# 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 hadnâ€™t really considered â€śiâ€ť normal until recently. (My favorite math formula contains all sorts of â€śnon-existentâ€ť numbers â€“ e^(pi*i)=-1.)

Anyway, need to point out a simple error in your article. (-1)^48 is 1, not -1. Itâ€™s a small issue, but didnâ€™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, â€śHOLY CRAP! Thatâ€™s freaking awesome!â€ť 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 donâ€™t understand why they donâ€™t teach imaginaries that way!

Nice article, but I always found the â€śbestâ€ť 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 â€śtake the square root of nothingâ€ť. But in the 1500s, Bombelli wanted to use one of Cardanoâ€™s formula to solve

x^3 = 15x + 4

, and get

x = cuberoot(2 + sqrt(-121)) + cuberoot(2 â€“ sqrt(â€“121))

After figuring that
cuberoot(2 + sqrt(â€“121)) = 2 + sqrt(â€“1)
cuberoot(2 - sqrt(â€“121)) = 2 - sqrt(â€“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 Eulerâ€™s formula. It is Eulerâ€™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. Iâ€™m a big fan of the e^i*pi = -1 formula as well.

@JB: Thanks! Yes, I had a similar â€śwowâ€ť moment and just wanted to share it. There are so many things we think we â€śknowâ€ť (because we learned them a decade ago), but never bother to revisit with a fresh set of eyes. Iâ€™ll keep the articles coming.

@Bryan: I agree â€“ I needed to see the diagram before it clicked. I donâ€™t know why itâ€™s not taught visually either â€“ 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)

Thereâ€™s more details here as well:

http://www.mth.kcl.ac.uk/events/summer_schools/summer_school2001/Alg013.html

[â€¦] 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. [â€¦]

Actually, your rotation calculation is wrong, depends on what you really mean by â€śheadingâ€ť. 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 â€śangleâ€ť, so the scale shouldnâ€™t matter in this case. Also, a triangle of sides 1/sqrt(2) + 7i/sqrt(2) is hard to draw

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 â€śmirror imagesâ€ť, then complex numbers are â€śrotationsâ€ť. I wish I had been taught that analogy first, instead of some arcane symbols which later get shown to have a geometric interpretation. Weâ€™re visual creatures!

Thanks for the link, Iâ€™ll have to check it out.

A very nice explanation, Iâ€™ve never thought of it that way before. So how would you describe an x,y plot where both x and y are complex? (Iâ€™m not trying to be a smart a, Iâ€™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?).

Dave

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 â€śliveâ€ť 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 hereâ€™s the octonions with seven imaginary axes and non-associative multiplicationâ€¦

George,

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 havenâ€™t read that book â€“ all analogies and mistakes came from my brain :). Iâ€™m a fan of using analogies to understand difficult topics, and theyâ€™ll continue appearing in my articles.

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

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

hi,
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, thereâ€™s a word missing here:

what confounded ancient mathematicians DIDNâ€™T.

@abc: Actually, itâ€™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. Thereâ€™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 Iâ€™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 didnâ€™t know the verb â€śto confoundâ€ť, so I thought â€śconfoundedâ€ť was an adjective meaning something like â€świseâ€ť :).