 align:start position:0%
with all this math wouldn't have any
real<00:30:36.240><c> application</c><00:30:36.880><c> until</c><00:30:37.600><c> the</c><00:30:37.919><c> digital</c><00:30:38.399><c> era.</c>

29:10.307 --> 29:10.317 align:start position:0%
real application until the digital era.
 

29:10.317 --> 29:13.426 align:start position:0%
real application until the digital era.
I<00:30:39.679><c> was</c><00:30:39.840><c> wrong.</c><00:30:40.799><c> For</c><00:30:41.120><c> most</c><00:30:41.360><c> of</c><00:30:41.600><c> human</c><00:30:41.919><c> history,</c>

29:13.426 --> 29:13.436 align:start position:0%
I was wrong. For most of human history,
 

29:13.436 --> 29:16.146 align:start position:0%
I was wrong. For most of human history,
trade,<00:30:43.120><c> travel,</c><00:30:43.600><c> and</c><00:30:43.919><c> war</c><00:30:44.399><c> depended</c><00:30:44.960><c> on</c>

29:16.146 --> 29:16.156 align:start position:0%
trade, travel, and war depended on
 

29:16.156 --> 29:19.426 align:start position:0%
trade, travel, and war depended on
ships,<00:30:46.159><c> and</c><00:30:46.399><c> ships</c><00:30:47.279><c> are</c><00:30:47.600><c> vulnerable</c><00:30:48.159><c> to</c>

29:19.426 --> 29:19.436 align:start position:0%
ships, and ships are vulnerable to
 

29:19.436 --> 29:21.426 align:start position:0%
ships, and ships are vulnerable to
tides.

29:21.426 --> 29:21.436 align:start position:0%
tides.
 

29:21.436 --> 29:23.266 align:start position:0%
tides.
Miscalculating<00:30:51.440><c> tides</c><00:30:51.840><c> could</c><00:30:52.000><c> mean</c><00:30:52.159><c> your</c>

29:23.266 --> 29:23.276 align:start position:0%
Miscalculating tides could mean your
 

29:23.276 --> 29:27.107 align:start position:0%
Miscalculating tides could mean your
ships<00:30:52.720><c> run</c><00:30:53.440><c> or</c><00:30:53.840><c> even</c><00:30:54.240><c> sinks.</c><00:30:55.679><c> But</c><00:30:55.919><c> wait,</c>

29:27.107 --> 29:27.117 align:start position:0%
ships run or even sinks. But wait,
 

29:27.117 --> 29:29.587 align:start position:0%
ships run or even sinks. But wait,
aren't<00:30:56.559><c> tides</c><00:30:56.960><c> easy</c><00:30:57.200><c> to</c><00:30:57.360><c> predict?</c><00:30:58.159><c> The</c><00:30:58.399><c> moon</c>

29:29.587 --> 29:29.597 align:start position:0%
aren't tides easy to predict? The moon
 

29:29.597 --> 29:32.546 align:start position:0%
aren't tides easy to predict? The moon
causes<00:30:59.200><c> two</c><00:30:59.600><c> high</c><00:30:59.840><c> tides</c><00:31:00.240><c> and</c><00:31:00.480><c> two</c><00:31:00.960><c> low</c><00:31:01.200><c> tides</c>

29:32.546 --> 29:32.556 align:start position:0%
causes two high tides and two low tides
 

29:32.556 --> 29:35.266 align:start position:0%
causes two high tides and two low tides
per<00:31:02.000><c> day,</c><00:31:02.480><c> right?</c><00:31:03.200><c> Well,</c><00:31:03.520><c> mostly</c><00:31:03.840><c> yes,</c><00:31:04.159><c> but</c>

29:35.266 --> 29:35.276 align:start position:0%
per day, right? Well, mostly yes, but
 

29:35.276 --> 29:38.067 align:start position:0%
per day, right? Well, mostly yes, but
the<00:31:04.640><c> exact</c><00:31:05.279><c> heights</c><00:31:05.600><c> and</c><00:31:05.840><c> the</c><00:31:06.000><c> times</c><00:31:06.720><c> change</c>

29:38.067 --> 29:38.077 align:start position:0%
the exact heights and the times change
 

29:38.077 --> 29:40.947 align:start position:0%
the exact heights and the times change
significantly<00:31:08.399><c> from</c><00:31:08.799><c> day</c><00:31:09.039><c> to</c><00:31:09.200><c> day,</c><00:31:09.520><c> making</c><00:31:09.840><c> it</c>

29:40.947 --> 29:40.957 align:start position:0%
significantly from day to day, making it
 

29:40.957 --> 29:44.467 align:start position:0%
significantly from day to day, making it
hard<00:31:10.480><c> to</c><00:31:10.799><c> predict</c><00:31:11.120><c> the</c><00:31:11.440><c> future.</c>

29:44.467 --> 29:44.477 align:start position:0%
hard to predict the future.
 

29:44.477 --> 29:47.187 align:start position:0%
hard to predict the future.
But<00:31:13.840><c> why</c><00:31:14.080><c> does</c><00:31:14.240><c> this</c><00:31:14.480><c> happen?</c><00:31:15.200><c> Well,</c><00:31:15.520><c> because</c>

29:47.187 --> 29:47.197 align:start position:0%
But why does this happen? Well, because
 

29:47.197 --> 29:49.747 align:start position:0%
But why does this happen? Well, because
moon<00:31:16.799><c> only</c><00:31:17.120><c> produces</c><00:31:17.840><c> one</c><00:31:18.399><c> major</c>

29:49.747 --> 29:49.757 align:start position:0%
moon only produces one major
 

29:49.757 --> 29:52.307 align:start position:0%
moon only produces one major
contribution.<00:31:20.159><c> There</c><00:31:20.320><c> is</c><00:31:20.559><c> also</c><00:31:20.799><c> the</c><00:31:21.039><c> sun</c>

29:52.307 --> 29:52.317 align:start position:0%
contribution. There is also the sun
 

29:52.317 --> 29:54.467 align:start position:0%
contribution. There is also the sun
which<00:31:21.760><c> produces</c><00:31:22.320><c> another</c><00:31:22.960><c> contribution</c><00:31:23.440><c> with</c>

29:54.467 --> 29:54.477 align:start position:0%
which produces another contribution with
 

29:54.477 --> 29:56.146 align:start position:0%
which produces another contribution with
a<00:31:23.760><c> slightly</c><00:31:24.080><c> different</c><00:31:24.320><c> period.</c><00:31:24.960><c> And</c><00:31:25.120><c> the</c>

29:56.146 --> 29:56.156 align:start position:0%
a slightly different period. And the
 

29:56.156 --> 29:59.266 align:start position:0%
a slightly different period. And the
moon's<00:31:25.679><c> changing</c><00:31:26.159><c> distance</c><00:31:27.279><c> and</c><00:31:27.679><c> angles</c>

29:59.266 --> 29:59.276 align:start position:0%
moon's changing distance and angles
 

29:59.276 --> 30:01.587 align:start position:0%
moon's changing distance and angles
introduce<00:31:28.960><c> more</c><00:31:29.200><c> and</c><00:31:29.440><c> more</c><00:31:29.679><c> such</c><00:31:30.000><c> components.</c>

30:01.587 --> 30:01.597 align:start position:0%
introduce more and more such components.
 

