title

What makes the natural log "natural"? | Ep. 7 Lockdown live math

description

All about ln(x).
Full playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDP5CVelJJ1bNDouqrAhVPev
Home page: https://www.3blue1brown.com
Brought to you by you: https://3b1b.co/ldm-thanks
Beautiful pictorial summary by @ThuyNganVu:
https://twitter.com/ThuyNganVu/status/1259288683489849344
Errors:
At minute 16, the sum should be written with a "..." to indicate going to infinity.
At minute 38, the exponent should have 1/(2s^2) instead of 1/s^2 for s to represent standard deviation.
At minute 54, an equal sign was mistakenly used in taking the derivative of x^3 / 3!.
At the end, it should be pointed out that the alternating series with x^n terms only converges for values of x between -1 and 1, so the values one can't be considered proven with values of x outside that range. Everything with the argument here is fine, as it only deals with the convergent input, but that fact should still be mentioned.
Related videos.
Calculus series:
https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr
The sum giving pi^2 / 6:
https://youtu.be/d-o3eB9sfls
The sum giving pi / 4:
https://youtu.be/NaL_Cb42WyY
https://youtu.be/00w8gu2aL-w (Mathologer)
-------------------
Video timeline (thanks to user "noonesperfect")
0:00:14 - Question 1
0:02:29 - Answer 1
0:06:27 - Prime nos. in Infinite Geometric Series (Basel problem) and their relationship with Natural logarithm
0:12:01 - More examples of prime numbers in infinite series and their relationship with ln
0:17:25 - Question 2
0:19:20 - Answer 2 and explanation using ln
0:22:25 - Question 3 and families of curves
0:26:37 - Answer 3 and explanation
0:28:50 - Imaginary exponential
0:30:57 - Derivatives of exponential terms
0:37:21 - Why derivative of e^t is the same as that e^t itself?
0:41:21 - Question 4
0:44:12 - Answer 4 and explanation using Python
0:46:02 - Taylor Series for e^x
0:48:29 - Derivatives of polynomial terms/Derivatives of e^x
0:50:56 - Derivative of natural logarithm using graph
0:56:07 - Question 5
0:57:37 - Answer 5 and explanation
1:02:15 - Eulerâ€“Mascheroni constant
1:08:37 - Question 6
1:12:41 - Connecting dots to the familiarity of different expression in math
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detail

{'title': 'What makes the natural log "natural"? | Ep. 7 Lockdown live math', 'heatmap': [{'end': 1618.227, 'start': 1573.225, 'weight': 0.731}, {'end': 2518.04, 'start': 2471.037, 'weight': 0.826}, {'end': 3243.008, 'start': 3186.866, 'weight': 0.917}, {'end': 4186.251, 'start': 4123.772, 'weight': 1}], 'summary': "Exploring prime number density, euler's formula, natural logarithms, exponentials, derivatives, limits, and calculus with 900 participants, showcasing the surprising proportion of prime numbers, the interrelationship between logarithms, and the convergence towards euler's constant.", 'chapters': [{'end': 650.095, 'segs': [{'end': 72.049, 'src': 'embed', 'start': 35.767, 'weight': 0, 'content': [{'end': 42.491, 'text': 'So if you were to go through the painstaking process of you know, looking at all of those numbers between a trillion and a trillion plus a thousand,', 'start': 35.767, 'duration': 6.724}, {'end': 46.973, 'text': 'considering which of them are prime, what do you guess is the relevant proportion there?', 'start': 42.491, 'duration': 4.482}, {'end': 48.534, 'text': "So don't worry about getting it right or wrong.", 'start': 47.033, 'duration': 1.501}, {'end': 51.636, 'text': "I'm mostly curious where people's intuitions are on this one.", 'start': 48.874, 'duration': 2.762}, {'end': 58.9, 'text': "So while we don't necessarily have to do those painstaking calculations, I have gone ahead and written up a simple program that can do that for us.", 'start': 52.656, 'duration': 6.244}, {'end': 67.146, 'text': "So if we hop on over to Python, it's not exactly the most sophisticated program for getting primes in the world, but it'll get the job done for us.", 'start': 59.301, 'duration': 7.845}, {'end': 72.049, 'text': 'So if I type something like get primes between 0 and 50, you know, we take a look.', 'start': 67.546, 'duration': 4.503}], 'summary': 'Exploring the proportion of prime numbers between a trillion and a trillion plus a thousand, using a simple python program.', 'duration': 36.282, 'max_score': 35.767, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw35767.jpg'}, {'end': 439.233, 'src': 'embed', 'start': 409.118, 'weight': 3, 'content': [{'end': 410.899, 'text': "And they'll actually approach a certain number.", 'start': 409.118, 'duration': 1.781}, {'end': 414.14, 'text': 'And it was this open question in Europe.', 'start': 411.679, 'duration': 2.461}, {'end': 418.582, 'text': 'It was posed in Basel by, I think, one of the Bernoullis for a while like what is the number that this equals?', 'start': 414.2, 'duration': 4.382}, {'end': 426.205, 'text': 'And eventually Euler genius of the day was able to prove that it equals pi squared divided by 6, which is very beautiful.', 'start': 419.182, 'duration': 7.023}, {'end': 432.068, 'text': 'The idea that pi is at all related to just adding up the reciprocals of squares.', 'start': 426.345, 'duration': 5.723}, {'end': 433.929, 'text': 'But it gets crazier than that.', 'start': 432.748, 'duration': 1.181}, {'end': 439.233, 'text': "I'm going to play this weird game that's going to kick out terms that don't look like prime numbers.", 'start': 434.629, 'duration': 4.604}], 'summary': 'Euler proved the sum of reciprocals of squares equals pi squared/6.', 'duration': 30.115, 'max_score': 409.118, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw409118.jpg'}, {'end': 591.405, 'src': 'embed', 'start': 557.251, 'weight': 2, 'content': [{'end': 558.552, 'text': "So it's definitely a smaller number.", 'start': 557.251, 'duration': 1.301}, {'end': 561.955, 'text': 'You might be able to guess where this is going based on the title of the video.', 'start': 559.112, 'duration': 2.843}, {'end': 568.721, 'text': 'What it ends up equaling is the natural log of what it was before, of pi squared over six.', 'start': 562.595, 'duration': 6.126}, {'end': 573.741, 'text': "And that's not just true for this particular sequence of sums of squares.", 'start': 570.1, 'duration': 3.641}, {'end': 581.363, 'text': "There's a number of other formulas that get us something related to prime, where we could, sorry, something related to pi,", 'start': 574.121, 'duration': 7.242}, {'end': 583.543, 'text': 'which is evidently related to primes.', 'start': 581.363, 'duration': 2.18}, {'end': 591.405, 'text': "in a way, that's, I mean you play the same game and you have this weird fashion of taking logarithms and not just any logarithm the log base e.", 'start': 583.543, 'duration': 7.862}], 'summary': 'Sequence of sums of squares equals natural log of pi squared over six, related to prime numbers.', 'duration': 34.154, 'max_score': 557.251, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw557251.jpg'}, {'end': 618.455, 'src': 'embed', 'start': 592.085, 'weight': 4, 'content': [{'end': 597.567, 'text': 'So, just to talk through what I mean in this other context, if you take one minus a third plus a fifth,', 'start': 592.085, 'duration': 5.482}, {'end': 603.369, 'text': 'minus a seventh plus a ninth and kind of alternate back and forth between the odd numbers, you get pi divided by four.', 'start': 597.567, 'duration': 5.802}, {'end': 604.91, 'text': 'I have a video all about this.', 'start': 603.869, 'duration': 1.041}, {'end': 607.211, 'text': "Mathologer also has a video about it if you're curious.", 'start': 604.97, 'duration': 2.241}, {'end': 608.931, 'text': "Very beautiful why it's true.", 'start': 607.491, 'duration': 1.44}, {'end': 614.493, 'text': 'But even stranger is when we play this game of keeping the primes and kicking out others.', 'start': 609.832, 'duration': 4.661}, {'end': 616.054, 'text': "So one, we're going to kick out.", 'start': 614.873, 'duration': 1.181}, {'end': 618.455, 'text': "If we keep the third term, that's negative three.", 'start': 616.454, 'duration': 2.001}], 'summary': 'Alternating sum of odd numbers converges to pi/4', 'duration': 26.37, 'max_score': 592.085, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw592085.jpg'}], 'start': 0.229, 'title': 'Prime number density and calculation', 'summary': "Discusses the density of prime numbers within a range of a thousand integers between one trillion and a trillion plus a thousand, revealing the surprising proportion of one in 25, while also highlighting the mathematician's ability to estimate this with remarkable accuracy.", 'chapters': [{'end': 258.233, 'start': 0.229, 'title': 'Prime number density and calculation', 'summary': "Discusses the density of prime numbers within a range of a thousand integers between one trillion and a trillion plus a thousand, revealing the surprising proportion of one in 25, while also highlighting the mathematician's ability to estimate this with remarkable accuracy.", 'duration': 258.004, 'highlights': ['The correct proportion of prime numbers within the range of a thousand integers between one trillion and a trillion plus a thousand is one in 25, contrary to the intuitive guesses of one in a thousand or one in 250, demonstrating the sparsity of primes as numbers get larger.', "The mathematician's remarkable ability to estimate the proportion of primes within the given range with impressive accuracy, being able to quickly recognize the correct answer as closer to one in 27 or 28, showcasing their expertise in prime number calculations.", 'The program written to calculate prime numbers within the specified range reveals that there are 37 prime numbers within the range of a thousand integers between one trillion and a trillion plus a thousand, resulting in a proportion of 0.37 or one in 27, demonstrating the actual density of primes within the given range.']