title

Complex number fundamentals | Ep. 3 Lockdown live math

description

Intro to the geometry complex numbers.
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/1258219199769440257
Errors:
- On the first sketch of a complex plane, there is a "2i" written instead of "-2i".
- At the end, in writing the angle sum identity, the last term should be sin(beta) instead of sin(alpha).
- During Q9, the terms in parentheses should include an i, (1/2 + sqrt(3)/2 i)
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The live question setup with stats on-screen is powered by Itempool.
https://itempool.com/
The graphing calculator used here is Desmos.
https://www.desmos.com/
The "Complex slide rule" came from Geogebra, via Ben Sparks.
https://www.geogebra.org/m/mbhbdvkr
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.
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Video Timeline (thanks to user "Just TIEriffic")
0:00:30 - W3 Results
0:01:00 - W4 Prompt
0:02:00 - Ask What would you call 'imaginary numbers'?
0:06:40 - Startingpoint & assumptions
0:10:25 - W4 Results
0:11:25 - Q1 Prompt
0:12:20 - Q1 Process
0:14:05 - RotatingCoordinates
0:16:40 - Q1 Result
0:17:40 - Q2
0:18:15 - Q3 Prompt
0:19:40 - Q3 Results
0:21:35 - RotationAnimation
0:22:35 - 3 facts about Multiplication
0:25:40 - Q4 Prompt
0:26:10 - Ask imaginary I vs physics i&j
0:28:15 - Q4 Result
0:31:00 - GeoGebraDemo
0:32:10 - Q5 Prompt
0:33:30 - Q5 Results
0:34:00 - Q5 Solution
0:35:55 - RotatingImages Example
0:37:10 - PythonExample
0:38:25 - PythonImage Rotation Example
0:40:35 - Ask Vectors & Matrices for rotation
0:42:40 - Q6 Prompt
0:46:55 - Q6 Results
0:47:25 - Q6 Solution
0:52:20 - RedefiningAngle Addition
0:57:20 - Q7 Prompt
0:57:55 - Ask Can we do without complex numbers?
1:00:10 - Q7 Results
1:00:55 - Q7 Solution
1:05:45 - Q8 Prompt
1:06:30 - Ask sum/difference of angles
1:09:25 - Q8 Results
1:10:25 - Q8 Solution
1:12:00 - DesmosExample
1:15:05 - Bringing it all together
1:16:25 - The "cis" shorthand explained
1:18:05 - Q9 Prompt
1:19:35 - Q9 Results
1:20:55 - ClosingRemarks
------------------
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
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detail

{'title': 'Complex number fundamentals | Ep. 3 Lockdown live math', 'heatmap': [{'end': 2564.132, 'start': 2514.651, 'weight': 0.719}, {'end': 4637.46, 'start': 4586.456, 'weight': 1}], 'summary': 'Series delves into the fundamentals of complex numbers, their applications in trigonometry, engineering, and mathematics, addressing common misconceptions and emphasizing the geometric interpretations and practical uses, with specific insights and quiz results provided.', 'chapters': [{'end': 289.893, 'segs': [{'end': 28.699, 'src': 'embed', 'start': 0.149, 'weight': 0, 'content': [{'end': 4.19, 'text': 'Today we are going to talk about one of my absolute all-time favorite pieces of math.', 'start': 0.149, 'duration': 4.041}, {'end': 9.312, 'text': "It's incredibly fundamental to engineering, to mathematics itself, to quantum mechanics.", 'start': 4.55, 'duration': 4.762}, {'end': 12.053, 'text': "But it's something that has a terrible, terrible name.", 'start': 9.852, 'duration': 2.201}, {'end': 14.034, 'text': 'We call them complex numbers.', 'start': 12.553, 'duration': 1.481}, {'end': 19.055, 'text': 'And worse than that, the things that bring about complex numbers we call imaginary numbers.', 'start': 14.634, 'duration': 4.421}, {'end': 23.637, 'text': 'And before we get into any of it, what I want to do is start with kind of a poll.', 'start': 19.735, 'duration': 3.902}, {'end': 28.699, 'text': 'Just to poll the audience on seeing what you guys consider to be Well, real.', 'start': 24.177, 'duration': 4.522}], 'summary': 'Complex numbers are fundamental to engineering and quantum mechanics, despite their terrible name; the audience is polled on their concept of reality.', 'duration': 28.55, 'max_score': 0.149, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo149.jpg'}, {'end': 84.78, 'src': 'embed', 'start': 60.71, 'weight': 2, 'content': [{'end': 70.494, 'text': 'um, among the values 2 square root of 2, square root of negative 1 and infinity which would you personally consider to really exist,', 'start': 60.71, 'duration': 9.784}, {'end': 72.395, 'text': 'whatever really existing means to you?', 'start': 70.494, 'duration': 1.901}, {'end': 76.677, 'text': 'So, in theory, if you guys go to 3b1b.co.', 'start': 73.515, 'duration': 3.162}, {'end': 79.058, 'text': 'slash live, you should be able to answer this.', 'start': 76.677, 'duration': 2.381}, {'end': 82.519, 'text': 'And then the statistics based on your answers are going to start populating the screen.', 'start': 79.218, 'duration': 3.301}, {'end': 84.78, 'text': "We won't know what those answers refer to.", 'start': 82.919, 'duration': 1.861}], 'summary': 'Discussion on the existence of mathematical values, with live statistics.', 'duration': 24.07, 'max_score': 60.71, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo60710.jpg'}], 'start': 0.149, 'title': 'Fundamental of complex numbers', 'summary': 'Explores the relevance of complex numbers in engineering, mathematics, and quantum mechanics, emphasizing the initial challenge of understanding the concept. it also discusses the subjective concept of the existence of mathematical numbers, including a poll on the existence of 2√2, √-1, and infinity, and a question on renaming complex and imaginary numbers.', 'chapters': [{'end': 60.71, 'start': 0.149, 'title': 'Complex numbers: fundamental in mathematics', 'summary': 'Explores the fundamental concept of complex numbers and their relevance to engineering, mathematics, and quantum mechanics, while also emphasizing the initial challenge of understanding the concept and seeking audience participation.', 'duration': 60.561, 'highlights': ['Complex numbers are fundamental to engineering, mathematics, and quantum mechanics. They play a crucial role in these fields, showcasing their importance and widespread application.', "Challenges with the terminology of 'complex numbers' and 'imaginary numbers' are highlighted. The speaker emphasizes the negative connotation associated with the terminology, setting the stage for the audience's initial challenge in understanding the concept.", "Audience participation is sought through a poll to gauge perceptions of 'real' numbers. The chapter begins with an interactive element, engaging the audience and setting the tone for the upcoming discussion on complex numbers."]}, {'end': 289.893, 'start': 60.71, 'title': 'Existence of mathematical numbers', 'summary': 'Discusses the subjective concept of the existence of mathematical numbers, including a poll on the existence of 2√2, √-1, and infinity, and a question on renaming complex and imaginary numbers for real-world applications.', 'duration': 229.183, 'highlights': ['The chapter discusses the subjective concept of the existence of mathematical numbers, including a poll on the existence of 2√2, √-1, and infinity. poll on the existence of 2√2, √-1, and infinity', 'It includes a question on renaming complex and imaginary numbers for real-world applications. question on renaming complex and imaginary numbers for real-world applications', 'The chapter emphasizes the subjective nature of considering the existence of numbers and the lack of strong consensus in one direction. subjective nature of considering the existence of numbers']}], 'duration': 289.744, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo149.jpg', 'highlights': ['Complex numbers are fundamental to engineering, mathematics, and quantum mechanics. They play a crucial role in these fields, showcasing their importance and widespread application.', "Challenges with the terminology of 'complex numbers' and 'imaginary numbers' are highlighted. The speaker emphasizes the negative connotation associated with the terminology, setting the stage for the audience's initial challenge in understanding the concept.", 'The chapter discusses the subjective concept of the existence of mathematical numbers, including a poll on the existence of 2√2, √-1, and infinity. It includes a question on renaming complex and imaginary numbers for real-world applications.']}, {'end': 725.7, 'segs': [{'end': 337.131, 'src': 'embed', 'start': 309.76, 'weight': 0, 'content': [{'end': 313.681, 'text': 'here I want to talk about two different trigonometric functions,', 'start': 309.76, 'duration': 3.921}, {'end': 320.874, 'text': "and this is kind of the thing that we're going to build to two identities from trigonometry, and I understand that maybe, ooh,", 'start': 313.681, 'duration': 7.193}, {'end': 327.18, 'text': 'these complicated identities from trigonometry is not going to be the best way to lure some people into understanding, oh yeah, complex numbers.', 'start': 320.874, 'duration': 6.306}, {'end': 327.881, 'text': "They're really useful.", 'start': 327.2, 'duration': 0.681}, {'end': 328.742, 'text': "You're really going to love them.", 'start': 327.921, 'duration': 0.821}, {'end': 334.668, 'text': "But I do think it's interesting that you can have a fact that has nothing to do with complex numbers or the square root of negative 1.", 'start': 329.302, 'duration': 5.366}, {'end': 335.589, 'text': "It's just trigonometry.", 'start': 334.668, 'duration': 0.921}, {'end': 337.131, 'text': "It's everything we were talking about last time.", 'start': 335.629, 'duration': 1.502}], 'summary': 'Discussion on building two trigonometric identities, emphasizing their usefulness and relevance.', 'duration': 27.371, 'max_score': 309.76, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo309760.jpg'}, {'end': 417.026, 'src': 'embed', 'start': 391.729, 'weight': 1, 'content': [{'end': 396.992, 'text': "they'll tell you that complex numbers are as real a part of their work and their life as real numbers are.", 'start': 391.729, 'duration': 5.263}, {'end': 400.694, 'text': 'But the starting point looks very strange.', 'start': 397.852, 'duration': 2.842}, {'end': 410.039, 'text': "okay?. When you start introducing this, the very first thing you do is to say assume that there's some number, i, so that i squared is equal to negative one.", 'start': 400.694, 'duration': 9.345}, {'end': 414.805, 'text': "And I think to a lot of students, there's maybe one of two possible reactions that you can have here.", 'start': 411.002, 'duration': 3.803}, {'end': 417.026, 'text': "One