Imaginary numbers have an intuitive explanation: they “rotate” numbers, just like negatives make a “mirror image” of a number. This insight makes arithmetic with complex numbers easier to understand, and is a great way to double-check your results. Here’s our cheatsheet:

Great explanation! Here is what i tried to understand the intuitution for multiplication:
We know that any comlplex number can be represented as z=r(cosA+isinA) where r is the magnitude and A is its angle.
Now, let z1=r(cosA+isinA) and z2=q(cosB+isinB)
z1z2=rq(cosA+isinA)(cosB+isinB)
=rq(cosAcosB-sinAsinB+i(sinAcosB+cosAsinB))
=rq(cos(A+B)+isin(A+B))
so magnitude of z1z2 is rq(product of magnitudes of z1 and z2) and the angle of z1*z2 is A+B(sum of angles of z1 and z2).
Again, great explanation, thanks!

Hi Ragavendar, thanks for the comment. You got it, by converting to polar coordinates you can see the sine and cosine addition formula emerge. I’ll be writing more on this in a later post, but thanks for providing the details!

If you’re interested in a fantastic text on complex analysis (with gorgeous proofs and arguments), you should check out Visual Complex Analysis, by Tristan Needham.

Hello, good article but I think it’s still missing the big A-ha idea with complex numbers, which is their relation to “e” for example when I multiply two exponentials I multiply the bases and add the exponents which is analogous to multiplying complex numbers. when you divide complex numbers you divide the magnitude and subtract the angles and when diving exponential you divide the bases and subtract the exponents.
I guess this connection is manifested in euler’s formula? I don’t get it either, but wish I did.

even though I know those rules work for all bases when multiplying or dividing exponentials not just “e” However e is used in euler’s formula
e^ix = cosx + isinx

The rotation is one way for us to visualize what’s happening, just like multiplication can be seen as “stretching” a number. We don’t need to do this, but it’s a nice learning tool.

After reading your article (which was much appreciated considering I’m in Calc 3 and still couldn’t comprehend complex numbers past their arithmetic consequences), I got to thinking of a real world example that could help substantiate my thoughts. My ‘a-ha moment’ occurred in the following fashion:
How can you describe an object’s (i.e. a bike) position from you?
Before negatives:
“The bike is 4 feet from me in that direction” and you point in the direction of the bike.
After negatives:
“The bike is -4 feet from me in that direction” and you point in the opposite direction.

My aha moment, considering the use-fulness of imaginary numbers occurred when I pointed in a direction other than the 2 aforementioned ones. If you pointed 45 degrees to the right of the bike, you could say “the bike is 1+i feet away from me in that direction”.

I think what throws people off is the addition sign. Perhaps saying “1i1” would make more sense conceptually even if it’d complicate operations performed on complex numbers.

Hi Tim, that’s great – thanks for sharing your insight!

Yes, negative and complex numbers are a different way to talk about where things are. Even decimals are like this: we write 2.3, not 2 + .3, even though we could. Similarly, writing 2i3 might be easier to make sense of than 2 + 3i.

Again, appreciate the notes! I love seeing the different ways people look at the same topic.

There is already a way to write complex numbers without addition in a nice form that avoids addition and even “i”, but it goes under the category of “interesting but useless”: the Quaternary Imaginary number system, base 2i. Donald Knuth created it, and it’s an interesting theoretical idea, but quite useless.

On another note, I’m all for rewriting “a+bi” as aib. It makes it more obvious that it’s one number, and not two separate numbers.

@Zac: Great point – I think I ran across Knuth’s system a while ago, and agree it’s interesting but not very practical. I’d love it if we’d rewrite a+bi into something more “combined” to show that they are really two parts of the same number (just like 3 + 1/2 is better expressed by 3.5).