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I'm wary of anyone who tries to explain complex numbers without pictures. The best explanation of complex numbers that I've ever seen is http://acko.net/blog/ho

by sxp 9y ago

I'm wary of anyone who tries to explain complex numbers without pictures. The best explanation of complex numbers that I've ever seen is http://acko.net/blog/how-to-fold-a-julia-fractal/ http://acko.net/blog/how-to-fold-a-julia-fractal/. I wish that explaination existed when I was struggling with my Electrical Engineering classes in college.

arkj 9y ago
Thank you for the brilliant link. I've been looking for a good explanation on complex numbers to share with a few struggling students.
- mercer 9y ago
  That site is a work of art. Also, apparently I unlocked an achievement for refreshing the page!
- Kagerjay 9y ago
  coolest math site I've ever seen
Grue3 9y ago
I'd rather be wary of explaining mathematical concepts with pictures. Pictures are known to be highly misleading and force the reader to imagine a thing in a certain way. For example, most pictures of triangles have all 3 angles being acute. This leads people into forgetting the case when one of the angles is obtuse.
- Falkens_Maze 9y ago
  Completely agree. It's what Prof. Susskind states in one of his lectures on QM explaining quantum fields. Eventually pictures and mental abstractions break down, and it's better to not visualize. At some point, one's limited biological brain will be unable to fully perceive or visualize some mathematical object. That's where intuition comes in.
  - jules 9y ago
    You can still reason by analogy with something you can visualise. You can't visualise 3+1 = 4 spacetime dimensions, but you can pretend that space has 2 dimensions so that you can visualise it as 2+1 dimensions. Blind algebra rarely works. I think that many have an optimistic view of blind algebra because proofs are often stated as algebra without geometric intuition, even when the author of the theorem almost certainly used geometric intuition to come up with it. This can give a false impression. I've seen people struggle tremendously with simple proofs in Hilbert spaces, when the corresponding proof in 2 dimensions is easy by drawing a picture and translating that into algebra that works just as well in the more general setting. For example, let S be a closed subspace of Hilbert space H and x in H. Prove |x-y| is minimised over y in S iff x-y is orthogonal to S.
    - AstralStorm 9y ago
      Analogies break down as easily as they are helpful. On average, they help mathematical intuition less than expected. (Check Bose-Einstein thought experiments and the mathematics they spawned for an example.) QM went into blind math side way past any analogies. Yet it is currently matching observations which is as good as it gets with true.
      - jules 9y ago
        QM did not go into blind math mode. The way physicists usually do QM is by glossing over a lot of the subtleties of Hilbert spaces and act as if the results from finite dimensional linear algebra work. Concepts physicists have been using for a long time, such as the Dirac delta function and path integrals, only got formalised much later. Many revolutions in QM, such as Feynman diagrams, are very much based on intuition. You can transfer all of that back into formal math, as Dyson did, but Dyson could not have done so without Feynman having invented them first. Especially in mathematics it is fashionable to write proofs in a style that completely erases the reasoning that lead to the proof. I think that's a pity. So much unwritten knowledge gets lost that way when the originators die. Differential forms, for example, have a beautiful geometric interpretation, but the way they're taught nowadays obscures that completely and makes it seem like they're a formal algebraic tool only. In fact, we're now several generations later, so even some of the instructors may be unaware of that, because their own instructors failed to transfer the original geometric intuition!
- mathgenius 9y ago
  "Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine." — Sir Michael Atiyah, 2002
- yummy 9y ago
  On the contrary, graphs is a good example, when pictures make perfect sense and any other way is rather meaningless for human brain. But yes, algebra specifically and pictures don't gel.
- wglb 9y ago
