I know that it’s true but why? I’ve never grasped this topic.
In: Mathematics
Exponential of a complex number “interprets” the imaginary part of that complex number as an angle — because this is the only natural way of defining the complex exponential function, so that it works (i.e. has a value, and respects the usual rules of exponentiation) for any complex numbers.
That angle is in radians, which is the natural angle in math (more on that below). 2π radians is a full circle, so π radians is half a circle. Therefore, e^iπ is rotated 180 degrees in the complex plane with respect to e^0 , which is 1. The point opposite to 1 in the complex plane is -1, QED.
Now why is it the case that the imaginary part of the exponent is an angle (in radians)? Well, it turns out that this is the (only) natural way to generalize “usual” (real) exponentiation to all complex numbers. Another poster derived the (not exactly ELI5-compliant) proof from the exponential definition as a sum of an infinite series; let me take another angle (no pun intended) from derivatives. A key property of what makes e the “natural” number for exponentiation, is that the derivative of e^x is itself. There is a similar property for the sine and cosine functions (but only if the angle is measured in radians); except you have to take the derivative four times to loop back to the original function — Just like you have to multiply i by itself four times to go back to 1.
Exponential and sine functions can therefore be seen as two aspects (in fact, they are two projections on orthogonal axes in the complex plane) of one and the same complex-exponential function, which is natural in the sense that it is the second-simplest function that is its own complex-derivative. (The simplest one is of course the zero function, but that’s boring.) This truth is expressed by the Euler formula, e^ix = cos x + i sin x, which gives you a way to compute the exponential of any given imaginary number — and knowing that e^(a + b) = e^a * e^b, which holds for any complex numbers a and b, you can compute separately the exponentials of the real part and imaginary part of any complex number z = x + i*y (the latter using the Euler formula), multiply them, and obtain the value of e^z = e^x * (cos y + i sin y).
(This is not the end of the depths of how sine and exponential functions are intertwined images of each other in the distorting mirror that is making a quarter of a turn in the complex plane. If you would like to know more, look up the Fourier transform and the Laplace transform.)
In conclusion, applying the Euler formula to π gives you e^iπ = cos π + i sin π = -1 + i * 0 = -1. Tada 🙂
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