title
Imaginary interest rates | Ep. 5 Lockdown live math

description
Compound interest, e, and how it relates to circles. 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 Great Mathologer video: https://youtu.be/-dhHrg-KbJ0 Beautiful pictorial summary by @ThuyNganVu: https://twitter.com/ThuyNganVu/status/1258221677990703105 https://twitter.com/ThuyNganVu/status/1258222002889875457 My other videos on imaginary exponents: https://youtu.be/v0YEaeIClKY https://youtu.be/mvmuCPvRoWQ Mistakes: In the off-handed remarks on quaternions, I mentioned rotation in 4d would require 10 degrees of freedom. That's wrong, what I should have said was it requires 6 degrees of freedom, and rotation in 5D is what requires 10 degrees of freedom. ------------------ Video Timeline (Thanks to user "Just TIEriffic") 0:00:00 Welcome 0:00:55 Q1: Prompt (Would you take an imaginary interest rate) 0:02:05 "e to the pi i for dummies" video shoutout 0:02:45 Q1: Results 0:03:30 Q2: Prompt (two banks, two rates) 0:04:55 Ask: Beauty of connections in math 0:06:00 Q2: Results 0:07:05 Desmos for Q2 0:09:10 Q3: Prompt (savings growth rate, 6% every 6mo) 0:10:35 Q3: Results 0:12:35 Desmos graph explored 0:14:45 Breaking down an interest rate 0:18:00 An interesting interest equation 0:19:20 Q4: Prompt (100*(1+0.12/n)^2 as n → ∞) 0:21:05 Ask: Quaternions 0:22:35 Q4: Results 0:24:50 Explaining Q4 0:26:40 Defining e 0:28:40 The definition of e from previous lectures 0:30:45 The imaginary interest rate 0:32:35 Graphing this relationship 0:33:50 The imaginary interest rate animation 0:37:55 Compounding continuously with i 0:40:45 The spring & Hooke's law 0:43:20 Q5: Prompt (Δx & Δv for a spring) 0:44:50 Ask: Rotation in for multiple dimensions 0:47:45 Q5: Results 0:49:50 Rewriting the spring's position 0:55:00 Bringing it all together 0:59:00 Ask: Hints on last lecture's homework 1:03:25 Closing Remarks ------------------ The live question setup with stats on-screen is powered by Itempool. https://itempool.com/ Curious about other animations? https://www.3blue1brown.com/faq#manim Music by Vincent Rubinetti. Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people. ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Instagram: https://www.instagram.com/3blue1brown_animations/ Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown

detail
{'title': 'Imaginary interest rates | Ep. 5 Lockdown live math', 'heatmap': [{'end': 1262.979, 'start': 1211.485, 'weight': 0.772}, {'end': 2882.121, 'start': 2837.424, 'weight': 1}], 'summary': 'Delves into the concept of imaginary interest rates and their relevance to physics and simple harmonic motion, exploring interest rate analysis, compound interest, squaring and compound growth, complex numbers, quaternions, simple harmonic motion in physics, and their connection to complex numbers and physical phenomena, while also discussing the implications of compounding interest and continuously compounded growth, with specific examples of time intervals and corresponding results.', 'chapters': [{'end': 139.59, 'segs': [{'end': 28.54, 'src': 'embed', 'start': 0.269, 'weight': 0, 'content': [{'end': 8.472, 'text': 'Welcome back to Lockdown Math, where we try to answer some of the important questions in life, some of the deeper things that are relevant.', 'start': 0.269, 'duration': 8.203}, {'end': 9.373, 'text': 'push your life forward.', 'start': 8.472, 'duration': 0.901}, {'end': 15.235, 'text': 'Like, for example, what if you have a bank that offers you an interest rate of the square root of negative one?', 'start': 9.953, 'duration': 5.282}, {'end': 18.156, 'text': 'A negative, not even a negative interest rate?', 'start': 15.795, 'duration': 2.361}, {'end': 19.616, 'text': 'we thought that was weird enough.', 'start': 18.156, 'duration': 1.46}, {'end': 21.097, 'text': 'an imaginary interest rate.', 'start': 19.616, 'duration': 1.481}, {'end': 28.54, 'text': "Should you take it? Now I almost guarantee, however interesting you think this question is, it's gonna be more interesting than that.", 'start': 21.777, 'duration': 6.763}], 'summary': 'Lockdown math explores the concept of an imaginary interest rate in a thought-provoking manner.', 'duration': 28.271, 'max_score': 0.269, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI269.jpg'}, {'end': 123.202, 'src': 'embed', 'start': 100.11, 'weight': 1, 'content': [{'end': 107.194, 'text': "So even if it feels like nonsense, if you're willing to engage with the weird storyline for a bit, there is a satisfying real-world endpoint for that.", 'start': 100.11, 'duration': 7.084}, {'end': 111.936, 'text': 'The other thing I want to mention is a brief plug for a video by Mathologer.', 'start': 107.874, 'duration': 4.062}, {'end': 115.498, 'text': "If you guys don't know Mathologer, you're certainly in for a treat.", 'start': 112.677, 'duration': 2.821}, {'end': 123.202, 'text': 'he did this one called e to the pi, i for dummies and in general the channel is just full of gyms.', 'start': 117.279, 'duration': 5.923}], 'summary': "Engage with the weird storyline for a satisfying real-world endpoint. check out mathologer's video 'e to the pi, i for dummies' for some mathematical insights.", 'duration': 23.092, 'max_score': 100.11, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI100110.jpg'}], 'start': 0.269, 'title': 'Imaginary interest rates in math', 'summary': 'Explores the concept of an imaginary interest rate and its relevance to physics and simple harmonic motion, while also mentioning a related video by mathologer and emphasizing the engaging nature of the topic.', 'chapters': [{'end': 139.59, 'start': 0.269, 'title': 'Imaginary interest rates in math', 'summary': 'Explores the concept of an imaginary interest rate and its relevance to physics and simple harmonic motion, while also mentioning a related video by mathologer and emphasizing the engaging nature of the topic.', 'duration': 139.321, 'highlights': ['The concept of an imaginary interest rate is explored and its relevance to physics and simple harmonic motion is discussed, highlighting the surprising real-world implications of this seemingly nonsensical question.', "The chapter mentions a video by Mathologer titled 'e to the pi, i for dummies,' emphasizing its relevance to the concepts discussed and recommending it as a valuable resource.", 'The engaging nature of the topic is emphasized, with a mention of the intriguing storyline and the satisfying real-world endpoint for considering the details of the seemingly nonsensical question.']}], 'duration': 139.321, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI269.jpg', 'highlights': ['The concept of an imaginary interest rate is explored and its relevance to physics and simple harmonic motion is discussed, highlighting the surprising real-world implications of this seemingly nonsensical question.', 'The engaging nature of the topic is emphasized, with a mention of the intriguing storyline and the satisfying real-world endpoint for considering the details of the seemingly nonsensical question.', "The chapter mentions a video by Mathologer titled 'e to the pi, i for dummies,' emphasizing its relevance to the concepts discussed and recommending it as a valuable resource."]}, {'end': 336.239, 'segs': [{'end': 191.907, 'src': 'embed', 'start': 161.364, 'weight': 0, 'content': [{'end': 166.387, 'text': 'So let me just see what this distribution is, because I am curious to know.', 'start': 161.364, 'duration': 5.023}, {'end': 170.371, 'text': 'All right, so it looks like 1,690, 1,700 of you said yes, you would take the imaginary interest rate.', 'start': 166.668, 'duration': 3.703}, {'end': 172.033, 'text': 'And then 1,645 of you said no.', 'start': 170.411, 'duration': 1.622}, {'end': 184.12, 'text': "All right, we'll see if you stand by that by the end in which we really understand the implications of imaginary interest.", 'start': 178.476, 'duration': 5.644}, {'end': 191.907, 'text': "Now, before that, let's ask a question that is a little bit more real world, because I think we should spend the first,", 'start': 184.961, 'duration': 6.946}], 'summary': '1,690-1,700 would take imaginary rate, 1,645 said no.', 'duration': 30.543, 'max_score': 161.364, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI161364.jpg'}, {'end': 240.131, 'src': 'embed', 'start': 216.228, 'weight': 2, 'content': [{'end': 222.814, 'text': 'What I want to gauge is what your gut reaction to this question is, okay? So there is a correct answer, but just answer what you think it might be.', 'start': 216.228, 'duration': 6.586}, {'end': 228.099, 'text': 'So the question asks, two banks offer different interest rates on your savings.', 'start': 223.775, 'duration': 4.324}, {'end': 232.323, 'text': 'Bank A will increase your savings by 12% every year.', 'start': 228.58, 'duration': 3.743}, {'end': 234.005, 'text': 'Okay, pretty good bank.', 'start': 232.344, 'duration': 1.661}, {'end': 237.929, 'text': 'Bank B will increase your savings by 1% every month.', 'start': 234.646, 'duration': 3.283}, {'end': 240.131, 'text': 'Again, I wish my bank worked like this.', 'start': 238.55, 'duration': 1.581}], 'summary': 'Comparing bank interest rates: 12% annually vs. 1% monthly.', 'duration': 23.903, 'max_score': 216.228, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI216228.jpg'}, {'end': 333.818, 'src': 'embed', 'start': 304.985, 'weight': 3, 'content': [{'end': 307.626, 'text': 'beautiful connections between seemingly unrelated areas.', 'start': 304.985, 'duration': 2.641}, {'end': 309.807, 'text': "A great example is Euler's identity.", 'start': 308.146, 'duration': 1.661}, {'end': 313.709, 'text': 'Do you think that such serendipity renders math a quintessential beauty?', 'start': 310.227, 'duration': 3.482}, {'end': 321.052, 'text': 'What a poetically written tweet which seems pretty fitting for someone with a profile of Yoda in there.', 'start': 314.309, 'duration': 6.743}, {'end': 323.413, 'text': "I mean, it's a subjective question.", 'start': 321.952, 'duration': 1.461}, {'end': 327.915, 'text': 'Personally, I do think the most beautiful things are the ones that have unexpected connections.', 'start': 323.453, 'duration': 4.462}, {'end': 333.818, 'text': "And I don't know if that's like a natural thing that humans just love to see things that seemed unrelated come together.", 'start': 328.856, 'duration': 4.962}], 'summary': 'Unexpected connections in math and human preference for such beauty.', 'duration': 28.833, 'max_score': 304.985, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI304985.jpg'}], 'start': 139.59, 'title': 'Interest rate comparison and poll results', 'summary': 'Presents an opinion poll on imaginary interest rates, comparing rates offered by two banks and audience responses, and includes a poetic tweet about the beauty of math.', 'chapters': [{'end': 336.239, 'start': 139.59, 'title': 'Interest rate comparison and poll results', 'summary': "Discusses an opinion poll on imaginary interest rates, presenting a real-world question on different interest rates offered by two banks and the audience's responses, while also including a poetic tweet about the beauty of math.", 'duration': 196.649, 'highlights': ["The chapter discusses an opinion poll on imaginary interest rates, presenting a real-world question on different interest rates offered by two banks and the audience's responses, while also including a poetic tweet about the beauty of math. The chapter includes an opinion poll on imaginary interest rates, a real-world question on different interest rates offered by two banks, and a poetic tweet about the beauty of math.", '1,700 of the audience said yes to taking the imaginary interest rate, while 1,645 said no, indicating an even split in the poll results. 