}, {'end': 650.095, 'start': 258.233, 'title': 'Density of prime numbers and natural logarithm', 'summary': 'Discusses the relationship between prime numbers and natural logarithm, highlighting how the density of prime numbers near a given value, like a trillion, is around the natural log, and how manipulating a series of reciprocals of squares results in the natural log of pi squared over six, revealing the strange relationship between primes and natural logarithms.', 'duration': 391.862, 'highlights': ['The density of prime numbers near a given value, like a trillion, is around the natural log. The natural log of a trillion is about 27, which is close to the ratio of 1,000 divided by the length of the list of primes.', "Manipulating a series of reciprocals of squares results in the natural log of pi squared over six, revealing the strange relationship between primes and natural logarithms. Euler's genius proved that the sum of reciprocals of squares equals pi squared over 6, and manipulating a series of primes and non-primes results in the natural log of pi squared over six, showing the bizarre relationship between primes and natural logarithms.", 'Playing a game of keeping the primes and kicking out others results in another strange relationship between primes and natural logarithm. Playing a game of keeping primes and kicking out others from a series of alternating odd numbers leads to a relationship where the resulting number is related to the natural log, further emphasizing the unusual connection between primes and natural logarithms.']}], 'duration': 649.866, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw229.jpg', 'highlights': ['The correct proportion of prime numbers within the range of a thousand integers between one trillion and a trillion plus a thousand is one in 25, contrary to the intuitive guesses of one in a thousand or one in 250, demonstrating the sparsity of primes as numbers get larger.', 'The program written to calculate prime numbers within the specified range reveals that there are 37 prime numbers within the range of a thousand integers between one trillion and a trillion plus a thousand, resulting in a proportion of 0.37 or one in 27, demonstrating the actual density of primes within the given range.', "The mathematician's remarkable ability to estimate the proportion of primes within the given range with impressive accuracy, being able to quickly recognize the correct answer as closer to one in 27 or 28, showcasing their expertise in prime number calculations.", 'The density of prime numbers near a given value, like a trillion, is around the natural log. The natural log of a trillion is about 27, which is close to the ratio of 1,000 divided by the length of the list of primes.', 'Manipulating a series of reciprocals of squares results in the natural log of pi squared over six, revealing the strange relationship between primes and natural logarithms.', 'Playing a game of keeping the primes and kicking out others results in another strange relationship between primes and natural logarithm. Playing a game of keeping primes and kicking out others from a series of alternating odd numbers leads to a relationship where the resulting number is related to the natural log, further emphasizing the unusual connection between primes and natural logarithms.']}, {'end': 1329.935, 'segs': [{'end': 812.282, 'src': 'embed', 'start': 787.694, 'weight': 3, 'content': [{'end': 794.816, 'text': "Don't you think that something like E would show up with relation to just on numbers oscillating back and forth like this?", 'start': 787.694, 'duration': 7.122}, {'end': 800.478, 'text': "And there's another relation that natural logarithms have to a sequence that looks like this", 'start': 796.077, 'duration': 4.401}, {'end': 806.3, 'text': "You might ask what happens if we don't alternate back and forth but we add them all together, right?", 'start': 800.879, 'duration': 5.421}, {'end': 807.261, 'text': 'What does that approach?', 'start': 806.501, 'duration': 0.76}, {'end': 812.282, 'text': 'Because, in the same way that you know, when we added all the squares,', 'start': 807.841, 'duration': 4.441}], 'summary': 'Discussing the relationship between natural logarithms and oscillating numbers.', 'duration': 24.588, 'max_score': 787.694, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw787694.jpg'}, {'end': 1112.695, 'src': 'embed', 'start': 1083.454, 'weight': 1, 'content': [{'end': 1087.635, 'text': 'And you can ask how long does it take before this gets larger than a million?', 'start': 1083.454, 'duration': 4.181}, {'end': 1097.562, 'text': "So I'll give you a little bit of time to think through what the answer to that will be, and I will say,", 'start': 1089.475, 'duration': 8.087}, {'end': 1105.289, 'text': "because we're kind of converting between e-related things and base-10 related things if you wanted a little reminder,", 'start': 1097.562, 'duration': 7.727}, {'end': 1112.695, 'text': 'I can pull up the fact that the natural log of 10 is around 2.3, if you wanted to use that for estimation purposes.', 'start': 1105.289, 'duration': 7.406}], 'summary': 'Estimate time to reach a million using e-related conversion, around 2.3.', 'duration': 29.241, 'max_score': 1083.454, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw1083454.jpg'}, {'end': 1329.935, 'src': 'embed', 'start': 1289.666, 'weight': 0, 'content': [{'end': 1295.373, 'text': "So if you're asking what's 1 divided by 2.3, I mean very roughly, it's like a half.", 'start': 1289.666, 'duration': 5.707}, {'end': 1303.882, 'text': "So we could think of n as being very, very roughly something that's like 10 to the 1 half to the 1 million.", 'start': 1295.954, 'duration': 7.928}, {'end': 1306.904, 'text': 'just to get us something kind of close.', 'start': 1305.484, 'duration': 1.42}, {'end': 1310.186, 'text': "So that looks like it's 10 to the 500,000.", 'start': 1307.404, 'duration': 2.782}, {'end': 1316.829, 'text': "And we know that that one half really should be a little bit smaller, because we're taking one divided by 2.3, not one divided by two.", 'start': 1310.186, 'duration': 6.643}, {'end': 1319.91, 'text': 'So the number should be something a little smaller than 500,000.', 'start': 1317.269, 'duration': 2.641}, {'end': 1326.473, 'text': "And indeed, of all the options here, there's one that's much closer to 10 to the 500,000 than anything else.", 'start': 1319.91, 'duration': 6.563}, {'end': 1328.614, 'text': 'So our very rough approximation would get us there.', 'start': 1326.493, 'duration': 2.121}, {'end': 1329.935, 'text': "All right, so that's pretty fun.", 'start': 1329.134, 'duration': 0.801}], 'summary': 'Roughly 1/2, n â‰ˆ 10^500,000, an approximation to 10^500,000.', 'duration': 40.269, 'max_score': 1289.666, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw1289666.jpg'}], 'start': 650.135, 'title': "Euler's formula, prime patterns, and natural logarithms", 'summary': 'Explores the interrelationship between logarithms with base e, prime patterns, and their powers, leading to a formula that relates all prime numbers and their powers to something related to pi. it also delves into the surprising properties of natural logarithms, showcasing their relation to a sequence that does not converge and determining the smallest value of n for which the sum exceeds a million, approximately 10 to the 400,000.', 'chapters': [{'end': 765.197, 'start': 650.135, 'title': "Euler's formula and prime patterns", 'summary': 'Discusses the interrelationship between logarithms with base e, prime patterns, and their powers, leading to a formula that relates all prime numbers and their powers, excluding the composite, to something related to pi. it also introduces an alternating game involving all numbers, demonstrating the beauty of math.', 'duration': 115.062, 'highlights': ['The relationship between taking logarithms with base e and prime patterns leads to a formula that relates all prime numbers and their powers, excluding the composite, to something related to pi.', 'The alternating game involving all numbers demonstrates the beauty of math and showcases the interplay between addition and subtraction in a visual manner.', 'The chapter introduces simpler facts related to series and primes, setting the stage for further exploration of mathematical concepts and their interrelationships.', 'The sequence involving minus, plus, and power operations showcases a unique interplay between numbers and their properties, providing insights into the elegance of mathematical patterns.']