  Ok, so how would you explain venn diagrams without pictures? How would you describe Mandelbrot sets without pictures? How would you explain hyperbolic parabaloids without pictures? How would you explain conic sections without pictures? How would you explain a trapezoid without pictures? How would you describe a sine wave without pictures? Eversion of a sphere? How would you illustrate the calculation of the slope of a curve when talking about derivatives without pictures? How about a cycloid? How would you show the representation of a signal in both time domain and frequency domain?
  - Grue3 9y ago
    >How would you describe Mandelbrot sets without pictures? It's literally easier to describe Mandelbrot set without the picture than with it. The set of complex numbers c where recursive equation "z_n = z_(n-1)^2 + c, z_0 = 0" does not diverge. In fact, the picture of a Mandelbrot set is actively misleading! You think you can see the set, but it's a fractal! It's infinitely complex in a way that cannot be shown in a picture. >Eversion of a sphere? Pretty sure that one was figured out way before anyone was able to visualize it. Same with Banach-Tarski paradox. These things resist visual intuition in the first place, so it's much safer to rely on equations when dealing with problems like these.
    - wglb 9y ago
      If you read of the early work that Mandelbrot did on this equation, you will recall that despite having those symbols in front of him, he had to plot it out, crudely at first on a line printer, before he fully understood the implications. Thus pictures are not only tutorial but a fundamental necessity for understanding mathematics.
    - Simon_says 9y ago
      You picked possibly the worst example for what you're trying to say. Nobody knew it was a fractal until it was plotted. Proving it's a fractal was done with a written, rigorous proof, but the intuition for where to look came from the pictures.
- i2om3r 9y ago
  You can explain things with more than a single picture. You can also play with pictures in your mind, and you can encourage students to do so by giving them more than one picture. Visualizing corner cases is often very useful. For triangles, you can change angles and side lengths and see which facts still hold and which don't and how certain values/function results develop. Moreover, it very much depends on who your target group is. My brain, for example, works entirely inductively (in the beginning). I won't be able to develop an intuition of something if I don't start with examples. Pictures are often good examples. During my undergrad studies, my linear algebra prof was as critical about pictures as you and other commenters here. I hated it. I was never able to get an intuition about the more abstract topics until I saw concrete examples including pictures in later lectures and projects. Moreover, not everyone is going to be a theoretical mathematician or quantum physicist. I suspect that by not showing pictures, you usually lose more students along the way than pictures would ruin students that need a fully abstract understanding (later). It would be interesting to see some data on this, but I guess its going to be difficult to collect.
crispyambulance 9y ago
Yep, although theorems and proofs are, of course, central to mathematics, pictures are absolutely essential for developing intuition and for introducing concepts. In the case of complex analysis, pictures are critical for explaining the contour integrals for example. The pictures do not mislead, they elucidate. That said, there's nothing wrong with the OP article. It is an introductory piece and folks seeing this stuff for the first time do well with a variety of approaches to the topic-- this one happens to not use pictures. That's OK.
roenxi 9y ago
Heartily agreeing. If the Reals are fundamentally the set of all points that make up a line, the "imaginary numbers" are the set of all 2d points in a plane. Turns out there was never a call to limit numbers to living in a straight line, but our notation was settled on before we realised that. I'm got grumpy just earlier this evening that the first 9 negative numbers didn't get their own glyphs. i is basically "rotate by 90" in the same way -1 is "head left from 0 instead of right"
- globuous 9y ago