1,700 audience members said yes to taking the imaginary interest rate, and 1,645 said no, showing an even split in the poll results.', "The chapter also delves into a real-world question about different interest rates offered by two banks, with a strong consensus but some contention in the audience's responses. The chapter presents a real-world question about different interest rates offered by two banks, with a strong consensus but some contention in the audience's responses.", 'A poetic tweet about the beauty of math is included in the discussion, raising a subjective question about the unexpected connections and beauty in mathematics. The discussion includes a poetic tweet about the beauty of math, raising a subjective question about the unexpected connections and beauty in mathematics.']}], 'duration': 196.649, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI139590.jpg', 'highlights': ["The chapter presents an opinion poll on imaginary interest rates and a real-world question on different interest rates offered by two banks, along with the audience's responses.", '1,700 audience members said yes to taking the imaginary interest rate, and 1,645 said no, showing an even split in the poll results.', "The chapter delves into a real-world question about different interest rates offered by two banks, with a strong consensus but some contention in the audience's responses.", 'The discussion includes a poetic tweet about the beauty of math, raising a subjective question about the unexpected connections and beauty in mathematics.']}, {'end': 629.431, 'segs': [{'end': 390.416, 'src': 'embed', 'start': 364.71, 'weight': 0, 'content': [{'end': 371.957, 'text': 'This might be the first time in lockdown math history that the plurality has not gotten what turns out to be the correct answer.', 'start': 364.71, 'duration': 7.247}, {'end': 377.283, 'text': 'So most of you said you would have more money with bank B and it would be by more than a dollar.', 'start': 372.618, 'duration': 4.665}, {'end': 385.451, 'text': "The second most common answer was to say you'd have more money with bank B by less than a dollar, which turns out to be correct.", 'start': 378.184, 'duration': 7.267}, {'end': 386.652, 'text': "We're going to walk through that in a moment.", 'start': 385.471, 'duration': 1.181}, {'end': 390.416, 'text': 'After that, more with bank A.', 'start': 387.453, 'duration': 2.963}], 'summary': 'In a math problem, most chose bank b for more money, but bank a actually has more.', 'duration': 25.706, 'max_score': 364.71, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI364710.jpg'}, {'end': 487.672, 'src': 'embed', 'start': 461.938, 'weight': 2, 'content': [{'end': 467.501, 'text': "We'll talk about more why I might want to emphasize that this is more of a significant move than you might think.", 'start': 461.938, 'duration': 5.563}, {'end': 473.164, 'text': "The fact that the rate of change is proportional to the thing that's changing means we can factor out this 100.", 'start': 467.961, 'duration': 5.203}, {'end': 477.827, 'text': "And we can just say, oh, that step that we take, we're multiplying it by a constant.", 'start': 473.164, 'duration': 4.663}, {'end': 481.189, 'text': 'In this case, that constant would be 1.12.', 'start': 478.167, 'duration': 3.022}, {'end': 487.672, 'text': "So, as you go from this step to the second year, at the end of that second year, your account balance doesn't jump by just $12,", 'start': 481.189, 'duration': 6.483}], 'summary': 'Emphasizing a significant move with a 1.12 constant multiplier.', 'duration': 25.734, 'max_score': 461.938, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI461938.jpg'}, {'end': 557.814, 'src': 'embed', 'start': 495.935, 'weight': 1, 'content': [{'end': 505.039, 'text': 'And this becomes particularly noticeable as we zoom out and see what would happen over the course of many years with such a fantastic interest rate in your bank.', 'start': 495.935, 'duration': 9.104}, {'end': 510.061, 'text': "It's not just growing like a straight line, it grows with this exponential curve.", 'start': 505.619, 'duration': 4.442}, {'end': 516.245, 'text': 'And in fact, if you look at 10 years in, it looks like the Y coordinate of that is around 310.', 'start': 510.562, 'duration': 5.683}, {'end': 523.368, 'text': "So you let your money sit in that savings account with 12% interest for 10 years, and you'd end up with three times as much money.", 'start': 516.245, 'duration': 7.123}, {'end': 524.409, 'text': 'Pretty interesting.', 'start': 523.828, 'duration': 0.581}, {'end': 533.443, 'text': 'So now, when we start thinking about having the interest accrue, not just at the end of the year, but at various chunks on the way in,', 'start': 525.656, 'duration': 7.787}, {'end': 535.725, 'text': 'let me just ask you another question.', 'start': 533.443, 'duration': 2.282}, {'end': 542.41, 'text': 'I think this is a fun way to start things off, is rather than having me go through the logic, have you guys think through the details.', 'start': 535.745, 'duration': 6.665}, {'end': 549.952, 'text': "So before we jump to the case of doing a step every month, let's just say there was two steps halfway through the year.", 'start': 543.511, 'duration': 6.441}, {'end': 557.814, 'text': 'So what does the question ask us? It says, a bank offers to increase the money in your savings account by 6% at the end of every six months.', 'start': 550.732, 'duration': 7.082}], 'summary': 'Savings with 12% interest triples in 10 years, 6% added every 6 months.', 'duration': 61.879, 'max_score': 495.935, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI495935.jpg'}], 'start': 336.599, 'title': 'Interest rate and compound interest', 'summary': 'Delves into interest rate analysis, highlighting participant misconceptions and providing visual and mathematical explanations. it also explores the concept of compound interest, demonstrating the exponential growth of money over time and the potential for increasing savings through periodic interest accrual.', 'chapters': [{'end': 481.189, 'start': 336.599, 'title': 'Interest rate analysis and bank comparisons', 'summary': "Discusses the analysis of interest rates and bank comparisons, where a majority of participants didn't choose the correct answer, illustrating the misconception regarding interest rates and their accumulative effects. the lesson provides a visual representation and mathematical explanation of the impact of interest rates on savings.", 'duration': 144.59, 'highlights': ["The majority of participants didn't choose the correct answer, indicating a misconception regarding interest rates and their effects. The majority of participants in the session didn't select the correct answer in a bank comparison scenario, indicating a lack of understanding of the impact of interest rates on savings.", "Participants' responses showed a common misconception about the effects of interest rates, with the second most common answer being correct. The second most common answer, indicating that one would have more money with bank B by less than a dollar, turned out to be correct, highlighting a common misconception about the effects of interest rates among the participants.", 'The lesson provides a visual representation and mathematical explanation of the impact of interest rates on savings. The lesson includes a visual representation through a Desmos graph and a mathematical explanation of how interest rates impact savings, aiming to clarify the concept for the participants.', "The rate of change being proportional to the thing that's changing is emphasized, illustrating a powerful idea in the context of interest rates. The concept that the rate of change is proportional to the thing that's changing is highlighted, demonstrating the significance of this principle in the context of interest rates and their impact on savings."]}, {'end': 629.431, 'start': 481.189, 'title': 'Understanding compound interest', 'summary': 'Explains how money grows exponentially with compound interest, showcasing a 12% interest rate leading to a threefold increase over 10 years, and introduces the concept of increasing savings through periodic interest accrual.', 'duration': 148.242, 'highlights': ['The Y coordinate of the exponential growth with a 12% interest rate after 10 years is approximately 310, showcasing the impact of compound interest over time.', 'A 12% interest rate would lead to a threefold increase in money over 10 years, demonstrating the exponential growth of savings with compound interest.', 'The concept of increasing savings through periodic interest accrual is introduced, discussing a scenario where a bank offers to increase the money in a savings account by 6% at the end of every six months.']}], 'duration': 292.832, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI336599.jpg', 'highlights': ["The majority of participants didn't choose the correct answer, indicating a misconception regarding interest rates and their effects.", 'The lesson provides a visual representation and mathematical explanation of the impact of interest rates on savings.', "The rate of change being proportional to the thing that's changing is emphasized, illustrating a powerful idea in the context of interest rates.", 'The Y coordinate of the exponential growth with a 12% interest rate after 10 years is approximately 310, showcasing the impact of compound interest over time.', 'A 12% interest rate would lead to a threefold increase in money over 10 years, demonstrating the exponential growth of savings with compound interest.', 'The concept of increasing savings through periodic interest accrual is introduced, discussing a scenario where a bank offers to increase the money in a savings account by 6% at the end of every six months.']