}, {'end': 1329.935, 'start': 765.197, 'title': 'Natural logarithms and their relationships', 'summary': 'Discusses the surprising properties of natural logarithms, including their relation to a sequence that does not converge, the grouping of terms to show that the sum approaches the natural log of n, and determining the smallest value of n for which the sum exceeds a million, which is approximately 10 to the 400,000.', 'duration': 564.738, 'highlights': ['The relation of natural logarithms to a sequence that does not converge, despite the numbers getting smaller, is explained through grouping terms to show that the sum approaches the natural log of n. The sequence of adding 1 plus a half plus a third plus a fourth, on and on, does not converge to a specific value despite the numbers getting smaller, as the sum approaches the natural log of n.', 'Determining the smallest value of n for which the sum 1 plus a half, plus a third plus a fourth, on and on, exceeds a million, is approximately 10 to the 400,000. The smallest value of n for which the sum 1 plus a half, plus a third plus a fourth, on and on, exceeds a million is approximately 10 to the 400,000, a significantly large number.', 'The surprising fact that a sequence of numbers, though getting smaller, eventually surpasses any chosen value, is explained through grouping terms appropriately. The sequence of numbers, though getting smaller, eventually surpasses any chosen value by grouping terms appropriately, demonstrating the surprising nature of the behavior of the sequence.']}], 'duration': 679.8, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw650135.jpg', 'highlights': ['The relationship between taking logarithms with base e and prime patterns leads to a formula that relates all prime numbers and their powers, excluding the composite, to something related to pi.', 'The sequence involving minus, plus, and power operations showcases a unique interplay between numbers and their properties, providing insights into the elegance of mathematical patterns.', 'The relation of natural logarithms to a sequence that does not converge, despite the numbers getting smaller, is explained through grouping terms to show that the sum approaches the natural log of n.', 'Determining the smallest value of n for which the sum 1 plus a half, plus a third plus a fourth, on and on, exceeds a million, is approximately 10 to the 400,000.']}, {'end': 1592.913, 'segs': [{'end': 1427.67, 'src': 'embed', 'start': 1394.326, 'weight': 1, 'content': [{'end': 1398.989, 'text': "Okay, once you're writing things in terms of a family, I think a lot of people have this instinct that,", 'start': 1394.326, 'duration': 4.663}, {'end': 1401.291, 'text': 'like these are all of the functions that e produces.', 'start': 1398.989, 'duration': 2.302}, {'end': 1404.353, 'text': 'like e, the number is producing this beautiful family of functions.', 'start': 1401.291, 'duration': 3.062}, {'end': 1411.818, 'text': "But it's important to realize this is the same statement as creating a family of functions that, just like, look like a to the x,", 'start': 1405.113, 'duration': 6.705}, {'end': 1416.001, 'text': 'with various different bases where it could tweak what that value of a is and say.', 'start': 1411.818, 'duration': 4.183}, {'end': 1418.603, 'text': 'you know, sometimes it looks like two to the power x.', 'start': 1416.001, 'duration': 2.602}, {'end': 1422.606, 'text': 'sometimes it looks like three to the power x, okay, or four to the power x.', 'start': 1418.603, 'duration': 4.003}, {'end': 1425.728, 'text': 'Tweaking that base gets us various different exponentials.', 'start': 1423.226, 'duration': 2.502}, {'end': 1427.67, 'text': "That's actually playing the same game.", 'start': 1426.229, 'duration': 1.441}], 'summary': 'Creating a family of functions with different bases and exponentials.', 'duration': 33.344, 'max_score': 1394.326, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw1394326.jpg'}, {'end': 1549.73, 'src': 'embed', 'start': 1523.11, 'weight': 0, 'content': [{'end': 1527.994, 'text': "That doesn't explain why this very specific curve comes up in statistics.", 'start': 1523.11, 'duration': 4.884}, {'end': 1531.737, 'text': 'but if you ever kind of want to remember, oh, what was the formula for a bell curve,', 'start': 1527.994, 'duration': 3.743}, {'end': 1535.679, 'text': 'again you can kind of think through the fact that this should have roughly that shape.', 'start': 1531.737, 'duration': 3.942}, {'end': 1538.501, 'text': 'And quite often it comes with some kind of parameters, though.', 'start': 1536.54, 'duration': 1.961}, {'end': 1541.003, 'text': 'For example, I could put in something, maybe a value.', 'start': 1538.702, 'duration': 2.301}, {'end': 1549.73, 'text': "I'll call s in there that will determine how wide and skinny this bell curve is, something like a standard deviation in the context of statistics.", 'start': 1541.003, 'duration': 8.727}], 'summary': 'The bell curve in statistics has a formula with parameters like standard deviation.', 'duration': 26.62, 'max_score': 1523.11, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw1523110.jpg'}], 'start': 1331.856, 'title': 'The role of e in exponential functions', 'summary': 'Discusses the significance of e in describing exponential growth with varying parameters, its role in creating families of functions, and the impact of tweaking the base and exponent values on the resulting exponentials and bell curve shapes.', 'chapters': [{'end': 1392.991, 'start': 1331.856, 'title': 'Understanding the role of e in math', 'summary': 'Discusses the role of e in math and its significance in describing exponential growth with varying parameters, commonly used in engineering, math, and physics.', 'duration': 61.135, 'highlights': ['The chapter explains the role of E in describing a family of exponential functions with varying parameters, commonly used in engineering, math, and physics.', "It emphasizes how different values of the parameter 'r' can lead to varying rates of exponential growth, providing a practical understanding of E's significance.", 'The discussion aims to clarify the presence of the natural log in certain circumstances, addressing potential misconceptions.']}, {'end': 1592.913, 'start': 1394.326, 'title': 'The role of exponentials in creating families of functions', 'summary': "Emphasizes that exponentials, including the bell curve, can be produced by various bases, not just e, and tweaking the base value results in different exponentials, challenging the notion that e is uniquely related to these functions. it also highlights the significance of choosing e for writing families of exponentials and the impact of tweaking the exponent on the bell curve's shape and standard deviation.", 'duration': 198.587, 'highlights': ['The significance of choosing e for writing families of exponentials is emphasized, as it impacts the various exponentials produced by tweaking the base value, challenging the notion that e is uniquely related to these functions.', "Tweaking the exponent on the bell curve's formula, e^(-x^2), affects the curve's shape and standard deviation, showcasing the impact of varying the exponent's value on the resulting bell curve.", "The chapter explains that various bases, not just e, can produce the same family of curves as the bell curve when adjusted, challenging the idea that e is solely responsible for the bell curve and emphasizing the influence of tweaking the base value on the curve's standard deviation."]}], 'duration': 261.057, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw1331856.jpg', 'highlights': ['The chapter explains the role of E in describing a family of exponential functions with varying parameters, commonly used in engineering, math, and physics.', 'The significance of choosing e for writing families of exponentials is emphasized, as it impacts the various exponentials produced by tweaking the base value, challenging the notion that e is uniquely related to these functions.', "Tweaking the exponent on the bell curve's formula, e^(-x^2), affects the curve's shape and standard deviation, showcasing the impact of varying the exponent's value on the resulting bell curve.", "It emphasizes how different values of the parameter 'r' can lead to varying rates of exponential growth, providing a practical understanding of E's significance.", "The chapter explains that various bases, not just e, can produce the same family of curves as the bell curve when adjusted, challenging the idea that e is solely responsible for the bell curve and emphasizing the influence of tweaking the base value on the curve's standard deviation.", 'The discussion aims to clarify the presence of the natural log in certain circumstances, addressing potential misconceptions.']