  > If the Reals are fundamentally the set of all points that make up a line, the "imaginary numbers" are the set of all 2d points in a plane I'm not sure that's true. It's a very good way to visualize complex numbers at first, but the set of points in a 2D plane does not define complex numbers. That's a very interesting question actually, what is the difference between R^2 != C ? Here's someone's attempt at showing this (top answer): https://math.stackexchange.com/questions/444475/whats-the-difference-between-mathbbr2-and-the-complex-plane https://math.stackexchange.com/questions/444475/whats-the-di... To me, trying to visualize complex numbers is an easy way to forget that R^2 != C.... They look the same but don't behave the same way (how do you multiply to members of R^2 with each other ? you need to define it, whereas multiplication in C is already defined). C doesn't behave like 2D vectors either. Since multiplying 2 complex is not the same as multiplying 2 vectors (whatever that means) although their components could be the same. structure structure structure.
  - zardo 9y ago
    >(how do you multiply to members of R^2 with each other ? you need to define it, whereas multiplication in C is already defined) You can use the geometric product, and C falls out as a result.
- yequalsx 9y ago
  Complex numbers along with the Quaternions are the only finite dimensional associative division algebras that properly contain the reals. The interesting thing is that the 2d real space can be equipped with a multiplication operation that works. This isn’t always possible to do on a finite dimensional real vector space. It happens to be true that this field called the complex numbers is a two dimensional real vector space. It took humans quite a long time to come up with 0 and negative numbers. There’s no way the jump to the complex numbers was going to be as easy and it isn’t as simple as, “numbers in 2d”. Why isn’t 3D a finite dimensional associative division algebra?
dsacco 9y ago
You want to be careful with that. Pictures can be helpful in mathematical education (especially in something like a first course in real analysis), but if you get accustomed to relying on learning with pictures your mathematical intuition can break down once you arrive at a place where pictures don’t make sense. Once you’re working in k-dimensions, where k > 3 (or maybe 4), you can’t rely on pictures anymore because there is no satisfying depiction of 4 or greater dimensions on a 2 dimensional medium. Complex numbers can be learned this way because they’re (simplified) the set R^2. Number systems in general are sort of okay to learn this way because they typically just extend other number systems, so you can successively build intuition on top of previous abstractions. But you won’t really “learn” a lot of much more complicated topological or geometric (and therein analytic) concepts if you can’t build intuition without “seeing” it. As a very simple example, contrast these two approaches to defining a trivial concept in topology, which you’d probably come across before complex numbers in an analysis course: 1. A neighborhood of a point p in a Euclidean space is the set of all q such that d(p, q) < r, where r is some radius greater than 0. 2. A neighborhood is a circle (or sphere) with a radius of r surrounding a point p. One of those feels more immediately intuitive, but that’s only because we’re dealing with the simple case of k = 2 or 3. You can’t “dodge” the complexity by visualizing it once you get into higher k, you have to just formalize it to develop intuition rigorously. Definition 1 is useful because it abstracts to conceptual domains we cannot practically visualize. Definition 2 would be useful if you could build a visual intuition of a 3-sphere, or 4-sphere, and so on. But building a visual intuition of k > 3 dimensions just shifts the problem away from what you’re trying to learn to something that is just as difficult: eventually you have to just get comfortable staring at a page of frustrating numbers, unfortunately.
- vorotato 9y ago
  That's actually untrue. 3 Blue 1 Brown has an excellent example of how you can use visuals to aid intuition in higher dimensional thinking. It's just not the naive representation.
  - mjburgess 9y ago
    Yes, I think this advice isn't quite right. A pictorial formalism is possible for most if not all mathematical concepts as much as a symbolic. It just won't be naive and may not have been systematically developed and communicated / taught anywhere. The issue here is mostly just an awareness that, naively, the pictures change fast than the formula. (r^1, r^2, r^3... look more a like than lines, circles and spheres).