30:01.597 --> 30:04.786 align:start position:0%
introduce more and more such components.
So<00:31:30.960><c> the</c><00:31:31.200><c> tides</c><00:31:31.600><c> we</c><00:31:31.919><c> actually</c><00:31:32.640><c> see</c><00:31:33.039><c> eventually</c>

30:04.786 --> 30:04.796 align:start position:0%
So the tides we actually see eventually
 

30:04.796 --> 30:08.627 align:start position:0%
So the tides we actually see eventually
is<00:31:34.399><c> all</c><00:31:34.640><c> of</c><00:31:34.880><c> them</c><00:31:35.360><c> added</c><00:31:36.080><c> together.</c><00:31:37.120><c> Now</c><00:31:37.440><c> the</c>

30:08.627 --> 30:08.637 align:start position:0%
is all of them added together. Now the
 

30:08.637 --> 30:10.627 align:start position:0%
is all of them added together. Now the
frequency<00:31:38.480><c> of</c><00:31:38.720><c> these</c><00:31:39.039><c> components</c><00:31:39.519><c> is</c>

30:10.627 --> 30:10.637 align:start position:0%
frequency of these components is
 

30:10.637 --> 30:11.827 align:start position:0%
frequency of these components is
something<00:31:39.919><c> that</c><00:31:40.080><c> we</c><00:31:40.159><c> know.</c><00:31:40.399><c> It</c><00:31:40.559><c> comes</c><00:31:40.720><c> from</c>

30:11.827 --> 30:11.837 align:start position:0%
something that we know. It comes from
 

30:11.837 --> 30:15.266 align:start position:0%
something that we know. It comes from
astronomy.<00:31:42.159><c> But</c><00:31:42.399><c> what</c><00:31:42.640><c> about</c><00:31:42.880><c> their</c><00:31:43.279><c> heights</c>

30:15.266 --> 30:15.276 align:start position:0%
astronomy. But what about their heights
 

30:15.276 --> 30:18.947 align:start position:0%
astronomy. But what about their heights
and<00:31:44.960><c> what</c><00:31:45.279><c> about</c><00:31:45.840><c> their</c><00:31:46.320><c> exact</c><00:31:46.960><c> phase</c><00:31:47.760><c> which</c>

30:18.947 --> 30:18.957 align:start position:0%
and what about their exact phase which
 

30:18.957 --> 30:21.026 align:start position:0%
and what about their exact phase which
definitely<00:31:48.559><c> affects</c><00:31:48.880><c> the</c><00:31:49.039><c> tides.</c><00:31:49.760><c> These</c>

30:21.026 --> 30:21.036 align:start position:0%
definitely affects the tides. These
 

30:21.036 --> 30:22.786 align:start position:0%
definitely affects the tides. These
depend<00:31:50.399><c> on</c><00:31:50.559><c> the</c><00:31:50.720><c> details,</c><00:31:51.120><c> the</c><00:31:51.279><c> nooks</c><00:31:51.600><c> and</c><00:31:51.760><c> the</c>

30:22.786 --> 30:22.796 align:start position:0%
depend on the details, the nooks and the
 

30:22.796 --> 30:24.947 align:start position:0%
depend on the details, the nooks and the
crannies<00:31:52.320><c> of</c><00:31:52.559><c> the</c><00:31:52.720><c> local</c><00:31:53.039><c> coastline</c><00:31:53.679><c> which</c><00:31:53.919><c> is</c>

30:24.947 --> 30:24.957 align:start position:0%
crannies of the local coastline which is
 

30:24.957 --> 30:26.947 align:start position:0%
crannies of the local coastline which is
practically<00:31:54.480><c> impossible</c><00:31:54.960><c> to</c><00:31:55.120><c> calculate.</c><00:31:55.840><c> So</c>

30:26.947 --> 30:26.957 align:start position:0%
practically impossible to calculate. So
 

30:26.957 --> 30:28.947 align:start position:0%
practically impossible to calculate. So
at<00:31:56.320><c> any</c><00:31:56.559><c> given</c><00:31:56.799><c> location,</c><00:31:57.440><c> how</c><00:31:57.600><c> do</c><00:31:57.760><c> we</c><00:31:57.919><c> find</c>

30:28.947 --> 30:28.957 align:start position:0%
at any given location, how do we find
 

30:28.957 --> 30:30.307 align:start position:0%
at any given location, how do we find
the<00:31:58.240><c> heights</c><00:31:58.480><c> and</c><00:31:58.640><c> the</c><00:31:58.799><c> phase</c><00:31:59.039><c> of</c><00:31:59.200><c> each</c>

30:30.307 --> 30:30.317 align:start position:0%
the heights and the phase of each
 

30:30.317 --> 30:34.307 align:start position:0%
the heights and the phase of each
component?<00:32:00.640><c> For</c><00:32:01.120><c> air</c>

30:34.307 --> 30:34.317 align:start position:0%
component? For air
 

30:34.317 --> 30:37.026 align:start position:0%
component? For air
William<00:32:03.840><c> Thompson,</c><00:32:04.399><c> who</c><00:32:04.799><c> later</c><00:32:05.200><c> became</c><00:32:05.760><c> Lord</c>

30:37.026 --> 30:37.036 align:start position:0%
William Thompson, who later became Lord
 

30:37.036 --> 30:39.187 align:start position:0%
William Thompson, who later became Lord
Kelvin,<00:32:06.799><c> realized</c><00:32:07.279><c> he</c><00:32:07.519><c> could</c><00:32:07.679><c> take</c>

30:39.187 --> 30:39.197 align:start position:0%
Kelvin, realized he could take
 

30:39.197 --> 30:41.827 align:start position:0%
Kelvin, realized he could take
historical<00:32:09.120><c> tide</c><00:32:09.440><c> data</c><00:32:09.760><c> from</c><00:32:10.000><c> any</c><00:32:10.240><c> location</c>

30:41.827 --> 30:41.837 align:start position:0%
historical tide data from any location
 

30:41.837 --> 30:44.867 align:start position:0%
historical tide data from any location
and<00:32:11.279><c> use</c><00:32:11.760><c> four</c><00:32:12.000><c> years</c><00:32:12.399><c> analysis</c><00:32:13.039><c> to</c><00:32:13.440><c> extract</c>

30:44.867 --> 30:44.877 align:start position:0%
and use four years analysis to extract
 

30:44.877 --> 30:48.546 align:start position:0%
and use four years analysis to extract
the<00:32:14.320><c> major</c><00:32:14.880><c> components,</c><00:32:16.000><c> then</c><00:32:16.720><c> simply</c><00:32:17.279><c> extend</c>

30:48.546 --> 30:48.556 align:start position:0%
the major components, then simply extend
 

30:48.556 --> 30:51.107 align:start position:0%
the major components, then simply extend
those<00:32:17.919><c> sinosoids</c><00:32:18.559><c> into</c><00:32:18.720><c> the</c><00:32:18.960><c> future</c><00:32:19.600><c> and</c><00:32:19.919><c> add</c>

30:51.107 --> 30:51.117 align:start position:0%
those sinosoids into the future and add
 

30:51.117 --> 30:54.307 align:start position:0%
those sinosoids into the future and add
them<00:32:20.559><c> back</c><00:32:21.200><c> together.</c><00:32:22.480><c> pretty</c><00:32:22.880><c> similar</c><00:32:23.200><c> to</c>

30:54.307 --> 30:54.317 align:start position:0%
them back together. pretty similar to
 

30:54.317 --> 30:56.786 align:start position:0%
them back together. pretty similar to
what<00:32:23.679><c> Foryer</c><00:32:24.240><c> initially</c><00:32:24.720><c> did</c><00:32:25.200><c> to</c><00:32:25.440><c> solve</c><00:32:25.760><c> the</c>

30:56.786 --> 30:56.796 align:start position:0%
what Foryer initially did to solve the
 