}, {'end': 2040.815, 'segs': [{'end': 1812.582, 'src': 'embed', 'start': 1786.832, 'weight': 0, 'content': [{'end': 1792.794, 'text': "It is true that e to the i t spins around, but that's not special to e.", 'start': 1786.832, 'duration': 5.962}, {'end': 1798.817, 'text': 'I could also take 2 to the i times t, and that will also produce values that walk around a circle.', 'start': 1792.794, 'duration': 6.023}, {'end': 1800.777, 'text': 'And we can think through more exactly.', 'start': 1799.417, 'duration': 1.36}, {'end': 1805.939, 'text': '2 is the same thing as e to the natural log of 2.', 'start': 1800.797, 'duration': 5.142}, {'end': 1812.582, 'text': 'So 2 to the i t is the same as e to the natural log of 2 times t, all of that times the imaginary number i.', 'start': 1805.939, 'duration': 6.643}], 'summary': 'E to the i t and 2 to the i t both rotate around a circle.', 'duration': 25.75, 'max_score': 1786.832, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw1786832.jpg'}, {'end': 1958.311, 'src': 'embed', 'start': 1925.69, 'weight': 1, 'content': [{'end': 1930.321, 'text': 'which looks like r, and we multiply by the derivative of the outside, which is e, to the rt.', 'start': 1925.69, 'duration': 4.631}, {'end': 1935.703, 'text': "And if anybody here doesn't know calculus, by the way, we're about to start doing a fair amount of it.", 'start': 1931.581, 'duration': 4.122}, {'end': 1941.945, 'text': 'I have a whole series on it that you can pop over and take a look at, lots of other places on YouTube and such to give a quick primer.', 'start': 1936.663, 'duration': 5.282}, {'end': 1947.707, 'text': "But if you're coming in and you're not familiar with calculus, like just be warned that that's where we're about to start going.", 'start': 1942.345, 'duration': 5.362}, {'end': 1952.849, 'text': 'Because, if you want to understand natural logarithms and, by extension, the number E,', 'start': 1948.007, 'duration': 4.842}, {'end': 1958.311, 'text': "the importance that they have has everything to do with rates of change and the inverse of that operation, as you'll see,", 'start': 1952.849, 'duration': 5.462}], 'summary': 'Introduction to calculus and its importance in understanding natural logarithms and the number e, which revolves around rates of change and inverse operations.', 'duration': 32.621, 'max_score': 1925.69, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw1925690.jpg'}, {'end': 2047.7, 'src': 'embed', 'start': 2019.958, 'weight': 4, 'content': [{'end': 2024.099, 'text': "where M this isn't specific to the E fact, but it's just a common thing.", 'start': 2019.958, 'duration': 4.141}, {'end': 2029.881, 'text': "you'll see, M gives you the mean of the distribution where this pile is, and S gives you the standard deviation.", 'start': 2024.099, 'duration': 5.782}, {'end': 2035.923, 'text': "And when we choose to write this family with E, it's giving those constants readable meanings.", 'start': 2030.621, 'duration': 5.302}, {'end': 2040.815, 'text': 'And a similar thing happens with how we describe complex exponentials.', 'start': 2037.352, 'duration': 3.463}, {'end': 2047.7, 'text': 'When we choose to write the idea of walking around a circle with e, it gives a very readable meaning to what this term t is.', 'start': 2041.115, 'duration': 6.585}], 'summary': 'M gives the mean, s gives the standard deviation, e gives readable meanings.', 'duration': 27.742, 'max_score': 2019.958, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw2019958.jpg'}], 'start': 1593.554, 'title': 'Number theory and exponentials', 'summary': 'Discusses the popular choice of 69 as an example number, simplifying 69 to the power x using exponentials, and the flexibility of using different bases such as pi. it also touches on the conversion of a to the x to e to the power of something times x and the discussion about imaginary exponentials.', 'chapters': [{'end': 1744.435, 'start': 1593.554, 'title': 'Number theory and exponentials', 'summary': 'Discusses the popular choice of 69 as an example number, the simplification of 69 to the power x using exponentials, and the flexibility of using different bases such as pi. it also touches on the conversion of a to the x to e to the power of something times x and the discussion about imaginary exponentials.', 'duration': 150.881, 'highlights': ['The most popular answer is 69, which adds up to 69 when you list all its divisors and add them up. 69 is the most popular choice due to its properties of adding up to 69 when listing all its divisors, making it a fun and common thing in number theory.', 'The simplification of 69 to the power x using exponentials, which can also be expressed as e to the power of the natural log of 69 times x. The conversion of 69 to the power x using exponentials and its representation as e to the power of the natural log of 69 times x, which shows the flexibility of exponentials.', 'The discussion about the flexibility of using different bases such as pi, including the example of writing the same function as pi raised to the log base pi of 69 times x. The flexibility of using different bases such as pi is demonstrated through the example of writing the same function as pi raised to the log base pi of 69 times x, showcasing the versatility in mathematical expressions.', 'The mention of the conversion of a to the x to e to the power of something times x and the discussion about imaginary exponentials. The mention of the conversion of a to the x to e to the power of something times x and the discussion about imaginary exponentials, highlighting the diversity and complexity of mathematical concepts.']}, {'end': 2040.815, 'start': 1744.435, 'title': 'Complex exponentials and the number e', 'summary': 'Discusses the concept of complex exponentials, their significance in electrical engineering, and their relation to the number e, emphasizing its role in describing oscillating patterns and rates of change.', 'duration': 296.38, 'highlights': ['The significance of complex exponentials lies in their ability to describe oscillating patterns in electrical engineering, with e^it providing a concise way to represent sine and cosine waves and signals that oscillate.', 'The number e is not inherently unique in producing values that walk around a circle, as other numbers like 2 also exhibit similar behavior with a different scaling of time, revealing that the concept of complex exponents walking around a circle is not exclusive to e.', 'The unique significance of e lies in its role in rates of change, as seen in its derivative e^t, where the slope of the graph is equal to its own height, conveying the concept of compound growth and proportionality in exponential functions.', 'The choice to express functions using e^rt provides a readable way to represent the rate of change, where the proportionality constant r determines the growth rate, making all constants involved more readable and understandable.']}], 'duration': 447.261, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw1593554.jpg', 'highlights': ['69 is the most popular choice due to its properties of adding up to 69 when listing all its divisors, making it a fun and common thing in number theory.', 'The simplification of 69 to the power x using exponentials, which can also be expressed as e to the power of the natural log of 69 times x, showcasing the flexibility of exponentials.', 'The flexibility of using different bases such as pi is demonstrated through the example of writing the same function as pi raised to the log base pi of 69 times x, showcasing the versatility in mathematical expressions.', 'The mention of the conversion of a to the x to e to the power of something times x and the discussion about imaginary exponentials, highlighting the diversity and complexity of mathematical concepts.', 'The significance of complex exponentials lies in their ability to describe oscillating patterns in electrical engineering, with e^it providing a concise way to represent sine and cosine waves and signals that oscillate.', 'The number e is not inherently unique in producing values that walk around a circle, as other numbers like 2 also exhibit similar behavior with a different scaling of time, revealing that the concept of complex exponents walking around a circle is not exclusive to e.', 'The unique significance of e lies in its role in rates of change, as seen in its derivative e^t, where the slope of the graph is equal to its own height, conveying the concept of compound growth and proportionality in exponential functions.', 'The choice to express functions using e^rt provides a readable way to represent the rate of change, where the proportionality constant r determines the growth rate, making all constants involved more readable and understandable.']}, {'end': 2650.597, 'segs': [{'end': 2092.024, 'src': 'embed', 'start': 2063.922, 'weight': 3, 'content': [{'end': 2068.228, 'text': 'By the chain rule, this is gonna look like i times itself.', 'start': 2063.922, 'duration': 4.306}, {'end': 2070.03, 'text': 'e to the i times t.', 'start': 2068.228, 'duration': 1.802}, {'end': 2071.072, 'text': 'Now, what would that actually mean?', 'start': 2070.03, 'duration': 1.042}, {'end': 2079.536, 'text': "That means that if you're sitting at some kind of number, If this is your current value for e to the i times t,", 'start': 2071.512, 'duration': 8.024}, {'end': 2086.501, 'text': 'the rate of change is i multiplied by that value, which is a 90 degree rotation of this vector.', 'start': 2079.536, 'duration': 6.965}, {'end': 2089.581, 'text': 'Maybe I would draw it like this.', 'start': 2087.561, 'duration': 2.02}, {'end': 2092.024, 'text': 'This right here would give you your rate of change.', 'start': 2090.103, 'duration': 1.921}], 'summary': 'Using the chain rule, the rate of change is i times the current value, resulting in a 90 degree rotation of the vector.', 'duration': 28.102, 'max_score': 2063.922, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw2063922.jpg'}, {'end': 2518.04, 'src': 'heatmap', 'start': 2471.037, 'weight': 0.826, 'content': [{'end': 2473.399, 'text': 'but we want to understand this proportionality constant.', 'start': 2471.037, 'duration': 2.362}, {'end': 2480.081, 'text': 'And I could ask you to guess, just to see if you can get a feel for it in the context of one particular example.', 'start': 2474.179, 'duration': 5.902}, {'end': 2487.083, 'text': "So let's say that I'm choosing a base of something like 2, and I want to understand rates of change of 2 to the x.", 'start': 2480.221, 'duration': 6.862}, {'end': 2491.745, 'text': 'Our question asks us, the limit below, I guess it tells us, it tells us a little bit about what it is.', 'start': 2487.083, 'duration': 4.662}, {'end': 2495.646, 'text': 'The limit below