I know that it’s true but why? I’ve never grasped this topic.
In: Mathematics
If you ever had to study some calculus you might vaguely remember the following identities*:
> Sin[x]=(1/2)*(I*Exp[I*x]-I*Exp[-I*x])
> Cos[x]=(1/2)*(Exp[I*x]+Exp[-I*x])
Well, if we do some re-arrangements we can show that
> Exp[I*x]=Cos[x]+I*Sin[x]
and we can see that we’re almost there, we just have to plug in x=Pi.
>Exp[I*Pi]=Cos[Pi]+I*Sin[Pi]
which gives us our answer once we remember Sin[Pi]=0 and Cos[Pi]=-1.
That’s all well and good, but even though we derived the formula (from those Sin[x] and Cos[x] identities up at least) it doesn’t tell us much about what is going on. Why does Exp[I*Pi] go to -1 when we’re taught that Exp[x] always just makes an “exponential growth” curve that starts at 0 when x=-Infinity, goes through 1 when x=0, and then gets super-crazy-big the more x grows?
Well, what happens if we plug in Pi/2 instead of Pi?
> Exp[I*Pi/2]=Cos[Pi/2]+I*Sin[Pi/2]
> Exp[I*Pi/2]=0+I*(1)
We see that Exp[I*Pi/2]=I…? What’s going on?!?! If we plug in more numbers, like 3Pi/2 and Pi/4 and a whole bunch in between, and then graph them out on the complex plane we might start to see what is happening. As x increases, Exp[I*x] just keeps pushing our result further and further around the unit circle. At x=Pi (180 degrees) we’re halfway around the circle at point -1+(0)*I, whereas x=0*Pi (0 degrees) is at 0+(0)*I, and x=Pi/2 (90 degrees) is at 0+(1)*I etc.
So what happens if we do something like put the complex number x+I*y into the exponential function? Well, then we can use regular exponent expansion rules to show:
> Exp[x+I*y]=Exp[x]*Exp[I*y]
Which means that the exponential of a complex number is equal to the exponential of the real part (which does the exponential growth stuff we’re used to seeing) multiplied by the exponential of the imaginary part (which does the rotatey stuff that we noticed when messing around with Exp[I*Pi]).
So why does Exp[I*Pi]=-1 work the way it does? Because, unlike taking the exponential of a purely real number (e.g. x+I*0) which shows monatonic “exponential growth”, taking the exponential of a purely imaginary number (e.g. 0+I*x) rotates a point around the origin of the complex plane (and for x=Pi being 180-degrees specifically we end up back on the real number axis on the other side of the origin)!
* Sidenote: I’m using the notation of Sin[x], Exp[x], I^2 =(-1), Pi=3.14159… so that anyone can copy/paste it into [WolframAlpha.com](https://www.wolframalpha.com/input/?i=Sin%5Bx%5D) if they want to see the graphs of these things.
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