6 ms·

Learning to Love Complex Numbers

90 points by adbge 12y ago

zercool 12y ago
Some fantastic visualizations to accompany his essay. I would love an ipython notebook version that I could download and play with.
gohrt 12y ago
Related: Kalid Azad's oft-posted article: http://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/ http://betterexplained.com/articles/a-visual-intuitive-guide... Tristn Needham's book Visual Complex Analysis http://usf.usfca.edu/vca// http://usf.usfca.edu/vca//
- gballan 12y ago
  There are interactive plots, along the lines of VCA, here http://puzlet.com/m/b00dd http://puzlet.com/m/b00dd (I am a dev).
- sytelus 12y ago
  +1 for VCA. This book is written with passion and is one of the truly amazing book on Mathematics. It's just too bad it's so priced heavily (hint: International editions are far more cheaper).
bcbrown 12y ago
I've loved complex numbers since first learning about them, and my first significant programming experience was writing these types of programs for my TI-89 graphing calculator in high school in the late 90s. This is a good introduction. Another good equation to play around with is x_next = a x (1 - x), for a from 0 to 4. You'll find that x settles to a single value for low a, eventually bifurcating as you increase a, then bifurcating again. At one point, though, it splits into a cycle of three values, then an erratic distribution. Someone once wrote a paper on how "period three implies chaos".
- darkmighty 12y ago
  aka logistic map http://en.wikipedia.org/wiki/File:LogisticCobwebChaos.gif http://en.wikipedia.org/wiki/File:LogisticCobwebChaos.gif
  - bcbrown 12y ago
    Yes, but the visualizations I programmed started with http://polygeek.com/images/chaos/fig_01_BifurcationGraph.png http://polygeek.com/images/chaos/fig_01_BifurcationGraph.png
cwhy 12y ago
I just hate them.. Why don't we just use vectors and make math easier to learn?
- cwhy 12y ago
  And basically, negative numbers can be avoided as well using the concept of vectors
- darkmighty 12y ago
  Complex numbers are the key to complex analysis which is different than just real multivariate calculus (it has certain constraints).
- onedognight 12y ago
  > Why don't we just use vectors and make math easier to learn? We do just use vectors; over complex numbers. You really want both, where the vectors themselves are made up of[1] complex numbers. Using a single complex number to represent a vector has gone out of style in some sense because it's restricted to two dimensions. For example historically it was quite common to represent the velocity of a fluid at each point with a complex number. While you can do this, and there are many of advantages, most don't generalize to higher dimensions. [1] technically the field is complex numbers https://en.wikipedia.org/wiki/Field_%28mathematics%29 https://en.wikipedia.org/wiki/Field_%28mathematics%29
- sobellian 12y ago
  Complicated math concepts (like differential equations) are much easier to learn with a solid background in complex numbers. Vectors are important, but so are complex numbers.
- tobinfricke 12y ago
  How would you compute exp(v) for a vector v? (for example)
makmanalp 12y ago
Great article! I've always intuited the rules with which we operate on imaginary numbers as a hack - we don't know what to do with i other than squaring it, so we avoid doing anything with it and algebra our way around the issue. Example: (i + 3) + (i + 3) == i^2 + 6i + 9 == 6i + 8 is about as logical to me as (x + 3) + (x + 3) == x^2 + 6x + 9. Sure, if we knew what the heck x was this operation would have been easier, but since we don't, we just use FOIL to work around it. In the former, somehow we know what i^2 is but not i, so we can reduce a little further but not completely. Of course, I don't know how sound this gutfeel impression is.
- dbaupp 12y ago
  Very sound. One formal model of complex numbers is R[i]/(i^2 + 1), which means all real polynomials in the variable i (R[i]) after "simplifying" (/) by taking i^2 + 1 = 0 (equivalently i^2 = -1)... which is exactly the process you are describing.
- DanielRibeiro 12y ago
  It is very sound. You can think of i as being a variable 'x', and therefore, you can see Complex numbers are polynomials over real numbers (denoted as R[x]). But you can simplify the polynomials by using the fact that x2 = 1 for this construction (since x = i on this situation). This is the exact construction of Complex numbers as an algebraic extension over the field of Real numbers[1] What is more amazing: there is no such way to do the same on the complex numbers. This is because the Complex numbers form an Algebraically closed field[2] [1] http://en.wikipedia.org/wiki/Field_extension#Examples http://en.wikipedia.org/wiki/Field_extension#Examples [2] http://en.wikipedia.org/wiki/Algebraically_closed_field http://en.wikipedia.org/wiki/Algebraically_closed_field
- acjohnson55 12y ago
  i's irreducible simply because it's a unit-sized basis vector for how much of a quantity is in the "i direction". It's just something to hang a coefficient on so that it doesn't mix with the real part. We could just as well invent a similar thing for the reals, and call it g. Then your result is 6i + 8g. Or we could use tuples. It doesn't matter, as long as you do proper bookkeeping of the real and imaginary parts.
pavelrub 12y ago
>One fact we all remember about numbers is that squaring a number gives you something non-negative. 7^2 = 49, (-2)^2 = 4, 0^2 = 0, and so on. But it certainly doesn’t have to be this way. What if we got sick of that stupid fact and decided to invent a new number whose square was negative? This presentation of complex numbers makes the whole article flawed and useless in my opinion. When I first learned about complex numbers, they were presented in a very similar fashion: let's just invent a number i such that i^2 = -1. This is inane: we have a multiplication operation that we are all familiar with and we know exactly what it does, and then somebody tells us that we can use it on some "imaginary" thing (??) such that it times itself equals -1? How is anybody supposed to make any sense of it? It's like saying: let's invent a "number" j such that j-j=the letter z. What does it even mean? Nothing! it's gibberish, and a similar definition of "i" is also gibberish. We cannot make sense, under our normal understanding of multiplication and under our normal understanding of numbers as including only the real line, of how can something times itself be -1, and neither should we, because there are much better ways to present the whole thing from the beginning. The correct way to present complex numbers is either in the context of abstract algebra - where there is a very obvious question of whether we can embed the real line as a field inside the real plane, or simply present them geometrically without going into fields. It is simply wrong, in my opinion, to present i as something we "invent" so that i^2=-1 (why would anybody do that??), and then go on and say that after we have invented this, there are ways to imagine this geometrically. No! If you want to talk about geometry, then define i geometrically, then extend the definition of multiplication geometrically to the plane, and then it becomes clear that i^2 = -1, and there is no mystery about anything. Edit: it should also be noted that historically, complex numbers didn't come into existence because somebody decided on a whim to "invent" a number i such that i^2=-1. Rather, it was a result of the fact that cubic equations such as x^3=15x+4 clearly had a solution (for example x=4), but using the cubic formula to solve them resulted in weird terms such as sqrt(-121). Bombelli, in the 16th century, decided to try and compute with those terms anyway, and through this process eventually succeeded in producing the right results (x=4), so it eventually became clear that the roots of negative numbers aren't just gibberish: they interacted somehow with the reals, and there was some way to "make them work" to produce real results, though the full realization of what was happening probably came much later.
- 12y ago