is a number between 0 and 1.', 'start': 2492.005, 'duration': 3.641}, {'end': 2500.889, 'text': "So this is, we're looking at 2 to a small value, minus 1, divided by that same small value.", 'start': 2495.646, 'duration': 5.243}, {'end': 2505.612, 'text': "Don't worry about calculating it exactly, I'm just kind of curious if you guessed.", 'start': 2502.15, 'duration': 3.462}, {'end': 2512.496, 'text': 'Enter some kind of guess for what this value is, and then round it to two decimal places so we can have some consistency.', 'start': 2506.012, 'duration': 6.484}, {'end': 2518.04, 'text': "So we'll give you a moment to just think of what it might be, but don't think too hard if you don't want to.", 'start': 2513.157, 'duration': 4.883}], 'summary': 'Exploring proportionality constant in the context of 2 to the x rates of change.', 'duration': 47.003, 'max_score': 2471.037, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw2471037.jpg'}, {'end': 2599.342, 'src': 'embed', 'start': 2575.538, 'weight': 0, 'content': [{'end': 2585.107, 'text': 'and then there was a truly terrible French pun about like an exponential and a logarithm walk into a bar and they order a beer and like who pays?', 'start': 2575.538, 'duration': 9.569}, {'end': 2593.836, 'text': 'and the answer is that the exponential has to pay, because the logarithm ne pay rien, which anyone who speaks French will like groan and laugh at,', 'start': 2585.107, 'duration': 8.729}, {'end': 2595.618, 'text': 'but that made me laugh a little bit.', 'start': 2593.836, 'duration': 1.782}, {'end': 2599.342, 'text': 'Do I have a personal vendetta against E? Yeah, yeah, I do.', 'start': 2596.659, 'duration': 2.683}], 'summary': 'A french pun about exponential and logarithm walk into a bar, with a personal vendetta against e.', 'duration': 23.804, 'max_score': 2575.538, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw2575538.jpg'}, {'end': 2644.913, 'src': 'embed', 'start': 2614.475, 'weight': 1, 'content': [{'end': 2621.099, 'text': 'We should just write it as what it is, which is a certain polynomial, and just be honest up front, rather than letting e,', 'start': 2614.475, 'duration': 6.624}, {'end': 2622.94, 'text': 'like e has nothing to do with e, to the pi i.', 'start': 2621.099, 'duration': 1.841}, {'end': 2623.901, 'text': "That's a frustrating fact.", 'start': 2622.94, 'duration': 0.961}, {'end': 2625.122, 'text': "It shouldn't be in there.", 'start': 2624.361, 'duration': 0.761}, {'end': 2631.305, 'text': 'Anyway, German here is normal for how you do maths on a ruled paper instead of graph paper.', 'start': 2625.862, 'duration': 5.443}, {'end': 2637.669, 'text': "I mean, graph paper is definitely nicer, but I don't know, this was the paper that I just had on hand.", 'start': 2632.686, 'duration': 4.983}, {'end': 2644.913, 'text': 'And in general, if you want to make any comments or questions about the lesson, you can do so on Twitter with the hashtag LockdownMath,', 'start': 2638.249, 'duration': 6.664}], 'summary': 'Discussion about a polynomial and math paper preferences.', 'duration': 30.438, 'max_score': 2614.475, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw2614475.jpg'}], 'start': 2041.115, 'title': 'Exponentials and derivatives', 'summary': 'Explains e and complex exponentials, highlighting e to the i times t as a 90-degree rotation leading to a speed of one unit per second, and discusses derivatives of e to the t, emphasizing its relation to the number e and the special value 2.718.', 'chapters': [{'end': 2236.017, 'start': 2041.115, 'title': 'Understanding e and complex exponentials', 'summary': 'Explains the concept of e and complex exponentials, emphasizing how e to the i times t represents a 90-degree rotation of the position vector, leading to a speed of one unit per second, while 2 to the i times t rotates and scales, resulting in a slower walk around the unit circle. additionally, it discusses the conversion of exponential functions to e-based form for easier differentiation and the continuous growth representation using the natural log of the base.', 'duration': 194.902, 'highlights': ['E to the i times t represents a 90-degree rotation of the position vector, leading to a speed of one unit per second, while 2 to the i times t rotates and scales, resulting in a slower walk around the unit circle.', 'Conversion of exponential functions to e-based form for easier differentiation, enabling the derivative of anything else and simplifying the differentiation process.', "Continuous growth representation using the natural log of the base for a more natural description of investments' growth rate in a continuous sense."]}, {'end': 2650.597, 'start': 2236.437, 'title': 'Derivatives of exponentials', 'summary': 'Discusses the derivative of e to the t, exploring its origin and its relation to the number e, and then delves into understanding derivatives of exponentials and the special value 2.718, with a focus on manipulating expressions and finding the proportionality constant.', 'duration': 414.16, 'highlights': ['The derivative of e to the t is itself, which is related to the definition of the number e. The chapter explores the reason behind the derivative of e to the t being equal to itself and its relation to the definition of the number e.', 'Manipulating expressions to calculate the derivative of exponentials without relying on pre-established facts. It discusses the approach of calculating the derivative of exponentials without depending on pre-established facts, focusing on manipulating expressions and understanding the derivative directly.', 'Exploring the proportionality constant for the derivatives of exponentials, such as 2 to the x. The chapter delves into understanding the proportionality constant for the derivatives of exponentials, specifically examining the example of 2 to the x and the associated proportionality constant.']}], 'duration': 609.482, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw2041115.jpg', 'highlights': ['E to the i times t represents a 90-degree rotation of the position vector, leading to a speed of one unit per second, while 2 to the i times t rotates and scales, resulting in a slower walk around the unit circle.', 'The derivative of e to the t is itself, which is related to the definition of the number e. The chapter explores the reason behind the derivative of e to the t being equal to itself and its relation to the definition of the number e.', 'Conversion of exponential functions to e-based form for easier differentiation, enabling the derivative of anything else and simplifying the differentiation process.', 'Exploring the proportionality constant for the derivatives of exponentials, such as 2 to the x. The chapter delves into understanding the proportionality constant for the derivatives of exponentials, specifically examining the example of 2 to the x and the associated proportionality constant.', "Continuous growth representation using the natural log of the base for a more natural description of investments' growth rate in a continuous sense.", 'Manipulating expressions to calculate the derivative of exponentials without relying on pre-established facts. It discusses the approach of calculating the derivative of exponentials without depending on pre-established facts, focusing on manipulating expressions and understanding the derivative directly.']}, {'end': 3450.901, 'segs': [{'end': 3049.966, 'src': 'embed', 'start': 3023.82, 'weight': 3, 'content': [{'end': 3030.542, 'text': 'then it feels a little bit more contentful and quite fun to say that e to the x ends up being its own derivative.', 'start': 3023.82, 'duration': 6.722}, {'end': 3034.762, 'text': 'And, like we showed earlier, that then lets you take the derivative of all sorts of other things,', 'start': 3030.922, 'duration': 3.84}, {'end': 3041.964, 'text': 'which in turn explains why we adopt the convention of writing all of our exponentials as e to something times t,', 'start': 3034.762, 'duration': 7.202}, {'end': 3049.966, 'text': 'as opposed to writing them all as a to something times t, even though those are equivalent and often weirdly hard to appreciate.', 'start': 3041.964, 'duration': 8.002}], 'summary': 'E to the x is its own derivative, explaining the convention of writing exponentials as e to something times t.', 'duration': 26.146, 'max_score': 3023.82, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw3023820.jpg'}, {'end': 3110.598, 'src': 'embed', 'start': 3065.011, 'weight': 4, 'content': [{'end': 3073.178, 'text': 'But if I have a deeper relationship with the natural log of x, in terms not just of how it relates to these series, but in all facets of math,', 'start': 3065.011, 'duration': 8.167}, {'end': 3074.88, 'text': 'maybe we can then start drawing connections.', 'start': 3073.178, 'duration': 1.702}, {'end': 3078.403, 'text': 'And if you build up that relationship by knowing things like its derivative,', 'start': 3075.26, 'duration': 3.143}, {'end': 3083.087, 'text': 'it actually helps you come back and understand things like the alternating series we were looking at before.', 'start': 3078.403, 'duration': 4.684}, {'end': 3094.366, 'text': 'So, can we use the fact that e to the x is its own derivative to figure out the slope of a natural log curve.', 'start': 3083.707, 'duration': 10.659}, {'end': 3101.851, 'text': 'Well, what that slope is asking us is to look at a given input x.', 'start': 3096.307, 'duration': 5.544}, {'end': 3105.074, 'text': 'we consider a tiny step dx