30:56.796 --> 30:59.187 align:start position:0%
what Foryer initially did to solve the
heat<00:32:26.240><c> equation.</c><00:32:26.799><c> Oh</c><00:32:27.039><c> my</c><00:32:27.200><c> god,</c><00:32:27.519><c> that</c><00:32:27.760><c> is</c><00:32:28.080><c> that</c>

30:59.187 --> 30:59.197 align:start position:0%
heat equation. Oh my god, that is that
 

30:59.197 --> 31:01.107 align:start position:0%
heat equation. Oh my god, that is that
is<00:32:28.480><c> pretty</c><00:32:28.720><c> cool.</c><00:32:29.360><c> So</c><00:32:29.519><c> he</c><00:32:29.679><c> and</c><00:32:29.840><c> his</c><00:32:30.000><c> brother</c>

31:01.107 --> 31:01.117 align:start position:0%
is pretty cool. So he and his brother
 

31:01.117 --> 31:03.347 align:start position:0%
is pretty cool. So he and his brother
built<00:32:30.559><c> two</c><00:32:30.720><c> machines.</c><00:32:31.440><c> An</c><00:32:31.600><c> integrator</c><00:32:32.240><c> to</c>

31:03.347 --> 31:03.357 align:start position:0%
built two machines. An integrator to
 

31:03.357 --> 31:06.386 align:start position:0%
built two machines. An integrator to
extract<00:32:33.039><c> the</c><00:32:33.360><c> components</c><00:32:34.080><c> and</c><00:32:34.399><c> another</c><00:32:34.799><c> one</c><00:32:35.279><c> a</c>

31:06.386 --> 31:06.396 align:start position:0%
extract the components and another one a
 

31:06.396 --> 31:08.627 align:start position:0%
extract the components and another one a
tide<00:32:35.840><c> predictor</c><00:32:36.320><c> machine</c><00:32:36.880><c> which</c><00:32:37.200><c> adds</c><00:32:37.519><c> those</c>

31:08.627 --> 31:08.637 align:start position:0%
tide predictor machine which adds those
 

31:08.637 --> 31:11.827 align:start position:0%
tide predictor machine which adds those
components<00:32:38.640><c> back.</c><00:32:39.679><c> A</c><00:32:39.919><c> few</c><00:32:40.159><c> hours</c><00:32:40.480><c> of</c><00:32:40.720><c> running</c>

31:11.827 --> 31:11.837 align:start position:0%
components back. A few hours of running
 

31:11.837 --> 31:14.467 align:start position:0%
components back. A few hours of running
these<00:32:41.279><c> machines</c><00:32:41.679><c> could</c><00:32:42.000><c> predict</c><00:32:42.480><c> an</c><00:32:42.960><c> year's</c>

31:14.467 --> 31:14.477 align:start position:0%
these machines could predict an year's
 

31:14.477 --> 31:17.906 align:start position:0%
these machines could predict an year's
worth<00:32:43.919><c> of</c><00:32:44.240><c> tides</c><00:32:44.880><c> data.</c><00:32:45.760><c> This</c><00:32:46.000><c> was</c><00:32:46.159><c> a</c><00:32:46.480><c> complete</c>

31:17.906 --> 31:17.916 align:start position:0%
worth of tides data. This was a complete
 

31:17.916 --> 31:21.507 align:start position:0%
worth of tides data. This was a complete
gamecher.<00:32:47.760><c> The</c><00:32:48.000><c> British</c><00:32:48.480><c> sea</c><00:32:48.799><c> trade</c><00:32:49.519><c> thrived</c>

31:21.507 --> 31:21.517 align:start position:0%
gamecher. The British sea trade thrived
 

31:21.517 --> 31:23.987 align:start position:0%
gamecher. The British sea trade thrived
because<00:32:50.960><c> of</c><00:32:51.279><c> this.</c>

31:23.987 --> 31:23.997 align:start position:0%
because of this.
 

31:23.997 --> 31:26.307 align:start position:0%
because of this.
And<00:32:53.360><c> decades</c><00:32:53.760><c> later,</c><00:32:54.159><c> the</c><00:32:54.399><c> Allies</c><00:32:54.799><c> used</c><00:32:55.200><c> a</c>

31:26.307 --> 31:26.317 align:start position:0%
And decades later, the Allies used a
 

31:26.317 --> 31:28.786 align:start position:0%
And decades later, the Allies used a
similar<00:32:55.919><c> machine</c><00:32:56.240><c> to</c><00:32:56.480><c> predict</c><00:32:56.799><c> the</c><00:32:56.960><c> tides</c><00:32:57.600><c> and</c>

31:28.786 --> 31:28.796 align:start position:0%
similar machine to predict the tides and
 

31:28.796 --> 31:30.707 align:start position:0%
similar machine to predict the tides and
plan<00:32:58.320><c> their</c><00:32:58.559><c> invasion</c><00:32:59.039><c> of</c><00:32:59.279><c> the</c><00:32:59.440><c> German</c>

31:30.707 --> 31:30.717 align:start position:0%
plan their invasion of the German
 

31:30.717 --> 31:33.187 align:start position:0%
plan their invasion of the German
occupied<00:33:00.320><c> France.</c><00:33:01.120><c> So</c><00:33:01.279><c> what's</c><00:33:01.600><c> crazy</c><00:33:01.840><c> is</c><00:33:02.080><c> that</c>

31:33.187 --> 31:33.197 align:start position:0%
occupied France. So what's crazy is that
 

31:33.197 --> 31:36.467 align:start position:0%
occupied France. So what's crazy is that
an<00:33:02.559><c> idea</c><00:33:02.799><c> which</c><00:33:03.120><c> was</c><00:33:03.279><c> developed</c><00:33:03.840><c> to</c><00:33:05.120><c> solve</c><00:33:05.440><c> the</c>

31:36.467 --> 31:36.477 align:start position:0%
an idea which was developed to solve the
 

31:36.477 --> 31:39.507 align:start position:0%
an idea which was developed to solve the
heat<00:33:05.840><c> equation</c><00:33:06.880><c> ended</c><00:33:07.279><c> up</c><00:33:07.440><c> helping</c><00:33:07.919><c> plan</c><00:33:08.320><c> the</c>

31:39.507 --> 31:39.517 align:start position:0%
heat equation ended up helping plan the
 

31:39.517 --> 31:42.227 align:start position:0%
heat equation ended up helping plan the
biggest<00:33:09.440><c> seabone</c><00:33:10.000><c> invasion</c><00:33:10.320><c> in</c><00:33:10.480><c> the</c><00:33:10.720><c> history.</c>

31:42.227 --> 31:42.237 align:start position:0%
biggest seabone invasion in the history.
 

31:42.237 --> 31:43.906 align:start position:0%
biggest seabone invasion in the history.
You<00:33:11.679><c> can't</c><00:33:12.080><c> make</c><00:33:12.320><c> this</c><00:33:12.480><c> up.</c>

31:43.906 --> 31:43.916 align:start position:0%
You can't make this up.
 

31:43.916 --> 31:47.347 align:start position:0%
You can't make this up.
&gt;&gt; Allied<00:33:13.360><c> armies</c><00:33:13.840><c> were</c><00:33:14.159><c> striking</c><00:33:14.640><c> back.</c>

31:47.347 --> 31:47.357 align:start position:0%
&gt;&gt; Allied armies were striking back.
 

31:47.357 --> 31:49.666 align:start position:0%
&gt;&gt; Allied armies were striking back.
&gt;&gt; And<00:33:16.640><c> today</c><00:33:16.960><c> this</c><00:33:17.200><c> idea</c><00:33:17.519><c> is</c><00:33:17.760><c> used</c><00:33:18.000><c> everywhere.</c>

31:49.666 --> 31:49.676 align:start position:0%
&gt;&gt; And today this idea is used everywhere.
 

