That was outstandigly pedagogical. I think this is incredibly useful after you have done complex numbers in the usual learning path (which means accepting the square-root of one with faith and just by the growing number of consequences). This was a beautiful aproach.
The square root of -1 is not by faith, but by definition. Complex numbers themselves of course, arise from including as solutions of the quadratic equation cases where the expression under the radical is negative. This is how we got complex numbers.
@@danielvonbose557 We extend the field of Reals to complex. C comes when we require a field containing reals, but like you said, has solutions to such equations. Miraculously, the complex field does far better than that : it is algebraically closed.
@Alric Shirsho Halder No, it should depend on which one comes first. I'm critiquing the flaws in our understanding and construction of Mathematics as a whole and not what the current facts are.
Every time someone says complex numbers don't exist or dont matter, i show them this video. I have never seen a better demonstration of not only why they work but also how they are actually required and work properly. It's not just an extra bit we added to solve some specific equations, it's as important and real as the integers or reals. No matter how hard you try to avoid it, you'll keep reinventing i.
THANK YOU! For the past few months, I was trying to find an explanation in the internet on why specifically √-1 is so useful. Most of the time, I get the usual "oh, it's because it is used in formulas in *[insert certain field of science]*" explanation, which doesn't really help or explain why THOSE numbers are used instead of the usual real numbers. It's really the only thing that's holding me back from fully understanding the potential of complex numbers and this video has perfectly helped me guide towards the right direction. :)
This gave me a lot to think about! My way of going about the effectiveness of _i_ lies in the definition of norm of a vector space (taking the square root of an vector dot-product'd with itself). The dot-product of any hypercomplex number is (a+bn)(a-bn) where n² is either -1, 0, or 1 which determines your flavor of hypercomplex number. In the case where n² = -1 you get that the norm is exactly the same as the 2D Euclidean norm, which gives you circles and rotations and all that jazz. I think the math is cooler in my approach, but yours is a much better explanation. It's more intuitive and beginner friendly. Definitely the best SoME2 I've seen so far!
If n² = 0, then the "rotations" instead become translations along a vertical line, with the "unit circle" being a pair of vertical lines at x = ±1. If instead n² = 1, then you get a unit hyperbola and hyperbolic rotations, which are very useful in Special Relativity among other things. Together with the complex numbers with i² = -1, you have access to elliptical geometry, hyperbolic geometry, and flat geometry.
Hey ! I'm playing with all this cool stuff too for some months ^^ It's really cool I think, but I struggle finding a good way to have hyperbolic geometry in 3D, do you know anything about that ?
Exactly this. The mystifying of i is fun for a time and helps us understand the beauty of it all, but it doesn't give further understanding. This was so clearly put and completely convincing. Bravo.
I think there's *one more* really important piece of the puzzle, which appears only in advanced calculus: Complex analyticity, and the integral theorems associated with it. Elementary functions on the real numbers--that is, the "nice" functions we spend most of our time learning about in primary and secondary school--all turn out to be analytic, or almost everywhere analytic, when you extend them to the complex numbers. That is, they are well-approximated by their Taylor series almost everywhere. That in turn means that they satisfy the conditions of certain absurdly strong theorems that make the complex versions really, really convenient to do calculus on. I've been staring at this for decades and it still seems close to wizard magic to me. It also seems to be closely related to a lot of deep facts in particle physics, in a way that I am not sure we entirely understand.
I have no idea about the complex analyticity. Honestly, it is the first time I heard that from your comment. Shamefully, I had a degree from electrical engineering. We used complex numbers billion times. But believe me, no one deeply explained what the complex numbers really are to us. However, I always felt the necessity of the complex numbers different than what we have been taught before. What do you suggest for? Any sources?
I think those explanations would’ve been way too deep into the weeds, the point of the video was to explain why it sees so many practical applications. In fact, going into heavy complex analysis would have made their application to something like Fourier series seem all the weirder imo.
@@maydin34 May I suggest “Visual Complex Analysis” by Tristan Needham, it is a treasure about complex numbers emphasizing the geometrical and visual aspects of complex numbers.
I don't understand why you think anything that you said is close to wizard magic. The functions we usually learn are well behaved. It shouldn't be a surprise that they or their extensions to complex numbers are also well behaved. This is far from being wizard magic. This is pretty normal, pretty expected.
I think another key property that makes complex numbers so useful other than rotations is that they are algebraically closed. They let polynomials have all their roots, enable analytic continuation, usual functions (sqrt, sin, exp, log, ...) have an almost entire domain & range (Picard's little theorem), and generally make math have less "undefined"s. This is also the reason for their discovery, they were discovered as an algebra trick to make things work out. I think rotations make them useful to introduce to a problem, but it's their algebraic necessity that makes them pop up everywhere. Btw great video, and awesome sound quality!
That's true, their initial discovery indeed was motivated by solving polynomials, specifically cubics! (Not sure if you've seen Veritasium's video on the discovery of imaginary numbers but it's absolutely fantastic) However, I would say that even solving polynomials has a geometric interpretation! For instance, take the equation x² = 1. Ignore the solution x=1 for now. You can think of it as saying "rotate something twice to get 1". Well that would be the number -1 since it has a phase of 180º; rotating it twice would get you to 360º=0º which is the number 1. The problem is that real numbers really only have two phases (as I mentioned in the video). If you wanna solve x³ = 1 you'll have no solutions other than x=1 because you need a number that has phase 120º and 240º. Basically, we need a concept of numbers with "phase" so that we can have n solutions to xⁿ = 1. In fact, the solutions to xⁿ = 1 are all equally spaced complex numbers on a unit circle, known as the roots of unity. I would say for most uses of complex numbers I've encountered, the magnitude-and-phase reasoning was the most intuitive. For Fourier transform/series, it's because those are the key properties of waves. For quantum mechanics, it's because two wavefunctions with the same probability amplitude need to be be able to destructively interfere. This all being said, I'm definitely not in the pure math space, so it's possible this interpretation hits a wall. It might be possible to still make the connection, but I would wager at a certain point it becomes the less elegant explanation. There's really no one "true" interpretation, but for at least engineering use cases, I feel more at peace with the geometric interpretation :)
@@RC2357 for both of those applications, though, you are talking about waves, where notions of magnitude and phase are built in. You can always think of multiplication by i as some sort of rotation but I don't know that that is the same as having notions of magnitude and phase. As a simple example, a complex structure on a real vector space is a transformation which squares to minus the identity. But if I haven't got an idea of length in my vector space then there is no idea of magnitude. Applied maths situations often have hidden assumptions built in such as an inner product to define lengths of things and angles between them.
@@RC2357 This was my rationalization of complex numbers too, but doing Linear algebra on an algebraically closed field makes life so much easier. Proofs of several key theorems are much simpler in such fields.
Just thought I'd add a slight nitpick on an otherwise great video. At 9:55 we define the nth root of (1 angle theta), but in reality we need to be a bit careful. For instance, (1 angle 0) and (1 angle 2*pi) are really the same, but their nth roots (as defined here) are different (1 and 1 angle pi, respectively). While I do understand the reason for not going over this, it is definitely a pitfall that can be a bit tricky to handle (basically we have to make the nth root into a multi-valued function and "choose" one of them as a "principal nth root", which is a bit more intricate than it might seem at first glance).
@@AlericResident I'm not sure if I get your point entirely. Do you mean that the field extensions R(i) and R(-i) are the same field C (so there is no natural way to say that either of i or -i is definitely the obvious candidate for the square root of -1)? While this is true, after a choice has been made between i and -i as the "standard" representative of the square root of -1, it is clear that there is a difference between i and -i, so I'm not sure what you're getting at exactly.
I've never learned about complex numbers (not even imaginary ones) but i understood everything in this (until the last part about the fourier series). i think it's my favorite some2 video yet!
Bravo! You have a very good grasp of the phenomenology of complex numbers and a true gift explaining it to others! Keep it up, you are doing a fine job!