to the right.', 'start': 3101.851, 'duration': 3.223}, {'end': 3110.598, 'text': 'look at the corresponding step dy up and we want to understand the ratio dy over dx.', 'start': 3105.074, 'duration': 5.524}], 'summary': 'Exploring the relationship between natural log and derivatives to understand math concepts and series.', 'duration': 45.587, 'max_score': 3065.011, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw3065011.jpg'}, {'end': 3171.635, 'src': 'embed', 'start': 3139.532, 'weight': 2, 'content': [{'end': 3150.14, 'text': 'If I ask about some tiny nudge to the value x and the corresponding tiny nudge to e to the y, well,', 'start': 3139.532, 'duration': 10.608}, {'end': 3154.343, 'text': 'what it means for e to the x or in this case e to the y, to be its own derivative,', 'start': 3150.14, 'duration': 4.203}, {'end': 3160.628, 'text': 'is that the size of that tiny nudge is e to whatever the y value at that point is times dy.', 'start': 3154.343, 'duration': 6.285}, {'end': 3164.529, 'text': "and we're saying that that equals dx.", 'start': 3162.708, 'duration': 1.821}, {'end': 3171.635, 'text': 'And what this lets us do then is express the slope that we want, dy over dx.', 'start': 3165.65, 'duration': 5.985}], 'summary': 'Expressing the slope as dy over dx using derivatives and tiny nudges.', 'duration': 32.103, 'max_score': 3139.532, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw3139532.jpg'}, {'end': 3243.008, 'src': 'heatmap', 'start': 3186.866, 'weight': 0.917, 'content': [{'end': 3192.59, 'text': "I can't immediately express it in terms of x, maybe, but I do know whatever this value of y is.", 'start': 3186.866, 'duration': 5.724}, {'end': 3196.253, 'text': 'if I take e to the power of that and then reciprocate, that gives me the slope.', 'start': 3192.59, 'duration': 3.663}, {'end': 3204.319, 'text': 'But of course, what it means to be on our graph is that y is the natural log of x, which is the same as saying e to the y equals x.', 'start': 3197.374, 'duration': 6.945}, {'end': 3208.402, 'text': 'so this whole thing is the same as taking 1 divided by x.', 'start': 3204.319, 'duration': 4.083}, {'end': 3215.547, 'text': 'So if I want to know that slope, I can say what is your x coordinate, take 1 divided by that, and that gets me the slope of the natural log.', 'start': 3208.402, 'duration': 7.145}, {'end': 3220.371, 'text': "which is, we've just gone through a process called implicit differentiation.", 'start': 3217.429, 'duration': 2.942}, {'end': 3230.759, 'text': "If you're not inclined to believe that this manipulation is legitimate, that we can just move around the dx's and dy's like that.", 'start': 3221.492, 'duration': 9.267}, {'end': 3234.802, 'text': 'I have a whole video about implicit differentiation in the calculus series that you can take a look at.', 'start': 3230.759, 'duration': 4.043}, {'end': 3243.008, 'text': 'But the point for us is that we have a very nice fact, that the derivative of ln of x looks like one divided by x.', 'start': 3235.283, 'duration': 7.725}], 'summary': 'The derivative of ln of x equals 1/x, shown through implicit differentiation.', 'duration': 56.142, 'max_score': 3186.866, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw3186866.jpg'}, {'end': 3292.167, 'src': 'embed', 'start': 3254.178, 'weight': 0, 'content': [{'end': 3256.34, 'text': 'And the graph of 1 over x.', 'start': 3254.178, 'duration': 2.162}, {'end': 3257.221, 'text': 'you know what does that look like?', 'start': 3256.34, 'duration': 0.881}, {'end': 3260.399, 'text': 'Well, at the input.', 'start': 3259.318, 'duration': 1.081}, {'end': 3264.721, 'text': "let's say we have the input one somewhere like here.", 'start': 3260.399, 'duration': 4.322}, {'end': 3265.361, 'text': "it'll be at one.", 'start': 3264.721, 'duration': 0.64}, {'end': 3268.763, 'text': "At the input two, it'll be sitting at a half.", 'start': 3265.381, 'duration': 3.382}, {'end': 3271.924, 'text': "At the input three, it'll be sitting at a third.", 'start': 3269.523, 'duration': 2.401}, {'end': 3276.606, 'text': 'And in general, it gets lower and lower and closer to zero.', 'start': 3271.944, 'duration': 4.662}, {'end': 3283.55, 'text': 'So the idea that this would describe the slope of that, something that gets lower and lower, closer to zero,', 'start': 3279.148, 'duration': 4.402}, {'end': 3285.351, 'text': 'seems to pass a little bit of a sanity check.', 'start': 3283.55, 'duration': 1.801}, {'end': 3292.167, 'text': 'Now, the relevance that this is going to have to us will involve the inverse operation to differentiation.', 'start': 3286.161, 'duration': 6.006}], 'summary': 'Graph of 1 over x represents decreasing values approaching zero, relevant to inverse differentiation.', 'duration': 37.989, 'max_score': 3254.178, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw3254178.jpg'}], 'start': 2651.437, 'title': 'Limits and derivatives of e', 'summary': 'Covers the limit of 2 raised to the negative power, which approaches around 0.69, and the definition of e and its derivative. it explores e as a limit, polynomial, and sequence, highlighting its connection to derivatives and the derivative of natural log of x.', 'chapters': [{'end': 2707.747, 'start': 2651.437, 'title': 'Limit of 2 raised to the negative power', 'summary': "Discusses the limit of 2 raised to the negative power, which approaches around 0.69, as experimentally verified. the majority correctly guessed the value, and it's connected to derivatives.", 'duration': 56.31, 'highlights': ['The limit of 2 raised to the negative power approaches around 0.6931, as experimentally verified.', 'The majority of participants correctly guessed the value of the limit.', 'The connection between the limit and derivatives is highlighted.']}, {'end': 3450.901, 'start': 2708.887, 'title': 'Defining e and its derivative', 'summary': 'Discusses different ways to define the number e and its relationship to its derivative, including the concept of e as a limit, e as a polynomial, and e as a sequence, leading to the recognition of e to the x as its own derivative and the derivative of natural log of x.', 'duration': 742.014, 'highlights': ['The number e is defined to be the constant such that the limit of (1 + 1/n)^n as n approaches infinity is 1, leading to e to the x being its own derivative. The discussion explores defining e as a limit, where the limit of (1 + 1/n)^n as n approaches infinity is 1, leading to e to the x being its own derivative.', 'E to the x is also seen as shorthand for a certain polynomial, which represents the role it plays more generally, providing a different avenue to understand its derivative. The concept of e to the x being shorthand for a certain polynomial is presented, offering a different approach to understanding its derivative and its role in mathematics.', 'The sequence x of n, defined by the property x of a + b = x of a * x of b, leads to the definition of e as this particular sequence evaluated at x equals one, providing a substantive approach to understanding e to the x as its own derivative. The discussion delves into the sequence x of n, defined by the property x of a + b = x of a * x of b, leading to the definition of e as this particular sequence evaluated at x equals one, offering a substantive approach to understanding e to the x as its own derivative.', 'The derivative of natural log of x is derived as 1 divided by x, showing its relationship to e and the concept of implicit differentiation. The process of deriving the derivative of natural log of x as 1 divided by x is outlined, showcasing its relationship to e and the concept of implicit differentiation.']}], 'duration': 799.464, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw2651437.jpg', 'highlights': ['The limit of 2 raised to the negative power approaches around 0.6931, as experimentally verified.', 'The majority of participants correctly guessed the value of the limit.', 'The connection between the limit and derivatives is highlighted.', 'The number e is defined to be the constant such that the limit of (1 + 1/n)^n as n approaches infinity is 1, leading to e to the x being its own derivative.', 'E to the x is also seen as shorthand for a certain polynomial, which represents the role it plays more generally, providing a different avenue to understand its derivative.', 'The sequence x of n, defined by the property x of a + b = x of a * x of b, leads to the definition of e as this particular sequence evaluated at x equals one, providing a substantive approach to understanding e to the x as its own derivative.', 'The derivative of natural log of x is derived as 1 divided by x, showing its relationship to e and the concept of implicit differentiation.']