90 points by adbge 12y ago

Some fantastic visualizations to accompany his essay. I would love an ipython notebook version that I could download and play with.

gohrt 12y ago

Related: Kalid Azad's oft-posted article: http://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/ http://betterexplained.com/articles/a-visual-intuitive-guide... Tristn Needham's book Visual Complex Analysis http://usf.usfca.edu/vca// http://usf.usfca.edu/vca//

gballan 12y ago
There are interactive plots, along the lines of VCA, here http://puzlet.com/m/b00dd http://puzlet.com/m/b00dd (I am a dev).
sytelus 12y ago
+1 for VCA. This book is written with passion and is one of the truly amazing book on Mathematics. It's just too bad it's so priced heavily (hint: International editions are far more cheaper).

bcbrown 12y ago

I've loved complex numbers since first learning about them, and my first significant programming experience was writing these types of programs for my TI-89 graphing calculator in high school in the late 90s. This is a good introduction. Another good equation to play around with is x_next = a x (1 - x), for a from 0 to 4. You'll find that x settles to a single value for low a, eventually bifurcating as you increase a, then bifurcating again. At one point, though, it splits into a cycle of three values, then an erratic distribution. Someone once wrote a paper on how "period three implies chaos".

darkmighty 12y ago
aka logistic map http://en.wikipedia.org/wiki/File:LogisticCobwebChaos.gif http://en.wikipedia.org/wiki/File:LogisticCobwebChaos.gif
- bcbrown 12y ago
  Yes, but the visualizations I programmed started with http://polygeek.com/images/chaos/fig_01_BifurcationGraph.png http://polygeek.com/images/chaos/fig_01_BifurcationGraph.png

cwhy 12y ago

I just hate them.. Why don't we just use vectors and make math easier to learn?