I never really got complex numbers until I learned about Geometric Algebra. It has an i (and more) First we have our mutually orthogonal basis vectors x and y. Then we can define any vector as a linear combination of our basis vectors multiplied by some scalars e.g. (ax + by). We also need to define three products: dot (a•b), wedge (a^b) and geometric (ab). If we have two 2D vectors a and b where a = a1x + a2y b = b1x + b2y Dot: a•b = a1x * b1x + a1y * b1y Wedge: a^b = (a1x + a2y) ^ (b1x + b2y) = (a1b1)x^x + (a1b2)x^y + (a2b1)y^x + (a2b2)y^y = (a1b2)x^y + (a2b1)y^x = (a1b2 - a2b1)x^y x^y is a new object called a bivector. This is the basis bivector. In 2D there is only one. Geometric: ab = a•b + a^b The basis vectors square to 1 under the geometric product: xx = x•x + x^x = 1 + 0 = 1 In 2D the basis vectors multiply together under the geometric product to form the basis bivector: xy = x•y + x^y = 0 + x^y = x^y The basis bivector squares to -1: xyxy = -xyyx = -xx = -1 Therefore: xy = x•y + x^y = x^y = i Geometric Algebra concerns itself with objects called multivectors and complex numbers are just a subset of these. In 2D the multivectors consist of 4 components. A scalar, two vectors and a bivector. A complex number is just a multivector with the vector components set to 0. a + bi = a + 0x + 0y + bi Multiplication using the geometric product produces the same results as complex multiplication. Geometric Algebra can be extended to n dimensions. In 3D multivectors have 8 components and the basis trivector fills the role of i there. The three basis bivectors act like quaternions: xyxy = yzyz = zxzx = -1 ii = jj = kk = -1 A vector is a direction with a magnitude A bivector is an area with an orientation A trivector is a volume with an orientation A quadvector is a hyper volume with an orientation And so on... I appreciate this is a bit of a brief description of geometric algebra and I've glossed over a lot of details. For a more in-depth introduction see ua-cam.com/video/60z_hpEAtD8/v-deo.html
Unfortunately, the trivector/pseudoscalar in G(3) doesn't give as nice of a geometric picture of the meaning of a sqrt(-1) as the bivectors. Things become more interesting however when you embed the algebra inside a larger one, turning the odd trivector into an even quadvector. Even elements like quadvectors are now rotations rather than reflections, much like how adding imaginary numbers turns reflections around 0 into rotations of a plane.
@@angeldude101 I've not really explored much beyond G(3). I've mostly been interested in how GA simplifies thinking about quaternions and I've been using rotors in my game projects. I'm still pretty new to GA, but so far it's made a lot more sense to me than the weird combination of linear algebra, matrices and quaternions. I guess I'll say here I have no formal education in maths and have just picked up bits and pieces as I've needed them. I develop CAD software for a living and have a few little game projects on the side.
Wow! Super excellent. In terms of math exposition, this is one of the best I've ever seen. And it's also about a topic that is so widespread and usually confusing, that I think it definitely deserves recognition in SoME2.
1:35 "I think it justifies the mechanics of using complex numbers but to me it doesn't really explain the core properties of those numbers." Thanks. That's the reason why so many explanations aren't at all explanations, you ask "why does a gyroscope stay up as it spins" and they explain "do the math on the vectors." Yeah, I'll know that the math says it will stay up, but I still won't know why. So, thanks a lot for doing the right work.
Wow! This is by far the best explanation of complex numbers I've ever seen, very clearly explained and logically built-up. Great job and thanks for posting.
If anyone has trouble connecting waves and polar coordinates because they look so different, it’s worth emphasizing what magnitude and phase are as numbers. Magnitude goes from 0 to infinity, like vector length or wave height. Phase is a periodic angle that circles from one point back to itself in 1 turn = 2pi radians = 360 degrees. Straightforward for vector angle, and you can see how a wave repeats when you shift by a fixed width, which can be mapped to that 1 turn. Magnitude and phase are a widely useful pair of quantities that can be represented by a pair of more intuitive real numbers. Refactoring the pair into real and imaginary is equivalent *and* a lot less work for addition, and only some more work for multiplication.
I haven't learned anything about complex numbers in school, but this video cleared up everything I was confused about regarding imaginary numbers- and now I feel ready to take them on in a couple weeks. Cheers!
Congratulations! You’ve made a great pedagogical material that’ll be used by many. Can you make another video explaining quaternions in the same manner ? I’m sure many would like get a new perspective on the subject. Thanks.
Thank you for the kind words! While I do plan on making more videos, what I’m most excited about is making a series of videos that are more math + computer science/engineering. It’ll be about bits, gates, integers, rings, Galois fields, and AES encryption. Regarding quaternions though, 3blue1brown made a couple of videos covering the quaternions and they’re really fantastic, I would definitely recommend checking them out!
I've gotten to the point that I no longer see i as sqrt(-1), but rather as either a 90° rotation, or as the fundamental number of elliptical geometry. The exponential meanwhile is defined by its derivative and the power laws a^(x+y) = (a^x)(a^y) and d/dx e^x = e^x. This implies that df/dx where f(x) = e^90°x = 90°f(x). The derivative of the complex exponential is the same as the exponential itself, but rotated by 90°. This instantly brought to mind physics lessons on centripetal forces, where the velocity is always at a right angle to the position. v = 90°x, but velocity is just the derivative of position. Much like centripetal forces, a derivative pointing perpendicular to the position with the same magnitude draws a circle. The function with a derivative proportional to itself is the exponential e^x = exp(x). There are some who prefer to say e^iτ = 1 because it's a full rotation, or e^iπ + 1 = 0 because of some BS about "the 5 most important numbers," but I think e^iπ = -1 is the most beautiful form and it's precisely because of the reasons given in the video. -1 is a reflection around 0, so sqrt(-1) is whatever operation that when done twice gives such a reflection. Translating doesn't work because then you get stuck at 0. When you expand your view to 2 dimensions, suddenly the reflection around 0 no longer looks like a reflection, but is instead a rotation by 180°. Half of a rotation of 180° is 90°, therefore, (90°)^2 = 180° = -1 (And as for why squaring doubles the angle rather than squaring it, I refer you back to a^(x+y) = (a^x)(a^y). This is where the exponential comes back in.) Some other people made comments about geometric algebra, but thinking about reflections as rotations through higher dimensions made me realize something about geometric algebras. Geometric Algebra has various geometric objects, all of which can be used for transformations. The odd grade elements product reflections, while the even grade ones product rotations. At the same time, the even elements of a geometric algebra always make up their own geometric algebra, leading to a nesting of algebras. What's strange is that when going down a level, some even elements get mapped to odd ones; rotations become reflections... or rather it would be strange if you hadn't already thought of complex numbers like in this video. The odd elements that produce reflections can be seen instead as even elements in a higher dimension where they act as rotations instead. This gives a geometric picture of what happens when you embed geometric algebras inside larger ones, or extract smaller ones from the larger ones.
Very nice. When you stated: "And as for why squaring doubles the angle rather than squaring it" I'd like to add the fact that if you convert the angles into radians instead of degrees consider this: (pi/2)^2 = pi^2/4 we know that pi/2 itself is right angle and within the unit circle that generates a right triangle. The area of a right triangle is defined as (b*h)/2. When we take our line y=mx+b and rotate it around the origin (0,0) on the unit circle we know this will always have a length of 1 and this is the hyp of the triangles and it remains constant. What varies here is b and h. We know that b is cos(t) and we know that h is sin(t). So to find the given area of any right triangle within the unit circle it can be calculated by A = (cos(t)*sin(t))/2 where t is the angle of rotation around the circle. We can use this along with the previous and watch the magic take place: A = (cos(pi/2)*sin(pi/2))/ 2 = (0*1)/2 = 0. As this gives the point (0,1) on the unit circle and since the right triangle has no area, this is why the tangent function has a vertical asymptote. This is the result of the phase changes within the sine and cosine functions as the limit of t approaches pi/2. Now let's see what happens when we substitute the squared angle into our area formula. A = (cos(pi^2/4) * sin(cos(pi^2/4))/2 =(-1*0)/2 = 0. Oh now, we have an area of 0 again... this doesn't seem to tell us much or does it? Well in this case let's consider the generated coordinates this time. The point on the circle here is (-1,0). Simply because cos(pi^2/4) = -1 and sin(pi^2/4) = 0. And this is why the the cosine is tightly coupled and related to the dot product. So just as you have said, squaring the angle or rotation does indeed double it. This is just another way to approach it. All of this would have been easier to understand visually via an animated graph though. As a side note at first it appears that A(pi/2) == A(pi^2/4) because both results are equal to 0, however A(pi/2) != A(pi^2/4) not because of its result but because on its internal vector states. In other words, (0,1) != (-1,0). Although the yield the same area of 0. The first is a vertical line and tan(t) here is undefined however the second is a horizontal line and tan(t) here is 0. Also we can consider the slope of the hyp from y=mx+b knowing that m = (y2-y1)/(x2-x1)=dy/dx = sin(t)/cos(t) = tan(t) shows these facts. For the two vectors we have tan(pi/2) = sin(pi/2)/cos(pi/2) = (1/0) = undefined and tan(pi^2/4) = sin(pi^2/4)/cos(pi^2/4) = (0/-1) = 0. So when we look at numbers in general especially values such as 1,2,0,-1, sqrt(-1), sqrt(2), pi, e, etc... they are all related and intertwined. How and why? Simply because 1+1 = 2. Yes it is simple arithmetic at first glance and it's the first mathematical equation we learn when we begin to count, however it's much more than just that. 1+1 = 2 is also a linear transformation both in translation and in scaling. It doesn't appear to imply rotation however 1+1 = 2^1 and thus rotation is there however, it's constant. Also 1+1 = 2 satisfies the Pythagorean Theorem. And knowing that the general equation of a circle is a special case of the Pythagorean Theorem the equation 1+1 = 2 is also a unit circle that is shifted or translated from (0,0) to (1,0). So all of these values as well as all of these formulas are literally embedded within the unit vector that can have some type of transformation applied to it. And we can see this simply because any and all mathematical operators such as +, -, *, /, ^, sqrt, etc... are transformations. And the beauty of it all especially when referring to the complex numbers can be seen within the Mandelbrot Set. And when we apply these concepts to physics, chemistry, nature, etc. is there any other reason why there is so much harmonic bliss when looking at the cosmos, when observing matter at its subatomic levels, etc. The greatest part of it all is that numbers don't actually exist in nature as they are all abstract concepts, they are all products of the mind. Yet the laws of physics and chemistry rely on the laws and axioms of mathematics and nature tends to obey those laws. It takes a consciousness to realize the beauty of numbers and all of their mathematical relationships. For example, take the Fibonacci sequence, the golden ratio, and its spiral and compare that to the Ram's horn. This is why I love and enjoy mathematics, physics, and chemistry. The more I observe it, the more it points to Intelligent Design. And the imagination is an infinite fractal of beauty and that in itself is divine.
