from Hacker News

Complex numbers have never been so intuitive to me

by AndrewMoffat on 6/30/11, 4:28 AM with 46 comments

• by T-hawk on 6/30/11, 7:15 PM

  Wow, I got this too.

  Here's my followup question and answer. We know i is the square root of -1, but what is the square root of -i ? I've always thought that you'd need another dimension to describe that, and another dimension for the square root of that unit, and so on.

  But no. (Let's tackle sqrt(i) as a simpler case first.) We can answer sqrt(i) in terms of rotation as described in the article. i is a rotation from the unit vector by 90°, so applying that twice turns 1 into -1. What operation applied twice would result in i? This just clicked: a 45° rotation. Thus the unit vector at a 45° angle is the square root of i: 0.5√2 + 0.5√2 * i.

  The mathematical approach bears that out. Follow the rules of complex number arithmetic to square 0.5√2 + 0.5√2 * i (multiply it by itself) and you do indeed get i.

  And we can solve sqrt(-i) the same way. -i is a 270° rotation from the unit vector 1. So the square root of -i is a 135° rotation, or -0.5√2 + 0.5√2 * i.

  Finally, a 270° rotation is equivalent to a -90° rotation. So -45° should also be a square root of -i, and indeed it is. Multiplying 0.5√2 + -0.5√2 * i by itself also gives you -i. We've arrived back at the axiom that all numbers have two square roots of opposite signs. 135° and -45° are the same vector pointing in exactly opposite directions.

  Last question: What's the cube root of i? Easy: a 30° rotation. The 30° unit vector is 0.5√3 + 0.5i, and cubing that does indeed get you i.

• by Deestan on 6/30/11, 10:47 AM

  So real numbers are 1-dimensional, and complex numbers are 2-dimensional. Going along the same lines, we also have Quaternions, 4-dimensional numbers: http://en.wikipedia.org/wiki/Quaternion Further again we have 8-dimensional Octonions, and 16-dimensional Sedenions.

  I'm curious as to why we don't have any useful numbers for the non-power-of-2 dimensions. E.g. 3-dimensional numbers.

• by aristus on 6/30/11, 4:34 PM

  "It’s a testament to our mental potential that today’s children are expected to understand ideas that once confounded ancient mathematicians."

  Ex-fucking-actly. Math is hard. Compsci is hard. But if it remains hard, then we adults have failed to do our job.

• by natemartin on 6/30/11, 6:47 AM

  This is a fantastic explanation. I've always had trouble "getting" imaginary numbers.... even though I've had to use the fairly often as an Electrical Engineer. This is the first time they've made intuitive sense to me.

• by iwwr on 6/30/11, 10:49 AM

  In the same vein, you can have split-complex numbers which represent the 2d hyperbolic plane. Also, there are the 4-dimensional generalizations like quaternions, split-quaternions or other related algebra systems.

  What's amazing about these systems is that there is usually an Euler relation that holds. Example: e^(i*t) = cosh(t) + sinh(t) for the split-complex.

• by vain on 6/30/11, 9:17 AM

  i wish someone had explained this to me about 15 years ago. i had sought, and settled on the unsatisfying explanation that it is a conventional notation with useful ways to do coordinate geometry. non essential, but a way of doing it. though i have looked at euler's identity with awe, it's mostly been a mystical sort of awe.

  I hate to admit it but i had started using a line of argument to certain theist friends, that if god helps you, as a concept, no need to be bothered, think no more of it than a concept such as the the mathematical concept of i, its a number that does not exist but has real consequences. now I feel stupid for doubting the existence of i.

  I keep trying to relearn my fundamentals, and this article does that beautifully. i would otherwise have died a disbeliever.

• by mvzink on 6/30/11, 6:54 AM

  Wow. I feel like if I had been taught this before trigonometry, high school would have been a breeze.

• by cormullion on 6/30/11, 10:44 AM

  If you'd like some sexy French graphics to accompany that, there's a good Dimensions episode devoted to imaginary numbers. I found it on YouTube at

  http://www.youtube.com/watch?v=egIPnwcJuZ8

  But the main website is

  http://www.dimensions-math.org/Dim_download_E.htm

  And I think it's chapter 5.

• by baddox on 6/30/11, 6:30 AM

  Given a complex number a + bi, the square root of a^2 + b^2 is called the norm. The norm shouldn't be thought of as a measure of the "size" of a complex number, since complex numbers are not well-ordered. It makes little sense to say that 2 + 3i is equal to 3 + 2i.

• by dkokelley on 6/30/11, 5:09 PM

  It's unfortunate that our schools in general haven't been able to convey with as much clarity and passion this concept. I suspect that any passion or enthusiasm for a subject quickly gets destroyed when it's turned into a job (particularly a job in a system run by bureaucrats).

  This is why I am excited about programs like Khan Academy. One of the things he's been able to do that has eluded most public schooling is explain a concept simply, and with enthusiasm.

• by PetrolMan on 6/30/11, 4:08 PM

  Why do I feel like my teachers intentionally made math harder than it needed to be? Some concepts just lend themselves to simple visual representations and just never seem to be taught that way...