The number $e$ can be written as a series:

$$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \cdots$$

Euler had the courage to put an imaginary number in the exponent. Changing $x$ to $i\theta$ and using $i^2 = -1$, $i^3 = -i$, $i^4 = 1$:

$$e^{i\theta} = 1 + i\theta - \frac{\theta^2}{2!} - i\frac{\theta^3}{3!} + \frac{\theta^4}{4!} + i\frac{\theta^5}{5!} - \cdots$$

Separating terms without $i$ from terms with $i$:

$$e^{i\theta} = \underbrace{\left(1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \cdots\right)}_{\cos\theta} + i\underbrace{\left(\theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \cdots\right)}_{\sin\theta}$$

The two series in parentheses are exactly those of cosine and sine. This is how the formula of Euler:

$$e^{i\theta} = \cos\theta + i\sin\theta.$$

Geometrically, $e^{i\theta}$ is a point that walks on a circle of radius 1: the real part is the cosine and the imaginary part is the sine. See the point rotate:

The point e^(iθ) goes around the circle. The blue projection is cos θ and the gold one is sin θ. Click “Go to θ = π”.

Now do $\theta = \pi$. As $\cos\pi = -1$ and $\sin\pi = 0$, we have $e^{i\pi} = -1$, that is:

$$e^{i\pi} + 1 = 0.$$

The five most important numbers in mathematics appear on a single line ($0$, $1$, $e$, $i$ and $\pi$) and the three basic operations (addition, multiplication and power). That’s why it is often called “the most beautiful equation in mathematics”. Euler’s formula is still used today in electrical engineering, signal processing and quantum physics.

All topics by Leonhard Euler · ← The number $e$ and compound interest · The spring and differential equations →

Credits and sources of images.