is, no, there isn't.", 'start': 415.305, 'duration': 1.721}], 'summary': 'Complex numbers are integral to work and life, with i²=-1. some may react skeptically.', 'duration': 25.297, 'max_score': 391.729, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo391729.jpg'}, {'end': 495.183, 'src': 'embed', 'start': 470.057, 'weight': 5, 'content': [{'end': 475.738, 'text': 'In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory.', 'start': 470.057, 'duration': 5.681}, {'end': 480.659, 'text': "It was meant to make fun of the fact that obviously there is no such answer and it shouldn't be taken as serious math.", 'start': 475.818, 'duration': 4.841}, {'end': 485.701, 'text': 'And then we stuck with that as a convention and we still call them imaginary numbers, which is genuinely absurd.', 'start': 481.159, 'duration': 4.542}, {'end': 488.441, 'text': "But that's not the only weird assumption that we make.", 'start': 486.561, 'duration': 1.88}, {'end': 495.183, 'text': "The second weird thing that you do when you start talking about complex numbers is to say there's not just such a number i,", 'start': 488.781, 'duration': 6.402}], 'summary': "Rene descartes coined the term 'imaginary' for numbers as a derogatory, but we still call them imaginary numbers, which is genuinely absurd.", 'duration': 25.126, 'max_score': 470.057, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo470057.jpg'}, {'end': 531.336, 'src': 'embed', 'start': 506.642, 'weight': 3, 'content': [{'end': 513.806, 'text': "i lives perpendicularly, there's one above and then there's one below, negative i, and you can have negative two, i.", 'start': 506.642, 'duration': 7.164}, {'end': 515.787, 'text': 'you scale it however you want.', 'start': 513.806, 'duration': 1.981}, {'end': 523.47, 'text': "essentially it's proposing that numbers be two-dimensional and that i has a very specific home one unit perpendicular, uh,", 'start': 515.787, 'duration': 7.683}, {'end': 525.531, 'text': 'perpendicularly above the real number line.', 'start': 523.47, 'duration': 2.061}, {'end': 531.336, 'text': 'And okay, if we want to extend our number system, I get it.', 'start': 526.514, 'duration': 4.822}], 'summary': 'The concept of imaginary numbers proposes a two-dimensional number system with i as a unit perpendicular to the real number line.', 'duration': 24.694, 'max_score': 506.642, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo506642.jpg'}, {'end': 671.576, 'src': 'embed', 'start': 641.494, 'weight': 4, 'content': [{'end': 646.738, 'text': 'And interesting, d is the one that says you should consider 2 square root of 2 and negative 1, but not infinity.', 'start': 641.494, 'duration': 5.244}, {'end': 653.823, 'text': "So there's a good contingent of you out there who would just reject infinity as being considered real but are very comfortable with the square root of negative 1..", 'start': 647.198, 'duration': 6.625}, {'end': 654.343, 'text': "That's awesome.", 'start': 653.823, 'duration': 0.52}, {'end': 659.067, 'text': 'And then after that, it looks like c, people who reject the square root of negative 1.', 'start': 654.884, 'duration': 4.183}, {'end': 662.149, 'text': 'Fascinating I actually would have thought that none of them would have come higher than that.', 'start': 659.067, 'duration': 3.082}, {'end': 664.491, 'text': 'None of them is much lower at a.', 'start': 662.99, 'duration': 1.501}, {'end': 671.576, 'text': "Okay, so it looks like we've got a cohort of people who are comfortable with negative one, a large cohort, are uncomfortable with infinity.", 'start': 665.211, 'duration': 6.365}], 'summary': 'A survey revealed preferences: 2√2 & -1>∞, √-1>∞, -1 comfortable, ∞ uncomfortable.', 'duration': 30.082, 'max_score': 641.494, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo641494.jpg'}, {'end': 714.156, 'src': 'embed', 'start': 689.437, 'weight': 6, 'content': [{'end': 697.863, 'text': "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward.", 'start': 689.437, 'duration': 8.426}, {'end': 703.367, 'text': "Addition is actually the least interesting part of this, but it's a good thing to know when you're learning about complex numbers.", 'start': 697.923, 'duration': 5.444}, {'end': 706.369, 'text': "It's definitely one of those operations that you are going to need to know.", 'start': 703.887, 'duration': 2.482}, {'end': 714.156, 'text': "Unfortunately and you can tell by the fact that I'm stalling and what I'm saying here it looks like the question is still not loading completely correctly.", 'start': 707.634, 'duration': 6.522}], 'summary': 'Understanding addition of complex numbers is essential for learning about them.', 'duration': 24.719, 'max_score': 689.437, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo689437.jpg'}], 'start': 291.534, 'title': 'Understanding complex numbers', 'summary': 'Discusses the utility and application of complex numbers in trigonometry, engineering, and serious math, as well as explores the historical context and two-dimensional nature of the number i, and perspectives on the realness of certain numbers.', 'chapters': [{'end': 469.397, 'start': 291.534, 'title': 'Understanding the utility of imaginary numbers', 'summary': 'Discusses the utility and application of complex numbers, particularly in trigonometry, highlighting their relevance and practicality in various fields, such as engineering and serious math.', 'duration': 177.863, 'highlights': ['The utility of complex numbers in trigonometry is demonstrated through the simplification and meaningful interpretation of trigonometric identities, making them less error-prone and more comprehensible.', 'Complex numbers, despite their seemingly abstract nature, are integrated into engineering and serious math as an essential component of work and life.', "The chapter addresses the skepticism and disbelief surrounding the concept of imaginary numbers, highlighting common reactions and viewpoints of students when introduced to the notion of 'i' and its square being equal to -1."]}, {'end': 725.7, 'start': 470.057, 'title': 'Complex numbers: the imaginary and real', 'summary': "Explores the concept of complex numbers, highlighting the historical context of 'imaginary' numbers and the two-dimensional nature of the number i, and delves into the addition of complex numbers and the perspectives of participants on the realness of certain numbers.", 'duration': 255.643, 'highlights': ["The term 'imaginary' for complex numbers was coined by Rene Descartes as a derogatory, yet it has persisted as a convention in mathematics. Rene Descartes coined the term 'imaginary' for complex numbers as a derogatory, which has persistently remained a convention in mathematics.", 'Complex numbers propose a two-dimensional number system with the number i having a specific home one unit perpendicular above the real number line, challenging traditional assumptions about numbers. Complex numbers propose a two-dimensional number system with the number i having a specific home one unit perpendicular above the real number line, challenging traditional assumptions about numbers.', "Participants' perspectives on the realness of certain numbers varied, with some rejecting the concept of infinity as a 'real' number while being comfortable with the square root of negative one. Participants' perspectives on the realness of certain numbers varied, with some rejecting the concept of infinity as a 'real' number while being comfortable with the square root of negative one.", 'The chapter explores the addition of complex numbers and emphasizes its importance in understanding complex numbers. The chapter explores the addition of complex numbers and emphasizes its importance in understanding complex numbers.']}], 'duration': 434.166, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo291534.jpg', 'highlights': ['Complex numbers simplify and interpret trigonometric identities, making them less error-prone and more comprehensible.', 'Complex numbers are integrated into engineering and serious math as an essential component of work and life.', "The chapter addresses skepticism and disbelief surrounding the concept of imaginary numbers, highlighting common reactions and viewpoints of students when introduced to the notion of 'i' and its square being equal to -1.", 'Complex numbers propose a two-dimensional number system with the number i having a specific home one unit perpendicular above the real number line, challenging traditional assumptions about numbers.', "Participants' perspectives on the realness of certain numbers varied, with some rejecting the concept of infinity as a 'real' number while being comfortable with the square root of negative one.", "The term 'imaginary' for complex numbers was coined by Rene Descartes as a derogatory, yet it has persisted as a convention in mathematics.", 'The chapter explores the addition of complex numbers and emphasizes its importance in understanding complex numbers.']