@@skilz8098 Something I wanted to bring up when you brought in areas is an alternate way to define angles. While the circumference of a unit circle is τ = 2π, the area of the circle is just π. This doesn't seem like much, but if you've dealt with quaternions, then you may have seen the two-sided rotation concept and how to turn θ radians, you actually need the quaternion exp(qθ/2) for some unit quaternion q to define the axis/plane of rotation. Thinking about things in terms of areas though, that same quaternion will do an area of θ/2; exactly the coefficient used. This can also be used with complex numbers, but it's not really needed, so this probably seems like unnecessary complexity. Where things get interesting is when you start looking at hyperbolic angles. There's no easy way to determine the arc length on the hyperbola for a given hyperbolic angle, but the area enclosed between the given vector and the hyperbola can be found without much issue, so hyperbolic angles are defined as twice this area. There are objects that work in hyperbolic geometry similarly to complex numbers and quaternions in spherical geometry. When applied the same way, with a two-sided rotation, they will again enclose a region above the unit hyperbola of exactly the coefficient given to the exponential. This same setup also works for a third type of object that works in flat geometry. Just a little something I noticed that I found interesting, especially since Geometric Algebra regularly uses all three kinds of "rotation" with the two-sided structure. Outside of abstract geometry, you may have heard of quantum spinors that do strange things when rotating 360°, but really they just follow the same kind of logic as the two-sided quaternion rotations.
@@angeldude101 Nice target points. Yes, I'm a bit familiar with Quaternions. I have done some 3D Graphics / Physics Engine / Animation programming. They are very interesting characters. What really intrigues me on this is another topic that is kind of special. The mathematics behind them is quite complex, yet their abstract concepts are very simple to incorporate into a given programmable language and within computer computations they are quite efficient and fast. Yet simple equations that have less complexity require more computational power. For example, consider the algorithms and time complexities for performing a square root.. even division in some cases. Another example would be that of quantum mechanics / physics. The mathematics behind it is quite complex but yet the abstract of it is relatively simple to write in a given programming language compared to its mathematical constructs. I'm starting to see a pattern here, but this pattern does not fit or represent all cases. For example, addition is simple in mathematical notation and it is very simple and fast to compute in a given language. So it doesn't always pan out. Now on the other hand, a majority of the times, a derivative is typically easier on paper than many integrals. The integrals can become quite a bit more complex. Yet it can be easier to program a definite integral solver than it is to program a derivative solver. Then again this does involve the nature of both the derivative and integral that you're trying to solve. However, if the solvers are to be completely abstract and generic to be able to take in any function or expression to either find its derivative or the integrate it yeah... oh, and the game changes even more when you try to incorporate indefinite integrals. Well, integrals for the most part are just repetitive sums (summations by parts), so it kind of makes sense. Again look how easy it is to code vector and matrix structures and compare that to encoding floating point arithmetic types at the compiler level. Again the thing that appears to be more complex on paper is easier to program and faster in computational instructions than something that appears to be simpler on paper. I'm just wondering if these type of patterns have any relation to the N vs NP problem... Just some fun observations.
Nice video! The way you presented the complex numbers is similar to how I learned about them for the first time. In Brazil we technical courses that can be done alongside high school and work like a middle ground between high school and engineering, in my case I took a course on electrotechnics. In one class we spent three months analyzing single phase RLC circuits using only geometry, thinking about Voltage and Current as vectors and looking how each component changed that vector. The next three months we did the same thing but with complex numbers and they made everything be solved faster, because the whole circuit could be described as a single equation instead of solving one component at a time, and we could apply the same techniques that we learned on a previous class on DC circuits to analyze AC circuits too. So for us complex numbers were more about having a convenient way to work with scaling and rotation than about the square root of -1. We even used the same notation presented in the video for the polar form of complex numbers.
Wonderful work- a compact and compelling presentation. I deal with phasors a lot for work and sometimes it's nice to take the time to sit back and refresh my mind on the philosophy behind the tools.
Bravo! Mathematics is just a language waiting to be translated into plain English. And you are the first one ( amongst many who keep trying ) who did it!
It allows you to take advantage of the nice parts of polar coordinates while not being on polar coordinates and seems to work like sin's and cos's without needing them either. Allows addition and multiplication to do these without all the complexities. Super awesome video
This was really brilliant. I'd always had a bit of a problem getting why it was reasonable to represent complex numbers geometrically (and all that entails, all the way up to Euler's formula) just because we create a definition of i = sqrt(-1). But this video has made it much clearer for me. As you said, if we think of complex numbers as just these polar objects, the i = sqrt(-1) thing emerges naturally, so it must work both ways! I still find the original definition of i slightly more "magical" (creating a square root of a negative number in order to solve certain cubics with real roots), but now I can better reconcile the two approaches in my head in a much more satisfying way. Thanks so much for this clear and concise explanation :)
I’m glad you liked it! I’ll say that the original square root of -1 definition of i also has a more intuitive explanation that I learned a while back thanks to one of Veritasiums video in imaginary numbers. The gist is that completing the square can be thought of as a geometry problem, but to solve it, you need a square that has negative area. That can only happen if the side lengths are imaginary.
Congratulations for a brilliant exposition! I hope you are planning to do more like this. It should be of great help to many students who want to understand the “why and how” of things. Many thanks for your effort.
Thanks! I'm glad you enjoyed it :) I actually am planning to make more videos! It'll be a series on integers in various levels of computing. It'll cover transistors/gates, C code, groups/rings/fields, and then the AES cipher
I've been doing a masters program for quantum science and technology (effectively anything to do with quantum computing). It's a lot of physics with complex numbers, as the amplitude of the state vectors are complex numbers and you can time evolve the state by using a matrix exponential to generate a phase shift and amplitude change on the state vector. I've been having trouble reconciling parts of the intuition for a lot of it and seeing how you approached this topic in this video has given me a much stronger intuition for what's really happening with these mathematical tools. Thanks for the video! Keep up the good work.
I remember learning a "trick" with using real and imaginary parts to make integration of trigonometric functions really easy, this video kind of explains why that technique is so effective.
Wow. This was very, very nice. Gives such an intuitive and natural grounding to complex numbers. For me the key takeaway is that we encounter a lot of polar objects in nature when one looks deeper. And complex numbers arise naturally in our attempt to describe these polar objects using numbers and in our attempt to work out how these polar objects add together / operate on each other in nature. The power of complex numbers then seems to follow logically/ intuitively from the fact that they describe nature better/ more fully than real numbers (because they try to describe polar objects). Very neat.
"We encounter a lot of polar objects in nature when one looks deeper, and complex numbers arise naturally in our attempt to describe these polar objects using numbers" I like that phrasing a lot, it's quite elegant!
Isomorphism may be the word you are looking for. There are many physical systems with an algebra isomorphic to the algebra of complex numbers. Complex numbers are in turn isomorphic to 2x2 matrices of the form a, b; -b, a. The square root of -1 is unknown, but the square roots of -1, 0; 0, -1 are 0, 1; -1, 0 and 0, -1; 1, 0 with the usual rule about matrix multiplication. This is practically useful.
I was actually was originally gonna talk about the isomorphism to matrices of the form [a, b; -b, a]. What's neat about that approach is you can show that matrices of that form can be decomposed into the product of a scaling and a rotation matrix, thus every complex number is a scaling and a rotation in disguise. The problem was that the script was running _way_ too long. A big part of that was I didn't wanna just drop in isomorphisms without explaining them. Then I was also left with the dilemma of do I actually prove the isomorphism or just say it's true? The latter saves time but also gets rid of any intuition (unless you've already encountered these topics before).
Excellent video! As an electronics engineer I think it's a shame that people find the use of complex numbers in quantum physics so mysterious. It's only because people weren't taught complex numbers the way did here. You would experience something similar in signal processing system if you limited the type of measurements you could do on it. You could have a system that require complex numbers to predict the outcome but where the only thing you actually measure are real number, and there's nothing magic about it. It's just that the signals or wave functions, have a phase relative to something else.. but we can't measure that phase directly.