}, {'end': 857.391, 'segs': [{'end': 671.168, 'src': 'embed', 'start': 629.431, 'weight': 0, 'content': [{'end': 633.814, 'text': "so i'm going to go ahead and lock this in, but, as always, feel free to pause and just think things through more yourself.", 'start': 629.431, 'duration': 4.383}, {'end': 634.635, 'text': 'if you want.', 'start': 633.814, 'duration': 0.821}, {'end': 635.735, 'text': "we're going to explain it in a moment.", 'start': 634.635, 'duration': 1.1}, {'end': 645.138, 'text': 'So the correct expression, which looks like 3,000 of you got, is that it should be $100 times 1 plus 0.6 squared.', 'start': 636.736, 'duration': 8.402}, {'end': 648.018, 'text': "Now let's think through why that might be the case.", 'start': 646.638, 'duration': 1.38}, {'end': 657.68, 'text': "So if we head back over to our Desmos expression that we were working with, if, instead of increasing by 12%, we're saying increase by 6%,", 'start': 648.538, 'duration': 9.142}, {'end': 663.762, 'text': 'when you factor it out, it looks like multiplying by 1.06..', 'start': 657.68, 'duration': 6.082}, {'end': 665.523, 'text': "And then there's two ways you can think about this.", 'start': 663.762, 'duration': 1.761}, {'end': 671.168, 'text': "If you're already comfortable with the idea that increasing by 6% is multiplying by this constant, you say, oh, we just square that.", 'start': 665.583, 'duration': 5.585}], 'summary': 'Explanation of a mathematical expression related to 6% increase', 'duration': 41.737, 'max_score': 629.431, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI629431.jpg'}, {'end': 764.395, 'src': 'embed', 'start': 735.342, 'weight': 3, 'content': [{'end': 742.346, 'text': 'The way you might think about that is by saying we multiply by 1.01, okay, by one plus that 1%, and then we do it after 12 months, okay?', 'start': 735.342, 'duration': 7.004}, {'end': 759.153, 'text': 'So instead of the $112 that you would have had from bank A woohoo it looks like compounding more frequently got us an extra 68 cents in our account.', 'start': 747.208, 'duration': 11.945}, {'end': 760.874, 'text': 'So wonderful.', 'start': 759.653, 'duration': 1.221}, {'end': 764.395, 'text': 'In our graph if we wanted to see what that would look like.', 'start': 762.395, 'duration': 2}], 'summary': 'Compounding more frequently resulted in an extra 68 cents in the account after 12 months.', 'duration': 29.053, 'max_score': 735.342, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI735342.jpg'}, {'end': 837.563, 'src': 'embed', 'start': 813.118, 'weight': 2, 'content': [{'end': 818.699, 'text': "And if you're curious, you know previously, after 10 years, we would have had $310,", 'start': 813.118, 'duration': 5.581}, {'end': 825.721, 'text': 'I think it was which we can verify for ourselves if I take that $100 times 1 plus 0.12 to the 10th.', 'start': 818.699, 'duration': 7.022}, {'end': 828.461, 'text': 'All right, this says $310.', 'start': 826.461, 'duration': 2}, {'end': 835.963, 'text': "When we compound every month instead, and we wait for 120 months, it looks like we've squeezed out an extra 20 bucks.", 'start': 828.461, 'duration': 7.502}, {'end': 837.563, 'text': 'So, not bad.', 'start': 836.543, 'duration': 1.02}], 'summary': 'Switching to monthly compounding yields $330 after 10 years, $20 more than annual compounding.', 'duration': 24.445, 'max_score': 813.118, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI813118.jpg'}], 'start': 629.431, 'title': 'Squaring growth and compound growth', 'summary': 'Explains squaring the growth rate with a 6% increase, resulting in $100 times 1 plus 0.6 squared. it also explores the impact of compounding frequency, showing an extra $20 earned over 10 years when compounding monthly instead of annually.', 'chapters': [{'end': 714.969, 'start': 629.431, 'title': 'Squaring growth rate calculation', 'summary': 'Explains the method of squaring the growth rate when increasing by 6% instead of 12%, with a visual demonstration and step-by-step explanation, leading to the conclusion that it should be $100 times 1 plus 0.6 squared.', 'duration': 85.538, 'highlights': ['Explaining the method of squaring the growth rate when increasing by 6% instead of 12% Visual demonstration and step-by-step explanation', 'Concluding that it should be $100 times 1 plus 0.6 squared Resulting conclusion from the explanation']}, {'end': 857.391, 'start': 717.113, 'title': 'The power of compound growth', 'summary': 'Explores the impact of compounding frequency on savings, demonstrating that more frequent compounding leads to higher returns, with an example showing an extra $20 earned over 10 years when compounding monthly instead of annually.', 'duration': 140.278, 'highlights': ['Compounding monthly instead of annually resulted in an extra $20 earned over 10 years. When compounding monthly instead of annually over 10 years, an extra $20 was earned, demonstrating the impact of compounding frequency on savings.', 'Compounding more frequently got us an extra 68 cents in our account over a year. More frequent compounding resulted in an additional 68 cents in the account over a year, showcasing the impact of compounding on savings growth.', 'The frequency with which you compound changes the amount, demonstrating the impact of compounding frequency on savings. The frequency of compounding directly affects the amount earned, highlighting the significance of compounding frequency in savings growth.']}], 'duration': 227.96, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI629431.jpg', 'highlights': ['Explaining the method of squaring the growth rate when increasing by 6% instead of 12% Visual demonstration and step-by-step explanation', 'Concluding that it should be $100 times 1 plus 0.6 squared Resulting conclusion from the explanation', 'Compounding monthly instead of annually resulted in an extra $20 earned over 10 years. When compounding monthly instead of annually over 10 years, an extra $20 was earned, demonstrating the impact of compounding frequency on savings.', 'More frequent compounding resulted in an additional 68 cents in the account over a year, showcasing the impact of compounding on savings growth.', 'The frequency of compounding directly affects the amount earned, highlighting the significance of compounding frequency in savings growth.']}, {'end': 1222.873, 'segs': [{'end': 942.646, 'src': 'embed', 'start': 919.11, 'weight': 2, 'content': [{'end': 926.214, 'text': "And just as another variable to throw down here, let's keep track of how many steps there are along the way if we were to count them all up.", 'start': 919.11, 'duration': 7.104}, {'end': 933.757, 'text': "And the total number would be the total amount of time, divided by the size of the time steps that we're taking.", 'start': 926.674, 'duration': 7.083}, {'end': 942.646, 'text': 'okay?. So what interest rates do is they say that the amount your money changes is going to equal that interest rate r,', 'start': 933.757, 'duration': 8.889}], 'summary': 'Interest rates dictate money changes based on time steps and total time.', 'duration': 23.536, 'max_score': 919.11, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI919110.jpg'}, {'end': 1003.633, 'src': 'embed', 'start': 965.982, 'weight': 3, 'content': [{'end': 969.404, 'text': 'It means you grow by 1 12th of what the annual rate was every month.', 'start': 965.982, 'duration': 3.422}, {'end': 975.768, 'text': "And then, what makes compound growth so powerful, it's also proportional to the amount of money you had.", 'start': 970.344, 'duration': 5.424}, {'end': 978.25, 'text': 'If you had $100, it grows by a certain amount.', 'start': 976.289, 'duration': 1.961}, {'end': 981.032, 'text': 'If you had $1,000 in there, it grows by 10 times as much.', 'start': 978.47, 'duration': 2.562}, {'end': 982.993, 'text': 'The more you have, the more it grows.', 'start': 981.352, 'duration': 1.641}, {'end': 988.797, 'text': 'So that, in our example, you might think of as being something like $100.', 'start': 983.754, 'duration': 5.043}, {'end': 989.017, 'text': 'All right.', 'start': 988.797, 'duration': 0.22}, {'end': 990.038, 'text': "so what's the magic of this?", 'start': 989.017, 'duration': 1.021}, {'end': 1003.633, 'text': 'It looks innocuous at first, but it means that our money changes by going to m plus r, delta t times m, and because m shows up in both of these terms,', 'start': 990.839, 'duration': 12.794}], 'summary': 'Compound growth increases by 1/12 of the annual rate monthly, proportional to the amount of money, making it more powerful.', 'duration': 37.651, 'max_score': 965.982, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI965982.jpg'}, {'end': 1178.246, 'src': 'embed', 'start': 1123.956, 'weight': 0, 'content': [{'end': 1127.518, 'text': "But you're not just curious about this for any particular n.", 'start': 1123.956, 'duration': 3.562}, {'end': 1131.981, 'text': 'What you want to know is what happens as you crank up the value of n.', 'start': 1127.518, 'duration': 4.463}, {'end': 1134.843, 'text': 'And the expression that mathematicians write for this.', 'start': 1131.981, 'duration': 2.862}, {'end': 1143.652, 'text': 'we write lim for limit, And then we write n with an arrow to infinity, saying I want to crank up that n and see what happens to this particular value.', 'start': 1134.843, 'duration': 8.809}, {'end': 1147.176, 'text': "Now, it's not obvious, I think, what happens to it.", 'start': 1144.393, 'duration': 2.783}, {'end': 1156.524, 'text': 'And playing with my toy again, I just want to do another half-opinion poll, half-real question about this one to see what your instinct is.', 'start': 1147.696, 'duration': 8.828}, {'end': 1159.407, 'text': "So let's say..", 'start': 1158.186, 'duration': 1.221}, {'end': 1161.806, 'text': 'Pull up the question here.', 'start': 1160.885, 'duration': 0.921}, {'end': 1172.583, 'text': 'What happens to the value 100 times one plus 0.12 over n, to the power n, as the value of n approaches infinity?', 'start': 1162.588, 'duration': 9.995}, {'end': 1178.246, 'text': "And in the back of your mind you can think this isn't just a purely mathematical expression.", 'start': 1174.545, 'duration': 3.701}], 'summary': 'Investigating the behavior of an expression as n approaches infinity.', 'duration': 54.29, 'max_score': 1123.956, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI1123956.jpg'}], 