cwhy 12y ago
And basically, negative numbers can be avoided as well using the concept of vectors
darkmighty 12y ago
Complex numbers are the key to complex analysis which is different than just real multivariate calculus (it has certain constraints).
onedognight 12y ago
> Why don't we just use vectors and make math easier to learn? We do just use vectors; over complex numbers. You really want both, where the vectors themselves are made up of[1] complex numbers. Using a single complex number to represent a vector has gone out of style in some sense because it's restricted to two dimensions. For example historically it was quite common to represent the velocity of a fluid at each point with a complex number. While you can do this, and there are many of advantages, most don't generalize to higher dimensions. [1] technically the field is complex numbers https://en.wikipedia.org/wiki/Field_%28mathematics%29 https://en.wikipedia.org/wiki/Field_%28mathematics%29
sobellian 12y ago
Complicated math concepts (like differential equations) are much easier to learn with a solid background in complex numbers. Vectors are important, but so are complex numbers.
tobinfricke 12y ago
How would you compute exp(v) for a vector v? (for example)

makmanalp 12y ago

Great article! I've always intuited the rules with which we operate on imaginary numbers as a hack - we don't know what to do with i other than squaring it, so we avoid doing anything with it and algebra our way around the issue. Example: (i + 3) + (i + 3) == i^2 + 6i + 9 == 6i + 8 is about as logical to me as (x + 3) + (x + 3) == x^2 + 6x + 9. Sure, if we knew what the heck x was this operation would have been easier, but since we don't, we just use FOIL to work around it. In the former, somehow we know what i^2 is but not i, so we can reduce a little further but not completely. Of course, I don't know how sound this gutfeel impression is.

dbaupp 12y ago
Very sound. One formal model of complex numbers is R[i]/(i^2 + 1), which means all real polynomials in the variable i (R[i]) after "simplifying" (/) by taking i^2 + 1 = 0 (equivalently i^2 = -1)... which is exactly the process you are describing.
DanielRibeiro 12y ago
It is very sound. You can think of i as being a variable 'x', and therefore, you can see Complex numbers are polynomials over real numbers (denoted as R[x]). But you can simplify the polynomials by using the fact that x2 = 1 for this construction (since x = i on this situation). This is the exact construction of Complex numbers as an algebraic extension over the field of Real numbers[1] What is more amazing: there is no such way to do the same on the complex numbers. This is because the Complex numbers form an Algebraically closed field[2] [1] http://en.wikipedia.org/wiki/Field_extension#Examples http://en.wikipedia.org/wiki/Field_extension#Examples [2] http://en.wikipedia.org/wiki/Algebraically_closed_field http://en.wikipedia.org/wiki/Algebraically_closed_field
acjohnson55 12y ago
i's irreducible simply because it's a unit-sized basis vector for how much of a quantity is in the "i direction". It's just something to hang a coefficient on so that it doesn't mix with the real part. We could just as well invent a similar thing for the reals, and call it g. Then your result is 6i + 8g. Or we could use tuples. It doesn't matter, as long as you do proper bookkeeping of the real and imaginary parts.

pavelrub 12y ago

>One fact we all remember about numbers is that squaring a number gives you something non-negative. 7^2 = 49, (-2)^2 = 4, 0^2 = 0, and so on. But it certainly doesn’t have to be this way. What if we got sick of that stupid fact and decided to invent a new number whose square was negative? This presentation of complex numbers makes the whole article flawed and useless in my opinion. When I first learned about complex numbers, they were presented in a very similar fashion: let's just invent a number i such that i^2 = -1. This is inane: we have a multiplication operation that we are all familiar with and we know exactly what it does, and then somebody tells us that we can use it on some "imaginary" thing (??) such that it times itself equals -1? How is anybody supposed to make any sense of it? It's like saying: let's invent a "number" j such that j-j=the letter z. What does it even mean? Nothing! it's gibberish, and a similar definition of "i" is also gibberish. We cannot make sense, under our normal understanding of multiplication and under our normal understanding of numbers as including only the real line, of how can something times itself be -1, and neither should we, because there are much better ways to present the whole thing from the beginning. The correct way to present complex numbers is either in the context of abstract algebra - where there is a very obvious question of whether we can embed the real line as a field inside the real plane, or simply present them geometrically without going into fields. It is simply wrong, in my opinion, to present i as something we "invent" so that i^2=-1 (why would anybody do that??), and then go on and say that after we have invented this, there are ways to imagine this geometrically. No! If you want to talk about geometry, then define i geometrically, then extend the definition of multiplication geometrically to the plane, and then it becomes clear that i^2 = -1, and there is no mystery about anything. Edit: it should also be noted that historically, complex numbers didn't come into existence because somebody decided on a whim to "invent" a number i such that i^2=-1. Rather, it was a result of the fact that cubic equations such as x^3=15x+4 clearly had a solution (for example x=4), but using the cubic formula to solve them resulted in weird terms such as sqrt(-121). Bombelli, in the 16th century, decided to try and compute with those terms anyway, and through this process eventually succeeded in producing the right results (x=4), so it eventually became clear that the roots of negative numbers aren't just gibberish: they interacted somehow with the reals, and there was some way to "make them work" to produce real results, though the full realization of what was happening probably came much later.

12y ago