This is an amazing video. I was always under the impression that the square root of -1 has useful properties that can be used in many situations. But your video showed me the square root of -1 can be easily discovered if you think about the polar geometry. Great stuff! I would love to see more videos from you
I think there is a pretty reasonable path from solving x^2 = -1 type problems to complex numbers to their uses, but it does make it all seem like an accident (which I think it was historically). However, to get a sense for why they are useful, I think the way this video approaches them, almost from a design point of view, is really enlightening. Though, crucially, we're not designing the cartesian from of complex numbers, but the polar form. That part is even more interesting to me. Starting from the polar form as the more fundamental representation, and working backwards.
I've thought about things like this before. Once I started with thinking 'what if we came up with multiplication first, when would we be forced to invent addition' and 'if we started with the complex numbers, but no initial notion of the reals, naturals or integers, how would we discover them' and things like that. (For example, rotations given by e^{i\theta} wrap every 2\pi, so behave like the real line factored by 2\pi\Z, and in turn the reals arise naturally as magnitudes of complex numbers).
I love the presentation. It is always fun to play with the framing of a subject, and some times what comes out on the other end is better, or at least complementary to the standard framing. I do wish you had at least acknowledged the obvious and glaring issue with writing sqrt(-1), though (and n-th roots in general). A 180 degree rotation has TWO DISTINCT halvings.
Ah true, (-i)² = -1. Not gonna lie I kinda just forgot about that while making the video haha. You're right though, i should specifically be defined as the *positive* square root of -1; or geometrically, the number with a phase of 90º. And then -i is the *negative* square root of -1, aka a number with a phase of -90º.
Magnitude, phase, scaling, rotating are adequately represented by Euler's representation if you consider it in the form re^(i * theta) where r is the magnitude of the complex number. Easily explains all of the properties.
Hi! Thank you very much for this great video! I have a pretty good question. Let's say the we have two waves, and the second one has a phase. How do I know that the amplitude of the wave which is the sum of the two waves is equal to the sum of the two complex numbers representing each wave?
I think the way you derived i is roughly the same i got taught in second class of engineering school almost 25+ years ago. Funnily enough i didn't understand the question you formulated in the video title correct😉. We learned to use them for calculation of alternating currents and complex wave forms (FFT and stuff). As you showed they are the best way of depicting/visualizing waves. So i never asked myself why they are usefull, i only asked myself how was someone intelligent enough to come up with these solutions/find that out...
Very good, sir. As a graduate of electrical and electronic engineering, I like my math to arise naturally from first principles. The first occurrence of i, and the hint of complex numbers, comes out of solving the good old right triangle. And we all prefer a more intuitive understanding of how such an object arises and what it means, in a real sense. We live in a three dimensional world and that the focus of scaling and rotation as the fundamental perspective of complex numbers does, indeed, seem to be a natural, intuitive approach. The fact of mathematics is it begins from the absolute and fundamental fact of nature that 1 meter + 1 meter = 2 meters, or 1 rock + 1 rock = 2 rocks. As long as further laws and derivations expand upon this real fundamental, it seems to suggest that all of mathematics is a real representation of some physical process. And the fundamental fact of nature is that we live in a 3D world, not one of singular dimension. Indeed, we don't really seem to live in one of negative direction. A photon, an electron, each from their own perspective, only move forward. A change in direction is due to a rotation of direction. Everything exists in a spherical universe of it's own center. It is as if, at the point where SQRT(-1) crops up, is an indicator that something has gone awry in our understanding. Like encountering the results of Special Relativity, that the scale of time and space are not "linear" and requires a more expansive perspective, it seems that SQRT(-1) necessitates a fundamental shift in perspective. The genius is in finding the right perspective. Below, 馬赫特 points us towards Geometric Algebra. As I have never learned about the Wedge, I think there may be something there.
Brilliantly explained. My early maths training perhaps followed others with algebraic equations being primary. Then you get to complex numbers when solving x^2+1=0, but algebraically the answer has little interpretation. Looking at it through geometry is a far better approach, more intuitive and practical. Love it 🙂
Another random thought is that there are two distinct ways to get from 1 to -1: multiplication by e^{i\pi}, and also z \mapsto e^{-i\pi/2}\conj{z}e^{i\pi/2} (i.e. conjugating, in the group sense, complex conjugation with a 90° rotation). With the complex numbers, we can distinguish between these two ways of getting 1 to -1, but if we only consider the real line, these two transformations are indistinguishable.
Good video. I like the fact that you stuck to the trigonometric formulas for the polar representation of complex numbers instead of bringing out Euler's formula, which wouldn't really make sense to a class full of students who've never seen Taylor series before. What I do think this video was missing was a bit of information on the rectangular representation of complex numbers. The polar representation seems advantageous when dealing with multiplication and division, but rectangular representation really wins out on addition and subtraction, since they are extremely straightforward and can directly be compared against the sum of binomials with constants. Also, I'd like to put out the fact that the product of two complex numbers (a+bi)(c+di) gives (ac-bd)+(bc+ad)i, the real part of which is the formula for the determinant of a 2x2 matrix. I think it's worth saying that determinants are closely tied to cross products, which are commonly used to represent rotations and twisting motions (for example, in torque). Even though the polar representation of complex numbers gives us a geometric interpretation that is inherently tied to circular motions (angles, turning around a circle, arcs), I believe it's worth saying that the rectangular representation can provide us with this connection to bending motions too.
It seem that the squre root of -1 only arises because we chose the basis in which we write our polar objects to be orthogonal. So asking why calculations with i make things simple is the same thing as asking why using orthogonal bases makes calculations easier. Thanks for the video, it was excellent and very clear.
The specific complex sinusoid I graphed was e^(it). This sinusoid has a frequency of 1 radian per second, aka 1/(2π) Hertz. The period is then 2π seconds. In the graph, the unit of the time axis is still seconds, but I tick-marked the axis in multiples of π because then visually the period is a whole number of ticks (specifically it's 2 ticks). If I had tick-marked the time axis as multiples of 1 second (i.e. 1, 2, 3, etc), then the period would look to be about 6.28 seconds, which in my opinion doesn't look as tidy.
Fascinating how you came to √-1 without any mysterious operations. It makes me think that to some super-being it would seem equally natural from the other direction, where √-1 would obviously be for magnitudes perpendicular to the number line. Makes me wonder whether there are equivalent "portholes" to third and higher dimensions from the real numbers.
You do have the quaternions that extend the dimensions to 4. They're a consistent number system with multiplication and addition, and the complex numbers are a subset of that system.
![The most beautiful equation in math, explained visually [Euler’s Formula]](http://i.ytimg.com/vi/f8CXG7dS-D0/mqdefault.jpg)
![The most beautiful equation in math, explained visually [Euler’s Formula]](/img/tr.png)
That was outstandigly pedagogical. I think this is incredibly useful after you have done complex numbers in the usual learning path (which means accepting the square-root of one with faith and just by the growing number of consequences). This was a beautiful aproach.
I think √-1 = -1
The square root of -1 is not by faith, but by definition. Complex numbers themselves of course, arise from including as solutions of the quadratic equation cases where the expression under the radical is negative. This is how we got complex numbers.
@@danielvonbose557 We extend the field of Reals to complex. C comes when we require a field containing reals, but like you said, has solutions to such equations. Miraculously, the complex field does far better than that : it is algebraically closed.
@@user_2793 yes if you mean with respect to the operations of addition/subtraction, multiplication/division then this is so complex in, complex out.
@Alric Shirsho Halder No, it should depend on which one comes first. I'm critiquing the flaws in our understanding and construction of Mathematics as a whole and not what the current facts are.
Every time someone says complex numbers don't exist or dont matter, i show them this video. I have never seen a better demonstration of not only why they work but also how they are actually required and work properly. It's not just an extra bit we added to solve some specific equations, it's as important and real as the integers or reals.
No matter how hard you try to avoid it, you'll keep reinventing i.
THANK YOU! For the past few months, I was trying to find an explanation in the internet on why specifically √-1 is so useful. Most of the time, I get the usual "oh, it's because it is used in formulas in *[insert certain field of science]*" explanation, which doesn't really help or explain why THOSE numbers are used instead of the usual real numbers.
It's really the only thing that's holding me back from fully understanding the potential of complex numbers and this video has perfectly helped me guide towards the right direction. :)
9:00 I like that insight: The important idea of complex numbers is the connection between negation and rotation.
This gave me a lot to think about!
My way of going about the effectiveness of _i_ lies in the definition of norm of a vector space (taking the square root of an vector dot-product'd with itself). The dot-product of any hypercomplex number is (a+bn)(a-bn) where n² is either -1, 0, or 1 which determines your flavor of hypercomplex number. In the case where n² = -1 you get that the norm is exactly the same as the 2D Euclidean norm, which gives you circles and rotations and all that jazz.