'start': 858.331, 'title': 'Compound interest concepts', 'summary': 'Covers compound interest, its impact on money growth, interest rate influence, negative and imaginary interest rates, compounding frequency, and the limit of a function as compounding periods approach infinity, with an example of 100 times 1 plus 0.12 over n to the power n.', 'chapters': [{'end': 988.797, 'start': 858.331, 'title': 'Understanding compound interest formulas', 'summary': 'Explains the concept of compound interest and its impact on the growth of money over time, with a focus on the formula for calculating the change in money over time and the influence of interest rates and time intervals, introducing the potential implications of negative interest rates and imaginary interest rates.', 'duration': 130.466, 'highlights': ['The formula for calculating the change in money over time involves breaking the total time into smaller intervals, with each interval resulting in an increase in money, denoted as delta M, influenced by the length of the time step, delta T, and the total number of steps taken over the total time. Involves breaking the total time into smaller intervals, with each interval resulting in an increase in money.', 'Interest rates impact the change in money over time, multiplying the rate by the size of the time step and being proportional to the amount of money, with the potential to be manipulated to be imaginary. Interest rates impact the change in money over time, being proportional to the amount of money.', 'The concept of compounding monthly involves growing by a fraction of the annual rate every month and is proportional to the amount of money invested. Compounding monthly involves growing by a fraction of the annual rate every month.']}, {'end': 1222.873, 'start': 988.797, 'title': 'Compound interest and frequency influence', 'summary': 'Explains the concept of compound interest, the influence of frequency on compounding, and the limit of a function as the number of compounding periods approaches infinity, with an example of 100 times 1 plus 0.12 over n to the power n, in the context of compound interest.', 'duration': 234.076, 'highlights': ["The expression for how much money you're gonna have after time t is the amount of money you had to start off with at time t=0 multiplied by 1 plus r times the total time divided by n raised to the nth power.", 'The impact of increasing the frequency of compounding on the amount of money generated is demonstrated using the example of 100 times 1 plus 0.12 over n to the power n, as the value of n approaches infinity.', 'Explanation of how the size of n influences the interest rate and the concept of taking the limit as n approaches infinity to observe the behavior of a function in the context of compound interest.']}], 'duration': 364.542, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI858331.jpg', 'highlights': ['The impact of increasing the frequency of compounding on the amount of money generated is demonstrated using the example of 100 times 1 plus 0.12 over n to the power n, as the value of n approaches infinity.', 'Explanation of how the size of n influences the interest rate and the concept of taking the limit as n approaches infinity to observe the behavior of a function in the context of compound interest.', 'Interest rates impact the change in money over time, being proportional to the amount of money.', 'The concept of compounding monthly involves growing by a fraction of the annual rate every month and is proportional to the amount of money invested.']}, {'end': 1914.462, 'segs': [{'end': 1277.904, 'src': 'embed', 'start': 1224.234, 'weight': 4, 'content': [{'end': 1226.115, 'text': "And I don't want you to calculate it necessarily.", 'start': 1224.234, 'duration': 1.881}, {'end': 1232.539, 'text': 'The spirit of this question is to see what your initial thought is, kind of intuitively based on the numbers.', 'start': 1226.155, 'duration': 6.384}, {'end': 1237.983, 'text': "So even though technically there's a right answer to this one, so I'll highlight it green, there's no right or wrong here.", 'start': 1232.8, 'duration': 5.183}, {'end': 1240.465, 'text': 'The spirit of it is for me to know where people are.', 'start': 1238.263, 'duration': 2.202}, {'end': 1243.052, 'text': "So I'll give you just a little bit of time on this.", 'start': 1241.327, 'duration': 1.725}, {'end': 1257.157, 'text': "And while I do, let's take a question from the audience.", 'start': 1255.256, 'duration': 1.901}, {'end': 1262.979, 'text': 'Considering that imaginary unit is defined in terms of operations on the real numbers?', 'start': 1257.917, 'duration': 5.062}, {'end': 1267.721, 'text': 'is there an analogous definition of quaternions in terms of operations on complex numbers?', 'start': 1262.979, 'duration': 4.742}, {'end': 1270.482, 'text': 'Okay, so a bit of an advanced question.', 'start': 1268.421, 'duration': 2.061}, {'end': 1277.904, 'text': "I don't expect most people watching the series to necessarily know what quaternions are, but it's a number system that,", 'start': 1270.502, 'duration': 7.402}], 'summary': 'The transcript discusses intuitive thinking and a question about quaternions.', 'duration': 53.67, 'max_score': 1224.234, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI1224234.jpg'}, {'end': 1339.385, 'src': 'embed', 'start': 1314.452, 'weight': 3, 'content': [{'end': 1320.874, 'text': 'But where it does come from is the idea, at least historically, is the idea of wanting to describe motion in three dimensions.', 'start': 1314.452, 'duration': 6.422}, {'end': 1326.015, 'text': 'And complex numbers can describe things like rotation in two dimensions,', 'start': 1321.434, 'duration': 4.581}, {'end': 1330.296, 'text': "but they don't have enough degrees of freedom to describe rotations in three dimensions.", 'start': 1326.015, 'duration': 4.281}, {'end': 1333.197, 'text': "So in that sense, it was a concrete problem they couldn't solve.", 'start': 1330.716, 'duration': 2.481}, {'end': 1339.385, 'text': "but the construct that you do to add on top of it doesn't look like a solution to a polynomial.", 'start': 1334.458, 'duration': 4.927}], 'summary': 'Historically, the challenge was describing motion in three dimensions using complex numbers, which lack degrees of freedom for 3d rotations.', 'duration': 24.933, 'max_score': 1314.452, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI1314452.jpg'}, {'end': 1623.481, 'src': 'embed', 'start': 1597.561, 'weight': 0, 'content': [{'end': 1601.727, 'text': "Let's call this constant e that empirically has a value around 2.71828.", 'start': 1597.561, 'duration': 4.166}, {'end': 1604.591, 'text': 'And then we can write our whole thing as e to the x.', 'start': 1601.727, 'duration': 2.864}, {'end': 1620.82, 'text': 'So, for example, if we wanted to know what happens in our interest example, when we started out with m of 0, which might be something like $100,', 'start': 1610.034, 'duration': 10.786}, {'end': 1623.481, 'text': 'what we do is we just focus on that rt term.', 'start': 1620.82, 'duration': 2.661}], 'summary': 'Constant e is approximately 2.71828; example uses m=0, $100.', 'duration': 25.92, 'max_score': 1597.561, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI1597561.jpg'}, {'end': 1754.177, 'src': 'embed', 'start': 1731.003, 'weight': 2, 'content': [{'end': 1738.009, 'text': 'what you would find is that the polynomial this expands to looks a lot like this polynomial here and in fact,', 'start': 1731.003, 'duration': 7.006}, {'end': 1746.032, 'text': 'As this value of n gets bigger and bigger and approaches infinity, this polynomial will get closer and closer to this one.', 'start': 1739.173, 'duration': 6.859}, {'end': 1751.956, 'text': "It's a challenging question, so I wouldn't expect everyone to necessarily be able to just bang it out, but if you want to,", 'start': 1746.712, 'duration': 5.244}, {'end': 1754.177, 'text': "it's a very elucidating exercise.", 'start': 1751.956, 'duration': 2.221}], 'summary': 'As n approaches infinity, the polynomial expands to look like the given one.', 'duration': 23.174, 'max_score': 1731.003, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI1731003.jpg'}, {'end': 1914.462, 'src': 'embed', 'start': 1880.876, 'weight': 1, 'content': [{'end': 1882.577, 'text': 'All of the operations there are quite fine.', 'start': 1880.876, 'duration': 1.701}, {'end': 1885.258, 'text': "Now let's draw it out to think about what it would look like.", 'start': 1883.397, 'duration': 1.861}, {'end': 1890.18, 'text': 'And the other thing I want to emphasize is I know this seems like utter nonsense.', 'start': 1886.979, 'duration': 3.201}, {'end': 1895.242, 'text': "We're talking about imaginary numbers in the context of interest rates,", 'start': 1890.76, 'duration': 4.482}, {'end': 1904.479, 'text': 'but in a couple minutes I really do hope to make this relevant to physics and hope to make you see that this is not a totally nonsensical circle of thoughts.', 'start': 1895.242, 'duration': 9.237}, {'end': 1914.462, 'text': "So, I've got my axes here, where this is going to be my real, this is my real money, and this will be all of my imaginary money.", 'start': 1905.999, 'duration': 8.463}], 'summary': 'Discussion on imaginary numbers and relevance to physics and interest rates.', 'duration': 33.586, 'max_score': 1880.876, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI1880876.jpg'}], 'start': 1224.234, 'title': 'Understanding quaternions and complex numbers, complex interest and the limiting expression', 'summary': 'Discusses the intuition behind mathematical questions, analogies between quaternions and complex numbers, and their historical context, while also exploring compound interest, the value of e, expression manipulation, and their relation to complex numbers, approaching infinity, and continuously compounded growth.', 'chapters': [{'end': 1333.197, 'start': 1224.234, 'title': 'Understanding quaternions and complex numbers', 'summary': "Discusses the intuitive understanding of a mathematical question, the analogy between quaternions and complex numbers, and the historical context of quaternions' development for describing motion in three dimensions.", 'duration': 108.963, 'highlights': ["The analogy between quaternions and complex numbers is discussed, highlighting the difference in dimensions and the historical context of quaternions' development.", 'The intuitive approach to a mathematical question is emphasized, focusing on initial thoughts and the absence of right or wrong answers for the question.', "The historical context of quaternions' development for describing motion in three dimensions is explained, emphasizing the concrete problem they couldn't solve with complex numbers."]