}, {'end': 1086.919, 'segs': [{'end': 784.154, 'src': 'embed', 'start': 743.228, 'weight': 0, 'content': [{'end': 750.751, 'text': "If you're moving four units to the right and then one unit up and you want to add the idea of moving two units to the left and then two units up,", 'start': 743.228, 'duration': 7.523}, {'end': 752.372, 'text': 'well, you just do each of those one at a time.', 'start': 750.751, 'duration': 1.621}, {'end': 754.173, 'text': "I'll go ahead and pull out black here.", 'start': 752.392, 'duration': 1.781}, {'end': 759.155, 'text': 'The real part is going to be those four to the right, then minus two to the left.', 'start': 754.873, 'duration': 4.282}, {'end': 760.756, 'text': 'Okay, straightforward enough.', 'start': 759.795, 'duration': 0.961}, {'end': 768.802, 'text': 'And then the imaginary part is gonna be this one unit up, and then these two units up, one plus two, times i.', 'start': 761.236, 'duration': 7.566}, {'end': 770.924, 'text': "So it's that one i plus two i.", 'start': 768.802, 'duration': 2.122}, {'end': 774.586, 'text': 'And then when you work that out, four minus two is two, one plus two is three.', 'start': 770.924, 'duration': 3.662}, {'end': 777.509, 'text': 'A nice simple introduction here.', 'start': 774.606, 'duration': 2.903}, {'end': 781.452, 'text': "Addition doesn't really have anything complicated going on, which is great.", 'start': 777.929, 'duration': 3.523}, {'end': 784.154, 'text': "That means that it's one fewer thing for us to worry about.", 'start': 781.712, 'duration': 2.442}], 'summary': 'Moving 4 right, 1 up, adding 2 left, 2 up, results in 2 right, 3 up.', 'duration': 40.926, 'max_score': 743.228, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo743228.jpg'}, {'end': 831.769, 'src': 'embed', 'start': 803.756, 'weight': 2, 'content': [{'end': 806.857, 'text': 'But the rules end up being very different from that in the number system.', 'start': 803.756, 'duration': 3.101}, {'end': 807.977, 'text': "You can't really do algebra.", 'start': 806.897, 'duration': 1.08}, {'end': 812.558, 'text': "You can't do things like assume that if two numbers multiply to make zero, then one of them has to be zero.", 'start': 808.217, 'duration': 4.341}, {'end': 817.62, 'text': 'But complex numbers are going to end up behaving much like the real numbers, so rules from algebra can carry over.', 'start': 812.979, 'duration': 4.641}, {'end': 820.518, 'text': 'But to understand what that rotation rule is?', 'start': 818.496, 'duration': 2.022}, {'end': 822.4, 'text': "oh no, I'm giving things away.", 'start': 820.518, 'duration': 1.882}, {'end': 823.801, 'text': 'what that multiplication rule is.', 'start': 822.4, 'duration': 1.401}, {'end': 831.769, 'text': 'I just want to ask you a simple question, which is basically suppose I have the point 3, 2..', 'start': 823.801, 'duration': 7.968}], 'summary': 'Complex numbers behave like real numbers in algebraic rules.', 'duration': 28.013, 'max_score': 803.756, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo803756.jpg'}], 'start': 726.46, 'title': 'Coordinate planes and complex numbers', 'summary': 'Introduces addition in coordinate planes with an example showing a final movement of 2 units left and 3 units up. it also explores complex numbers, their multiplication rule, and 90-degree rotation, incorporating student insights.', 'chapters': [{'end': 784.154, 'start': 726.46, 'title': 'Introduction to addition in coordinate planes', 'summary': 'Explains the addition in coordinate planes using a simple example, demonstrating the process of adding movements in x and y directions, resulting in a final movement of 2 units to the left and 3 units up, making it a straightforward concept to grasp.', 'duration': 57.694, 'highlights': ['The addition in coordinate planes involves moving units in the x and y directions, resulting in a final movement of 2 units to the left and 3 units up.', 'The process involves sequentially adding the movements in x and y directions, which simplifies the concept and makes it easier to understand.', 'The real part of the addition entails moving four units to the right and then two units to the left, resulting in a net movement of 2 units to the left.', 'The imaginary part of the addition involves moving one unit up and then two units up, resulting in a net movement of 3 units up.', 'The simplicity of addition in coordinate planes eliminates complexities, making it an easier concept to comprehend.']}, {'end': 1086.919, 'start': 784.474, 'title': 'Complex numbers and rotations', 'summary': 'Explores the concept of complex numbers, their multiplication rule and the 90-degree rotation, demonstrating the application of the rotation rule and grading of complex addition and product with insights from student responses.', 'duration': 302.445, 'highlights': ['The chapter explores the concept of complex numbers, their multiplication rule, and the 90-degree rotation. The discussion delves into the fundamental concepts of complex numbers, their multiplication rule, and the application of the 90-degree rotation to understand the behavior of coordinates under rotation.', 'Demonstrating the application of the rotation rule and grading of complex addition and product with insights from student responses. The instructor demonstrates the application of the rotation rule through the example of rotating the point (3, 2) and evaluates student responses for complex addition and introduces the grading process for complex product.', 'Insights from student responses to complex addition and product. The instructor provides insights into student responses for complex addition and product, highlighting common errors and misunderstandings in the answers submitted by the students.']}], 'duration': 360.459, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo726460.jpg', 'highlights': ['The addition in coordinate planes involves moving units in the x and y directions, resulting in a final movement of 2 units to the left and 3 units up.', 'The process involves sequentially adding the movements in x and y directions, which simplifies the concept and makes it easier to understand.', 'The chapter explores the concept of complex numbers, their multiplication rule, and the 90-degree rotation.', 'The real part of the addition entails moving four units to the right and then two units to the left, resulting in a net movement of 2 units to the left.', 'The imaginary part of the addition involves moving one unit up and then two units up, resulting in a net movement of 3 units up.']}, {'end': 1682.573, 'segs': [{'end': 1126.326, 'src': 'embed', 'start': 1086.919, 'weight': 0, 'content': [{'end': 1089.08, 'text': "can't you make the live questions work a little bit better for us?", 'start': 1086.919, 'duration': 2.161}, {'end': 1091.761, 'text': "Okay, I think we're finally there.", 'start': 1090.02, 'duration': 1.741}, {'end': 1098.062, 'text': 'Everybody ready? Aha! Wonderful! Very simple question.', 'start': 1093.601, 'duration': 4.461}, {'end': 1103.036, 'text': 'I want you to take the number I, and I want you to multiply it by 3 plus 2i.', 'start': 1098.322, 'duration': 4.714}, {'end': 1106.397, 'text': "And even though I haven't really talked about the rules for multiplication,", 'start': 1103.676, 'duration': 2.721}, {'end': 1110.179, 'text': 'what I can say is pretend like it operates just like it does for normal numbers.', 'start': 1106.397, 'duration': 3.782}, {'end': 1113.741, 'text': "You've got things like the distributive property, where you can distribute this throughout.", 'start': 1110.479, 'duration': 3.262}, {'end': 1118.923, 'text': 'And then the defining feature of i is this idea that i squared is negative 1.', 'start': 1114.521, 'duration': 4.402}, {'end': 1120.944, 'text': "That's the only special thing you need to know about that.", 'start': 1118.923, 'duration': 2.021}, {'end': 1126.326, 'text': "Other than that, just treat it like it's a normal number, and then proceed forward with the product.", 'start': 1121.424, 'duration': 4.902}], 'summary': 'Instruction on multiplying complex numbers using i, following distributive property and i squared equals -1.', 'duration': 39.407, 'max_score': 1086.919, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo1086919.jpg'}, {'end': 1272.561, 'src': 'embed', 'start': 1248.861, 'weight': 4, 'content': [{'end': 1257.466, 'text': '3 is the imaginary part, but that 2 got multiplied by a negative 1 because i has this defining feature of squaring to become negative 1..', 'start': 1248.861, 'duration': 8.605}, {'end': 1263.369, 'text': 'So that should give you some indication that, okay, multiplying by i has this action of rotating things by 90 degrees.', 'start': 1257.466, 'duration': 5.903}, {'end': 1272.561, 'text': "Maybe that means that it's not a crazy thing to do, to geometrically position i at a 90 degree angle with the real number line.", 'start': 1264.029, 'duration': 8.532}], 'summary': 'Multiplying by i rotates by 90 degrees, positioning i at a 90 degree angle with the real number line.', 'duration': 23.7, 'max_score': 1248.861, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo1248861.jpg'}, {'end': 1319.439, 'src': 'embed', 'start': 1292.751, 'weight': 3, 'content': [{'end': 1302.335, 'text': "So let's say we have any number z, and in this case, z is gonna be, let's see, where do I have it? Z is gonna be at two plus i, great.", 'start': 1292.751, 'duration': 9.584}, {'end': 1310.358, 'text': "And let's say I wanna understand what is multiplying by z do to every other possible complex number? Well, we can go one by one.", 'start': 1302.735, 'duration': 7.623}, {'end': 1319.439, 'text': "The very first one that's relatively simple is if I ask, what is z times one? Where does it take the number one? Well, Z times 1 is going to be Z.", 'start': 1310.938, 'duration': 8.501}], 'summary': 'Analyzing the effect of multiplying z by 1 on complex numbers.', 'duration': 26.688, 'max_score': 1292.751, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo1292751.jpg'}], 'start': 1086.919, 'title': 'Complex numbers', 'summary': 'Introduces complex numbers multiplication and discusses the concept, geometric representation, and application of complex numbers with emphasis on the significance of multiplying by i and its relation to geometric and mechanistic processes.', 'chapters': [{'end': 1126.326, 'start': 1086.919, 'title': 'Complex numbers multiplication', 'summary': 'Introduces complex numbers multiplication by demonstrating the multiplication of the imaginary unit i by 3+2i using the distributive property and the defining feature of i squared being negative 1.', 'duration': 39.407, 'highlights': ['The defining feature of i is that i squared is negative 1, which is crucial for understanding complex number multiplication.', 'The chapter demonstrates the multiplication of i by 3+2i using the distributive property, providing a foundational understanding of complex number multiplication.', 'The chapter emphasizes treating i as a normal number when multiplying, simplifying the process for learners.']}, {'end': 1682.573, 'start': 1127.026, 'title': 'Understanding complex numbers', 'summary': 'Discusses the concept of complex numbers, their geometric representation, and their application in rotations, emphasizing the significance of multiplying by i and its relation to geometric and mechanistic processes.', 'duration': 555.547, 'highlights': ['Complex numbers and their geometric representation are explained, emphasizing the significance of multiplying by i in rotations. The chapter delves into the geometric interpretation of complex numbers, highlighting the significance of multiplying by i in rotating numbers by 90 degrees and its implication in positioning i at a 90-degree angle with the real number line.', 'The concept of multiplying by a complex number is illustrated through an example, demonstrating its effect on other complex numbers. The presentation provides an example of multiplying a complex number by another, showcasing its impact on rotating and stretching other complex numbers, exemplifying the constrained rule applied to the whole plane.', 'The discussion on multiplying by a complex number is linked to its mechanistic process and geometric intuition, stressing the relationship between the two approaches. The chapter emphasizes the correlation between the geometric rule of rotating and stretching numbers and the mechanistic process of expanding a sum, highlighting the integration of geometric intuition with algebraic manipulation.']