The Complex Exponential and Euler's Formula

The complex exponential is defined so that it agrees with its real Taylor series, giving Euler's formula:

Because $\cos^2\theta + \sin^2\theta = 1$, the point $e^{i\theta}$ sits on the unit circle: $|e^{i\theta}| = 1$ for every real $\theta$. More generally $e^{a + bi} = e^{a}(\cos b + i\sin b)$, so $|e^{a+bi}| = e^{a}$.

Setting $\theta = \pi$ gives the celebrated Euler identity $e^{i\pi} = -1$, linking $e$, $i$ and $\pi$ in one line.

Worked example. $e^{i\pi/2} = \cos\tfrac\pi2 + i\sin\tfrac\pi2 = 0 + i(1) = i$. And $e^{i\pi} = \cos\pi + i\sin\pi = -1 + 0 = -1$.

Euler's formula turns any complex number into polar form $z = r\,e^{i\theta}$, where $r = |z|$ and $\theta$ is the argument (angle). In polar form multiplication becomes "multiply the moduli, add the angles" — the cleanest way to see rotation.