the complex exponential
On the real line e^x is the familiar growth curve. We would like to feed it a complex number z = x + i y and still get something sensible. The honest question is: what could e^z even mean when the exponent has an imaginary part? The answer turns out to be both forced and beautiful, and it ties exponential growth together with rotation.
We define e^z = e^x (cos y + i sin y). Read that as: the real part x controls the size through the ordinary real e^x, and the imaginary part y controls the angle through Euler's formula e^(i y) = cos y + i sin y. So |e^z| = e^x and arg(e^z) = y. For example e^(1 + i pi/2) = e^1 (cos(pi/2) + i sin(pi/2)) = e * i = e i, a number of modulus e pointing straight up. This definition is not arbitrary: it is the only continuous extension that keeps e^(a+b) = e^a e^b and agrees with the real e^x on the real axis, and it is exactly what the power series sum z^n / n! gives.
The complex exponential is the engine room of complex analysis. Sine, cosine, the hyperbolic functions, and the logarithm are all defined from it, and it is the model holomorphic function: differentiable everywhere with derivative equal to itself. Its single most surprising feature, periodicity in the imaginary direction, is what makes its inverse (the logarithm) multi-valued.
e^(i pi) = cos pi + i sin pi = -1 + 0 i = -1, so e^(i pi) + 1 = 0.
Euler's identity is just e^z evaluated at z = i pi: size e^0 = 1, angle pi, landing on -1.
Do not read e^(i y) as repeated multiplication; for an imaginary exponent that has no meaning. It is defined by the cos y + i sin y formula (equivalently by the power series), and from that the laws of exponents are proved.