by sxp 9y ago

arkj 9y ago

Thank you for the brilliant link. I've been looking for a good explanation on complex numbers to share with a few struggling students.

mercer 9y ago
That site is a work of art. Also, apparently I unlocked an achievement for refreshing the page!
Kagerjay 9y ago
coolest math site I've ever seen

Grue3 9y ago

I'd rather be wary of explaining mathematical concepts with pictures. Pictures are known to be highly misleading and force the reader to imagine a thing in a certain way. For example, most pictures of triangles have all 3 angles being acute. This leads people into forgetting the case when one of the angles is obtuse.

Falkens_Maze 9y ago
Completely agree. It's what Prof. Susskind states in one of his lectures on QM explaining quantum fields. Eventually pictures and mental abstractions break down, and it's better to not visualize. At some point, one's limited biological brain will be unable to fully perceive or visualize some mathematical object. That's where intuition comes in.
- jules 9y ago
  You can still reason by analogy with something you can visualise. You can't visualise 3+1 = 4 spacetime dimensions, but you can pretend that space has 2 dimensions so that you can visualise it as 2+1 dimensions. Blind algebra rarely works. I think that many have an optimistic view of blind algebra because proofs are often stated as algebra without geometric intuition, even when the author of the theorem almost certainly used geometric intuition to come up with it. This can give a false impression. I've seen people struggle tremendously with simple proofs in Hilbert spaces, when the corresponding proof in 2 dimensions is easy by drawing a picture and translating that into algebra that works just as well in the more general setting. For example, let S be a closed subspace of Hilbert space H and x in H. Prove |x-y| is minimised over y in S iff x-y is orthogonal to S.
  - AstralStorm 9y ago
    Analogies break down as easily as they are helpful. On average, they help mathematical intuition less than expected. (Check Bose-Einstein thought experiments and the mathematics they spawned for an example.) QM went into blind math side way past any analogies. Yet it is currently matching observations which is as good as it gets with true.
    - jules 9y ago
      QM did not go into blind math mode. The way physicists usually do QM is by glossing over a lot of the subtleties of Hilbert spaces and act as if the results from finite dimensional linear algebra work. Concepts physicists have been using for a long time, such as the Dirac delta function and path integrals, only got formalised much later. Many revolutions in QM, such as Feynman diagrams, are very much based on intuition. You can transfer all of that back into formal math, as Dyson did, but Dyson could not have done so without Feynman having invented them first. Especially in mathematics it is fashionable to write proofs in a style that completely erases the reasoning that lead to the proof. I think that's a pity. So much unwritten knowledge gets lost that way when the originators die. Differential forms, for example, have a beautiful geometric interpretation, but the way they're taught nowadays obscures that completely and makes it seem like they're a formal algebraic tool only. In fact, we're now several generations later, so even some of the instructors may be unaware of that, because their own instructors failed to transfer the original geometric intuition!
mathgenius 9y ago
"Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine." — Sir Michael Atiyah, 2002
yummy 9y ago
On the contrary, graphs is a good example, when pictures make perfect sense and any other way is rather meaningless for human brain. But yes, algebra specifically and pictures don't gel.
wglb 9y ago
Ok, so how would you explain venn diagrams without pictures? How would you describe Mandelbrot sets without pictures? How would you explain hyperbolic parabaloids without pictures? How would you explain conic sections without pictures? How would you explain a trapezoid without pictures? How would you describe a sine wave without pictures? Eversion of a sphere? How would you illustrate the calculation of the slope of a curve when talking about derivatives without pictures? How about a cycloid? How would you show the representation of a signal in both time domain and frequency domain?
- Grue3 9y ago
  >How would you describe Mandelbrot sets without pictures? It's literally easier to describe Mandelbrot set without the picture than with it. The set of complex numbers c where recursive equation "z_n = z_(n-1)^2 + c, z_0 = 0" does not diverge. In fact, the picture of a Mandelbrot set is actively misleading! You think you can see the set, but it's a fractal! It's infinitely complex in a way that cannot be shown in a picture. >Eversion of a sphere? Pretty sure that one was figured out way before anyone was able to visualize it. Same with Banach-Tarski paradox. These things resist visual intuition in the first place, so it's much safer to rely on equations when dealing with problems like these.
  - wglb 9y ago
    If you read of the early work that Mandelbrot did on this equation, you will recall that despite having those symbols in front of him, he had to plot it out, crudely at first on a line printer, before he fully understood the implications. Thus pictures are not only tutorial but a fundamental necessity for understanding mathematics.
  - Simon_says 9y ago
    You picked possibly the worst example for what you're trying to say. Nobody knew it was a fractal until it was plotted. Proving it's a fractal was done with a written, rigorous proof, but the intuition for where to look came from the pictures.
i2om3r 9y ago
You can explain things with more than a single picture. You can also play with pictures in your mind, and you can encourage students to do so by giving them more than one picture. Visualizing corner cases is often very useful. For triangles, you can change angles and side lengths and see which facts still hold and which don't and how certain values/function results develop. Moreover, it very much depends on who your target group is. My brain, for example, works entirely inductively (in the beginning). I won't be able to develop an intuition of something if I don't start with examples. Pictures are often good examples. During my undergrad studies, my linear algebra prof was as critical about pictures as you and other commenters here. I hated it. I was never able to get an intuition about the more abstract topics until I saw concrete examples including pictures in later lectures and projects. Moreover, not everyone is going to be a theoretical mathematician or quantum physicist. I suspect that by not showing pictures, you usually lose more students along the way than pictures would ruin students that need a fully abstract understanding (later). It would be interesting to see some data on this, but I guess its going to be difficult to collect.

crispyambulance 9y ago

Yep, although theorems and proofs are, of course, central to mathematics, pictures are absolutely essential for developing intuition and for introducing concepts. In the case of complex analysis, pictures are critical for explaining the contour integrals for example. The pictures do not mislead, they elucidate. That said, there's nothing wrong with the OP article. It is an introductory piece and folks seeing this stuff for the first time do well with a variety of approaches to the topic-- this one happens to not use pictures. That's OK.