}, {'end': 1914.462, 'start': 1334.458, 'title': 'Complex interest and the limiting expression', 'summary': 'Explores the concept of compound interest, the value of e, and the manipulation of expressions as it relates to continuously compounded growth, with a focus on approaching infinity and the relevance to complex numbers.', 'duration': 580.004, 'highlights': ['The chapter delves into the concept of compound interest, particularly the idea of continuously compounded growth, and its relevance to a value around 2.71828, known as the original definition of e. The exploration of continuously compounded growth and the value of e, around 2.71828, is a central theme in the chapter, providing insights into the fundamental constant of nature.', 'The manipulation of expressions and the approach towards infinity are discussed with a focus on the relationship between the polynomial representation and the limiting expression, particularly as the value of n increases. The chapter discusses the manipulation of expressions and the relationship between the polynomial representation and the limiting expression, emphasizing the implications as the value of n approaches infinity.', 'The relevance of complex numbers in the context of interest rates is explored, emphasizing the operations involving complex numbers and their connection to physics. The chapter explores the relevance of complex numbers in the context of interest rates, highlighting the operations involving complex numbers and their relevance to physics.']}], 'duration': 690.228, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI1224234.jpg', 'highlights': ['The exploration of continuously compounded growth and the value of e, around 2.71828, is a central theme in the chapter, providing insights into the fundamental constant of nature.', 'The relevance of complex numbers in the context of interest rates is explored, emphasizing the operations involving complex numbers and their connection to physics.', 'The manipulation of expressions and the approach towards infinity are discussed with a focus on the relationship between the polynomial representation and the limiting expression, particularly as the value of n increases.', "The historical context of quaternions' development for describing motion in three dimensions is explained, emphasizing the concrete problem they couldn't solve with complex numbers.", "The analogy between quaternions and complex numbers is discussed, highlighting the difference in dimensions and the historical context of quaternions' development.", 'The intuitive approach to a mathematical question is emphasized, focusing on initial thoughts and the absence of right or wrong answers for the question.']}, {'end': 2397.184, 'segs': [{'end': 1941.679, 'src': 'embed', 'start': 1915.602, 'weight': 5, 'content': [{'end': 1928.531, 'text': 'What it means, if your interest rate is i, is that the change to the money looks like i times, whatever the time step is, delta t times,', 'start': 1915.602, 'duration': 12.929}, {'end': 1930.013, 'text': 'whatever the money is to begin with.', 'start': 1928.531, 'duration': 1.482}, {'end': 1933.617, 'text': 'Now I believe it was lecture three.', 'start': 1931.317, 'duration': 2.3}, {'end': 1939.638, 'text': 'we talked all about complex numbers and one of the fundamental facts was that when you multiply i by something,', 'start': 1933.617, 'duration': 6.021}, {'end': 1941.679, 'text': 'it has the effect of a 90 degree rotation.', 'start': 1939.638, 'duration': 2.041}], 'summary': 'Interest rate i causes money change in i*delta t times. multiplying i results in 90-degree rotation.', 'duration': 26.077, 'max_score': 1915.602, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI1915602.jpg'}, {'end': 2009.412, 'src': 'embed', 'start': 1979.609, 'weight': 4, 'content': [{'end': 1983.432, 'text': 'And now from that point, when you do another time step, it does the same thing.', 'start': 1979.609, 'duration': 3.823}, {'end': 1991.582, 'text': "It takes the vector that you've newly landed on and it rotates it 90 degrees And then you take a little step based on that.", 'start': 1983.552, 'duration': 8.03}, {'end': 1995.464, 'text': 'And then again, you take a little step based on that.', 'start': 1992.442, 'duration': 3.022}, {'end': 2001.208, 'text': "Where at each point, you're looking at what is your money number that has some real part and some imaginary part.", 'start': 1995.784, 'duration': 5.424}, {'end': 2004.93, 'text': 'You rotate it 90 degrees and you add a little step on that.', 'start': 2001.688, 'duration': 3.242}, {'end': 2009.412, 'text': 'And now the question is, what happens to this? This is the original question.', 'start': 2006.05, 'duration': 3.362}], 'summary': 'Iterative rotation and stepping process to analyze complex numbers.', 'duration': 29.803, 'max_score': 1979.609, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI1979609.jpg'}, {'end': 2048.79, 'src': 'embed', 'start': 2023.419, 'weight': 2, 'content': [{'end': 2030.402, 'text': "So, to illustrate this, let's say you compounded it annually, meaning at the end of every year you take a big step.", 'start': 2023.419, 'duration': 6.983}, {'end': 2032.323, 'text': "that's based on this interest rate multiple.", 'start': 2030.402, 'duration': 1.921}, {'end': 2035.004, 'text': 'So for every dollar that you have.', 'start': 2033.023, 'duration': 1.981}, {'end': 2042.728, 'text': 'if you started out just putting your real money into the bank At the end of the year, you would add i times that amount,', 'start': 2035.004, 'duration': 7.724}, {'end': 2045.429, 'text': 'which is a 90 degree rotation of your money.', 'start': 2042.728, 'duration': 2.701}, {'end': 2048.79, 'text': 'Again, I know this is utter nonsense, but follow along with me.', 'start': 2045.809, 'duration': 2.981}], 'summary': 'Compounding annually with interest rate multiple, 90 degree rotation of money.', 'duration': 25.371, 'max_score': 2023.419, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI2023419.jpg'}, {'end': 2192.051, 'src': 'embed', 'start': 2164.074, 'weight': 0, 'content': [{'end': 2170.722, 'text': "and you were willing to just hold for eight total years, you would have 16 times as much, so that's pretty good.", 'start': 2164.074, 'duration': 6.648}, {'end': 2179.005, 'text': 'i would say We might call this the venture capitalist approach, where you put in your money.', 'start': 2170.722, 'duration': 8.283}, {'end': 2186.028, 'text': "you can't see it for a long time and most of the time it's completely imaginary, but every now and then you get a giant multiple at the end.", 'start': 2179.005, 'duration': 7.023}, {'end': 2192.051, 'text': "Now, before you excitedly do this, though, let's say that your bank doesn't compound it annually, right?", 'start': 2187.589, 'duration': 4.462}], 'summary': 'Hold for 8 years for 16x growth, like venture capitalist approach.', 'duration': 27.977, 'max_score': 2164.074, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI2164074.jpg'}, {'end': 2241.929, 'src': 'embed', 'start': 2219.283, 'weight': 1, 'content': [{'end': 2227.246, 'text': "But as that time step gets smaller and smaller, smaller and smaller and smaller, to the point where it's genuinely continuously compounding interest,", 'start': 2219.283, 'duration': 7.963}, {'end': 2229.767, 'text': "what it means is that you're simply walking around a circle.", 'start': 2227.246, 'duration': 2.521}, {'end': 2233.864, 'text': 'So if you were writing this out as an expression,', 'start': 2230.822, 'duration': 3.042}, {'end': 2241.929, 'text': 'it might seem halfway reasonable to look at the fact that we were using e to the x as a shorthand for this limiting value here.', 'start': 2233.864, 'duration': 8.065}], 'summary': 'As the time step decreases, interest compounds continuously using e to the x as a shorthand.', 'duration': 22.646, 'max_score': 2219.283, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI2219283.jpg'}, {'end': 2381.303, 'src': 'embed', 'start': 2359.234, 'weight': 3, 'content': [{'end': 2367.178, 'text': "But if you're willing to stick it out and wait a total of 6.28 years, 2 pi years you come back and for all of your stress,", 'start': 2359.234, 'duration': 7.944}, {'end': 2368.178, 'text': "you're back to where you started.", 'start': 2367.178, 'duration': 1}, {'end': 2371.26, 'text': 'You just have $1 for every dollar that you put in originally.', 'start': 2368.518, 'duration': 2.742}, {'end': 2374.506, 'text': 'So should you accept this interest rate from your bank??', 'start': 2372.08, 'duration': 2.426}, {'end': 2380.14, 'text': "Depends on how much emotional turmoil you want, but if it's continuously compounding, certainly not.", 'start': 2374.988, 'duration': 5.152}, {'end': 2381.303, 'text': 'I think is the appropriate answer.', 'start': 2380.14, 'duration': 1.163}], 'summary': 'Waiting 6.28 years with continuous compounding interest will only yield $1 for every dollar invested.', 'duration': 22.069, 'max_score': 2359.234, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI2359234.jpg'}], 'start': 1915.602, 'title': 'Imaginary money and interest rates', 'summary': 'Delves into the concept of imaginary money, its connection to interest rates, and the 90-degree rotation effect, illustrating implications on money over time. it also discusses rotating money vectors, compounding interest, and the potential 16-fold increase in the original amount after eight years. additionally, it explores continuously compounding interest and its unexpected outcomes, including negative returns, with specific examples of time intervals and corresponding results.', 'chapters': [{'end': 1978.108, 'start': 1915.602, 'title': 'Imaginary money and interest rates', 'summary': 'Explains the concept of imaginary money and its connection to interest rates, illustrating a 90-degree rotation effect, and its implications on the change in money over time.', 'duration': 62.506, 'highlights': ['The change to the money due to an interest rate of i is represented as i times the time step delta t times the initial money, showcasing a connection between a square root of one property and the mechanistic idea of a 90-degree rotation, providing a conceptual understanding of the effect of interest rates on money.', 'The concept of imaginary money is introduced, emphasizing that the change in money results in a perpendicular movement from the original money, leading to an accumulation of imaginary money over time.']