}], 'duration': 595.654, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo1086919.jpg', 'highlights': ['The chapter emphasizes treating i as a normal number when multiplying, simplifying the process for learners.', 'The defining feature of i is that i squared is negative 1, crucial for understanding complex number multiplication.', 'The chapter demonstrates the multiplication of i by 3+2i using the distributive property, providing a foundational understanding of complex number multiplication.', 'The discussion on multiplying by a complex number is linked to its mechanistic process and geometric intuition, stressing the relationship between the two approaches.', 'Complex numbers and their geometric representation are explained, emphasizing the significance of multiplying by i in rotations.', 'The concept of multiplying by a complex number is illustrated through an example, demonstrating its effect on other complex numbers.']}, {'end': 2766.345, 'segs': [{'end': 1713.996, 'src': 'embed', 'start': 1683.717, 'weight': 1, 'content': [{'end': 1686.96, 'text': "And I think that's probably been enough time to work through our little bit of algebra here.", 'start': 1683.717, 'duration': 3.243}, {'end': 1690.784, 'text': "So let's go ahead and grade it for everyone to see.", 'start': 1687.461, 'duration': 3.323}, {'end': 1692.906, 'text': 'And it looks like the vast majority of you..', 'start': 1691.104, 'duration': 1.802}, {'end': 1697.59, 'text': "Well, I told you the answer, so I'm going to assume the vast majority of you are giving the correct answer.", 'start': 1692.906, 'duration': 4.684}, {'end': 1698.871, 'text': 'Oh, interesting.', 'start': 1697.61, 'duration': 1.261}, {'end': 1702.495, 'text': 'Okay So it looks like five..', 'start': 1698.991, 'duration': 3.504}, {'end': 1705.61, 'text': "5 is correct and that's the most common answer.", 'start': 1703.989, 'duration': 1.621}, {'end': 1713.996, 'text': 'but then the ones where it missed were 3 and 4, and maybe we can try to understand why 3 and 4 would have been common misconceptions on this one.', 'start': 1705.61, 'duration': 8.386}], 'summary': 'Most common answer for the algebra problem is 5, 3 and 4 were common misconceptions.', 'duration': 30.279, 'max_score': 1683.717, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo1683717.jpg'}, {'end': 1848.575, 'src': 'embed', 'start': 1811.489, 'weight': 2, 'content': [{'end': 1813.59, 'text': 'Very common mistake, no harm in that.', 'start': 1811.489, 'duration': 2.101}, {'end': 1819.714, 'text': 'And then if you answered 4, it comes from forgetting the fact that i squared can be simplified into a real number.', 'start': 1814.05, 'duration': 5.664}, {'end': 1826.238, 'text': 'so that you know that very much counts for the the real component of your final answer.', 'start': 1820.935, 'duration': 5.303}, {'end': 1827.499, 'text': "now, what's interesting, i think,", 'start': 1826.238, 'duration': 1.261}, {'end': 1839.025, 'text': 'is that that process of just walking through the algebra feels very different from this geometric idea that we had before of saying multiplying by z has the action of rotating and stretching things,', 'start': 1827.499, 'duration': 11.526}, {'end': 1848.575, 'text': "and how it rotates and stretches things is in the way that's necessary to get the number one to sit on the number z, And in fact I have a good friend,", 'start': 1839.025, 'duration': 9.55}], 'summary': 'Common mistake: forgetting to simplify i squared. algebraic vs geometric approach.', 'duration': 37.086, 'max_score': 1811.489, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo1811489.jpg'}, {'end': 2021.326, 'src': 'embed', 'start': 1994.144, 'weight': 0, 'content': [{'end': 1998.306, 'text': 'whatever complex number z this is is gonna have the appropriate action.', 'start': 1994.144, 'duration': 4.162}, {'end': 2002.568, 'text': "And then from there, it's all of the trigonometry that you and I were working through last time.", 'start': 1999.086, 'duration': 3.482}, {'end': 2010.251, 'text': "So let's look at our answers and let's go ahead and lock them into place now before we talk through the explanation.", 'start': 2003.108, 'duration': 7.143}, {'end': 2016.782, 'text': 'And the correct answer in this context is d, which is cosine of pi sixth plus i sine of pi sixth.', 'start': 2011.198, 'duration': 5.584}, {'end': 2021.326, 'text': 'Congratulations to the majority of you who correctly answered that.', 'start': 2017.023, 'duration': 4.303}], 'summary': 'Solving for complex number z led to majority choosing d: cosine(pi/6) + i sine(pi/6).', 'duration': 27.182, 'max_score': 1994.144, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo1994144.jpg'}, {'end': 2483.714, 'src': 'embed', 'start': 2454.138, 'weight': 3, 'content': [{'end': 2456.579, 'text': "In 3D graphics, that actually wouldn't work.", 'start': 2454.138, 'duration': 2.441}, {'end': 2459.28, 'text': 'Complex numbers only describe rotations in two dimensions.', 'start': 2456.759, 'duration': 2.521}, {'end': 2465.864, 'text': 'However, if you want to use a number system to describe rotations in three dimensions, instead of using matrices,', 'start': 2460.361, 'duration': 5.503}, {'end': 2470.026, 'text': 'you could use these things called quaternions, which are basically the complex numbers on steroids.', 'start': 2465.864, 'duration': 4.162}, {'end': 2477.11, 'text': "And in fact, they expose you to fewer errors, like there's errors that come about when you try to use matrices for 3D rotation.", 'start': 2470.586, 'duration': 6.524}, {'end': 2483.714, 'text': "This thing called gimbal lock, talk to any roboticist or any computer graphics programmers, they'll tell you that gimbal lock is a real pain.", 'start': 2477.47, 'duration': 6.244}], 'summary': 'Quaternions are used for 3d rotations, reducing errors like gimbal lock.', 'duration': 29.576, 'max_score': 2454.138, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo2454138.jpg'}, {'end': 2564.132, 'src': 'heatmap', 'start': 2514.651, 'weight': 0.719, 'content': [{'end': 2519.794, 'text': 'And this is particularly visceral for me when I think in terms of trig identities.', 'start': 2514.651, 'duration': 5.143}, {'end': 2524.516, 'text': 'So if you remember, at the very beginning, I opened up by talking about one particular trig identity.', 'start': 2520.314, 'duration': 4.202}, {'end': 2527.618, 'text': "And without pulling it up again, I don't want to go back and just read off of it.", 'start': 2524.816, 'duration': 2.802}, {'end': 2532.02, 'text': "I'm gonna ask you one more question, which this is gonna be a tricky question.", 'start': 2528.218, 'duration': 3.802}, {'end': 2534.261, 'text': 'This one will actually take some time to work through.', 'start': 2532.04, 'duration': 2.221}, {'end': 2545.247, 'text': "Which is I want you to calculate for me the cosine of 75 degrees, okay? And let's go ahead and pull this up as a quiz question.", 'start': 2535.462, 'duration': 9.785}, {'end': 2546.807, 'text': "So let's pull up our quiz.", 'start': 2545.867, 'duration': 0.94}, {'end': 2549.609, 'text': "Let's move on to the next question.", 'start': 2548.368, 'duration': 1.241}, {'end': 2554.225, 'text': "Let's cross our fingers and hope that Cam and Eder have not broken anything again.", 'start': 2551.243, 'duration': 2.982}, {'end': 2557.487, 'text': "And that's all it's asking.", 'start': 2555.686, 'duration': 1.801}, {'end': 2564.132, 'text': 'What is the cosine of 75 degrees? Now let me get you started here on how you might think about it.', 'start': 2557.627, 'duration': 6.505}], 'summary': 'The transcript discusses a trigonometry problem asking to calculate the cosine of 75 degrees.', 'duration': 49.481, 'max_score': 2514.651, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo2514651.jpg'}], 'start': 1683.717, 'title': 'Complex numbers applications', 'summary': 'Covers complex number multiplication, misconceptions, and geometric interpretations, including its application in rotating vectors and objects in computer graphics. it also highlights specific correct answers and common mistakes, with 5 being the correct answer for a specific problem and the majority correctly identifying the answer as cosine of pi sixth plus i sine of pi sixth for a 30-degree counterclockwise rotation.', 'chapters': [{'end': 1864.069, 'start': 1683.717, 'title': 'Complex numbers: multiplication and misconceptions', 'summary': 'Discusses the process of multiplying complex numbers, highlighting common misconceptions and emphasizing the correct approach, with 5 being the correct answer for a specific problem and the reasons behind common mistakes.', 'duration': 180.352, 'highlights': ['The correct answer for the specific problem is 5, with the majority of respondents providing the correct answer.', 'Common misconceptions include answering 3 by subtracting 1 instead of subtracting negative 1, and answering 4 by forgetting that i squared can be simplified into a real number.', 'The process of multiplying complex numbers feels different from the geometric idea of rotating and stretching, and a GeoGebra tool was created to illustrate the complex number equivalent of a slide rule.']}, {'end': 2139.519, 'start': 1864.569, 'title': 'Complex numbers & geometry', 'summary': 'Discusses the connection between complex numbers and geometry, including how to use complex numbers to rotate vectors, and the geometric interpretation of the correct complex number for a 30-degree counterclockwise rotation, with the majority of the audience correctly identifying the answer as cosine of pi sixth plus i sine of pi sixth.', 'duration': 274.95, 'highlights': ['The majority of the audience correctly identified the complex number z for a 30-degree counterclockwise rotation as cosine of pi sixth plus i sine of pi sixth.', 'The x component of the complex number z for the 30-degree counterclockwise rotation is the cosine of pi sixth, and the y component is the sine of pi sixth, as derived from trigonometry.', 'The second most common answer for the complex number z, 1 plus pi sixth over i, was interesting as it represents a slight misconception, being close to the correct answer in terms of geometric reasoning, despite being computationally inaccurate.', 'The use of complex numbers to rotate vectors and the alignment of geometric reasoning with truth, even in cases of misconceptions, is highlighted, showcasing the strong connection between complex numbers and geometry.']