We have Dopers who are equipped to explain this; I am not one of them. But while we wait for someone to stop by, I can at least give you a name and an article to peruse:
BTW, before anyone else points it out, I meant imaginary answer. Irrational numbers are real. They just never end and never repeat (which is something else I could never understand–but that’s for a different thread ).
I often think it was a mistake for Descartes to call them ‘imaginary’ and treat them as a hack or as fictional numbers. There are a non-negligible number of people who can’t get past the ‘imaginary’ label alone. They are as real (as in ‘actual’ number, not “real numbers” in math jargon) as any number.
It’s not even an original thought on my part. Gauss believed this and said that naming them something different would have avoided a lot of confusion over the years, and he (as usual) was probably right about that.
Basically, a comlex number has a real part and an imaginary part. If you write an expression that causes the imaginary part to be zero, you only have the real part.
In this case, we have Euler’s formula which states that e ^ (i * theta) = cos (theta) + i sin (theta). Where theta is an angle expressed in radians. It so happens that sin(pi) is 0 and cos(pi) is -1, so you get the famous identity from the OP.
An even simpler one is the somewhat obvious e ^ (i + 0) = 1.
Agreed that “imaginary” is just a terrible word. They are simple numbers in another dimension. Real numbers are on the number line and the other number line that i is on runs perpendicular to it.
In fact, if you think of the “imaginary” numbers as “perpendicular” numbers, then a basic folk understanding of two-dimensional math/physics gets you to understanding the basic concept of e^(i*pi). Exponentiation is growth. If you grow in a straight line, you accelerate in that direction. What happens when you grow at a right angle to your heading? You spin in a circle.
Gelfand’s theorem does not apply since neither e nor pi is algebraic. And if pi were algebraic so would pi*i be.
To give a quick idea of a proof, take the power series for e^x
1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! + …
and replace x by ix. Collecting the real and imaginary terms and recalling the power series for sin x and cos x (written for x in radians) gives you
e^{ix} = cos x + i*sin x. Now let x = pi, whence cos x = -1 and sin x = 0.
These power series are all derived in calc 2, based on the facts that the derivative of e^x is e^x, the derivative of sin x is cos x and the derivative of cos x is -sin x (when x is in radians).
My understanding (which is admittedly laughably small) is that imaginary numbers are legitimate numbers because you can apply mathematical operations to them and produce consistent answers. Which you can’t do with a zillion or a bunch or hun bundred.
Not only is it legitimate, you can hardly avoid considering complex numbers in number theory once you consider something like the fundamental theorem of algebra, or more simply any equation like x² + 1 = 0