}, {'end': 2186.028, 'start': 1979.609, 'title': 'Imaginary money and 90 degree rotations', 'summary': 'Discusses the concept of rotating money vectors by 90 degrees, compounding interest, and the resulting accumulation of imaginary and real money over time, ultimately leading to a potential 16-fold increase in the original amount after eight years.', 'duration': 206.419, 'highlights': ['The concept of rotating money vectors by 90 degrees and adding a step based on that is discussed, leading to the accumulation of imaginary and real money. The process of rotating money vectors by 90 degrees and taking steps based on that, resulting in the accumulation of both imaginary and real money, is explained.', 'The discussion on compounding interest annually, leading to the accumulation of imaginary and real money, with a potential 16-fold increase in the original amount after eight years. The compounding of interest annually, leading to the accumulation of both imaginary and real money, with a potential 16-fold increase in the original amount after eight years, is illustrated.', 'The implication of holding the investment for eight years resulting in a 16-fold increase in the original amount, showcasing the venture capitalist approach. The implication of holding the investment for eight years resulting in a 16-fold increase in the original amount, demonstrating the venture capitalist approach, is highlighted.']}, {'end': 2397.184, 'start': 2187.589, 'title': 'Continuous compounding interest', 'summary': 'Explores the concept of continuously compounding interest, demonstrating how it leads to walking around a circle and results in unexpected outcomes, such as negative returns, with specific examples of time intervals and their corresponding outcomes.', 'duration': 209.595, 'highlights': ['The concept of continuously compounding interest is illustrated by demonstrating how it leads to walking around a circle, which results in unexpected outcomes such as negative returns, with specific examples of time intervals and their corresponding outcomes.', 'The example shows that waiting a total of 6.28 years (2 pi years) results in having $1 for every dollar originally invested, indicating that continuously compounding interest can lead to a return to the initial investment amount after a prolonged period.', 'Specific time intervals, such as waiting a total of pi halves years (a little over a year and a half) and pi years (around 3.14 years), lead to unexpected outcomes, including having an imaginary dollar and negative returns for each dollar originally invested, highlighting the non-intuitive nature of continuously compounding interest.']}], 'duration': 481.582, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI1915602.jpg', 'highlights': ['The implication of holding the investment for eight years resulting in a 16-fold increase in the original amount, demonstrating the venture capitalist approach, is highlighted.', 'The concept of continuously compounding interest is illustrated by demonstrating how it leads to walking around a circle, which results in unexpected outcomes such as negative returns, with specific examples of time intervals and their corresponding outcomes.', 'The compounding of interest annually, leading to the accumulation of both imaginary and real money, with a potential 16-fold increase in the original amount after eight years, is illustrated.', 'The example shows that waiting a total of 6.28 years (2 pi years) results in having $1 for every dollar originally invested, indicating that continuously compounding interest can lead to a return to the initial investment amount after a prolonged period.', 'The concept of rotating money vectors by 90 degrees and taking steps based on that, resulting in the accumulation of both imaginary and real money, is explained.', 'The change to the money due to an interest rate of i is represented as i times the time step delta t times the initial money, showcasing a connection between a square root of one property and the mechanistic idea of a 90-degree rotation, providing a conceptual understanding of the effect of interest rates on money.']}, {'end': 2853.555, 'segs': [{'end': 2426.468, 'src': 'embed', 'start': 2397.184, 'weight': 0, 'content': [{'end': 2402.911, 'text': "what if we did something else to it? What if each time step wasn't in the direction but it was perpendicular to the direction?", 'start': 2397.184, 'duration': 5.727}, {'end': 2410.805, 'text': "And the answer is that this is actually incredibly relevant to anything that involves what's called simple harmonic motion in physics.", 'start': 2404.278, 'duration': 6.527}, {'end': 2413.648, 'text': 'And so a great example of this is a spring.', 'start': 2411.345, 'duration': 2.303}, {'end': 2417.032, 'text': "Let's say we wanted to understand the motion of a spring.", 'start': 2414.389, 'duration': 2.643}, {'end': 2419.274, 'text': 'So let me pull this up here.', 'start': 2417.052, 'duration': 2.222}, {'end': 2426.468, 'text': "I think I have an actual spring sitting somewhere here if it didn't roll off my table.", 'start': 2422.267, 'duration': 4.201}], 'summary': 'Exploring perpendicular time steps in physics, relevant to simple harmonic motion like a spring.', 'duration': 29.284, 'max_score': 2397.184, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI2397184.jpg'}, {'end': 2524.372, 'src': 'embed', 'start': 2493.697, 'weight': 1, 'content': [{'end': 2498.039, 'text': 'Because intuitively, we know that the spring is pulling back with some kind of force,', 'start': 2493.697, 'duration': 4.342}, {'end': 2503.46, 'text': "and what Hooke's Law suggests is that it's a force proportional to that size of x.", 'start': 2498.039, 'duration': 5.421}, {'end': 2508.722, 'text': "And when we use k as a proportionality constant, a negative to emphasize that it's pointed in the opposite direction.", 'start': 2503.46, 'duration': 5.262}, {'end': 2516.227, 'text': "So in this context, what we're saying is if we double the amount that we're pulling it, it's going to pull us back with twice the force.", 'start': 2509.622, 'duration': 6.605}, {'end': 2518.708, 'text': 'If we triple it, it pulls us back with three times the force.', 'start': 2516.407, 'duration': 2.301}, {'end': 2524.372, 'text': 'As with any kind of physical models, this is only true to an approximation.', 'start': 2519.929, 'duration': 4.443}], 'summary': "Hooke's law states force is proportional to displacement, with a constant k.", 'duration': 30.675, 'max_score': 2493.697, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI2493697.jpg'}, {'end': 2578.422, 'src': 'embed', 'start': 2549.59, 'weight': 2, 'content': [{'end': 2552.751, 'text': "This is what force is, this is what Newton's second law tells us.", 'start': 2549.59, 'duration': 3.161}, {'end': 2556.733, 'text': "It says, what is your mass? I'm telling you how much you're gonna accelerate.", 'start': 2553.171, 'duration': 3.562}, {'end': 2561.335, 'text': 'Acceleration is how much your velocity changes, velocity being how much the value of x changes.', 'start': 2556.973, 'duration': 4.362}, {'end': 2566.517, 'text': 'Now, this is quite interesting because what it means is that acceleration,', 'start': 2562.255, 'duration': 4.262}, {'end': 2574.84, 'text': 'which is influencing the displacement kind of in a second order way via how it influences velocity, is itself influenced by the value of x.', 'start': 2566.517, 'duration': 8.323}, {'end': 2578.422, 'text': "So I'm gonna go ahead and ask you a quiz question here.", 'start': 2576.38, 'duration': 2.042}], 'summary': "Newton's second law relates mass to acceleration and velocity changes.", 'duration': 28.832, 'max_score': 2549.59, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI2549590.jpg'}, {'end': 2710.984, 'src': 'embed', 'start': 2682.679, 'weight': 3, 'content': [{'end': 2688.644, 'text': "While you're thinking about that, we've got another question in about quaternions, not entirely relevant to the lesson,", 'start': 2682.679, 'duration': 5.965}, {'end': 2691.506, 'text': 'but certainly a very interesting idea and question.', 'start': 2688.644, 'duration': 2.862}, {'end': 2698.692, 'text': "So if you're enjoying this kind of side plot to the full lecture, rotation in 2D requires one extra number, I.", 'start': 2691.606, 'duration': 7.086}, {'end': 2705.579, 'text': 'So why is it that it requires three extra numbers i, j, k and not just two i and j for 3D rotation?', 'start': 2699.733, 'duration': 5.846}, {'end': 2710.984, 'text': 'Is there an equivalent way for rotation in four dimensions, and how many extra numbers are required there?', 'start': 2705.999, 'duration': 4.985}], 'summary': 'Discussion on quaternions and their relevance to 3d and 4d rotation.', 'duration': 28.305, 'max_score': 2682.679, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI2682679.jpg'}, {'end': 2766.067, 'src': 'embed', 'start': 2739.649, 'weight': 4, 'content': [{'end': 2744.093, 'text': 'So the way that complex numbers describe rotation is entirely based on the numbers that sit on a circle.', 'start': 2739.649, 'duration': 4.444}, {'end': 2747.516, 'text': "It's only one dimension's worth of numbers in some sense.", 'start': 2744.193, 'duration': 3.323}, {'end': 2753.48, 'text': "Awesome So for three-dimensional rotation, it's not just two extra degrees of freedom you have.", 'start': 2747.976, 'duration': 5.504}, {'end': 2756.661, 'text': 'You actually have three degrees of freedom to describe any 3D rotation.', 'start': 2753.52, 'duration': 3.141}, {'end': 2765.006, 'text': "You might think of it as saying, choose an axis for your rotation, a latitude and longitude that you're going to poke a hole through the entire Earth.", 'start': 2757.081, 'duration': 7.925}, {'end': 2766.067, 'text': 'And then,', 'start': 2765.686, 'duration': 0.381}], 'summary': 'Complex numbers describe rotation on a circle, while 3d rotation involves three degrees of freedom and an axis for rotation.', 'duration': 26.418, 'max_score': 2739.649, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI2739649.jpg'}, {'end': 2864.462, 'src': 'embed', 'start': 2833.221, 'weight': 5, 'content': [{'end': 2837.043, 'text': "and if it's what, if it's what you're chatting about on Twitter, I'm happy to chime in on that.", 'start': 2833.221, 'duration': 3.822}, {'end': 2842.007, 'text': "So yeah, thinking about degrees of freedom and the fact that it's nice to live one dimension higher,", 'start': 2837.424, 'duration': 4.583}, {'end': 2846.39, 'text': 'to have basically have a more interesting surface than flat Euclidean space.', 'start': 2842.007, 'duration': 4.383}, {'end': 2853.555, 