}, {'end': 2766.345, 'start': 2140.36, 'title': 'Complex numbers in computer graphics', 'summary': 'Discusses the use of complex numbers for rotating objects in computer graphics, demonstrating how complex numbers are used to represent points, the application of complex multiplication for rotation, and the comparison of using complex numbers, vectors/matrices, and quaternions for 2d and 3d rotations.', 'duration': 625.985, 'highlights': ["Complex numbers are used to represent points in computer graphics, and complex multiplication is applied for rotation. The chapter discusses how complex numbers are used to represent points in computer graphics and how complex multiplication is applied for rotation. It demonstrates the process of rotating a 'pie creature' by using complex numbers to represent its coordinates and applying complex multiplication to calculate the new position after rotation.", "Comparison of using complex numbers, vectors/matrices, and quaternions for 2D and 3D rotations. The chapter compares the use of complex numbers, vectors/matrices, and quaternions for 2D and 3D rotations in computer graphics. It explains that complex numbers are suitable for 2D rotations, while quaternions, described as 'complex numbers on steroids,' are used for 3D rotations to avoid issues like gimbal lock associated with matrices.", 'Discussion on the application of complex numbers in trigonometry and the computation of trig identities. The chapter delves into the application of complex numbers in trigonometry and the computation of trig identities. It challenges the audience to calculate the cosine of 75 degrees by using the concept of rotating points and the x-coordinate.']}], 'duration': 1082.628, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo1683717.jpg', 'highlights': ['The majority correctly identified the complex number z for a 30-degree counterclockwise rotation as cosine of pi sixth plus i sine of pi sixth.', 'The correct answer for the specific problem is 5, with the majority of respondents providing the correct answer.', 'Common misconceptions include answering 3 by subtracting 1 instead of subtracting negative 1, and answering 4 by forgetting that i squared can be simplified into a real number.', 'Complex numbers are used to represent points in computer graphics, and complex multiplication is applied for rotation.', 'The process of multiplying complex numbers feels different from the geometric idea of rotating and stretching, and a GeoGebra tool was created to illustrate the complex number equivalent of a slide rule.']}, {'end': 3598.22, 'segs': [{'end': 2819.684, 'src': 'embed', 'start': 2766.345, 'weight': 0, 'content': [{'end': 2776.828, 'text': 'I want you to take a moment to seriously guess what you think the second most common answer is going to be and where exactly the misconception could have taken place,', 'start': 2766.345, 'duration': 10.483}, {'end': 2778.369, 'text': 'or to double check your work.', 'start': 2776.828, 'duration': 1.541}, {'end': 2786.611, 'text': "You very well might be one of those sitting in the misconception territory, which, you know, it's a lot to work out.", 'start': 2778.489, 'duration': 8.122}, {'end': 2789.052, 'text': "There's a lot of different failure points along the way.", 'start': 2786.651, 'duration': 2.401}, {'end': 2805.955, 'text': "Alright, so with that, I'm going to go ahead and lock in the answers.", 'start': 2801.223, 'duration': 4.732}, {'end': 2810.298, 'text': "And if you wanted to still work it through, even if the answer isn't going to be recorded, that's totally fine.", 'start': 2806.395, 'duration': 3.903}, {'end': 2811.619, 'text': "We're going to walk through it in a moment.", 'start': 2810.318, 'duration': 1.301}, {'end': 2816.322, 'text': "But I'm going to lock them in just so that we can see what the correct answer turns out to be.", 'start': 2812.019, 'duration': 4.303}, {'end': 2819.684, 'text': 'And the correct answer is D, which it looks like a majority of you got.', 'start': 2817.022, 'duration': 2.662}], 'summary': 'The majority guessed d as the second most common answer.', 'duration': 53.339, 'max_score': 2766.345, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo2766345.jpg'}, {'end': 2887.667, 'src': 'embed', 'start': 2859.132, 'weight': 4, 'content': [{'end': 2863.274, 'text': 'I know that 45 degrees is a friendly angle for trigonometry, and I know that 30 degrees is.', 'start': 2859.132, 'duration': 4.142}, {'end': 2866.875, 'text': "What I don't have memorized, though, are the addition formulas.", 'start': 2863.694, 'duration': 3.181}, {'end': 2870.817, 'text': "I learned them once, slipped out of my head, definitely don't have them memorized.", 'start': 2867.616, 'duration': 3.201}, {'end': 2873.358, 'text': 'But I do know how to re-derive them on the fly.', 'start': 2871.017, 'duration': 2.341}, {'end': 2879.101, 'text': 'Basically, the idea is that we take the complex number associated with the 45 degree rotation,', 'start': 2874.099, 'duration': 5.002}, {'end': 2887.667, 'text': 'which is gonna be cosine of 45 degrees plus i times the sine of 45 degrees.', 'start': 2879.101, 'duration': 8.566}], 'summary': '45 degrees and 30 degrees are friendly angles for trigonometry, addition formulas not memorized, but can re-derive on the fly.', 'duration': 28.535, 'max_score': 2859.132, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo2859132.jpg'}, {'end': 3163.758, 'src': 'embed', 'start': 3104.62, 'weight': 1, 'content': [{'end': 3112.466, 'text': "And if we pull up the question, I think it's very interesting that the second most common answer, the most common mistake you might say,", 'start': 3104.62, 'duration': 7.846}, {'end': 3114.268, 'text': 'is only different because of that plus sign.', 'start': 3112.466, 'duration': 1.802}, {'end': 3121.734, 'text': 'And if we look at the math, where that came from, as it happens, is the idea that this was a minus sign,', 'start': 3114.548, 'duration': 7.186}, {'end': 3125.217, 'text': 'which came about from the idea that there was an i sitting in front of each sign.', 'start': 3121.734, 'duration': 3.483}, {'end': 3128.7, 'text': 'So the most common misconception came from maybe forgetting to apply.', 'start': 3125.817, 'duration': 2.883}, {'end': 3134.504, 'text': 'the fact that i squared is equal to negative one, at least if you were going about it using complex numbers which,', 'start': 3128.7, 'duration': 5.804}, {'end': 3136.926, 'text': 'from the whole context of the lecture, of course we want to.', 'start': 3134.504, 'duration': 2.422}, {'end': 3141.384, 'text': 'So, this actually corresponds to a much more general formula.', 'start': 3137.867, 'duration': 3.517}, {'end': 3145.026, 'text': "And I think it's edifying to write out that general formula for ourselves,", 'start': 3141.844, 'duration': 3.182}, {'end': 3150.729, 'text': "because anytime that I've forgotten what the sum of two angles in trigonometry should be,", 'start': 3145.026, 'duration': 5.703}, {'end': 3156.853, 'text': "it's totally okay because you can re-derive it relatively quickly if you're comfortable with complex multiplication.", 'start': 3150.729, 'duration': 6.124}, {'end': 3163.758, 'text': 'And the more math that you do, because it shows up all over the place, you actually get very comfortable with complex multiplication, even if you,', 'start': 3156.933, 'duration': 6.825}], 'summary': 'The most common math mistake involves forgetting to apply the fact that i squared is equal to negative one.', 'duration': 59.138, 'max_score': 3104.62, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo3104620.jpg'}, {'end': 3447.216, 'src': 'embed', 'start': 3417.178, 'weight': 7, 'content': [{'end': 3421.08, 'text': "There's a much more standard way to write these, but it's weird, so I don't want to just throw it down there.", 'start': 3417.178, 'duration': 3.902}, {'end': 3423.481, 'text': 'But before we do that,', 'start': 3421.9, 'duration': 1.581}, {'end': 3430.223, 'text': 'I want to illustrate how you can do a little bit of magic when you have this proper intuition for complex numbers and how they multiply.', 'start': 3423.481, 'duration': 6.742}, {'end': 3431.984, 'text': 'So let me pull up our quiz again.', 'start': 3430.903, 'duration': 1.081}, {'end': 3435.187, 'text': "And let's move on to another question.", 'start': 3433.305, 'duration': 1.882}, {'end': 3447.216, 'text': 'And this question asks, which of the following values of Z satisfies Z squared equals I? Okay.', 'start': 3435.207, 'duration': 12.009}], 'summary': 'Illustrating magic with complex numbers, solving z^2 = i.', 'duration': 30.038, 'max_score': 3417.178, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo3417178.jpg'}], 'start': 2766.345, 'title': 'Trigonometry misconceptions and complex number multiplication', 'summary': 'Discusses common misconceptions in trigonometry, highlights the second most common answer, explains complex number associated with trigonometric angles, explains complex number multiplication, highlights the most common mistake and its root cause, the general formula for the sum of two angles in trigonometry, and the significance of complex multiplication in working out rotations.', 'chapters': [{'end': 3079.682, 'start': 2766.345, 'title': 'Trigonometry misconceptions', 'summary': 'Discusses common misconceptions in trigonometry, highlights the second most common answer, and explains the complex number associated with trigonometric angles.', 'duration': 313.337, 'highlights': ['The correct answer to the trigonometry question is D, which the majority of participants got.', 'The second most common answer was C, differing only by a minus sign.', 'The distribution of answers indicates that the question is not obvious and requires deeper understanding.', 'The complex number associated with a 45-degree rotation involves cosine and sine, which can be used to derive addition formulas on the fly.']