A Visual, Intuitive Guide to Imaginary Numbers

Imaginary numbers always confused me. Like understanding e, most explanations fell into one of two categories:

This is a companion discussion topic for the original entry at http://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/

I enjoy reading your intuitive approach to math and hadn’t really considered “i” normal until recently. (My favorite math formula contains all sorts of “non-existent” numbers – e^(pi*i)=-1.)

Anyway, need to point out a simple error in your article. (-1)^48 is 1, not -1. It’s a small issue, but didn’t want others to be confused.

Happy Holidays,
. John

I must thank you for this wonderful site. It has opened up my eyes to many things that I knew how to use, but never truly understood. This article in particular made me say, “HOLY CRAP! That’s freaking awesome!” Thank you very much for your work, and please keep it up.

It was a real breakthrough when I came to visualize that model for the first time. I really don’t understand why they don’t teach imaginaries that way!

Nice article, but I always found the “best” way to understand math is by its history, especially how mathematical idea came into being. No one actually wanted to solve

x^2 = -9

, nor want to “take the square root of nothing”. But in the 1500s, Bombelli wanted to use one of Cardano’s formula to solve

x^3 = 15x + 4

, and get

x = cuberoot(2 + sqrt(-121)) + cuberoot(2 – sqrt(–121))

After figuring that
cuberoot(2 + sqrt(–121)) = 2 + sqrt(–1)
cuberoot(2 - sqrt(–121)) = 2 - sqrt(–1)
, he found the real solution
x = 4
The idea was that this number sqrt(-1) was actually useful!
And yeah, everyone should also see the (simple) proof of Euler’s formula. It is Euler’s formula that links trigonometry to arithmetic (and allows for a geometric interpretation of complex numbers as a result).

@John: Thanks for the catch, I just fixed it. I’m a big fan of the e^i*pi = -1 formula as well.

@JB: Thanks! Yes, I had a similar “wow” moment and just wanted to share it. There are so many things we think we “know” (because we learned them a decade ago), but never bother to revisit with a fresh set of eyes. I’ll keep the articles coming.

@Bryan: I agree – I needed to see the diagram before it clicked. I don’t know why it’s not taught visually either – it makes students think imaginaries are entirely made up and unintuitive.

@Chick: Thanks for the background info! I had to plug in the numbers myself to see myself:

(2 + i)^3 = 2 + 11i = 2 + sqrt(-121)

There’s more details here as well:


[…] When studying linear algebra (matrices), you can view multiplication as a type of transformation (scaling, rotating, skewing), instead of a bunch of operations that change a matrix around. This approach will help when we cover imaginary numbers, that foul beast which has befuddled many students. […]

Actually, your rotation calculation is wrong, depends on what you really mean by “heading”. If you only want to rotate by 45 and not to scale, you have to multiply by a complex value with length 1. 1+i has length sqrt(2) so the final answer is thus -1/sqrt(2)+7i/sqrt(2).

Yeah, I wanted to leave out the discussion of scaling until the next article. The meaning of heading was just the “angle”, so the scale shouldn’t matter in this case. Also, a triangle of sides 1/sqrt(2) + 7i/sqrt(2) is hard to draw :slight_smile:

I really like how you explicitly relate rotation to complex numbers. It really does feel like a whole new angle (heh) to START with rotation.

I have occasionally pointed people towards http://mathforum.org/johnandbetty/ which is good for the early stages of complex numbers.

Thanks Chaz! Yeah, if negatives are “mirror images”, then complex numbers are “rotations”. I wish I had been taught that analogy first, instead of some arcane symbols which later get shown to have a geometric interpretation. We’re visual creatures! :slight_smile:

Thanks for the link, I’ll have to check it out.

A very nice explanation, I’ve never thought of it that way before. So how would you describe an x,y plot where both x and y are complex? (I’m not trying to be a smart a, I’m sincerely curious). Or maybe the question should be if you add rotation to an x,y co-ordinates you then get something else (quaternions?).


I see you’ve read
Where Mathematics Comes From by George Lakoff and Rafael Nunez.

Everything said here and more, except errors, is in that book!8-))

Your pages are good publicity for these ideas. But you need to post more about analogy.

@Dave: a pair (z,s) of two complex numbers would “live” in 4 dimensional space. They are not quaternion, however, although both are 4 dimensional. Quaternion have three imaginary axes i,j, and k; with non-commutative multiplication. But they are actually used in your favorite FPS games: Halo, Doom, etc., in, 3D rotation. Surprise!

And then here’s the octonions with seven imaginary axes and non-associative multiplication…


I loved your book but you never answered the question posed by the title.

As far as analogies are concerned, thinking about imaginary numbers as rotation is a good start but I think periodicity goes deeper. Your book kind of touched on that in the e^i*pi = -1 section.

@George: Actually, I haven’t read that book – all analogies and mistakes came from my brain :). I’m a fan of using analogies to understand difficult topics, and they’ll continue appearing in my articles.

@Chick: Thanks for the details, I’m not familiar with quaternion but am looking forward to learning.

@bayareaguy: Yep, the rotation analogies go much deeper with Euler’s formula. But all that would be too much for one sitting :). It’ll be in a future article.

suppose x^2=a
then x can have two values sqrt(a) or - sqrt(a)
will this same rule not apply to imaginary no.s??

ie. i^2 can be equal to + or - 1

i= sqrt(-1)

i^2 = sqrt(-1)* sqrt(-1)
= sqrt( -1 * -1 )
= sqrt(1)
= 1

I never had imaginary numbers in school, but I think I can deal with them now, thanks to your explanation :).

By the way, there’s a word missing here:

what confounded ancient mathematicians DIDN’T.

@abc: Actually, it’s the other way around: if you have x^2 = a, then sqrt(a) is either +x or -x. For example, sqrt(9) is either +3 or -3.

So, there are 2 values of sqrt(-1): +i and -i. There’s only one value of i^2, which is -1. (Just like there is only one value of -3^2, which is -9).

@Robin: Thanks, glad you found it useful! Yes, imaginary numbers are weird at first but I’m getting a handle on them also. Btw, I also fixed up the sentence to be more clear.

@Kalid: Sorry, the sentence was right. I just didn’t know the verb “to confound”, so I thought “confounded” was an adjective meaning something like “wise” :).