'text': 'Okay! With that as our changing scenes abruptly from the side plot of quaternions,', 'start': 2847.29, 'duration': 6.265}, {'end': 2858.898, 'text': "we return back to our regular programming of physics and how it's relevant to imaginary interest rates.", 'start': 2853.555, 'duration': 5.343}, {'end': 2864.462, 'text': 'Locking in our answers here, it looks like most of you correctly answered A.', 'start': 2859.599, 'duration': 4.863}], 'summary': 'Discussion on degrees of freedom and higher dimensions in physics and twitter engagement.', 'duration': 31.241, 'max_score': 2833.221, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI2833221.jpg'}], 'start': 2397.184, 'title': 'Simple harmonic motion and degrees of freedom in rotation', 'summary': "Covers the concept of simple harmonic motion, exploring its relevance in physics, particularly in relation to a spring's motion and its connection to hooke's law and newton's second law. additionally, it delves into the concept of degrees of freedom in rotation, highlighting the need for additional descriptors in 2d and 3d rotations and briefly addressing higher-dimensional spaces and quaternions.", 'chapters': [{'end': 2650.57, 'start': 2397.184, 'title': 'Simple harmonic motion in physics', 'summary': "Discusses the concept of simple harmonic motion, focusing on the relevance of perpendicular force in physics, specifically in the context of a spring's motion, and its relationship with hooke's law and newton's second law, emphasizing the influence of displacement on force and acceleration.", 'duration': 253.386, 'highlights': ["The concept of simple harmonic motion is incredibly relevant to anything that involves what's called simple harmonic motion in physics, such as the motion of a spring. The concept of simple harmonic motion is relevant in physics, especially in the context of a spring's motion.", "Hooke's Law suggests that the force exerted by a spring is proportional to the displacement, with the force being proportional to the size of the displacement. Hooke's Law states that the force exerted by a spring is proportional to the displacement, following a specific proportionality constant.", "Newton's second law states that the force exerted on an object is equal to the mass of the object multiplied by its acceleration, emphasizing the influence of displacement on force and acceleration. Newton's second law highlights the relationship between force, mass, and acceleration, with a focus on the influence of displacement on force and acceleration."]}, {'end': 2853.555, 'start': 2682.679, 'title': 'Degrees of freedom in rotation', 'summary': 'Discusses the concept of degrees of freedom in rotation, explaining the need for extra numbers to describe 2d and 3d rotations and the concept of living in a higher-dimensional space, with a brief mention of quaternions and the potential degrees of freedom for four-dimensional rotations.', 'duration': 170.876, 'highlights': ['Rotation in 2D requires one extra number, I, while 3D rotation requires three extra numbers i, j, k to describe the three degrees of freedom. 3D rotation requires three extra numbers i, j, k to describe the three degrees of freedom.', "Complex numbers describe rotation based on the numbers that sit on a circle, representing one dimension's worth of numbers in some sense. Complex numbers describe rotation based on the numbers that sit on a circle, representing one dimension's worth of numbers.", "For four-dimensional rotations, there would be ten degrees of freedom, but there's no number system for it. Four-dimensional rotations would have ten degrees of freedom, but there's no number system for it.", 'The concept of living in a higher-dimensional space is explained, emphasizing the idea of having a more interesting surface than flat Euclidean space. The concept of living in a higher-dimensional space is explained, emphasizing the idea of having a more interesting surface than flat Euclidean space.']}], 'duration': 456.371, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI2397184.jpg', 'highlights': ["The concept of simple harmonic motion is incredibly relevant to anything that involves what's called simple harmonic motion in physics, such as the motion of a spring.", "Hooke's Law suggests that the force exerted by a spring is proportional to the displacement, with the force being proportional to the size of the displacement.", "Newton's second law states that the force exerted on an object is equal to the mass of the object multiplied by its acceleration, emphasizing the influence of displacement on force and acceleration.", '3D rotation requires three extra numbers i, j, k to describe the three degrees of freedom.', "Complex numbers describe rotation based on the numbers that sit on a circle, representing one dimension's worth of numbers.", 'The concept of living in a higher-dimensional space is explained, emphasizing the idea of having a more interesting surface than flat Euclidean space.']}, {'end': 3785.995, 'segs': [{'end': 2906.017, 'src': 'embed', 'start': 2878.678, 'weight': 3, 'content': [{'end': 2882.121, 'text': 'What is the change in distance per unit time? So you multiply it by the change in time.', 'start': 2878.678, 'duration': 3.443}, {'end': 2890.066, 'text': 'and then the only tricky part is knowing that the velocity changes based on acceleration times, delta, t and acceleration.', 'start': 2882.981, 'duration': 7.085}, {'end': 2894.249, 'text': 'in this case we can work out based on the various equations.', 'start': 2890.066, 'duration': 4.183}, {'end': 2897.251, 'text': "not sure if you can hear, but there's some sirens in the background.", 'start': 2894.249, 'duration': 3.002}, {'end': 2898.992, 'text': "in general whenever i'm recording.", 'start': 2897.251, 'duration': 1.741}, {'end': 2901.534, 'text': "it's so annoying when there's background noise.", 'start': 2898.992, 'duration': 2.542}, {'end': 2906.017, 'text': 'so one of the nice things about live is that i just have no option.', 'start': 2901.534, 'duration': 4.483}], 'summary': 'Calculating velocity based on acceleration and time.', 'duration': 27.339, 'max_score': 2878.678, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI2878678.jpg'}, {'end': 3095.466, 'src': 'embed', 'start': 3059.454, 'weight': 5, 'content': [{'end': 3064.495, 'text': "I'm going to package this pair of numbers as a single, complex number,", 'start': 3059.454, 'duration': 5.041}, {'end': 3068.576, 'text': 'where the real part is the position of the spring and the imaginary part is the velocity.', 'start': 3064.495, 'duration': 4.081}, {'end': 3070.717, 'text': 'Again, totally crazy thing to do.', 'start': 3069.036, 'duration': 1.681}, {'end': 3074.177, 'text': "If that feels weird or you don't understand it, that's fine.", 'start': 3070.897, 'duration': 3.28}, {'end': 3075.658, 'text': "It's not a natural thing.", 'start': 3074.378, 'duration': 1.28}, {'end': 3080.379, 'text': 'But if we play this out and we start walking through the math, hopefully it starts to justify itself.', 'start': 3076.278, 'duration': 4.101}, {'end': 3095.466, 'text': 'Because what I can write is that the change to this state, this complex number, is actually equal to negative i times itself,', 'start': 3081.159, 'duration': 14.307}], 'summary': 'Packaging numbers as complex numbers for spring position and velocity. change is equal to negative i times itself.', 'duration': 36.012, 'max_score': 3059.454, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI3059454.jpg'}, {'end': 3191.076, 'src': 'embed', 'start': 3163.775, 'weight': 4, 'content': [{'end': 3170.16, 'text': "And I could write this even more compactly by saying, let's describe our whole state with a number z.", 'start': 3163.775, 'duration': 6.385}, {'end': 3179.542, 'text': 'The way that z changes is equal to negative i times itself times a small step in time.', 'start': 3171.653, 'duration': 7.889}, {'end': 3188.293, 'text': 'Geometrically, what that looks like is taking wherever you started, multiplying it by negative i, which rotates at 90 degrees,', 'start': 3180.764, 'duration': 7.529}, {'end': 3191.076, 'text': 'so this is a 90 degree angle and then taking a little step.', 'start': 3188.293, 'duration': 2.783}], 'summary': 'Describing state with z, z changes = -i*z*dt, geometrically rotates at 90 degrees.', 'duration': 27.301, 'max_score': 3163.775, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI3163775.jpg'}, {'end': 3317.656, 'src': 'embed', 'start': 3286.832, 'weight': 1, 'content': [{'end': 3291.695, 'text': "this is a spring where we're not taking into account friction, so it'll just oscillate back and forth forever.", 'start': 3286.832, 'duration': 4.863}, {'end': 3294.657, 'text': 'that oscillation corresponds with circular motion.', 'start': 3291.695, 'duration': 2.962}, {'end': 3299.772, 'text': 'And we can see this maybe even more concretely if we write out the math of it.', 'start': 3295.751, 'duration': 4.021}, {'end': 3308.934, 'text': "So what we just saw is that imaginary compound interest invites us to write the solution, the place that we're gonna be after an amount of time,", 'start': 3300.492, 'duration': 8.442}, {'end': 3317.656, 'text': 't as wherever we were in the beginning times e, to the power of that interest rate, where, in this case, the interest rate is negative i.', 'start': 3308.934, 'duration': 8.722}], 'summary': 'In a frictionless spring, oscillation corresponds to circular motion.', 'duration': 30.824, 'max_score': 3286.832, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI3286832.jpg'}, {'end': 3440.019, 'src': 'embed', 'start': 3412.788, 'weight': 2, 'content': [{'end': 3423.929, 'text': 'hopefully it feels more concrete when you realize all that saying is that the position of the spring oscillates back and forth in a nice little cosine wave which matches physical intuition.', 'start': 3412.788, 'duration': 11.141}, {'end': 3426.831, 'text': 'It kind of oscillates back and forth in this gentle smooth wave.', 'start': 3423.969, 'duration': 2.862}, {'end': 3430.993, 'text': "And then similarly, the velocity, let's say it started off at zero.", 'start': 3427.731, 'duration': 3.262}, {'end': 3440.019, 'text': "What our Euler's formula is telling us is that this will end up being a negative sine wave.", 'start': 3432.474, 'duration': 7.545}], 'summary': 'Spring position oscillates in a cosine wave, velocity in a negative sine wave.', 'duration': 27.231, 'max_score': 3412.788, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI3412788.jpg'}, {'end': 3526.428, 'src': 'embed', 'start': 3493.299, 'weight': 0, 'content': [{'end': 3495.981, 