I think the math is cooler in my approach, but yours is a much better explanation. It's more intuitive and beginner friendly. Definitely the best SoME2 I've seen so far!
If n² = 0, then the "rotations" instead become translations along a vertical line, with the "unit circle" being a pair of vertical lines at x = ±1. If instead n² = 1, then you get a unit hyperbola and hyperbolic rotations, which are very useful in Special Relativity among other things. Together with the complex numbers with i² = -1, you have access to elliptical geometry, hyperbolic geometry, and flat geometry.
@@angeldude101 yup!
Hey ! I'm playing with all this cool stuff too for some months ^^ It's really cool I think, but I struggle finding a good way to have hyperbolic geometry in 3D, do you know anything about that ?
Exactly this. The mystifying of i is fun for a time and helps us understand the beauty of it all, but it doesn't give further understanding. This was so clearly put and completely convincing. Bravo.
I think there's *one more* really important piece of the puzzle, which appears only in advanced calculus: Complex analyticity, and the integral theorems associated with it. Elementary functions on the real numbers--that is, the "nice" functions we spend most of our time learning about in primary and secondary school--all turn out to be analytic, or almost everywhere analytic, when you extend them to the complex numbers. That is, they are well-approximated by their Taylor series almost everywhere. That in turn means that they satisfy the conditions of certain absurdly strong theorems that make the complex versions really, really convenient to do calculus on. I've been staring at this for decades and it still seems close to wizard magic to me. It also seems to be closely related to a lot of deep facts in particle physics, in a way that I am not sure we entirely understand.
I have no idea about the complex analyticity. Honestly, it is the first time I heard that from your comment. Shamefully, I had a degree from electrical engineering. We used complex numbers billion times. But believe me, no one deeply explained what the complex numbers really are to us. However, I always felt the necessity of the complex numbers different than what we have been taught before. What do you suggest for? Any sources?
I think those explanations would’ve been way too deep into the weeds, the point of the video was to explain why it sees so many practical applications. In fact, going into heavy complex analysis would have made their application to something like Fourier series seem all the weirder imo.
@@maydin34 May I suggest “Visual Complex Analysis” by Tristan Needham, it is a treasure about complex numbers emphasizing the geometrical and visual aspects of complex numbers.
@@amr1fahmy Thank you.
I don't understand why you think anything that you said is close to wizard magic. The functions we usually learn are well behaved. It shouldn't be a surprise that they or their extensions to complex numbers are also well behaved. This is far from being wizard magic. This is pretty normal, pretty expected.
This is by far the best math video I have seen… Please make more
I think another key property that makes complex numbers so useful other than rotations is that they are algebraically closed. They let polynomials have all their roots, enable analytic continuation, usual functions (sqrt, sin, exp, log, ...) have an almost entire domain & range (Picard's little theorem), and generally make math have less "undefined"s.
This is also the reason for their discovery, they were discovered as an algebra trick to make things work out.
I think rotations make them useful to introduce to a problem, but it's their algebraic necessity that makes them pop up everywhere.
Btw great video, and awesome sound quality!
That's true, their initial discovery indeed was motivated by solving polynomials, specifically cubics! (Not sure if you've seen Veritasium's video on the discovery of imaginary numbers but it's absolutely fantastic)
However, I would say that even solving polynomials has a geometric interpretation! For instance, take the equation x² = 1. Ignore the solution x=1 for now. You can think of it as saying "rotate something twice to get 1". Well that would be the number -1 since it has a phase of 180º; rotating it twice would get you to 360º=0º which is the number 1. The problem is that real numbers really only have two phases (as I mentioned in the video). If you wanna solve x³ = 1 you'll have no solutions other than x=1 because you need a number that has phase 120º and 240º. Basically, we need a concept of numbers with "phase" so that we can have n solutions to xⁿ = 1. In fact, the solutions to xⁿ = 1 are all equally spaced complex numbers on a unit circle, known as the roots of unity.
I would say for most uses of complex numbers I've encountered, the magnitude-and-phase reasoning was the most intuitive. For Fourier transform/series, it's because those are the key properties of waves. For quantum mechanics, it's because two wavefunctions with the same probability amplitude need to be be able to destructively interfere.
This all being said, I'm definitely not in the pure math space, so it's possible this interpretation hits a wall. It might be possible to still make the connection, but I would wager at a certain point it becomes the less elegant explanation. There's really no one "true" interpretation, but for at least engineering use cases, I feel more at peace with the geometric interpretation :)
@@RC2357 for both of those applications, though, you are talking about waves, where notions of magnitude and phase are built in.
You can always think of multiplication by i as some sort of rotation but I don't know that that is the same as having notions of magnitude and phase. As a simple example, a complex structure on a real vector space is a transformation which squares to minus the identity. But if I haven't got an idea of length in my vector space then there is no idea of magnitude. Applied maths situations often have hidden assumptions built in such as an inner product to define lengths of things and angles between them.
@@RC2357 This was my rationalization of complex numbers too, but doing Linear algebra on an algebraically closed field makes life so much easier. Proofs of several key theorems are much simpler in such fields.
Just thought I'd add a slight nitpick on an otherwise great video. At 9:55 we define the nth root of (1 angle theta), but in reality we need to be a bit careful. For instance, (1 angle 0) and (1 angle 2*pi) are really the same, but their nth roots (as defined here) are different (1 and 1 angle pi, respectively). While I do understand the reason for not going over this, it is definitely a pitfall that can be a bit tricky to handle (basically we have to make the nth root into a multi-valued function and "choose" one of them as a "principal nth root", which is a bit more intricate than it might seem at first glance).
In other words, Galois Theory gives more insight ;). i and -i are indistinguishable.
@@AlericResident I'm not sure if I get your point entirely. Do you mean that the field extensions R(i) and R(-i) are the same field C (so there is no natural way to say that either of i or -i is definitely the obvious candidate for the square root of -1)? While this is true, after a choice has been made between i and -i as the "standard" representative of the square root of -1, it is clear that there is a difference between i and -i, so I'm not sure what you're getting at exactly.
I've never learned about complex numbers (not even imaginary ones) but i understood everything in this (until the last part about the fourier series). i think it's my favorite some2 video yet!
1 2 3… is a complex number
Bravo! You have a very good grasp of the phenomenology of complex numbers and a true gift explaining it to others! Keep it up, you are doing a fine job!
you have no idea of how long i've been waiting for a video like this!! excellent
I never really got complex numbers until I learned about Geometric Algebra. It has an i (and more)
First we have our mutually orthogonal basis vectors x and y. Then we can define any vector as a linear combination of our basis vectors multiplied by some scalars e.g. (ax + by). We also need to define three products: dot (a•b), wedge (a^b) and geometric (ab).
If we have two 2D vectors a and b where
a = a1x + a2y
b = b1x + b2y
Dot:
a•b = a1x * b1x + a1y * b1y
Wedge:
a^b = (a1x + a2y) ^ (b1x + b2y)
= (a1b1)x^x + (a1b2)x^y + (a2b1)y^x + (a2b2)y^y
= (a1b2)x^y + (a2b1)y^x
= (a1b2 - a2b1)x^y
x^y is a new object called a bivector. This is the basis bivector. In 2D there is only one.
Geometric:
ab = a•b + a^b
The basis vectors square to 1 under the geometric product:
xx = x•x + x^x
= 1 + 0
= 1
In 2D the basis vectors multiply together under the geometric product to form the basis bivector:
xy = x•y + x^y
= 0 + x^y
= x^y
The basis bivector squares to -1:
xyxy = -xyyx
= -xx
= -1
Therefore:
xy = x•y + x^y = x^y = i
Geometric Algebra concerns itself with objects called multivectors and complex numbers are just a subset of these. In 2D the multivectors consist of 4 components. A scalar, two vectors and a bivector. A complex number is just a multivector with the vector components set to 0.
a + bi = a + 0x + 0y + bi
Multiplication using the geometric product produces the same results as complex multiplication.
Geometric Algebra can be extended to n dimensions. In 3D multivectors have 8 components and the basis trivector fills the role of i there. The three basis bivectors act like quaternions:
xyxy = yzyz = zxzx = -1
ii = jj = kk = -1
A vector is a direction with a magnitude
A bivector is an area with an orientation
A trivector is a volume with an orientation
A quadvector is a hyper volume with an orientation
And so on...
I appreciate this is a bit of a brief description of geometric algebra and I've glossed over a lot of details. For a more in-depth introduction see ua-cam.com/video/60z_hpEAtD8/v-deo.html
This sounds interesting, thanks for the brief description and the link :)
Unfortunately, the trivector/pseudoscalar in G(3) doesn't give as nice of a geometric picture of the meaning of a sqrt(-1) as the bivectors. Things become more interesting however when you embed the algebra inside a larger one, turning the odd trivector into an even quadvector. Even elements like quadvectors are now rotations rather than reflections, much like how adding imaginary numbers turns reflections around 0 into rotations of a plane.