}, {'end': 3598.22, 'start': 3079.983, 'title': 'Complex number multiplication', 'summary': 'Explains the process of complex number multiplication, highlighting the most common mistake and its root cause, the general formula for the sum of two angles in trigonometry, and the significance of complex multiplication in working out rotations.', 'duration': 518.237, 'highlights': ['The most common mistake in complex number multiplication is due to forgetting to apply the fact that i squared is equal to negative one, with the second most common answer differing by a plus sign. The lecture emphasizes that the second most common mistake in complex number multiplication is only different due to a plus sign, highlighting the significance of remembering that i squared equals negative one.', 'The lecture provides a general formula for the sum of two angles in trigonometry, demonstrating the ease of re-deriving it using complex multiplication, fostering comfort in dealing with complex numbers. The general formula for the sum of two angles in trigonometry is introduced, promoting the re-derivation of trigonometric identities using complex multiplication, enhancing comfort with complex numbers.', 'The lecture underscores the significance of complex multiplication in working out rotations, illustrating the non-trivial relationship between the algebra and geometry of complex multiplication. The lecture highlights the non-trivial relationship between the algebra and geometry of complex multiplication, emphasizing its role in working out rotations.']}], 'duration': 831.875, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo2766345.jpg', 'highlights': ['The correct answer to the trigonometry question is D, which the majority of participants got.', 'The second most common answer was C, differing only by a minus sign.', 'The most common mistake in complex number multiplication is due to forgetting to apply the fact that i squared is equal to negative one, with the second most common answer differing by a plus sign.', 'The distribution of answers indicates that the question is not obvious and requires deeper understanding.', 'The complex number associated with a 45-degree rotation involves cosine and sine, which can be used to derive addition formulas on the fly.', 'The lecture emphasizes that the second most common mistake in complex number multiplication is only different due to a plus sign, highlighting the significance of remembering that i squared equals negative one.', 'The lecture provides a general formula for the sum of two angles in trigonometry, demonstrating the ease of re-deriving it using complex multiplication, fostering comfort in dealing with complex numbers.', 'The lecture underscores the significance of complex multiplication in working out rotations, illustrating the non-trivial relationship between the algebra and geometry of complex multiplication.']}, {'end': 3930.221, 'segs': [{'end': 3625.92, 'src': 'embed', 'start': 3599.585, 'weight': 0, 'content': [{'end': 3604.388, 'text': "So coming back to our quiz here, it looks like, again, we've got a lot of strong consensus, which is good.", 'start': 3599.585, 'duration': 4.803}, {'end': 3607.049, 'text': 'And let me go ahead and grade this.', 'start': 3605.829, 'duration': 1.22}, {'end': 3612.973, 'text': 'All right, so the correct answer is A.', 'start': 3608.99, 'duration': 3.983}, {'end': 3615.374, 'text': "It's root two over two plus root two over two I.", 'start': 3612.973, 'duration': 2.401}, {'end': 3617.295, 'text': 'It looks like the majority of you got that.', 'start': 3615.374, 'duration': 1.921}, {'end': 3625.92, 'text': 'And okay, so very interesting is that the second most common answer was D, that no solution exists without extending beyond the complex numbers.', 'start': 3618.956, 'duration': 6.964}], 'summary': 'Majority got correct answer a: root two over two plus root two over two i.', 'duration': 26.335, 'max_score': 3599.585, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo3599585.jpg'}, {'end': 3724.615, 'src': 'embed', 'start': 3696.007, 'weight': 2, 'content': [{'end': 3701.168, 'text': 'The real part will also include the products of the imaginary parts because of that i squared.', 'start': 3696.007, 'duration': 5.161}, {'end': 3708.27, 'text': "So we're going to have root 2 over 2 times root 2 over 2, which, like we just saw, oh, excuse me, that was supposed to be 2 over 4.", 'start': 3701.188, 'duration': 7.082}, {'end': 3715.933, 'text': "You can tell I'm a little all over the place today, which, you know, just because sometimes complex numbers get me excited.", 'start': 3708.27, 'duration': 7.663}, {'end': 3724.615, 'text': "Root 2 over 2 times root 2 over 2 is going to be another 2 over 4, but this time it's times i times i, so we're going to have a negative in there.", 'start': 3716.613, 'duration': 8.002}], 'summary': 'Discussion on complex numbers, including calculations and a mistake, resulting in a negative value.', 'duration': 28.608, 'max_score': 3696.007, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo3696007.jpg'}, {'end': 3849.731, 'src': 'embed', 'start': 3821.94, 'weight': 1, 'content': [{'end': 3827.282, 'text': 'All right, well, what number is that? We could think of it in terms of sines and cosines if we want.', 'start': 3821.94, 'duration': 5.342}, {'end': 3830.203, 'text': 'You know, this is gonna be cosine of 45 degrees.', 'start': 3827.422, 'duration': 2.781}, {'end': 3835.105, 'text': 'This vertical part is gonna be the sine of 45 degrees.', 'start': 3831.404, 'duration': 3.701}, {'end': 3839.207, 'text': "And it's root two over two for each of them.", 'start': 3837.286, 'duration': 1.921}, {'end': 3844.829, 'text': 'This is something we were working out just by looking at the isosceles right triangle.', 'start': 3839.907, 'duration': 4.922}, {'end': 3849.731, 'text': 'Each one of these is equal to root two over two.', 'start': 3845.509, 'duration': 4.222}], 'summary': 'Explaining sines and cosines using 45 degrees, resulting in root two over two.', 'duration': 27.791, 'max_score': 3821.94, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo3821940.jpg'}], 'start': 3599.585, 'title': 'Complex numbers and square roots', 'summary': 'Discusses quiz results on finding the square root of i, where the majority answered correctly as root two over two plus root two over two i. it also illustrates the process of finding the square root of i by multiplying it by itself, identifying the two square roots using rotation and trigonometry.', 'chapters': [{'end': 3668.469, 'start': 3599.585, 'title': 'Complex numbers quiz and square roots', 'summary': 'Discusses the quiz results on finding the square root of i, where the majority answered correctly as root two over two plus root two over two i, while the second most common answer was d, indicating the complexity of extending beyond complex numbers.', 'duration': 68.884, 'highlights': ['The majority of the participants correctly answered A as the square root of i, which is root two over two plus root two over two I.', 'The second most common answer was D, suggesting the complexity of finding a solution without extending beyond complex numbers.', 'The concept of extending the rabbit hole of square roots beyond i is discussed, highlighting the surprising nature of having i and its implications in solving polynomials.']}, {'end': 3930.221, 'start': 3669.915, 'title': 'Finding square root of i', 'summary': 'Illustrates the process of finding the square root of i by multiplying it by itself, demonstrating the geometric interpretation and identifying the two square roots using rotation and trigonometry.', 'duration': 260.306, 'highlights': ['The geometric interpretation of finding the square root of i is demonstrated, showcasing the use of rotation and trigonometry to identify the two square roots. Geometric interpretation, rotation by 45 degrees, trigonometry', 'The process of finding the square root of i by multiplying it by itself is illustrated, showcasing the real and imaginary parts and the resulting square root. Real and imaginary parts calculation, illustrating the result', 'The concept of square roots in complex numbers is explained, highlighting the existence of two square roots for every number except zero. Existence of two square roots for all numbers except zero']}], 'duration': 330.636, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo3599585.jpg', 'highlights': ['The majority correctly answered root two over two plus root two over two I as the square root of i.', 'Geometric interpretation, rotation by 45 degrees, trigonometry.', 'The process of finding the square root of i by multiplying it by itself is illustrated, showcasing the real and imaginary parts and the resulting square root.']}, {'end': 4315.402, 'segs': [{'end': 3955.553, 'src': 'embed', 'start': 3930.221, 'weight': 0, 'content': [{'end': 3936.965, 'text': "but if you're thinking about the geometry of it, you're able to come to a pretty, uh, a pretty meaningful answer relatively quickly.", 'start': 3930.221, 'duration': 6.744}, {'end': 3940.727, 'text': "All right, what's our question here?", 'start': 3939.787, 'duration': 0.94}, {'end': 3946.31, 'text': 'The equation x cubed equals one has one real number.', 'start': 3942.288, 'duration': 4.022}, {'end': 3947.67, 'text': 'solution x equals one.', 'start': 3946.31, 'duration': 1.36}, {'end': 3949.791, 'text': 'Makes sense, you cube one, you get one.', 'start': 3948.21, 'duration': 1.581}, {'end': 3953.473, 'text': 'Among the complex numbers, there are two more solutions.', 'start': 3950.712, 'duration': 2.761}, {'end': 3955.553, 'text': 'Which of the following is one of them?', 'start': 3953.993, 'duration': 1.56}], 'summary': 'Solving x cubed equals one yields one real and two complex solutions.', 'duration': 25.332, 'max_score': 3930.221, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo3930221.jpg'}, {'end': 4163.867, 'src': 'embed', 'start': 4129.008, 'weight': 2, 'content': [{'end': 4132.29, 'text': 'But the overall answer there is that division is the same as multiplication.', 'start': 4129.008, 'duration': 3.282}, {'end': 4134.111, 'text': "So that's absolutely the right instinct.", 'start': 4132.55, 'duration': 1.561}, {'end': 4140.453, 'text': 'You can derive for yourself the angled, what would you call them, the trigonometric