'text': "Because this Hooke's law let us describe things at a 90 degree angle.", 'start': 3493.299, 'duration': 2.682}, {'end': 3498.983, 'text': 'And all of that is actually deeper than you might think it is.', 'start': 3496.701, 'duration': 2.282}, {'end': 3508.273, 'text': "There's certain formalizations of classical Newtonian mechanics that introduce complex numbers for pretty similar reasons to what's going on here.", 'start': 3499.544, 'duration': 8.729}, {'end': 3509.734, 'text': "It's all a bit advanced.", 'start': 3508.813, 'duration': 0.921}, {'end': 3511.256, 'text': "I won't necessarily go into it,", 'start': 3509.734, 'duration': 1.522}, {'end': 3518.983, 'text': "but what I want to emphasize is that this original question we asked of what happens if your bank offers you an imaginary interest rate isn't totally insane,", 'start': 3511.256, 'duration': 7.727}, {'end': 3526.428, 'text': "and in fact at least it's insane for money, but it's not insane as an abstract idea to pursue and it corresponds to some very real things.", 'start': 3518.983, 'duration': 7.445}], 'summary': "Hooke's law applies at a 90-degree angle, with connections to complex numbers in classical mechanics, suggesting that an imaginary interest rate isn't entirely absurd and has real-world implications.", 'duration': 33.129, 'max_score': 3493.299, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI3493299.jpg'}, {'end': 3639.461, 'src': 'embed', 'start': 3613.681, 'weight': 8, 'content': [{'end': 3619.985, 'text': "The thing I want you to notice is, when you go through parts one and two of the homework, where you're expanding it out,", 'start': 3613.681, 'duration': 6.304}, {'end': 3622.987, 'text': "and especially in the case of part two, where you're expanding things out,", 'start': 3619.985, 'duration': 3.002}, {'end': 3633.895, 'text': 'to think critically about whether the commutative property x times y equals y times x is needed or relevant.', 'start': 3622.987, 'duration': 10.908}, {'end': 3639.461, 'text': 'okay, because often when we expand out binomial terms, it assumes that this is true,', 'start': 3634.395, 'duration': 5.066}], 'summary': 'When expanding binomial terms, critically evaluate the relevance of commutative property x times y equals y times x.', 'duration': 25.78, 'max_score': 3613.681, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI3613681.jpg'}, {'end': 3726.581, 'src': 'embed', 'start': 3700.923, 'weight': 7, 'content': [{'end': 3705.324, 'text': "For complex numbers this is true, for real numbers this is true, but for matrices that's actually not true.", 'start': 3700.923, 'duration': 4.401}, {'end': 3707.985, 'text': 'and this is one of the reasons.', 'start': 3706.864, 'duration': 1.121}, {'end': 3714.571, 'text': 'by the way, i actually think writing e to the x for this is a bad convention, because you have this bizarre situation where,', 'start': 3707.985, 'duration': 6.586}, {'end': 3718.154, 'text': "when you have matrices and this does come up there's real math.", 'start': 3714.571, 'duration': 3.583}, {'end': 3718.594, 'text': "that's done.", 'start': 3718.154, 'duration': 0.44}, {'end': 3726.581, 'text': 'where you are exponentiating matrices, the notation inspires you to think oh, I should be able to write this.', 'start': 3718.594, 'duration': 7.987}], 'summary': 'Exponentiation rules differ for complex, real numbers, and matrices, challenging conventional notation.', 'duration': 25.658, 'max_score': 3700.923, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI3700923.jpg'}], 'start': 2853.555, 'title': 'Physics and imaginary interest rates', 'summary': 'Explores the relevance of physics to imaginary interest rates through discussions on velocity, displacement, acceleration, circular motion, and exponential functions, emphasizing the deep connection between complex numbers and physical phenomena.', 'chapters': [{'end': 3211.532, 'start': 2853.555, 'title': 'Physics relevance to imaginary interest rates', 'summary': 'Explores the relevance of physics to imaginary interest rates by discussing the relationship between velocity, displacement, and acceleration, and ultimately describes the system using complex numbers.', 'duration': 357.977, 'highlights': ['The change to x is whatever your velocity was, times, a small step in time, which represents the change in distance per unit time. The change to x is determined by velocity multiplied by a small step in time, representing the change in distance per unit time.', 'The acceleration is negative x, leading to a description of the system using complex numbers and a compact representation of the state with a complex number z. The acceleration is described as negative x, which leads to a compact representation of the system using a complex number z.', 'The system is described using complex numbers, with the real part representing the position of the spring and the imaginary part representing the velocity. The system is described using complex numbers, where the real part represents the position of the spring and the imaginary part represents the velocity.']}, {'end': 3526.428, 'start': 3212.333, 'title': 'Circular motion and imaginary compound interest', 'summary': "Explains the relationship between circular motion and imaginary compound interest, connecting the oscillation of a spring to circular motion in a complex plane, showcasing the use of euler's formula, and emphasizing the deep connection between complex numbers and physical phenomena.", 'duration': 314.095, 'highlights': ['The oscillation of a spring corresponds to circular motion in a complex plane, with the x and y components representing the displacement and velocity, respectively, showcasing a deep connection between circular motion and complex numbers.', "The use of Euler's formula demonstrates that the displacement and velocity of the spring can be represented by sine and cosine waves, providing a concrete visualization of the oscillatory behavior of the spring.", 'The concept of imaginary compound interest is connected to the oscillation of a spring through the use of complex numbers, showing that the idea of an imaginary interest rate, while unconventional for money, corresponds to real physical phenomena.', 'The discussion highlights the deeper implications of using complex numbers in describing physical phenomena, emphasizing the advanced formalizations of classical mechanics that introduce complex numbers for similar reasons.', 'The explanation illustrates the deep connection between complex numbers and physical phenomena, showcasing the relevance of abstract mathematical ideas in describing real-world concepts.']}, {'end': 3785.995, 'start': 3527.352, 'title': 'Understanding exponential function and matrices', 'summary': 'Discusses the concept of expanding exponential functions and the relevance of the commutative property in matrices, highlighting the advanced nuances and limitations associated with exponentiating matrices.', 'duration': 258.643, 'highlights': ['The commutative property x times y equals y times x is crucial in expanding binomial terms, especially in the case of part two where the terms end up looking like n choose k, and this prerequisite knowledge is essential for understanding the expansion.', 'The convention of writing e to the x for exponentiating matrices is confusing and not entirely applicable, particularly in cases involving quaternions, revealing the advanced complexities associated with matrices and the limitations of the notation.']}], 'duration': 932.44, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IAEASE5GjdI/pics/IAEASE5GjdI2853555.jpg', 'highlights': ['The explanation illustrates the deep connection between complex numbers and physical phenomena, showcasing the relevance of abstract mathematical ideas in describing real-world concepts.', 'The concept of imaginary compound interest is connected to the oscillation of a spring through the use of complex numbers, showing that the idea of an imaginary interest rate, while unconventional for money, corresponds to real physical phenomena.', 'The oscillation of a spring corresponds to circular motion in a complex plane, with the x and y components representing the displacement and velocity, respectively, showcasing a deep connection between circular motion and complex numbers.', 'The change to x is whatever your velocity was, times, a small step in time, which represents the change in distance per unit time. The change to x is determined by velocity multiplied by a small step in time, representing the change in distance per unit time.', 'The acceleration is negative x, leading to a description of the system using complex numbers and a compact representation of the state with a complex number z.', 'The system is described using complex numbers, with the real part representing the position of the spring and the imaginary part representing the velocity.', 'The discussion highlights the deeper implications of using complex numbers in describing physical phenomena, emphasizing the advanced formalizations of classical mechanics that introduce complex numbers for similar reasons.', 'The convention of writing e to the x for exponentiating matrices is confusing and not entirely applicable, particularly in cases involving quaternions, revealing the advanced complexities associated with matrices and the limitations of the notation.', 'The commutative property x times y equals y times x is crucial in expanding binomial terms, especially in the case of part two where the terms end up looking like n choose k, and this prerequisite knowledge is essential for understanding the expansion.']}], 'highlights': ['The concept of an imaginary interest rate is explored and its relevance to physics and simple harmonic motion is discussed, highlighting the surprising real-world implications of this seemingly nonsensical question.', 'The exploration of continuously compounded growth and the value of e, around 2.71828, is a central theme in the chapter, providing insights into the fundamental constant of nature.', "The concept of simple harmonic motion is incredibly relevant to anything that involves what's called simple harmonic motion in physics, such as the motion of a spring.", 'The explanation illustrates the deep connection between complex numbers and physical phenomena, showcasing the relevance of abstract mathematical ideas in describing real-world concepts.', 'The concept of imaginary compound interest is connected to the oscillation of a spring through the use of complex numbers, showing that the idea of an imaginary interest rate, while unconventional for money, corresponds to real physical phenomena.']}