@@angeldude101 I've not really explored much beyond G(3). I've mostly been interested in how GA simplifies thinking about quaternions and I've been using rotors in my game projects. I'm still pretty new to GA, but so far it's made a lot more sense to me than the weird combination of linear algebra, matrices and quaternions. I guess I'll say here I have no formal education in maths and have just picked up bits and pieces as I've needed them. I develop CAD software for a living and have a few little game projects on the side.
@@Hector-bj3ls I actually have no formal math education either. :P Just a hobbyist programmer who loves learning cool stuff.
@@angeldude101 More power to you!
Wow! Super excellent. In terms of math exposition, this is one of the best I've ever seen. And it's also about a topic that is so widespread and usually confusing, that I think it definitely deserves recognition in SoME2.
1:35 "I think it justifies the mechanics of using complex numbers but to me it doesn't really explain the core properties of those numbers."
Thanks. That's the reason why so many explanations aren't at all explanations, you ask "why does a gyroscope stay up as it spins" and they explain "do the math on the vectors."
Yeah, I'll know that the math says it will stay up, but I still won't know why.
So, thanks a lot for doing the right work.
Wow! This is by far the best explanation of complex numbers I've ever seen, very clearly explained and logically built-up. Great job and thanks for posting.
Cool, now I need the complex reason of why real numbers are useful
This is how I was taught complex numbers in high school, and I've kind of missed it since then. Getting a refresher was great!
This video gave such a clear explanation of complex numbers in a way that I could almost fundamentally understand! Phenomenal work!
If anyone has trouble connecting waves and polar coordinates because they look so different, it’s worth emphasizing what magnitude and phase are as numbers. Magnitude goes from 0 to infinity, like vector length or wave height. Phase is a periodic angle that circles from one point back to itself in 1 turn = 2pi radians = 360 degrees. Straightforward for vector angle, and you can see how a wave repeats when you shift by a fixed width, which can be mapped to that 1 turn. Magnitude and phase are a widely useful pair of quantities that can be represented by a pair of more intuitive real numbers. Refactoring the pair into real and imaginary is equivalent *and* a lot less work for addition, and only some more work for multiplication.
Having Just submitted my Thesis for computer science, I love seeing fresh or new ways to look at the concepts of mathematics. Great Video.
I haven't learned anything about complex numbers in school, but this video cleared up everything I was confused about regarding imaginary numbers- and now I feel ready to take them on in a couple weeks. Cheers!
8:45 you got me, I subbed because that kind of insight was never explained to me so clearly and simply
lovely!
Congratulations! You’ve made a great pedagogical material that’ll be used by many. Can you make another video explaining quaternions in the same manner ? I’m sure many would like get a new perspective on the subject. Thanks.
Thank you for the kind words!
While I do plan on making more videos, what I’m most excited about is making a series of videos that are more math + computer science/engineering. It’ll be about bits, gates, integers, rings, Galois fields, and AES encryption.
Regarding quaternions though, 3blue1brown made a couple of videos covering the quaternions and they’re really fantastic, I would definitely recommend checking them out!
Thanks so much dude! This kind of in-depth content is GOLD and I absolutely love it🙏
I've gotten to the point that I no longer see i as sqrt(-1), but rather as either a 90° rotation, or as the fundamental number of elliptical geometry. The exponential meanwhile is defined by its derivative and the power laws a^(x+y) = (a^x)(a^y) and d/dx e^x = e^x. This implies that df/dx where f(x) = e^90°x = 90°f(x). The derivative of the complex exponential is the same as the exponential itself, but rotated by 90°. This instantly brought to mind physics lessons on centripetal forces, where the velocity is always at a right angle to the position. v = 90°x, but velocity is just the derivative of position. Much like centripetal forces, a derivative pointing perpendicular to the position with the same magnitude draws a circle. The function with a derivative proportional to itself is the exponential e^x = exp(x).
There are some who prefer to say e^iτ = 1 because it's a full rotation, or e^iπ + 1 = 0 because of some BS about "the 5 most important numbers," but I think e^iπ = -1 is the most beautiful form and it's precisely because of the reasons given in the video. -1 is a reflection around 0, so sqrt(-1) is whatever operation that when done twice gives such a reflection. Translating doesn't work because then you get stuck at 0. When you expand your view to 2 dimensions, suddenly the reflection around 0 no longer looks like a reflection, but is instead a rotation by 180°. Half of a rotation of 180° is 90°, therefore, (90°)^2 = 180° = -1 (And as for why squaring doubles the angle rather than squaring it, I refer you back to a^(x+y) = (a^x)(a^y). This is where the exponential comes back in.)
Some other people made comments about geometric algebra, but thinking about reflections as rotations through higher dimensions made me realize something about geometric algebras. Geometric Algebra has various geometric objects, all of which can be used for transformations. The odd grade elements product reflections, while the even grade ones product rotations. At the same time, the even elements of a geometric algebra always make up their own geometric algebra, leading to a nesting of algebras. What's strange is that when going down a level, some even elements get mapped to odd ones; rotations become reflections... or rather it would be strange if you hadn't already thought of complex numbers like in this video. The odd elements that produce reflections can be seen instead as even elements in a higher dimension where they act as rotations instead. This gives a geometric picture of what happens when you embed geometric algebras inside larger ones, or extract smaller ones from the larger ones.
Very nice. When you stated: "And as for why squaring doubles the angle rather than squaring it" I'd like to add the fact that if you convert the angles into radians instead of degrees consider this: (pi/2)^2 = pi^2/4 we know that pi/2 itself is right angle and within the unit circle that generates a right triangle. The area of a right triangle is defined as (b*h)/2. When we take our line y=mx+b and rotate it around the origin (0,0) on the unit circle we know this will always have a length of 1 and this is the hyp of the triangles and it remains constant. What varies here is b and h. We know that b is cos(t) and we know that h is sin(t). So to find the given area of any right triangle within the unit circle it can be calculated by A = (cos(t)*sin(t))/2 where t is the angle of rotation around the circle. We can use this along with the previous and watch the magic take place: A = (cos(pi/2)*sin(pi/2))/ 2 = (0*1)/2 = 0. As this gives the point (0,1) on the unit circle and since the right triangle has no area, this is why the tangent function has a vertical asymptote. This is the result of the phase changes within the sine and cosine functions as the limit of t approaches pi/2. Now let's see what happens when we substitute the squared angle into our area formula. A = (cos(pi^2/4) * sin(cos(pi^2/4))/2 =(-1*0)/2 = 0. Oh now, we have an area of 0 again... this doesn't seem to tell us much or does it? Well in this case let's consider the generated coordinates this time. The point on the circle here is (-1,0). Simply because cos(pi^2/4) = -1 and sin(pi^2/4) = 0. And this is why the the cosine is tightly coupled and related to the dot product. So just as you have said, squaring the angle or rotation does indeed double it. This is just another way to approach it. All of this would have been easier to understand visually via an animated graph though. As a side note at first it appears that A(pi/2) == A(pi^2/4) because both results are equal to 0, however A(pi/2) != A(pi^2/4) not because of its result but because on its internal vector states. In other words, (0,1) != (-1,0). Although the yield the same area of 0. The first is a vertical line and tan(t) here is undefined however the second is a horizontal line and tan(t) here is 0. Also we can consider the slope of the hyp from y=mx+b knowing that m = (y2-y1)/(x2-x1)=dy/dx = sin(t)/cos(t) = tan(t) shows these facts. For the two vectors we have tan(pi/2) = sin(pi/2)/cos(pi/2) = (1/0) = undefined and tan(pi^2/4) = sin(pi^2/4)/cos(pi^2/4) = (0/-1) = 0.
So when we look at numbers in general especially values such as 1,2,0,-1, sqrt(-1), sqrt(2), pi, e, etc... they are all related and intertwined. How and why? Simply because 1+1 = 2. Yes it is simple arithmetic at first glance and it's the first mathematical equation we learn when we begin to count, however it's much more than just that. 1+1 = 2 is also a linear transformation both in translation and in scaling. It doesn't appear to imply rotation however 1+1 = 2^1 and thus rotation is there however, it's constant. Also 1+1 = 2 satisfies the Pythagorean Theorem. And knowing that the general equation of a circle is a special case of the Pythagorean Theorem the equation 1+1 = 2 is also a unit circle that is shifted or translated from (0,0) to (1,0). So all of these values as well as all of these formulas are literally embedded within the unit vector that can have some type of transformation applied to it. And we can see this simply because any and all mathematical operators such as +, -, *, /, ^, sqrt, etc... are transformations. And the beauty of it all especially when referring to the complex numbers can be seen within the Mandelbrot Set. And when we apply these concepts to physics, chemistry, nature, etc. is there any other reason why there is so much harmonic bliss when looking at the cosmos, when observing matter at its subatomic levels, etc. The greatest part of it all is that numbers don't actually exist in nature as they are all abstract concepts, they are all products of the mind. Yet the laws of physics and chemistry rely on the laws and axioms of mathematics and nature tends to obey those laws. It takes a consciousness to realize the beauty of numbers and all of their mathematical relationships. For example, take the Fibonacci sequence, the golden ratio, and its spiral and compare that to the Ram's horn. This is why I love and enjoy mathematics, physics, and chemistry. The more I observe it, the more it points to Intelligent Design. And the imagination is an infinite fractal of beauty and that in itself is divine.