difference formulas if you wanted to.', 'start': 4134.331, 'duration': 6.122}, {'end': 4143.41, 'text': "Great, so I think that's probably enough time.", 'start': 4141.889, 'duration': 1.521}, {'end': 4152.078, 'text': "If we punch back to our quiz and let's tone down the pause and ponder music here, it seems like answers have sort of stopped rolling in.", 'start': 4143.831, 'duration': 8.247}, {'end': 4155.622, 'text': 'So this is as good a time as any to grade things.', 'start': 4152.139, 'duration': 3.483}, {'end': 4163.867, 'text': 'So the correct answer here of the four options that we were given ends up being drumroll please,', 'start': 4156.763, 'duration': 7.104}], 'summary': 'The transcript discusses division, multiplication, and grading a quiz.', 'duration': 34.859, 'max_score': 4129.008, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo4129008.jpg'}, {'end': 4194.934, 'src': 'embed', 'start': 4167.45, 'weight': 3, 'content': [{'end': 4174.316, 'text': "Now I just want you to imagine for a moment that if you didn't know any of the geometry of complex numbers, someone asks you this question,", 'start': 4167.45, 'duration': 6.866}, {'end': 4178.58, 'text': 'and if you were able to spit it back decently quickly, it makes you look like a machine.', 'start': 4174.316, 'duration': 4.264}, {'end': 4187.688, 'text': 'Right?. Because you would imagine that the only way to answer this is to look at each number and then cube it that you have to go through and do the whole first inside,', 'start': 4179, 'duration': 8.688}, {'end': 4194.934, 'text': "outside, last but twice, because you're cubing it for each one of those, and then deduce that answer B turns out to have this property that we want.", 'start': 4187.688, 'duration': 7.246}], 'summary': 'Understanding complex numbers efficiently can make you look like a machine in problem-solving.', 'duration': 27.484, 'max_score': 4167.45, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo4167450.jpg'}, {'end': 4303.813, 'src': 'embed', 'start': 4275.343, 'weight': 1, 'content': [{'end': 4281.646, 'text': 'You can see that the real part is negative 1 half, and then the imaginary part is the square root of 3 over 2.', 'start': 4275.343, 'duration': 6.303}, {'end': 4283.407, 'text': 'So, even before working it out,', 'start': 4281.646, 'duration': 1.761}, {'end': 4294.907, 'text': "I can know with confidence that if I were to go through the painstaking process of taking negative 1 half plus the square root of 3 over 2 and cubing that I'm going to get back 1..", 'start': 4283.407, 'duration': 11.5}, {'end': 4298.57, 'text': "I don't have to go through that process to know that that's going to happen, which is pretty magical.", 'start': 4294.907, 'duration': 3.663}, {'end': 4303.813, 'text': 'This also answers for us a mystery that we had in the last lecture.', 'start': 4299.831, 'duration': 3.982}], 'summary': 'Complex number -1/2 - sqrt(3)/2, when cubed, equals 1.', 'duration': 28.47, 'max_score': 4275.343, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo4275343.jpg'}], 'start': 3930.221, 'title': 'Complex numbers and their applications', 'summary': 'Covers solutions to x cubed equals one, dividing by complex numbers, and the connection between complex numbers and rotations. it also discusses trigonometric difference formulas, geometric interpretation of complex numbers, and complex number cubing using trigonometry, with a specific result mentioned.', 'chapters': [{'end': 4128.428, 'start': 3930.221, 'title': 'Complex numbers and rotations', 'summary': 'Discusses the solutions to the equation x cubed equals one, introduces the concept of dividing by complex numbers, and illustrates the inverse action of rotating complex numbers, emphasizing the connection between complex numbers and rotations.', 'duration': 198.207, 'highlights': ['The equation x cubed equals one has one real number solution x equals one, and among the complex numbers, there are two more solutions. The chapter introduces the equation x cubed equals one, highlighting the one real number solution and two complex number solutions.', 'Dividing by a complex number is the same as multiplying by its multiplicative inverse, and the inverse action of rotating complex numbers is illustrated using the example of rotating alpha the other way. The concept of dividing by complex numbers is explained as the same as multiplying by its multiplicative inverse, and the inverse action of rotating complex numbers is illustrated using an example.', "The connection between complex numbers and exponentials, specifically Euler's formula, is alluded to, but the chapter emphasizes understanding the concept of cis before delving into Euler's formula. The chapter alludes to the connection between complex numbers and exponentials, particularly Euler's formula, but emphasizes the importance of understanding the concept of cis before delving into Euler's formula."]}, {'end': 4315.402, 'start': 4129.008, 'title': 'Complex numbers and trigonometric formulas', 'summary': 'Discusses the derivation of trigonometric difference formulas, the geometric interpretation of complex number properties, and the calculation of complex number cubing using trigonometry, with negative 1 half plus the root three over 2 times i as the correct answer.', 'duration': 186.394, 'highlights': ['The correct answer for the complex number cubing is negative one half plus the root three over two times i, demonstrating the geometric interpretation of complex number properties and the use of trigonometry in calculating complex number cubing.', 'The chapter discusses the derivation of trigonometric difference formulas and the revelation that division is the same as multiplication, which showcases the fundamental principles of trigonometry and complex numbers.', 'The explanation of the geometric interpretation of complex number properties and the calculation of complex number cubing using trigonometry, emphasizing the elegant approach of understanding equations and properties without direct numerical computation.', 'The revelation that the real part is negative 1 half and the imaginary part is the square root of 3 over 2, demonstrating the application of trigonometry in understanding complex number properties and providing a confident prediction of the result without the need for direct computation.', 'The discussion on using trigonometry to calculate complex number cubing and the revelation that the trigonometric difference formulas can be derived, showcasing the practical application and derivation capabilities of trigonometric principles.']}], 'duration': 385.181, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo3930221.jpg', 'highlights': ['The equation x cubed equals one has one real number solution x equals one, and among the complex numbers, there are two more solutions. The chapter introduces the equation x cubed equals one, highlighting the one real number solution and two complex number solutions.', 'The correct answer for the complex number cubing is negative one half plus the root three over two times i, demonstrating the geometric interpretation of complex number properties and the use of trigonometry in calculating complex number cubing.', 'The chapter discusses the derivation of trigonometric difference formulas and the revelation that division is the same as multiplication, which showcases the fundamental principles of trigonometry and complex numbers.', 'The explanation of the geometric interpretation of complex number properties and the calculation of complex number cubing using trigonometry, emphasizing the elegant approach of understanding equations and properties without direct numerical computation.', 'The revelation that the real part is negative 1 half and the imaginary part is the square root of 3 over 2, demonstrating the application of trigonometry in understanding complex number properties and providing a confident prediction of the result without the need for direct computation.']}, {'end': 4930.386, 'segs': [{'end': 4385.146, 'src': 'embed', 'start': 4353.115, 'weight': 0, 'content': [{'end': 4364.904, 'text': 'We know that if you rotate around by theta degrees, if we call that number z, then when we rotate around another theta degrees,', 'start': 4353.115, 'duration': 11.789}, {'end': 4366.806, 'text': 'this new number is z squared.', 'start': 4364.904, 'duration': 1.902}, {'end': 4380.922, 'text': 'And Z is what I was calling CIS of theta, which we could expand out as saying cosine of theta plus i sine of theta.', 'start': 4369.031, 'duration': 11.891}, {'end': 4385.146, 'text': 'And we can work out that square merely algebraically.', 'start': 4382.403, 'duration': 2.743}], 'summary': 'Rotation by theta degrees results in z squared, where z is cis of theta expressed as cosine of theta plus i sine of theta.', 'duration': 32.031, 'max_score': 4353.115, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo4353115.jpg'}, {'end': 4637.46, 'src': 'heatmap', 'start': 4586.456, 'weight': 1, 'content': [{'end': 4594.639, 'text': "I was saying that CIS of alpha is not really standard notation when we're talking about rotating alpha around the unit circle.", 'start': 4586.456, 'duration': 8.183}, {'end': 4599.561, 'text': 'As a reminder, this was my shorthand for cosine plus i sine.', 'start': 4595.62, 'duration': 3.941}, {'end': 4609.063, 'text': "The reason is because what you're always going to see instead through electrical engineering, through quantum mechanics, through math,", 'start': 4601.277, 'duration': 7.786}, {'end': 4612.726, 'text': 'through all sorts of things, is instead the way that this is written is.', 'start': 4609.063, 'duration': 3.663}, {'end': 4614.827, 'text': "Actually, before I show you how it's written..", 'start': 4612.726, 'duration': 2.101}, {'end': 4617.93, 'text': "No, no, I'll just show you how it's written.", 'start': 4614.827, 'duration': 3.103}, {'end': 4620.752, 'text': "It's weird, and we're going to talk about it at length next time.", 'start': 4618.11, 'duration': 2.642}, {'end': 4626.672, 'text': "but you imagine that you're raising something to a power, okay? e to the i times alpha.", 'start': 4622.349, 'duration': 4.323}, {'end': 4630.235, 'text': 'Now this obviously makes a lot of people very uncomfortable.', 'start': 4627.853, 'duration': 2.382}, {'end': 4633.097, 'text': "First of all, if you don't know about e, what is e??", 'start': 4630.575, 'duration': 2.522}, {'end': 4637.46, 