31:49.676 --> 31:51.426 align:start position:0%
&gt;&gt; And today this idea is used everywhere.
For<00:33:18.960><c> example,</c><00:33:19.440><c> when</c><00:33:19.679><c> I'm</c><00:33:19.919><c> responding</c><00:33:20.240><c> to</c><00:33:20.399><c> my</c>

31:51.426 --> 31:51.436 align:start position:0%
For example, when I'm responding to my
 

31:51.436 --> 31:53.587 align:start position:0%
For example, when I'm responding to my
wife<00:33:20.799><c> with</c><00:33:21.039><c> a</c><00:33:21.279><c> voice</c><00:33:21.519><c> message,</c><00:33:22.080><c> say,</c><00:33:22.399><c> "Hey</c>

31:53.587 --> 31:53.597 align:start position:0%
wife with a voice message, say, "Hey
 

31:53.597 --> 31:56.627 align:start position:0%
wife with a voice message, say, "Hey
wifey,<00:33:23.440><c> I</c><00:33:23.679><c> love</c><00:33:23.840><c> you</c><00:33:24.080><c> too."</c><00:33:24.880><c> My</c><00:33:25.120><c> phone</c><00:33:25.360><c> is</c><00:33:25.600><c> not</c>

31:56.627 --> 31:56.637 align:start position:0%
wifey, I love you too." My phone is not
 

31:56.637 --> 31:58.947 align:start position:0%
wifey, I love you too." My phone is not
going<00:33:25.919><c> to</c><00:33:26.240><c> save</c><00:33:26.559><c> the</c><00:33:26.880><c> entire</c><00:33:27.360><c> thing</c><00:33:27.600><c> as</c><00:33:27.760><c> is</c>

31:58.947 --> 31:58.957 align:start position:0%
going to save the entire thing as is
 

31:58.957 --> 32:00.786 align:start position:0%
going to save the entire thing as is
because<00:33:28.559><c> it's</c><00:33:28.880><c> going</c><00:33:28.960><c> to</c><00:33:29.039><c> take</c><00:33:29.200><c> up</c><00:33:29.440><c> a</c><00:33:29.679><c> lot</c><00:33:29.760><c> of</c>

32:00.786 --> 32:00.796 align:start position:0%
because it's going to take up a lot of
 

32:00.796 --> 32:03.347 align:start position:0%
because it's going to take up a lot of
unnecessary<00:33:30.640><c> space.</c><00:33:31.279><c> Instead,</c><00:33:31.760><c> it</c><00:33:32.000><c> breaks</c><00:33:32.320><c> it</c>

32:03.347 --> 32:03.357 align:start position:0%
unnecessary space. Instead, it breaks it
 

32:03.357 --> 32:05.747 align:start position:0%
unnecessary space. Instead, it breaks it
down<00:33:32.720><c> into</c><00:33:33.039><c> its</c><00:33:33.360><c> constituent</c><00:33:34.000><c> frequencies.</c>

32:05.747 --> 32:05.757 align:start position:0%
down into its constituent frequencies.
 

32:05.757 --> 32:07.426 align:start position:0%
down into its constituent frequencies.
Now,<00:33:35.120><c> of</c><00:33:35.279><c> course,</c><00:33:35.679><c> this</c><00:33:35.840><c> is</c><00:33:35.919><c> a</c><00:33:36.080><c> 4year</c>

32:07.426 --> 32:07.436 align:start position:0%
Now, of course, this is a 4year
 

32:07.436 --> 32:09.666 align:start position:0%
Now, of course, this is a 4year
transform,<00:33:37.200><c> so</c><00:33:37.519><c> there</c><00:33:37.679><c> will</c><00:33:37.919><c> be</c><00:33:38.000><c> a</c><00:33:38.240><c> continuum</c>

32:09.666 --> 32:09.676 align:start position:0%
transform, so there will be a continuum
 

32:09.676 --> 32:11.587 align:start position:0%
transform, so there will be a continuum
because<00:33:38.960><c> it's</c><00:33:39.120><c> a</c><00:33:39.279><c> non-periodic</c><00:33:40.000><c> signal,</c><00:33:40.399><c> but</c>

32:11.587 --> 32:11.597 align:start position:0%
because it's a non-periodic signal, but
 

32:11.597 --> 32:13.266 align:start position:0%
because it's a non-periodic signal, but
never<00:33:40.880><c> mind</c><00:33:41.120><c> all</c><00:33:41.279><c> of</c><00:33:41.440><c> that.</c><00:33:41.840><c> What's</c><00:33:42.159><c> important</c>

32:13.266 --> 32:13.276 align:start position:0%
never mind all of that. What's important
 

32:13.276 --> 32:15.507 align:start position:0%
never mind all of that. What's important
is<00:33:42.640><c> that</c><00:33:42.880><c> a</c><00:33:43.200><c> lot</c><00:33:43.279><c> of</c><00:33:43.360><c> these</c><00:33:43.679><c> frequencies</c><00:33:44.399><c> like</c>

32:15.507 --> 32:15.517 align:start position:0%
is that a lot of these frequencies like
 

32:15.517 --> 32:17.426 align:start position:0%
is that a lot of these frequencies like
very<00:33:44.880><c> high,</c><00:33:45.200><c> very</c><00:33:45.440><c> low</c><00:33:45.679><c> frequencies</c><00:33:46.159><c> or</c><00:33:46.320><c> the</c>

32:17.426 --> 32:17.436 align:start position:0%
very high, very low frequencies or the
 

32:17.436 --> 32:19.426 align:start position:0%
very high, very low frequencies or the
ones<00:33:46.720><c> that</c><00:33:46.880><c> have</c><00:33:47.120><c> very</c><00:33:47.360><c> feeble</c><00:33:47.760><c> sound,</c><00:33:48.320><c> the</c>

32:19.426 --> 32:19.436 align:start position:0%
ones that have very feeble sound, the
 

32:19.436 --> 32:21.266 align:start position:0%
ones that have very feeble sound, the
human<00:33:48.720><c> ears</c><00:33:49.039><c> can't</c><00:33:49.200><c> detect</c><00:33:49.519><c> them</c><00:33:49.679><c> anyways.</c><00:33:50.240><c> So</c>

32:21.266 --> 32:21.276 align:start position:0%
human ears can't detect them anyways. So
 

32:21.276 --> 32:23.587 align:start position:0%
human ears can't detect them anyways. So
there's<00:33:50.720><c> no</c><00:33:50.960><c> point</c><00:33:51.200><c> in</c><00:33:51.600><c> keeping</c><00:33:51.919><c> them.</c><00:33:52.320><c> So</c><00:33:52.480><c> we</c>

32:23.587 --> 32:23.597 align:start position:0%
there's no point in keeping them. So we
 

32:23.597 --> 32:26.146 align:start position:0%
there's no point in keeping them. So we
can<00:33:52.880><c> get</c><00:33:52.960><c> rid</c><00:33:53.200><c> of</c><00:33:53.440><c> a</c><00:33:53.760><c> lot</c><00:33:54.000><c> of</c><00:33:54.399><c> that</c><00:33:54.720><c> useless</c>

32:26.146 --> 32:26.156 align:start position:0%
can get rid of a lot of that useless
 

32:26.156 --> 32:28.307 align:start position:0%
can get rid of a lot of that useless
information<00:33:55.679><c> and</c><00:33:55.919><c> only</c><00:33:56.159><c> keep</c><00:33:56.399><c> the</c><00:33:56.559><c> ones</c><00:33:57.039><c> that</c>

32:28.307 --> 32:28.317 align:start position:0%
information and only keep the ones that
 