}, {'end': 3750.712, 'segs': [{'end': 3478.023, 'src': 'embed', 'start': 3451.702, 'weight': 0, 'content': [{'end': 3457.767, 'text': "And, in fact, even if it's not been enough time to thoroughly think through, I'm gonna go ahead and grade it,", 'start': 3451.702, 'duration': 6.065}, {'end': 3459.849, 'text': 'just so that we can see what it happens to be.', 'start': 3457.767, 'duration': 2.082}, {'end': 3460.95, 'text': 'And a lot of these.', 'start': 3460.249, 'duration': 0.701}, {'end': 3462.991, 'text': 'the spirit of it is that you kind of hazard a guess.', 'start': 3460.95, 'duration': 2.041}, {'end': 3467.355, 'text': "so feel no shame if you entered an answer and then it's not what turns out to be correct.", 'start': 3462.991, 'duration': 4.364}, {'end': 3473.22, 'text': 'So in this case, the sum does in fact end up being bigger than the integral.', 'start': 3468.035, 'duration': 5.185}, {'end': 3478.023, 'text': 'And it looks like 900 of you got that correct, which is awesome.', 'start': 3474, 'duration': 4.023}], 'summary': '900 participants answered correctly that the sum is bigger than the integral.', 'duration': 26.321, 'max_score': 3451.702, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw3451702.jpg'}, {'end': 3547.818, 'src': 'embed', 'start': 3526.162, 'weight': 3, 'content': [{'end': 3534.527, 'text': 'so its upper left corner hits that and then the area of this bar, whose height is 1 half, is well 1 half, because its width is 1..', 'start': 3526.162, 'duration': 8.365}, {'end': 3538.049, 'text': 'Similarly, this bar has an area of 1 third, this bar has an area of 1 fourth.', 'start': 3534.527, 'duration': 3.522}, {'end': 3547.818, 'text': 'and so what you have is a sequence of rectangles whose total area is going to be similar to the area under the curve, definitely similar,', 'start': 3538.869, 'duration': 8.949}], 'summary': 'Total area of rectangles approximates area under the curve.', 'duration': 21.656, 'max_score': 3526.162, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw3526162.jpg'}, {'end': 3611.486, 'src': 'embed', 'start': 3583.061, 'weight': 2, 'content': [{'end': 3585.884, 'text': 'If we know calculus, we do know the area under the curve.', 'start': 3583.061, 'duration': 2.823}, {'end': 3586.785, 'text': "It's very nice.", 'start': 3586.104, 'duration': 0.681}, {'end': 3589.927, 'text': "It involves the antiderivative of 1 over x, like we'll show in a moment.", 'start': 3587.025, 'duration': 2.902}, {'end': 3592.99, 'text': "What we don't know is the sum of the areas of the rectangles.", 'start': 3590.228, 'duration': 2.762}, {'end': 3596.293, 'text': 'That was the sum that we were looking at earlier and trying to understand.', 'start': 3593.35, 'duration': 2.943}, {'end': 3601.818, 'text': "So here we're going backwards and using the area under a curve to approximate the area of a bunch of rectangles.", 'start': 3596.673, 'duration': 5.145}, {'end': 3603.039, 'text': 'which I think is fun.', 'start': 3602.338, 'duration': 0.701}, {'end': 3605.421, 'text': 'It shows that calculus has this back and forth.', 'start': 3603.119, 'duration': 2.302}, {'end': 3611.486, 'text': "It's not just geometry informing understanding of curves, but it's an understanding of curves informing,", 'start': 3605.481, 'duration': 6.005}], 'summary': 'Calculus involves area under the curve and rectangles, showing back and forth relationship.', 'duration': 28.425, 'max_score': 3583.061, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw3583061.jpg'}, {'end': 3693.244, 'src': 'embed', 'start': 3660.304, 'weight': 1, 'content': [{'end': 3664.727, 'text': 'And we evaluate it at the bounds at n and 1.', 'start': 3660.304, 'duration': 4.423}, {'end': 3674.073, 'text': 'And this notation where I kind of put brackets around it and then a number in the upper right corner and lower right corner means I take that expression evaluated at the top,', 'start': 3664.727, 'duration': 9.346}, {'end': 3677.335, 'text': 'minus that expression evaluated at the bottom.', 'start': 3674.073, 'duration': 3.262}, {'end': 3693.244, 'text': "Okay? And that, well, natural log of 1, what is that? e to the what equals 1? Well, it's 0, right? Pretty much anything to the 0 will equal 1.", 'start': 3677.355, 'duration': 15.889}], 'summary': 'Evaluating the expression at bounds n and 1, using natural log and exponentiation properties.', 'duration': 32.94, 'max_score': 3660.304, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw3660304.jpg'}], 'start': 3451.702, 'title': 'Calculus and approximation', 'summary': "Compares sum and integral, with 900 participants demonstrating the concept using the example of 1 over x curve and rectangles. it also discusses the relationship between calculus, geometry, and number theory, showcasing the convergence towards euler's constant.", 'chapters': [{'end': 3564.911, 'start': 3451.702, 'title': 'Comparison of summation and integration', 'summary': 'Discusses the comparison between the sum and integral, with 900 participants getting the correct answer, demonstrating the concept using the example of 1 over x curve and a sequence of rectangles.', 'duration': 113.209, 'highlights': ['The concept of comparing the sum and integral is explained using the example of 1 over x curve and a sequence of rectangles, demonstrating how the total area of the rectangles is similar to the area under the curve, but some area leaks out, providing a visual understanding of the comparison.', 'A total of 900 participants provided the correct answer, showcasing a comprehensive understanding of the comparison between the sum and integral.']}, {'end': 3750.712, 'start': 3565.431, 'title': 'Calculus and rectangles', 'summary': "Discusses the concept of using the area under a curve to approximate the area of rectangles, showcasing the back-and-forth relationship between calculus, geometry, and number theory, and highlights the convergence of the leaked out area towards euler's constant.", 'duration': 185.281, 'highlights': ["The leaked out area approaches Euler's constant, which is approximately 0.577, showcasing the convergence of the area as n tends towards infinity. The leaked out area approaches a certain constant, Euler's constant, which is approximately 0.577, as n tends towards infinity.", 'Utilizing the integral to approximate the sum of the rectangles demonstrates the relationship between calculus, geometry, and number theory. Using the integral to approximate the sum of the rectangles demonstrates the relationship between calculus, geometry, and number theory, showcasing a back-and-forth relationship.', 'The inverse derivative of the function 1/x is the natural log, which is evaluated at the bounds n and 1 to obtain the natural log of n. The inverse derivative of the function 1/x is the natural log, evaluated at the bounds n and 1 to obtain the natural log of n, demonstrating the process of evaluating the natural log at the specified bounds.']}], 'duration': 299.01, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw3451702.jpg', 'highlights': ['A total of 900 participants provided the correct answer, showcasing a comprehensive understanding of the comparison between the sum and integral.', 'The concept of comparing the sum and integral is explained using the example of 1 over x curve and a sequence of rectangles, demonstrating how the total area of the rectangles is similar to the area under the curve, but some area leaks out, providing a visual understanding of the comparison.', 'Utilizing the integral to approximate the sum of the rectangles demonstrates the relationship between calculus, geometry, and number theory, showcasing a back-and-forth relationship.', "The leaked out area approaches Euler's constant, which is approximately 0.577, showcasing the convergence of the area as n tends towards infinity."]}, {'end': 4493.425, 'segs': [{'end': 3866.894, 'src': 'embed', 'start': 3834.949, 'weight': 0, 'content': [{'end': 3840.734, 'text': 'we can try to introspect and think about the reasonable ways that someone would come up with the following line of reasoning', 'start': 3834.949, 'duration': 5.785}, {'end': 3843.877, 'text': 'But it is not unique to this situation.', 'start': 3841.555, 'duration': 2.322}, {'end': 3847.32, 'text': "It's kind of a useful set of tricks to be familiar with.", 'start': 3843.897, 'duration': 3.423}, {'end': 3849.542, 'text': "And there's a couple general principles in there.", 'start': 3848.041, 'duration': 1.501}, {'end': 3856.648, 'text': 'The first general principle is that if we have a hard question in this case figuring out what this sum approaches, bizarrely,', 'start': 3849.882, 'duration': 6.766}, {'end': 3858.991, 'text': 'it can become easier if we make it more general.', 'start': 3856.648, 'duration': 2.343}, {'end': 3866.894, 'text': 'You might think that making things more general would make it harder because you have to answer a more powerful fact.', 'start': 3859.651, 'duration': 7.243}], 'summary': 'Analyzing reasoning through generalization can simplify problem-solving.', 'duration': 31.945, 'max_score': 3834.949, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw3834949.jpg'}, {'end': 4135.296, 'src': 'embed', 'start': 4103.45, 'weight': 2, 'content': [{'end': 4108.192, 'text': 'we might have an alternate expression for what the initial sequence was.', 'start': 4103.45, 'duration': 4.742}, {'end': 4112.149, 'text': "Okay, so from here, I'm going to go ahead and pose a quiz.", 'start': 4108.968, 'duration': 3.181}, {'end': 4119.451, 'text': 'And part of this quiz is seeing who in the audience is comfortable with calculus.', 'start': 4113.169, 'duration': 6.282}, {'end': 4122.752, 'text': "And again, if you're not, calculus series, go and check it out.", 'start': 4120.051, 'duration': 2.701}, {'end': 4135.296, 'text': 'But what we have here is the question, what is the integral from 0 up to 1 of 1 divided by 1 plus x dx? Okay, I want you to evaluate that integral.', 'start': 4123.772, 'duration': 11.524}], 'summary': 'Quiz on evaluating the integral from 0 to 1 of 1/(1+x) dx in a calculus session.', 'duration': 31.846, 'max_score': 4103.45, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw4103450.jpg'}, {'end': 4186.251, 'src': 'heatmap', 'start': 4123.772, 'weight': 1, 'content': [{'end': 4135.296, 'text': 