@@skilz8098 Something I wanted to bring up when you brought in areas is an alternate way to define angles. While the circumference of a unit circle is τ = 2π, the area of the circle is just π. This doesn't seem like much, but if you've dealt with quaternions, then you may have seen the two-sided rotation concept and how to turn θ radians, you actually need the quaternion exp(qθ/2) for some unit quaternion q to define the axis/plane of rotation. Thinking about things in terms of areas though, that same quaternion will do an area of θ/2; exactly the coefficient used. This can also be used with complex numbers, but it's not really needed, so this probably seems like unnecessary complexity.
Where things get interesting is when you start looking at hyperbolic angles. There's no easy way to determine the arc length on the hyperbola for a given hyperbolic angle, but the area enclosed between the given vector and the hyperbola can be found without much issue, so hyperbolic angles are defined as twice this area. There are objects that work in hyperbolic geometry similarly to complex numbers and quaternions in spherical geometry. When applied the same way, with a two-sided rotation, they will again enclose a region above the unit hyperbola of exactly the coefficient given to the exponential. This same setup also works for a third type of object that works in flat geometry.
Just a little something I noticed that I found interesting, especially since Geometric Algebra regularly uses all three kinds of "rotation" with the two-sided structure. Outside of abstract geometry, you may have heard of quantum spinors that do strange things when rotating 360°, but really they just follow the same kind of logic as the two-sided quaternion rotations.
@@angeldude101 Nice target points. Yes, I'm a bit familiar with Quaternions. I have done some 3D Graphics / Physics Engine / Animation programming. They are very interesting characters. What really intrigues me on this is another topic that is kind of special. The mathematics behind them is quite complex, yet their abstract concepts are very simple to incorporate into a given programmable language and within computer computations they are quite efficient and fast. Yet simple equations that have less complexity require more computational power. For example, consider the algorithms and time complexities for performing a square root.. even division in some cases.
Another example would be that of quantum mechanics / physics. The mathematics behind it is quite complex but yet the abstract of it is relatively simple to write in a given programming language compared to its mathematical constructs.
I'm starting to see a pattern here, but this pattern does not fit or represent all cases. For example, addition is simple in mathematical notation and it is very simple and fast to compute in a given language. So it doesn't always pan out. Now on the other hand, a majority of the times, a derivative is typically easier on paper than many integrals. The integrals can become quite a bit more complex. Yet it can be easier to program a definite integral solver than it is to program a derivative solver. Then again this does involve the nature of both the derivative and integral that you're trying to solve. However, if the solvers are to be completely abstract and generic to be able to take in any function or expression to either find its derivative or the integrate it yeah... oh, and the game changes even more when you try to incorporate indefinite integrals. Well, integrals for the most part are just repetitive sums (summations by parts), so it kind of makes sense.
Again look how easy it is to code vector and matrix structures and compare that to encoding floating point arithmetic types at the compiler level. Again the thing that appears to be more complex on paper is easier to program and faster in computational instructions than something that appears to be simpler on paper.
I'm just wondering if these type of patterns have any relation to the N vs NP problem...
Just some fun observations.
The best complex number description youtube I've came across yet, thanks
Spot on!👊
Nice video! The way you presented the complex numbers is similar to how I learned about them for the first time.
In Brazil we technical courses that can be done alongside high school and work like a middle ground between high school and engineering, in my case I took a course on electrotechnics.
In one class we spent three months analyzing single phase RLC circuits using only geometry, thinking about Voltage and Current as vectors and looking how each component changed that vector.
The next three months we did the same thing but with complex numbers and they made everything be solved faster, because the whole circuit could be described as a single equation instead of solving one component at a time, and we could apply the same techniques that we learned on a previous class on DC circuits to analyze AC circuits too.
So for us complex numbers were more about having a convenient way to work with scaling and rotation than about the square root of -1. We even used the same notation presented in the video for the polar form of complex numbers.
Thanks. Never understood this before. Very well explained.
Great video. I have not seen your definition of the multiplication transformation before. Du you introduce this transformation here?
Beautiful. This is a great video to show to anyone who feels fine about polar coordinates and objects but doesn't feel great about complex numbers.
This is one of the single greatest videos ever made
You possess wonderful insight on this fascinating topic. Complex numbers don't seem to be such an elusive concept now!
Probably the best video on the subject I've ever seen.
Outstanding. Thank you. This is very insightful and very helpful.
Wonderful work- a compact and compelling presentation.
I deal with phasors a lot for work and sometimes it's nice to take the time to sit back and refresh my mind on the philosophy behind the tools.
Bravo!
Mathematics is just a language waiting to be translated into plain English. And you are the first one ( amongst many who keep trying ) who did it!
It allows you to take advantage of the nice parts of polar coordinates while not being on polar coordinates and seems to work like sin's and cos's without needing them either. Allows addition and multiplication to do these without all the complexities. Super awesome video
This was really brilliant. I'd always had a bit of a problem getting why it was reasonable to represent complex numbers geometrically (and all that entails, all the way up to Euler's formula) just because we create a definition of i = sqrt(-1).
But this video has made it much clearer for me. As you said, if we think of complex numbers as just these polar objects, the i = sqrt(-1) thing emerges naturally, so it must work both ways!
I still find the original definition of i slightly more "magical" (creating a square root of a negative number in order to solve certain cubics with real roots), but now I can better reconcile the two approaches in my head in a much more satisfying way.
Thanks so much for this clear and concise explanation :)
I’m glad you liked it! I’ll say that the original square root of -1 definition of i also has a more intuitive explanation that I learned a while back thanks to one of Veritasiums video in imaginary numbers.
The gist is that completing the square can be thought of as a geometry problem, but to solve it, you need a square that has negative area. That can only happen if the side lengths are imaginary.
Congratulations for a brilliant exposition! I hope you are planning to do more like this. It should be of great help to many students who want to understand the “why and how” of things. Many thanks for your effort.
Thanks! I'm glad you enjoyed it :)
I actually am planning to make more videos! It'll be a series on integers in various levels of computing. It'll cover transistors/gates, C code, groups/rings/fields, and then the AES cipher
A video that perfectly explains the subject covered in 1 year of university in 17 minutes.
Dude, wtf!! This was the most intuitive way iof why we use complex numbers in my Electrical Engineering classes!!!!
My man...what's unreasonably effective is the lettuce. And its insistence on exploring all degrees of rotation. Impressive. Unreasonably impressive.
I've been doing a masters program for quantum science and technology (effectively anything to do with quantum computing). It's a lot of physics with complex numbers, as the amplitude of the state vectors are complex numbers and you can time evolve the state by using a matrix exponential to generate a phase shift and amplitude change on the state vector. I've been having trouble reconciling parts of the intuition for a lot of it and seeing how you approached this topic in this video has given me a much stronger intuition for what's really happening with these mathematical tools. Thanks for the video! Keep up the good work.
You just cleared my year doubt about why complex numbers are necessary for quantum mechanics!
In 10 minutes!!Woah.
Thanks a lot❤🙏
I remember learning a "trick" with using real and imaginary parts to make integration of trigonometric functions really easy, this video kind of explains why that technique is so effective.
this is the best explanation for complex numbers I've seen so far
Wow. This was very, very nice.
Gives such an intuitive and natural grounding to complex numbers.
For me the key takeaway is that we encounter a lot of polar objects in nature when one looks deeper. And complex numbers arise naturally in our attempt to describe these polar objects using numbers and in our attempt to work out how these polar objects add together / operate on each other in nature. The power of complex numbers then seems to follow logically/ intuitively from the fact that they describe nature better/ more fully than real numbers (because they try to describe polar objects).
Very neat.
"We encounter a lot of polar objects in nature when one looks deeper, and complex numbers arise naturally in our attempt to describe these polar objects using numbers"
I like that phrasing a lot, it's quite elegant!
Wow, you really have the talent to teach
Isomorphism may be the word you are looking for. There are many physical systems with an algebra isomorphic to the algebra of complex numbers. Complex numbers are in turn isomorphic to 2x2 matrices of the form a, b; -b, a. The square root of -1 is unknown, but the square roots of -1, 0; 0, -1 are 0, 1; -1, 0 and 0, -1; 1, 0 with the usual rule about matrix multiplication. This is practically useful.
I was actually was originally gonna talk about the isomorphism to matrices of the form [a, b; -b, a]. What's neat about that approach is you can show that matrices of that form can be decomposed into the product of a scaling and a rotation matrix, thus every complex number is a scaling and a rotation in disguise.