roenxi 9y ago

Heartily agreeing. If the Reals are fundamentally the set of all points that make up a line, the "imaginary numbers" are the set of all 2d points in a plane. Turns out there was never a call to limit numbers to living in a straight line, but our notation was settled on before we realised that. I'm got grumpy just earlier this evening that the first 9 negative numbers didn't get their own glyphs. i is basically "rotate by 90" in the same way -1 is "head left from 0 instead of right"

globuous 9y ago
> If the Reals are fundamentally the set of all points that make up a line, the "imaginary numbers" are the set of all 2d points in a plane I'm not sure that's true. It's a very good way to visualize complex numbers at first, but the set of points in a 2D plane does not define complex numbers. That's a very interesting question actually, what is the difference between R^2 != C ? Here's someone's attempt at showing this (top answer): https://math.stackexchange.com/questions/444475/whats-the-difference-between-mathbbr2-and-the-complex-plane https://math.stackexchange.com/questions/444475/whats-the-di... To me, trying to visualize complex numbers is an easy way to forget that R^2 != C.... They look the same but don't behave the same way (how do you multiply to members of R^2 with each other ? you need to define it, whereas multiplication in C is already defined). C doesn't behave like 2D vectors either. Since multiplying 2 complex is not the same as multiplying 2 vectors (whatever that means) although their components could be the same. structure structure structure.
- zardo 9y ago
  >(how do you multiply to members of R^2 with each other ? you need to define it, whereas multiplication in C is already defined) You can use the geometric product, and C falls out as a result.
yequalsx 9y ago
Complex numbers along with the Quaternions are the only finite dimensional associative division algebras that properly contain the reals. The interesting thing is that the 2d real space can be equipped with a multiplication operation that works. This isn’t always possible to do on a finite dimensional real vector space. It happens to be true that this field called the complex numbers is a two dimensional real vector space. It took humans quite a long time to come up with 0 and negative numbers. There’s no way the jump to the complex numbers was going to be as easy and it isn’t as simple as, “numbers in 2d”. Why isn’t 3D a finite dimensional associative division algebra?

dsacco 9y ago

You want to be careful with that. Pictures can be helpful in mathematical education (especially in something like a first course in real analysis), but if you get accustomed to relying on learning with pictures your mathematical intuition can break down once you arrive at a place where pictures don’t make sense. Once you’re working in k-dimensions, where k > 3 (or maybe 4), you can’t rely on pictures anymore because there is no satisfying depiction of 4 or greater dimensions on a 2 dimensional medium. Complex numbers can be learned this way because they’re (simplified) the set R^2. Number systems in general are sort of okay to learn this way because they typically just extend other number systems, so you can successively build intuition on top of previous abstractions. But you won’t really “learn” a lot of much more complicated topological or geometric (and therein analytic) concepts if you can’t build intuition without “seeing” it. As a very simple example, contrast these two approaches to defining a trivial concept in topology, which you’d probably come across before complex numbers in an analysis course: 1. A neighborhood of a point p in a Euclidean space is the set of all q such that d(p, q) < r, where r is some radius greater than 0. 2. A neighborhood is a circle (or sphere) with a radius of r surrounding a point p. One of those feels more immediately intuitive, but that’s only because we’re dealing with the simple case of k = 2 or 3. You can’t “dodge” the complexity by visualizing it once you get into higher k, you have to just formalize it to develop intuition rigorously. Definition 1 is useful because it abstracts to conceptual domains we cannot practically visualize. Definition 2 would be useful if you could build a visual intuition of a 3-sphere, or 4-sphere, and so on. But building a visual intuition of k > 3 dimensions just shifts the problem away from what you’re trying to learn to something that is just as difficult: eventually you have to just get comfortable staring at a page of frustrating numbers, unfortunately.

vorotato 9y ago
That's actually untrue. 3 Blue 1 Brown has an excellent example of how you can use visuals to aid intuition in higher dimensional thinking. It's just not the naive representation.
- mjburgess 9y ago
  Yes, I think this advice isn't quite right. A pictorial formalism is possible for most if not all mathematical concepts as much as a symbolic. It just won't be naive and may not have been systematically developed and communicated / taught anywhere. The issue here is mostly just an awareness that, naively, the pictures change fast than the formula. (r^1, r^2, r^3... look more a like than lines, circles and spheres).