@Harish: Glad you enjoyed it – yes, I really hope that in 100 years (or 50, or 20!) imaginary numbers will be seen just as “real” as 0 or -1.
[…] your kids The closest to such an enlightened creature that I have found on-line is this guy Intuitive Arithmetic With Complex Numbers | BetterExplained Personally, that is his best written article and provides an entirely sensible, intuitive […]
@Alex: Interesting question – I think you’re asking whether x must be real, but y = f(x) can be complex (real / imaginary), which gives 3 dimensions? I hadn’t thought of that before – unfortunately, I’m not familiar with the rules of defining functions, but that seems interesting. At the minimum, you could start talking about matrixes and say you are considering matrixes of the form [x, y(Re), y(Im)] and do transformations on that. It would be cool to merge vectors, complex numbers, and polar as they are all ways of describing multi-dimensional coordinates.
You might be interested in Quaternions – 4-dimensional numbers. From what I’ve read offhand, it seems when you extend complex numbers directly, you end up needing 2, 4 or 8 dimensions for reasons of symmetry.
Really interesting thoughts!
Not knowing how anything works above 2 dimensions I’ve tried imagining the 3rd dimension as the imaginary dimension to help visualize things. I’m only in high school so this might not make much sense but…
say y = f(x) = square root (x)
if x can be any real we would have 3 dimensions and this seemed to satisfy my curiosity for 3 dimensional graphs. How ever if x is also allowed to be a real AND an imaginary/complex number I have to add a 4th dimension. x-real, x imag, y real, y imag are the 4 dimensions. The real x “axis” becomes a x “plane” and same thing for the y axis.
My question is can I actually define a 3 dimensional function just with something along the lines of
"y plane position"= f(x) = some function where x is uni-dimensional
or y = f(x) where x is the plane and y is uni-dimensional
simply with the help of complex numbers?
For me this is much easier to understand than something like z = yx
As far as I know, vectors, complex numbers, parametric/ polar and 3d functions are usually taught separately. I’d love to see an article where you can somehow unify them all (or anything in 3 dimensions really).
Sorry for the wall of text and thanks for explaining things so awesomely.
Thank you from India
Thank you so much … Reading your article has given me a good reason to start liking complex numbers … Math would be real fun if all concepts are taught this way in school …
Yes, it’s pretty cool. If you looked in a phase-conjugate mirror, all you would see is your eye.
@Tim: Wow, that’s an awesome visualization, thanks for the example. I love real-world use cases that aren’t super contrived. Especially ones that get a new metaphor in place (a reflection or rotation).
Big K–
I used to be in the laser business where I learned about something called optical phase conjugation, which provides a good analogy to the math concept. The conjugate of a light wave is basically a reflection, but it doesn’t bounce off at an angle. Rather, it is “an exact, time-reversed replica” of the incoming wave. That is, it comes back on itself, a backwards reflection being analogous to a backward rotation.
–Tim
@Cool_kau: Thanks for the kind words! 
Hi Kalid, I may have realized an answer to the first question I wrote you about in my previous e-mail. If i^2 = -1, then this says (something1 x something1) has a magnitude of 1 (and it just so happens to be in a negative direction). If I recall, the only time when (something1 x something1) would have a magnitude of 1 is when the (something1) itself has a magnitude of 1 . Since (something1) = i, then i has a also has a magnitude of 1. Aside from some weird situation that I am unaware of where a multiplication of two numbers results in a magnitude of 1, yet at least one of the two numbers is not 1, would this be a fair explanation ?
Thanks
P.S. I am still curious on your thoughts to my second question.
Dang it ! That only gives the magnitude for i^2…not i. So I give up. No more postings from me…on this question that is :-).
Thanks
Alright, how about this one. A friend mentioned that the a x b = 1 so ‘a’ and ‘b’ must be 1 applies to real numbers, and said “but we are not dealing with real numbers”. Sigh…of course not. So what about the sqrt(a^2 + b^2) ? If i^2 = 0 + 1i^2, and a= 0 and b=1, then sqrt(0^2 + 1^2) = sqrt(1) = 1. Any better ?
Thanks
@Joe: Awesome! I totally hear you, complex numbers bothered me for years ever since first encountering them.
Thank you for this brilliant article. Can magnitudes of “real” things be imaginary? Like, number of apples. If so, what does it mean?
Thank you!!! I’ve banged my head against the wall for years over complex numbers. I now see the light!!!
@Alonzo: Thanks for the comments! I left a detailed reply on the other article [on imaginary numbers], hope it helps!
excellent explanation! can complex numbers be used to find things like sqrt(X+2)?
This may be obvious to many but it was an Aha! moment for me. The same way that e^x is growth and (1/e)^x is growth in reverse, i.e., contraction, i^x means rotate counterclockwise and (1/i)^x means rotate backwards, i.e., clockwise
Hi Pablo, that’s an awesome insight, and not obvious at all. If multiplication is forward, then division should reverse it. And, interestingly, 1/i = -i :).