32:28.317 --> 32:31.507 align:start position:0%
information and only keep the ones that
matter.<00:33:58.159><c> So</c><00:33:58.320><c> we'll</c><00:33:58.640><c> only</c><00:33:59.120><c> save</c><00:34:00.080><c> the,</c><00:34:00.480><c> you</c>

32:31.507 --> 32:31.517 align:start position:0%
matter. So we'll only save the, you
 

32:31.517 --> 32:32.947 align:start position:0%
matter. So we'll only save the, you
know,<00:34:00.799><c> information</c><00:34:01.200><c> of</c><00:34:01.440><c> the</c><00:34:01.600><c> frequencies</c>

32:32.947 --> 32:32.957 align:start position:0%
know, information of the frequencies
 

32:32.957 --> 32:35.827 align:start position:0%
know, information of the frequencies
that<00:34:02.399><c> matter</c><00:34:03.200><c> and</c><00:34:03.440><c> that</c><00:34:04.000><c> will</c><00:34:04.240><c> take</c><00:34:04.399><c> up</c><00:34:04.559><c> a</c><00:34:04.799><c> lot</c>

32:35.827 --> 32:35.837 align:start position:0%
that matter and that will take up a lot
 

32:35.837 --> 32:38.946 align:start position:0%
that matter and that will take up a lot
less<00:34:05.279><c> space.</c><00:34:06.159><c> That's</c><00:34:06.480><c> how</c><00:34:06.799><c> audio</c><00:34:07.360><c> compression</c>

32:38.946 --> 32:38.956 align:start position:0%
less space. That's how audio compression
 

32:38.956 --> 32:41.026 align:start position:0%
less space. That's how audio compression
works.<00:34:08.560><c> It's</c><00:34:08.800><c> an</c><00:34:08.960><c> oversimplified</c><00:34:09.679><c> version,</c>

32:41.026 --> 32:41.036 align:start position:0%
works. It's an oversimplified version,
 

32:41.036 --> 32:43.347 align:start position:0%
works. It's an oversimplified version,
but<00:34:10.399><c> that's</c><00:34:10.720><c> basically</c><00:34:11.119><c> it.</c><00:34:11.839><c> And</c><00:34:12.000><c> now</c><00:34:12.159><c> when</c><00:34:12.320><c> I</c>

32:43.347 --> 32:43.357 align:start position:0%
but that's basically it. And now when I
 

32:43.357 --> 32:45.827 align:start position:0%
but that's basically it. And now when I
send<00:34:12.639><c> this</c><00:34:12.800><c> to</c><00:34:12.960><c> my</c><00:34:13.200><c> wife,</c><00:34:13.760><c> her</c><00:34:14.079><c> phone</c><00:34:14.639><c> will</c>

32:45.827 --> 32:45.837 align:start position:0%
send this to my wife, her phone will
 

32:45.837 --> 32:49.026 align:start position:0%
send this to my wife, her phone will
basically<00:34:15.760><c> reconstruct</c><00:34:16.720><c> that</c><00:34:17.119><c> message</c><00:34:17.760><c> by</c>

32:49.026 --> 32:49.036 align:start position:0%
basically reconstruct that message by
 

32:49.036 --> 32:51.427 align:start position:0%
basically reconstruct that message by
adding<00:34:18.560><c> those</c><00:34:19.119><c> frequencies</c><00:34:19.760><c> together.</c><00:34:20.399><c> And</c>

32:51.427 --> 32:51.437 align:start position:0%
adding those frequencies together. And
 

32:51.437 --> 32:53.107 align:start position:0%
adding those frequencies together. And
if<00:34:20.720><c> everything</c><00:34:21.040><c> works</c><00:34:21.359><c> well,</c><00:34:21.679><c> that</c><00:34:21.919><c> message</c>

32:53.107 --> 32:53.117 align:start position:0%
if everything works well, that message
 

32:53.117 --> 32:54.946 align:start position:0%
if everything works well, that message
will<00:34:22.480><c> sound</c><00:34:22.960><c> identical.</c>

32:54.946 --> 32:54.956 align:start position:0%
will sound identical.
 

32:54.956 --> 32:57.986 align:start position:0%
will sound identical.
&gt;&gt; I<00:34:24.399><c> love</c><00:34:24.800><c> YouTube.</c><00:34:25.679><c> Oh</c><00:34:26.079><c> crap.</c><00:34:26.480><c> The</c><00:34:26.720><c> same</c><00:34:26.879><c> idea</c>

32:57.986 --> 32:57.996 align:start position:0%
&gt;&gt; I love YouTube. Oh crap. The same idea
 

32:57.996 --> 33:00.387 align:start position:0%
&gt;&gt; I love YouTube. Oh crap. The same idea
works<00:34:27.359><c> with</c><00:34:27.599><c> photos.</c><00:34:28.159><c> A</c><00:34:28.399><c> photo</c><00:34:28.639><c> is</c><00:34:28.960><c> basically</c>

33:00.387 --> 33:00.397 align:start position:0%
works with photos. A photo is basically
 

33:00.397 --> 33:02.627 align:start position:0%
works with photos. A photo is basically
a<00:34:30.000><c> grid</c><00:34:30.240><c> of</c><00:34:30.480><c> pixels,</c><00:34:31.040><c> which</c><00:34:31.280><c> can</c><00:34:31.440><c> be</c><00:34:31.599><c> thought</c>

33:02.627 --> 33:02.637 align:start position:0%
a grid of pixels, which can be thought
 

33:02.637 --> 33:06.387 align:start position:0%
a grid of pixels, which can be thought
of<00:34:32.320><c> as</c><00:34:32.639><c> a</c><00:34:33.040><c> signal,</c><00:34:33.919><c> a</c><00:34:34.159><c> brightness</c><00:34:34.720><c> signal.</c><00:34:35.280><c> Of</c>

33:06.387 --> 33:06.397 align:start position:0%
of as a signal, a brightness signal. Of
 

33:06.397 --> 33:07.827 align:start position:0%
of as a signal, a brightness signal. Of
course,<00:34:35.679><c> this</c><00:34:35.919><c> should</c><00:34:36.000><c> be</c><00:34:36.159><c> a</c><00:34:36.320><c> two-dimensional</c>

33:07.827 --> 33:07.837 align:start position:0%
course, this should be a two-dimensional
 

33:07.837 --> 33:09.587 align:start position:0%
course, this should be a two-dimensional
signal,<00:34:37.440><c> but</c><00:34:37.679><c> I'm</c><00:34:37.919><c> just</c><00:34:38.000><c> representing</c><00:34:38.399><c> it</c><00:34:38.560><c> as</c>

33:09.587 --> 33:09.597 align:start position:0%
signal, but I'm just representing it as
 

33:09.597 --> 33:12.387 align:start position:0%
signal, but I'm just representing it as
1D.<00:34:39.359><c> Again,</c><00:34:39.760><c> oversimplification.</c><00:34:41.040><c> But</c><00:34:41.200><c> the</c>

33:12.387 --> 33:12.397 align:start position:0%
1D. Again, oversimplification. But the
 

33:12.397 --> 33:14.706 align:start position:0%
1D. Again, oversimplification. But the
idea<00:34:41.839><c> is</c><00:34:42.000><c> the</c><00:34:42.159><c> same.</c><00:34:42.399><c> If</c><00:34:42.480><c> you</c><00:34:42.639><c> try</c><00:34:42.800><c> to</c><00:34:42.960><c> store</c><00:34:43.520><c> it</c>

33:14.706 --> 33:14.716 align:start position:0%
idea is the same. If you try to store it
 