'text': 'And then, secondly, what on earth does it mean to raise something to an imaginary power?', 'start': 4633.457, 'duration': 4.003}], 'summary': 'Cis of alpha is not standard notation for rotating alpha around the unit circle; instead, e to the i times alpha is commonly used in various fields like electrical engineering, quantum mechanics, and math.', 'duration': 51.004, 'max_score': 4586.456, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo4586456.jpg'}, {'end': 4688.745, 'src': 'embed', 'start': 4663.114, 'weight': 2, 'content': [{'end': 4669.342, 'text': 'Which means our CIS function has this kind of interesting property that adding the inputs is the same as multiplying the outputs.', 'start': 4663.114, 'duration': 6.228}, {'end': 4676.218, 'text': 'So with that, I want to ask one final quiz question to try to relate this to other functions that you might be familiar with.', 'start': 4670.455, 'duration': 5.763}, {'end': 4679.88, 'text': "And at this point, it might be halfway obvious where we're going with this.", 'start': 4676.578, 'duration': 3.302}, {'end': 4688.745, 'text': "But as the very final question to cue you up for where we're going to go next time and maybe appreciate the relationship at play here and why it's not that crazy,", 'start': 4680.28, 'duration': 8.465}], 'summary': 'Cis function has interesting property: adding inputs equals multiplying outputs.', 'duration': 25.631, 'max_score': 4663.114, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo4663114.jpg'}, {'end': 4781.981, 'src': 'embed', 'start': 4746.164, 'weight': 3, 'content': [{'end': 4748.265, 'text': 'This is math presentation raw.', 'start': 4746.164, 'duration': 2.101}, {'end': 4754.049, 'text': 'Alright, so as we do this it looks like a pretty strong consensus.', 'start': 4748.285, 'duration': 5.764}, {'end': 4757.491, 'text': "I'll give you a little bit more time to submit the answers.", 'start': 4754.489, 'duration': 3.002}, {'end': 4771.913, 'text': "All right, and with that, let's go ahead and take a look.", 'start': 4768.41, 'duration': 3.503}, {'end': 4777.457, 'text': 'So the question is what function has this property, where adding the inputs looks like multiplying by the outputs?', 'start': 4771.933, 'duration': 5.524}, {'end': 4781.981, 'text': 'So 2,700 of you correctly answered that two to the x.', 'start': 4778.038, 'duration': 3.943}], 'summary': '2,700 people correctly answered the function as 2 to the x.', 'duration': 35.817, 'max_score': 4746.164, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo4746164.jpg'}, {'end': 4839.004, 'src': 'embed', 'start': 4806.181, 'weight': 4, 'content': [{'end': 4813.128, 'text': "and when we're thinking in terms of repeated multiplication, this makes a lot of sense, because one of these is saying multiply in a copies of 2,", 'start': 4806.181, 'duration': 6.947}, {'end': 4819.313, 'text': 'then multiply in b copies of 2, and the result should be well a plus b copies of 2.', 'start': 4813.128, 'duration': 6.185}, {'end': 4825.428, 'text': "the rule with logarithms is exactly the opposite, and it's because logarithms are the inverse of multiplication.", 'start': 4819.313, 'duration': 6.115}, {'end': 4839.004, 'text': 'On that front, it looks like taking the log of a plus b ends up being the log of a, sorry, the log of a times b, multiply log of a plus log of b.', 'start': 4825.949, 'duration': 13.055}], 'summary': 'Logarithms are the inverse of multiplication, where log(a + b) = log(a) * log(b)', 'duration': 32.823, 'max_score': 4806.181, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo4806181.jpg'}, {'end': 4875.807, 'src': 'embed', 'start': 4848.5, 'weight': 5, 'content': [{'end': 4853.925, 'text': "It's because it's a very similar property, but it's just getting mixed up whether the outputs are being added or the inputs are.", 'start': 4848.5, 'duration': 5.425}, {'end': 4857.247, 'text': "Now, what you'll notice is, how strange.", 'start': 4854.744, 'duration': 2.503}, {'end': 4868.439, 'text': 'This factor that comes about because 2 to the x is defined as repeated multiplication looks very similar to the rule that we found for complex numbers that sit on the unit circle.', 'start': 4857.767, 'duration': 10.672}, {'end': 4871.762, 'text': 'That multiplying the outputs is the same as adding the inputs.', 'start': 4868.899, 'duration': 2.863}, {'end': 4875.807, 'text': 'And that should be very suggestive that there might be some kind of unifying theory between them.', 'start': 4872.223, 'duration': 3.584}], 'summary': '2 to the x is defined as repeated multiplication, similar to the rule for complex numbers on the unit circle, suggesting a unifying theory.', 'duration': 27.307, 'max_score': 4848.5, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo4848500.jpg'}], 'start': 4316.943, 'title': 'Complex numbers and exponential functions', 'summary': 'Explores the relationship between complex numbers and trigonometric functions, including the effect of squaring a cosine wave on its frequency. it also discusses the property of exponential functions, logarithms, and their relationship to repeated multiplication, with 2,700 respondents correctly identifying the function as two to the x.', 'chapters': [{'end': 4745.744, 'start': 4316.943, 'title': 'Complex numbers and trig identities', 'summary': 'Explores the relationship between complex numbers and trigonometric functions, demonstrating how squaring a cosine wave doubles its frequency and discussing the significance of the cis function. it also introduces the concept of raising something to an imaginary power and its connection to rotating around the unit circle.', 'duration': 428.801, 'highlights': ['The chapter explores the relationship between complex numbers and trigonometric functions, demonstrating how squaring a cosine wave doubles its frequency and discussing the significance of the CIS function. The concept of squaring a cosine wave and its resemblance to doubling its frequency is explored, showcasing the relationship between complex numbers and trigonometric functions.', 'The chapter introduces the concept of raising something to an imaginary power and its connection to rotating around the unit circle. The concept of raising something to an imaginary power and its connection to rotating around the unit circle is introduced, discussing the properties and significance of the CIS function.', 'The chapter discusses the significance of the CIS function and its interesting property that adding the inputs is the same as multiplying the outputs. The significance of the CIS function and its unique property of adding the inputs being equivalent to multiplying the outputs is highlighted, emphasizing its relevance in understanding complex numbers and trigonometric functions.']}, {'end': 4930.386, 'start': 4746.164, 'title': 'Exponential functions and logarithms', 'summary': 'Discusses the property of exponential functions and logarithms where adding the inputs looks like multiplying by the outputs, with 2,700 respondents correctly identifying the function as two to the x, highlighting the relationship between repeated multiplication and the rules of logarithms, and hinting at the unifying theory between complex numbers and exponential functions.', 'duration': 184.222, 'highlights': ['The question regarding the function property was correctly answered by 2,700 respondents as two to the x, showcasing a strong consensus among the participants.', 'The relationship between repeated multiplication and the rules of logarithms is highlighted, demonstrating that the rules of logarithms are the inverse of multiplication.', "The similarity between the property of exponential functions and the rule for complex numbers on the unit circle is mentioned, suggesting a potential unifying theory between them and teasing the discussion of Euler's formula in the next lecture."]}], 'duration': 613.443, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/5PcpBw5Hbwo/pics/5PcpBw5Hbwo4316943.jpg', 'highlights': ['The chapter explores the relationship between complex numbers and trigonometric functions, demonstrating how squaring a cosine wave doubles its frequency and discussing the significance of the CIS function.', 'The chapter introduces the concept of raising something to an imaginary power and its connection to rotating around the unit circle, discussing the properties and significance of the CIS function.', 'The significance of the CIS function and its unique property that adding the inputs is the same as multiplying the outputs is highlighted, emphasizing its relevance in understanding complex numbers and trigonometric functions.', 'The question regarding the function property was correctly answered by 2,700 respondents as two to the x, showcasing a strong consensus among the participants.', 'The relationship between repeated multiplication and the rules of logarithms is highlighted, demonstrating that the rules of logarithms are the inverse of multiplication.', "The similarity between the property of exponential functions and the rule for complex numbers on the unit circle is mentioned, suggesting a potential unifying theory between them and teasing the discussion of Euler's formula in the next lecture."]}], 'highlights': ['Complex numbers are fundamental to engineering, mathematics, and quantum mechanics, showcasing their importance and widespread application.', 'Complex numbers simplify and interpret trigonometric identities, making them less error-prone and more comprehensible.', 'The addition in coordinate planes involves moving units in the x and y directions, simplifying the concept and making it easier to understand.', 'The chapter emphasizes treating i as a normal number when multiplying, simplifying the process for learners.', 'The majority correctly identified the complex number z for a 30-degree counterclockwise rotation as cosine of pi sixth plus i sine of pi sixth.', 'The correct answer to the trigonometry question is D, which the majority of participants got.', 'The majority correctly answered root two over two plus root two over two I as the square root of i.', 'The equation x cubed equals one has one real number solution x equals one, and among the complex numbers, there are two more solutions.', 'The chapter explores the relationship between complex numbers and trigonometric functions, demonstrating how squaring a cosine wave doubles its frequency and discussing the significance of the CIS function.']}