The problem was that the script was running _way_ too long. A big part of that was I didn't wanna just drop in isomorphisms without explaining them. Then I was also left with the dilemma of do I actually prove the isomorphism or just say it's true? The latter saves time but also gets rid of any intuition (unless you've already encountered these topics before).
Excellent video! As an electronics engineer I think it's a shame that people find the use of complex numbers in quantum physics so mysterious. It's only because people weren't taught complex numbers the way did here. You would experience something similar in signal
processing system if you limited the type of measurements you could do on it. You could have a system that require complex numbers to predict the outcome but where the only thing you actually measure are real number, and there's nothing magic about it. It's just that the signals or wave functions, have a phase relative to something else.. but we can't measure that phase directly.
Incredible video. Exactly what I was looking foe
An=sgn
14:43
QED
This is an amazing video. I was always under the impression that the square root of -1 has useful properties that can be used in many situations. But your video showed me the square root of -1 can be easily discovered if you think about the polar geometry. Great stuff! I would love to see more videos from you
Incredible video. Congratulations
A fresh perspective. Thanx.
Throwback to circuit analysis as an electrical engineering undergraduate! Great explanations of i (j as I like it 😊) and phasers!!!
I completely agree with everything you pointed out. Cool!
This is genuinely so good
I think there is a pretty reasonable path from solving x^2 = -1 type problems to complex numbers to their uses, but it does make it all seem like an accident (which I think it was historically). However, to get a sense for why they are useful, I think the way this video approaches them, almost from a design point of view, is really enlightening. Though, crucially, we're not designing the cartesian from of complex numbers, but the polar form. That part is even more interesting to me. Starting from the polar form as the more fundamental representation, and working backwards.
Outstanding exposition! Subscribed.
You are a great speaker and teacher.
I've thought about things like this before. Once I started with thinking 'what if we came up with multiplication first, when would we be forced to invent addition' and 'if we started with the complex numbers, but no initial notion of the reals, naturals or integers, how would we discover them' and things like that. (For example, rotations given by e^{i\theta} wrap every 2\pi, so behave like the real line factored by 2\pi\Z, and in turn the reals arise naturally as magnitudes of complex numbers).
Whoa dude! This explanation alone could win the Fields Medal! I enjoyed your presentation so much my bro. Keep it up!👏
Great video, and to think, you managed to convey all that without writing anything down at all!
I really like the spirit of the video. I hope you continue to make videos. Thank you.
Holy Molly, i ve never felt so excited to learn Math in a long time.
10:29 that was beautiful! Thank you for this video, it clarified a lot.
This is a topic which has been on my mind for a while. Great video!!
Great explanation.
You nailed good BGM. a lot other youtubers use toooooo loud BGM that I cannot hear speaking.
I love the presentation. It is always fun to play with the framing of a subject, and some times what comes out on the other end is better, or at least complementary to the standard framing.
I do wish you had at least acknowledged the obvious and glaring issue with writing sqrt(-1), though (and n-th roots in general). A 180 degree rotation has TWO DISTINCT halvings.
Ah true, (-i)² = -1. Not gonna lie I kinda just forgot about that while making the video haha. You're right though, i should specifically be defined as the *positive* square root of -1; or geometrically, the number with a phase of 90º. And then -i is the *negative* square root of -1, aka a number with a phase of -90º.
Great video and excellent instruction. I'm saving this to my collection of supplementary videos I send to my Physics students. Thank you!
Agree, this is how I have always thought of complex numbers
With people like you out there making content that is this great, I genuinly believe we can make the next generation of kids think math is cool!
Amazing exposition my man!
Magnitude, phase, scaling, rotating are adequately represented by Euler's representation if you consider it in the form re^(i * theta) where r is the magnitude of the complex number. Easily explains all of the properties.
That's a really well-explained video.
Hi! Thank you very much for this great video!
I have a pretty good question.
Let's say the we have two waves, and the second one has a phase. How do I know that the amplitude of the wave which is the sum of the two waves is equal to the sum of the two complex numbers representing each wave?
Nice video and explanation!
I think the way you derived i is roughly the same i got taught in second class of engineering school almost 25+ years ago.
Funnily enough i didn't understand the question you formulated in the video title correct😉. We learned to use them for calculation of alternating currents and complex wave forms (FFT and stuff). As you showed they are the best way of depicting/visualizing waves. So i never asked myself why they are usefull, i only asked myself how was someone intelligent enough to come up with these solutions/find that out...
Wow this is so cool. It reminds me of a Socratica video about transformations. This is really cool stuff.
Very good, sir. As a graduate of electrical and electronic engineering, I like my math to arise naturally from first principles. The first occurrence of i, and the hint of complex numbers, comes out of solving the good old right triangle. And we all prefer a more intuitive understanding of how such an object arises and what it means, in a real sense. We live in a three dimensional world and that the focus of scaling and rotation as the fundamental perspective of complex numbers does, indeed, seem to be a natural, intuitive approach.
The fact of mathematics is it begins from the absolute and fundamental fact of nature that 1 meter + 1 meter = 2 meters, or 1 rock + 1 rock = 2 rocks. As long as further laws and derivations expand upon this real fundamental, it seems to suggest that all of mathematics is a real representation of some physical process. And the fundamental fact of nature is that we live in a 3D world, not one of singular dimension. Indeed, we don't really seem to live in one of negative direction. A photon, an electron, each from their own perspective, only move forward. A change in direction is due to a rotation of direction. Everything exists in a spherical universe of it's own center. It is as if, at the point where SQRT(-1) crops up, is an indicator that something has gone awry in our understanding. Like encountering the results of Special Relativity, that the scale of time and space are not "linear" and requires a more expansive perspective, it seems that SQRT(-1) necessitates a fundamental shift in perspective. The genius is in finding the right perspective.
Below, 馬赫特 points us towards Geometric Algebra. As I have never learned about the Wedge, I think there may be something there.
Brilliantly explained. My early maths training perhaps followed others with algebraic equations being primary. Then you get to complex numbers when solving x^2+1=0, but algebraically the answer has little interpretation. Looking at it through geometry is a far better approach, more intuitive and practical. Love it 🙂
I have nothing much to say... Just the honest to god best video in this SoME2. My mind was blown.
Rafiki u r an amazing communicator
Another random thought is that there are two distinct ways to get from 1 to -1: multiplication by e^{i\pi}, and also z \mapsto e^{-i\pi/2}\conj{z}e^{i\pi/2} (i.e. conjugating, in the group sense, complex conjugation with a 90° rotation). With the complex numbers, we can distinguish between these two ways of getting 1 to -1, but if we only consider the real line, these two transformations are indistinguishable.
Please do more math videos
Good video. I like the fact that you stuck to the trigonometric formulas for the polar representation of complex numbers instead of bringing out Euler's formula, which wouldn't really make sense to a class full of students who've never seen Taylor series before.
What I do think this video was missing was a bit of information on the rectangular representation of complex numbers. The polar representation seems advantageous when dealing with multiplication and division, but rectangular representation really wins out on addition and subtraction, since they are extremely straightforward and can directly be compared against the sum of binomials with constants.
Also, I'd like to put out the fact that the product of two complex numbers (a+bi)(c+di) gives (ac-bd)+(bc+ad)i, the real part of which is the formula for the determinant of a 2x2 matrix. I think it's worth saying that determinants are closely tied to cross products, which are commonly used to represent rotations and twisting motions (for example, in torque). Even though the polar representation of complex numbers gives us a geometric interpretation that is inherently tied to circular motions (angles, turning around a circle, arcs), I believe it's worth saying that the rectangular representation can provide us with this connection to bending motions too.
You will do more of these 😮 please 😅
Thank you. Intuitive and very helpfull... please do more such
It seem that the squre root of -1 only arises because we chose the basis in which we write our polar objects to be orthogonal. So asking why calculations with i make things simple is the same thing as asking why using orthogonal bases makes calculations easier. Thanks for the video, it was excellent and very clear.
underrated video, great job
15:21 why is time given in pi? is it the angle of the minute hand of a clock?
The specific complex sinusoid I graphed was e^(it). This sinusoid has a frequency of 1 radian per second, aka 1/(2π) Hertz. The period is then 2π seconds.
In the graph, the unit of the time axis is still seconds, but I tick-marked the axis in multiples of π because then visually the period is a whole number of ticks (specifically it's 2 ticks). If I had tick-marked the time axis as multiples of 1 second (i.e. 1, 2, 3, etc), then the period would look to be about 6.28 seconds, which in my opinion doesn't look as tidy.
Love this video
Great presentation!
Very impressive.
I’m subbing at 105 subscribers, fantastic video. I hope you will do more with style along these lines.
Fascinating how you came to √-1 without any mysterious operations. It makes me think that to some super-being it would seem equally natural from the other direction, where √-1 would obviously be for magnitudes perpendicular to the number line. Makes me wonder whether there are equivalent "portholes" to third and higher dimensions from the real numbers.
You do have the quaternions that extend the dimensions to 4. They're a consistent number system with multiplication and addition, and the complex numbers are a subset of that system.