33:14.716 --> 33:16.946 align:start position:0%
idea is the same. If you try to store it
as<00:34:44.159><c> is,</c><00:34:44.879><c> that's</c><00:34:45.200><c> going</c><00:34:45.280><c> to</c><00:34:45.359><c> take</c><00:34:45.520><c> up</c><00:34:45.679><c> a</c><00:34:45.839><c> lot</c><00:34:45.919><c> of</c>

33:16.946 --> 33:16.956 align:start position:0%
as is, that's going to take up a lot of
 

33:16.956 --> 33:19.187 align:start position:0%
as is, that's going to take up a lot of
unnecessary<00:34:46.639><c> space.</c><00:34:47.280><c> Instead,</c><00:34:47.919><c> break</c><00:34:48.159><c> it</c>

33:19.187 --> 33:19.197 align:start position:0%
unnecessary space. Instead, break it
 

33:19.197 --> 33:21.026 align:start position:0%
unnecessary space. Instead, break it
down<00:34:48.480><c> into</c><00:34:48.720><c> its</c><00:34:48.879><c> constituent</c><00:34:49.359><c> frequencies.</c>

33:21.026 --> 33:21.036 align:start position:0%
down into its constituent frequencies.
 

33:21.036 --> 33:22.946 align:start position:0%
down into its constituent frequencies.
Get<00:34:50.240><c> rid</c><00:34:50.399><c> of</c><00:34:50.560><c> the</c><00:34:50.720><c> ones</c><00:34:50.960><c> that</c><00:34:51.200><c> our</c><00:34:51.440><c> eyes</c><00:34:51.839><c> will</c>

33:22.946 --> 33:22.956 align:start position:0%
Get rid of the ones that our eyes will
 

33:22.956 --> 33:25.107 align:start position:0%
Get rid of the ones that our eyes will
not<00:34:52.240><c> be</c><00:34:52.399><c> able</c><00:34:52.480><c> to</c><00:34:52.639><c> detect</c><00:34:53.040><c> anyways,</c><00:34:53.760><c> and</c><00:34:54.000><c> then</c>

33:25.107 --> 33:25.117 align:start position:0%
not be able to detect anyways, and then
 

33:25.117 --> 33:27.187 align:start position:0%
not be able to detect anyways, and then
just<00:34:54.480><c> store</c><00:34:54.720><c> the</c><00:34:54.960><c> information</c><00:34:55.440><c> of</c><00:34:55.760><c> the</c><00:34:56.000><c> ones</c>

33:27.187 --> 33:27.197 align:start position:0%
just store the information of the ones
 

33:27.197 --> 33:30.226 align:start position:0%
just store the information of the ones
that<00:34:56.800><c> matter.</c><00:34:57.920><c> That</c><00:34:58.400><c> will</c><00:34:58.640><c> take</c><00:34:58.800><c> up</c><00:34:58.960><c> a</c><00:34:59.200><c> lot</c>

33:30.226 --> 33:30.236 align:start position:0%
that matter. That will take up a lot
 

33:30.236 --> 33:34.226 align:start position:0%
that matter. That will take up a lot
less<00:34:59.839><c> space.</c><00:35:00.960><c> That</c><00:35:01.359><c> is</c><00:35:01.760><c> image</c><00:35:02.400><c> compression.</c>

33:34.226 --> 33:34.236 align:start position:0%
less space. That is image compression.
 

33:34.236 --> 33:35.827 align:start position:0%
less space. That is image compression.
And<00:35:03.520><c> then</c><00:35:03.680><c> when</c><00:35:03.839><c> I</c><00:35:04.079><c> double</c><00:35:04.240><c> click</c><00:35:04.480><c> and</c><00:35:04.720><c> open</c>

33:35.827 --> 33:35.837 align:start position:0%
And then when I double click and open
 

33:35.837 --> 33:37.986 align:start position:0%
And then when I double click and open
that<00:35:05.440><c> photo</c><00:35:05.760><c> again,</c><00:35:06.400><c> it's</c><00:35:06.720><c> going</c><00:35:06.800><c> to</c>

33:37.986 --> 33:37.996 align:start position:0%
that photo again, it's going to
 

33:37.996 --> 33:40.706 align:start position:0%
that photo again, it's going to
reconstruct<00:35:08.000><c> that</c><00:35:08.400><c> image</c><00:35:09.040><c> from</c><00:35:09.520><c> those</c>

33:40.706 --> 33:40.716 align:start position:0%
reconstruct that image from those
 

33:40.716 --> 33:43.587 align:start position:0%
reconstruct that image from those
constituent<00:35:10.400><c> frequencies.</c><00:35:11.440><c> And</c><00:35:11.760><c> my</c><00:35:12.000><c> image</c><00:35:12.320><c> is</c>

33:43.587 --> 33:43.597 align:start position:0%
constituent frequencies. And my image is
 

33:43.597 --> 33:50.627 align:start position:0%
constituent frequencies. And my image is
pretty<00:35:13.040><c> much</c><00:35:13.280><c> going</c><00:35:13.520><c> to</c><00:35:13.680><c> look</c><00:35:14.720><c> the</c><00:35:14.960><c> same.</c>

33:50.627 --> 33:50.637 align:start position:0%
 
 

33:50.637 --> 33:52.946 align:start position:0%
 
The<00:35:20.000><c> same</c><00:35:20.160><c> idea</c><00:35:20.480><c> works</c><00:35:20.800><c> with</c><00:35:21.040><c> videos.</c><00:35:21.839><c> And</c>

33:52.946 --> 33:52.956 align:start position:0%
The same idea works with videos. And
 

33:52.956 --> 33:55.507 align:start position:0%
The same idea works with videos. And
think<00:35:22.320><c> about</c><00:35:22.640><c> removing</c><00:35:23.119><c> noise</c><00:35:23.599><c> from</c><00:35:24.000><c> audio</c><00:35:24.320><c> or</c>

33:55.507 --> 33:55.517 align:start position:0%
think about removing noise from audio or
 

33:55.517 --> 33:57.907 align:start position:0%
think about removing noise from audio or
blurring<00:35:24.960><c> and</c><00:35:25.200><c> sharpening</c><00:35:25.680><c> photos.</c><00:35:26.480><c> It's</c><00:35:26.800><c> all</c>

33:57.907 --> 33:57.917 align:start position:0%
blurring and sharpening photos. It's all
 

33:57.917 --> 33:59.427 align:start position:0%
blurring and sharpening photos. It's all
about<00:35:27.280><c> just</c><00:35:27.520><c> fiddling</c><00:35:27.920><c> with</c><00:35:28.079><c> the</c><00:35:28.240><c> right</c>

33:59.427 --> 33:59.437 align:start position:0%
about just fiddling with the right
 

33:59.437 --> 34:01.667 align:start position:0%
about just fiddling with the right
frequency<00:35:29.119><c> components.</c><00:35:29.920><c> Foryear</c><00:35:30.320><c> transforms</c>

34:01.667 --> 34:01.677 align:start position:0%
frequency components. Foryear transforms
 

34:01.677 --> 34:04.787 align:start position:0%
frequency components. Foryear transforms
are<00:35:30.960><c> used</c><00:35:31.280><c> in</c><00:35:31.599><c> Wi-Fi,</c><00:35:32.320><c> MRIs,</c><00:35:32.960><c> radars,</c><00:35:33.680><c> you</c>

34:04.787 --> 34:04.797 align:start position:0%
are used in Wi-Fi, MRIs, radars, you
 

34:04.797 --> 34:07.587 align:start position:0%
are used in Wi-Fi, MRIs, radars, you
name<00:35:34.079><c> it.</c><00:35:34.560><c> Finally,</c><00:35:35.280><c> just</c><00:35:35.520><c> like</c><00:35:35.680><c> how</c><00:35:35.920><c> a</c><00:35:36.079><c> prism</c>

