Paul Calnan's Blog
Published December 21, 2020

Euler's formula states that for any real number \(x\):

\[ e^{ix} = \cos x + i \sin x \]

When \(x = \pi\), Euler's formula evaluates to:

\[ e^{i\pi} + 1 = 0 \]

This is known as Euler's identity and it links five fundamental mathematical constants (\(\pi\), \(e\), \(i\), \(0\), and \(1\)) in a beautifully simple equation. Richard Feynman called the equation "the most remarkable formula in mathematics."

The formula can be interpreted as saying that the function \(e^{i\phi}\) is a unit complex number, i.e. it traces out the unit circle in the complex plane as \(\phi\) ranges through the real numbers.

We can derive Euler's formula from the Maclaurin series expansions of \(\sin x\), \(\cos x\), and \(e^x\):

\[ \begin{align} \sin x &= \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!} x^{2n+1} \\ &= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \\ \cos x &= \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!} x^{2n} \\ &= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \\ e^x &= \sum_{n = 0}^{\infty} \frac{x^n}{n!} \\ &= 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \end{align} \]

Thus:

\[ \begin{align} e^{ix} &= \sum_{n = 0}^{\infty} \frac{(ix)^n}{n!} \\ e^{ix} &= 1 + ix + \frac{(ix)^2}{2!} + \frac{(ix)^3}{3!} + \frac{(ix)^4}{4!} + \frac{(ix)^5}{5!} + \frac{(ix)^6}{6!} + \frac{(ix)^7}{7!} + \cdots \\ &= 1 + ix - \frac{x^2}{2!} - \frac{ix^3}{3!} + \frac{x^4}{4!} + \frac{ix^5}{5!} - \frac{x^6}{6!} - \frac{ix^7}{7!} + \cdots \\ &= \left( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} - \cdots \right) + \left( ix - \frac{ix^3}{3!} + \frac{ix^5}{5!} - \frac{ix^7}{7!} + \cdots \right) \\ &= \left( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} - \cdots \right) + i\left( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \right) \\ &= \cos x + i\sin x \end{align} \]

This yields Euler's identity. Substitute \(\pi\) for \(x\). Since \(\cos \pi = -1\) and \(\sin \pi = 0\), it follows that:

\[ e^{i\pi} = -1 + 0i \]

or

\[ e^{i\pi} + 1 = 0 \]