'But what we have here is the question, what is the integral from 0 up to 1 of 1 divided by 1 plus x dx? Okay, I want you to evaluate that integral.', 'start': 4123.772, 'duration': 11.524}, {'end': 4138.577, 'text': "And I'll give you a little moment for this one.", 'start': 4135.895, 'duration': 2.682}, {'end': 4186.251, 'text': "And you know, tell you what, while answers are rolling in, before locking it in, I'm going to go ahead and just start describing the answer here.", 'start': 4181.069, 'duration': 5.182}], 'summary': 'Evaluate the integral from 0 to 1 of 1/(1+x) dx.', 'duration': 62.479, 'max_score': 4123.772, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw4123772.jpg'}, {'end': 4419.521, 'src': 'embed', 'start': 4392.34, 'weight': 5, 'content': [{'end': 4395.683, 'text': 'And then you build up familiarity with other things, like derivatives of natural logs.', 'start': 4392.34, 'duration': 3.343}, {'end': 4401.508, 'text': 'And the more familiarity you have with a lot of different pieces of math, then sometimes, when you see one particular pattern,', 'start': 4396.144, 'duration': 5.364}, {'end': 4405.572, 'text': "you're able to draw in your mind a connection to things that make you look like a genius.", 'start': 4401.508, 'duration': 4.064}, {'end': 4407.493, 'text': 'I think Euler did this all the time.', 'start': 4405.952, 'duration': 1.541}, {'end': 4411.897, 'text': 'If you look at some of the great discoveries of Euler, they just come really out of nowhere.', 'start': 4407.553, 'duration': 4.344}, {'end': 4415.178, 'text': 'I mean the very opening thing that I talked about.', 'start': 4412.157, 'duration': 3.021}, {'end': 4419.521, 'text': "I guess it wasn't the very opening, but early on, where are we?", 'start': 4415.178, 'duration': 4.343}], 'summary': "Familiarity with math allows connections, seen in euler's discoveries.", 'duration': 27.181, 'max_score': 4392.34, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw4392340.jpg'}, {'end': 4468.486, 'src': 'embed', 'start': 4438.066, 'weight': 3, 'content': [{'end': 4439.687, 'text': 'things can become a little bit of a mess over here.', 'start': 4438.066, 'duration': 1.621}, {'end': 4446.031, 'text': "But if we look at this long sum, where we're taking one over one squared, one over two squared and equals pi squared over six,", 'start': 4440.047, 'duration': 5.984}, {'end': 4448.752, 'text': 'the way that Euler found this,', 'start': 4446.031, 'duration': 2.721}, {'end': 4456.377, 'text': 'I mean it involves this very strange thing where you start looking at an infinite product associated with sine of pi times x.', 'start': 4448.752, 'duration': 7.625}, {'end': 4463.883, 'text': 'And if you think of it as starting with this sum and then dreaming up out of your head an infinite product associated with sine of pi times x,', 'start': 4456.377, 'duration': 7.506}, {'end': 4466.245, 'text': 'or it might have been cotangent of pi times x, or something like that.', 'start': 4463.883, 'duration': 2.362}, {'end': 4468.486, 'text': 'It really does seem like it came out of nowhere.', 'start': 4466.905, 'duration': 1.581}], 'summary': "Euler's discovery of pi squared over six through infinite products seemed unexpected.", 'duration': 30.42, 'max_score': 4438.066, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw4438066.jpg'}], 'start': 3751.792, 'title': "Euler's influence and generalizing math", 'summary': "Discusses euler's influence on analyzing the harmonic sum and its relationship with the natural log of x, along with the satisfying nature of understanding the alternating series and its connection to the natural log of two. it also explores the concept of generalizing mathematical problems to make them more tractable, simplifying expressions using derivatives, integrating to find alternate expressions, and using familiarity with different mathematical concepts to solve problems, illustrated through the example of evaluating the integral of 1/(1+x) from 0 to 1.", 'chapters': [{'end': 3812.604, 'start': 3751.792, 'title': "Euler's contribution to harmonic sum", 'summary': "Discusses euler's influence on analyzing the harmonic sum and its relationship with the natural log of x, along with the satisfying nature of understanding the alternating series and its connection to the natural log of two.", 'duration': 60.812, 'highlights': ["Euler's influence on analyzing the harmonic sum and its relationship with the natural log of x.", 'Understanding the alternating series and its connection to the natural log of two.', 'The series that grows like the natural log and the alternated series were discussed.']}, {'end': 4493.425, 'start': 3816.946, 'title': 'Generalizing and simplifying math through derivatives and integrals', 'summary': 'Explores the concept of generalizing mathematical problems to make them more tractable, simplifying expressions using derivatives, integrating to find alternate expressions, and using familiarity with different mathematical concepts to solve problems, illustrated through the example of evaluating the integral of 1/(1+x) from 0 to 1.', 'duration': 676.479, 'highlights': ['The general principle that making a hard question more general can make it easier is illustrated through the example of generalizing a sum to a function of x and evaluating it for various values of x. Illustrates the principle of making a problem more general to make it more tractable; showcases the concept of generalizing a sum to a function of x and evaluating it for various values of x.', 'The concept of simplifying expressions using derivatives is discussed, demonstrating how taking the derivative of a series can lead to a more tractable expression. Explains the concept of simplifying expressions using derivatives; showcases the process of taking the derivative of a series to simplify it.', 'The application of integrating to find alternate expressions is shown through the example of evaluating the integral of 1/(1+x) from 0 to 1 and obtaining the result as the natural log of 2. Illustrates the process of integrating to find alternate expressions; demonstrates the evaluation of the integral of 1/(1+x) from 0 to 1 and obtaining the result as the natural log of 2.', 'The importance of familiarity with different mathematical concepts in problem-solving is emphasized, highlighting how it can help draw connections and simplify complex problems. Emphasizes the importance of familiarity with different mathematical concepts in problem-solving; demonstrates how familiarity with various mathematical concepts can help simplify complex problems.']}], 'duration': 741.633, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4PDoT7jtxmw/pics/4PDoT7jtxmw3751792.jpg', 'highlights': ["Euler's influence on analyzing the harmonic sum and its relationship with the natural log of x.", 'Understanding the alternating series and its connection to the natural log of two.', 'The general principle that making a hard question more general can make it easier is illustrated through the example of generalizing a sum to a function of x and evaluating it for various values of x.', 'The concept of simplifying expressions using derivatives is discussed, demonstrating how taking the derivative of a series can lead to a more tractable expression.', 'The application of integrating to find alternate expressions is shown through the example of evaluating the integral of 1/(1+x) from 0 to 1 and obtaining the result as the natural log of 2.', 'The importance of familiarity with different mathematical concepts in problem-solving is emphasized, highlighting how it can help draw connections and simplify complex problems.']}], 'highlights': ['The correct proportion of prime numbers within the range of a thousand integers between one trillion and a trillion plus a thousand is one in 25, contrary to the intuitive guesses of one in a thousand or one in 250, demonstrating the sparsity of primes as numbers get larger.', 'The program written to calculate prime numbers within the specified range reveals that there are 37 prime numbers within the range of a thousand integers between one trillion and a trillion plus a thousand, resulting in a proportion of 0.37 or one in 27, demonstrating the actual density of primes within the given range.', "The mathematician's remarkable ability to estimate the proportion of primes within the given range with impressive accuracy, being able to quickly recognize the correct answer as closer to one in 27 or 28, showcasing their expertise in prime number calculations.", 'The density of prime numbers near a given value, like a trillion, is around the natural log. The natural log of a trillion is about 27, which is close to the ratio of 1,000 divided by the length of the list of primes.', 'The relationship between taking logarithms with base e and prime patterns leads to a formula that relates all prime numbers and their powers, excluding the composite, to something related to pi.', 'The sequence involving minus, plus, and power operations showcases a unique interplay between numbers and their properties, providing insights into the elegance of mathematical patterns.', 'The derivative of e to the t is itself, which is related to the definition of the number e. The chapter explores the reason behind the derivative of e to the t being equal to itself and its relation to the definition of the number e.', "The leaked out area approaches Euler's constant, which is approximately 0.577, showcasing the convergence of the area as n tends towards infinity."]}