34:07.587 --> 34:07.597 align:start position:0%
name it. Finally, just like how a prism
 

34:07.597 --> 34:09.507 align:start position:0%
name it. Finally, just like how a prism
physically<00:35:37.280><c> separates</c><00:35:37.839><c> light</c><00:35:38.160><c> into</c><00:35:38.400><c> its</c>

34:09.507 --> 34:09.517 align:start position:0%
physically separates light into its
 

34:09.517 --> 34:11.267 align:start position:0%
physically separates light into its
consuming<00:35:39.119><c> colors,</c><00:35:39.599><c> I</c><00:35:39.839><c> was</c><00:35:39.920><c> wondering,</c><00:35:40.240><c> could</c>

34:11.267 --> 34:11.277 align:start position:0%
consuming colors, I was wondering, could
 

34:11.277 --> 34:13.986 align:start position:0%
consuming colors, I was wondering, could
we<00:35:40.560><c> build</c><00:35:40.800><c> something</c><00:35:41.040><c> similar</c><00:35:41.599><c> for</c><00:35:42.000><c> sound?</c>

34:13.986 --> 34:13.996 align:start position:0%
we build something similar for sound?
 

34:13.996 --> 34:16.946 align:start position:0%
we build something similar for sound?
Turns<00:35:43.440><c> out</c><00:35:43.760><c> you</c><00:35:44.160><c> can</c><00:35:44.960><c> just</c><00:35:45.280><c> play</c><00:35:45.520><c> the</c><00:35:45.760><c> sound</c>

34:16.946 --> 34:16.956 align:start position:0%
Turns out you can just play the sound
 

34:16.956 --> 34:20.067 align:start position:0%
Turns out you can just play the sound
near<00:35:46.480><c> a</c><00:35:47.200><c> row</c><00:35:47.280><c> of</c><00:35:47.680><c> tiny</c><00:35:48.240><c> oscillators</c><00:35:48.880><c> of</c>

34:20.067 --> 34:20.077 align:start position:0%
near a row of tiny oscillators of
 

34:20.077 --> 34:21.667 align:start position:0%
near a row of tiny oscillators of
different<00:35:49.599><c> lengths.</c><00:35:50.320><c> What</c><00:35:50.480><c> do</c><00:35:50.560><c> you</c><00:35:50.640><c> think</c>

34:21.667 --> 34:21.677 align:start position:0%
different lengths. What do you think
 

34:21.677 --> 34:23.986 align:start position:0%
different lengths. What do you think
will<00:35:51.040><c> happen?</c><00:35:51.680><c> Well,</c><00:35:52.000><c> the</c><00:35:52.240><c> short</c><00:35:52.560><c> oscillators</c>

34:23.986 --> 34:23.996 align:start position:0%
will happen? Well, the short oscillators
 

34:23.996 --> 34:25.667 align:start position:0%
will happen? Well, the short oscillators
are<00:35:53.280><c> pretty</c><00:35:53.520><c> stiff,</c><00:35:53.839><c> so</c><00:35:54.000><c> they</c><00:35:54.240><c> only</c><00:35:54.400><c> respond</c>

34:25.667 --> 34:25.677 align:start position:0%
are pretty stiff, so they only respond
 

34:25.677 --> 34:28.147 align:start position:0%
are pretty stiff, so they only respond
to<00:35:55.200><c> high</c><00:35:55.440><c> frequencies.</c><00:35:56.240><c> But</c><00:35:56.480><c> the</c><00:35:56.640><c> long</c><00:35:56.880><c> ones</c>

34:28.147 --> 34:28.157 align:start position:0%
to high frequencies. But the long ones
 

34:28.157 --> 34:30.946 align:start position:0%
to high frequencies. But the long ones
are<00:35:57.520><c> quite</c><00:35:57.839><c> flexible.</c><00:35:58.640><c> So</c><00:35:58.720><c> they</c><00:35:59.040><c> respond</c><00:35:59.680><c> to</c>

34:30.946 --> 34:30.956 align:start position:0%
are quite flexible. So they respond to
 

34:30.956 --> 34:33.507 align:start position:0%
are quite flexible. So they respond to
low<00:36:00.320><c> frequencies.</c><00:36:01.520><c> So</c><00:36:01.760><c> depending</c><00:36:02.160><c> on</c><00:36:02.400><c> the</c>

34:33.507 --> 34:33.517 align:start position:0%
low frequencies. So depending on the
 

34:33.517 --> 34:35.907 align:start position:0%
low frequencies. So depending on the
frequencies<00:36:03.200><c> that</c><00:36:03.520><c> make</c><00:36:03.760><c> this</c><00:36:04.000><c> sound</c><00:36:04.400><c> up,</c>

34:35.907 --> 34:35.917 align:start position:0%
frequencies that make this sound up,
 

34:35.917 --> 34:37.907 align:start position:0%
frequencies that make this sound up,
different<00:36:05.440><c> oscillators</c><00:36:06.160><c> will</c><00:36:06.480><c> start</c><00:36:06.720><c> getting</c>

34:37.907 --> 34:37.917 align:start position:0%
different oscillators will start getting
 

34:37.917 --> 34:41.667 align:start position:0%
different oscillators will start getting
driven.<00:36:07.920><c> So</c><00:36:08.320><c> look</c><00:36:08.880><c> which</c><00:36:09.119><c> ones</c><00:36:09.760><c> vibrate</c><00:36:10.640><c> tell</c>

34:41.667 --> 34:41.677 align:start position:0%
driven. So look which ones vibrate tell
 

34:41.677 --> 34:44.547 align:start position:0%
driven. So look which ones vibrate tell
us<00:36:11.200><c> which</c><00:36:11.599><c> frequencies</c><00:36:12.240><c> are</c><00:36:12.480><c> present</c><00:36:13.119><c> and</c><00:36:13.359><c> how</c>

34:44.547 --> 34:44.557 align:start position:0%
us which frequencies are present and how
 

34:44.557 --> 34:46.706 align:start position:0%
us which frequencies are present and how
strongly<00:36:14.079><c> they</c><00:36:14.320><c> vibrate</c><00:36:14.960><c> tells</c><00:36:15.280><c> us</c><00:36:15.440><c> the</c>

34:46.706 --> 34:46.716 align:start position:0%
strongly they vibrate tells us the
 

34:46.716 --> 34:49.187 align:start position:0%
strongly they vibrate tells us the
strength<00:36:16.240><c> of</c><00:36:16.480><c> each</c><00:36:16.720><c> one.</c><00:36:17.280><c> A</c><00:36:17.599><c> physical</c>

34:49.187 --> 34:49.197 align:start position:0%
strength of each one. A physical
 

34:49.197 --> 34:52.547 align:start position:0%
strength of each one. A physical
fouryear<00:36:18.800><c> analyzer.</c><00:36:20.079><c> And</c><00:36:20.320><c> this</c><00:36:20.720><c> is</c><00:36:21.119><c> kind</c><00:36:21.440><c> of</c>

34:52.547 --> 34:52.557 align:start position:0%
fouryear analyzer. And this is kind of
 

34:52.557 --> 34:56.787 align:start position:0%
fouryear analyzer. And this is kind of
how<00:36:22.880><c> our</c><00:36:23.280><c> ears</c><00:36:24.160><c> separate</c><00:36:24.800><c> frequencies.</c><00:36:25.359><c> I</c><00:36:25.680><c> kid</c>

34:56.787 --> 34:56.797 align:start position:0%
how our ears separate frequencies. I kid
 

34:56.797 --> 34:59.427 align:start position:0%
how our ears separate frequencies. I kid
you<00:36:26.079><c> not,</c><00:36:26.880><c> the</c><00:36:27.119><c> coccia</c><00:36:27.760><c> which</c><00:36:27.920><c> is</c><00:36:28.079><c> part</c><00:36:28.320><c> of</c><00:36:28.400><c> the</c>

34:59.427 --> 34:59.437 align:start position:0%
you not, the coccia which is part of the
 

34:59.437 --> 35:03.187 align:start position:0%
you not, the coccia which is part of the
inner<00:36:28.800><c> ear</c><00:36:29.200><c> has</c><00:36:29.520><c> a</c><00:36:29.839><c> similar</c><00:36:30.880><c> setup.</c><00:36:31.920><c> One</c><00:36:32.160><c> end</c>

35:03.187 --> 35:03.197 align:start position:0%
inner ear has a similar setup. One end
 

35:03.197 --> 35:05.027 align:start position:0%
inner ear has a similar setup. One end
is<00:36:32.480><c> stiff</c><00:36:32.720><c> and</c><00:36:32.960><c> it</c><00:36:33.119><c> responds</c><00:36:33.440><c> to</c><00:36:33.680><c> only</c><00:36:33.920><c> high</c>

35:05.027 --> 35:05.037 align:start position:0%
is stiff and it responds to only high
 

35:05.037 --> 35:07.187 align:start position:0%
is stiff and it responds to only high
frequencies<00:36:35.040><c> whereas</c><00:36:35.440><c> the</c><00:36:35.680><c> other</c><00:36:35.839><c> end</c><00:36:36.079><c> which</c>

35:07.187 --> 35:07.197 align:start position:0%
frequencies whereas the other end which
 

35:07.197 --> 35:09.347 align:start position:0%
frequencies whereas the other end which
is<00:36:36.400><c> the</c><00:36:36.480><c> apex</c><00:36:36.960><c> is</c><00:36:37.200><c> quite</c><00:36:37.520><c> flexible</c><00:36:38.000><c> and</c><00:36:38.240><c> that</c>

35:09.347 --> 35:09.357 align:start position:0%
is the apex is quite flexible and that
 

35:09.357 --> 35:12.067 align:start position:0%
is the apex is quite flexible and that
responds<00:36:38.960><c> to</c><00:36:39.200><c> the</c><00:36:39.520><c> lower</c><00:36:39.839><c> ones.</c><00:36:40.720><c> And</c><00:36:40.880><c> so</c>

35:12.067 --> 35:12.077 align:start position:0%
responds to the lower ones. And so
 

35:12.077 --> 35:13.747 align:start position:0%
responds to the lower ones. And so
depending<00:36:41.520><c> upon</c><00:36:41.760><c> which</c><00:36:42.000><c> ones</c><00:36:42.240><c> vibrate</c><00:36:42.640><c> and</c>

35:13.747 --> 35:13.757 align:start position:0%
depending upon which ones vibrate and
 

35:13.757 --> 35:15.747 align:start position:0%
depending upon which ones vibrate and
how<00:36:43.040><c> strongly</c><00:36:43.440><c> they</c><00:36:43.680><c> vibrate</c><00:36:44.400><c> they</c><00:36:44.640><c> get</c>

35:15.747 --> 35:15.757 align:start position:0%
how strongly they vibrate they get
 

35:15.757 --> 35:17.347 align:start position:0%
how strongly they vibrate they get
converted<00:36:45.280><c> into</c><00:36:45.520><c> the</c><00:36:45.760><c> electrical</c><00:36:46.160><c> signals</c>

35:17.347 --> 35:17.357 align:start position:0%
converted into the electrical signals
 

35:17.357 --> 35:19.827 align:start position:0%
converted into the electrical signals
that<00:36:46.720><c> get</c><00:36:46.960><c> sent</c><00:36:47.200><c> to</c><00:36:47.359><c> the</c><00:36:47.520><c> brain.</c><00:36:48.400><c> This</c><00:36:48.640><c> is</c><00:36:48.720><c> one</c>

35:19.827 --> 35:19.837 align:start position:0%
that get sent to the brain. This is one
 

35:19.837 --> 35:22.387 align:start position:0%
that get sent to the brain. This is one
of<00:36:49.040><c> the</c><00:36:49.359><c> first</c><00:36:49.680><c> steps</c><00:36:50.000><c> that</c><00:36:50.320><c> lets</c><00:36:50.640><c> our</c><00:36:50.880><c> brain</c>

35:22.387 --> 35:22.397 align:start position:0%
of the first steps that lets our brain
 

35:22.397 --> 35:24.946 align:start position:0%
of the first steps that lets our brain
distinguish<00:36:52.240><c> sounds</c><00:36:53.040><c> and</c><00:36:53.359><c> understand</c>

35:24.946 --> 35:24.956 align:start position:0%
distinguish sounds and understand
 

35:24.956 --> 35:26.147 align:start position:0%
distinguish sounds and understand
speech.

35:26.147 --> 35:26.157 align:start position:0%
speech.
 

35:26.157 --> 35:28.067 align:start position:0%
speech.
So<00:36:55.520><c> although</c><00:36:55.839><c> it</c><00:36:56.000><c> took</c><00:36:56.240><c> us</c><00:36:56.320><c> a</c><00:36:56.480><c> while</c><00:36:56.640><c> to</c><00:36:56.880><c> get</c><00:36:57.040><c> to</c>

35:28.067 --> 35:28.077 align:start position:0%
So although it took us a while to get to
 

35:28.077 --> 35:29.587 align:start position:0%
So although it took us a while to get to
a<00:36:57.359><c> point</c><00:36:57.520><c> where</c><00:36:57.680><c> we</c><00:36:57.920><c> understand</c><00:36:58.320><c> for</c>

35:29.587 --> 35:29.597 align:start position:0%
a point where we understand for
 

35:29.597 --> 35:32.787 align:start position:0%
a point where we understand for
transforms,<00:36:59.760><c> our</c><00:37:00.079><c> bodies</c><00:37:00.720><c> have</c><00:37:01.119><c> been</c><00:37:01.599><c> doing</c>

35:32.787 --> 35:32.797 align:start position:0%
transforms, our bodies have been doing
 

35:32.797 --> 35:35.986 align:start position:0%
transforms, our bodies have been doing
it<00:37:02.400><c> since</c><00:37:02.880><c> we</c><00:37:03.119><c> were</c><00:37:03.359><c> born.</c>

35:35.986 --> 35:35.996 align:start position:0%
it since we were born.
 

35:35.996 --> 35:38.706 align:start position:0%
it since we were born.
That's<00:37:05.359><c> pretty</c><00:37:05.599><c> awesome.</c><00:37:06.400><c> I'll</c><00:37:06.640><c> see</c><00:37:06.800><c> you.</c>

35:38.706 --> 35:38.716 align:start position:0%
That's pretty awesome. I'll see you.
 

35:38.716 --> 35:40.787 align:start position:0%
That's pretty awesome. I'll see you.
Special<00:37:08.160><c> thanks</c><00:37:08.400><c> to</c><00:37:08.560><c> my</c><00:37:08.720><c> top</c><00:37:09.040><c> patronons.</c><00:37:09.680><c> You</c>

35:40.787 --> 35:40.797 align:start position:0%
Special thanks to my top patronons. You
 

35:40.797 --> 35:45.476 align:start position:0%
Special thanks to my top patronons. You
folks<00:37:10.240><c> are</c><00:37:10.480><c> truly</c><00:37:11.040><c> truly</c><00:37:11.599><c> awesome.</c>