Euler's identity

What is Euler's identity?

There is a famous short equation that ties together the five most important numbers in maths: 0, 1, π, e, and i, the square root of minus one. It is written \( e^{i\pi} + 1 = 0 \), and a lot of mathematicians will tell you it is the most beautiful line in the subject.

Here is the idea behind it. Think of numbers as points on a flat sheet. Multiplying by i does not make a number bigger or smaller. Instead it turns it a quarter of the way round, ninety degrees. Do it twice and you have turned halfway round, which flips a number to its negative. That is exactly what it means to say i times i is minus one.

Raising e to an imaginary power is a way of going round a circle. The number in the exponent tells you how far round to walk, measured as an angle. Go exactly halfway round, an angle of π, and you land on the point -1. Add 1 to that and you are back where you started, at 0. That is the whole equation.

So the line is not a coincidence or a piece of magic. It is a short way of saying that walking half a lap around a circle takes you to the opposite side.

Why does e to the i pi equal minus one?

To see the identity properly you need the complex plane. A complex number has two parts, a real part and an imaginary part, so you can draw it as a point on a flat grid: the real part measured left to right, the imaginary part measured up and down. The ordinary number 1 sits one step to the right. The number i sits one step up.

Now watch what multiplying by i does. It sends 1 (one step right) to i (one step up), and it sends i to -1 (one step left). Every multiplication by i rotates a point ninety degrees anticlockwise about the origin. Multiplying by i twice rotates you a full one hundred and eighty degrees, which is why \( i^2 = -1 \). The imaginary unit is a quarter turn.

Euler's formula pulls this together into one clean statement:

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

Read the right-hand side as a point on the plane. Its horizontal coordinate is \( \cos\theta \) and its vertical coordinate is \( \sin\theta \), and by Pythagoras those always add in square to 1. So \( e^{i\theta} \) is a point at distance 1 from the origin, sitting on the unit circle, at angle \( \theta \) round from the positive real axis. As \( \theta \) grows, the point sweeps around the circle.

Set \( \theta = \pi \), half a full turn. Then \( \cos\pi = -1 \) and \( \sin\pi = 0 \), so

\[ e^{i\pi} = \cos\pi + i\sin\pi = -1. \]

Add 1 to both sides and you have \( e^{i\pi} + 1 = 0 \), the identity itself.

Why should an exponential turn into a rotation at all? The honest answer is in the rate of change. If you track the point \( e^{i\theta} \) as \( \theta \) increases, its velocity is \( \frac{d}{d\theta} e^{i\theta} = i\,e^{i\theta} \). Multiplying by i is that quarter turn, so the velocity is always at right angles to the arrow pointing out from the origin. A point whose motion stays perpendicular to its own radius cannot move nearer or further out. It can only circle. And because the arrow has length 1 and the velocity has length 1 too, it circles at a steady unit speed, sweeping exactly \( \theta \) radians of arc by angle \( \theta \). That is the circle, falling straight out of the derivative.

Euler's formula and the complex exponential

Deriving the formula from power series

The cleanest route to Euler's formula runs through the Taylor series, which converge everywhere on the complex plane. Start from the exponential and substitute \( x = i\theta \):

\[ e^{i\theta} = \sum_{n=0}^{\infty} \frac{(i\theta)^n}{n!} = 1 + i\theta - \frac{\theta^2}{2!} - i\frac{\theta^3}{3!} + \frac{\theta^4}{4!} + \cdots \]

The powers of i cycle with period four: \( i^0 = 1,\ i^1 = i,\ i^2 = -1,\ i^3 = -i \), then back to 1. Sort the terms by whether they carry a factor of i. The terms without i are the even powers, and the terms with i are the odd powers:

\[ e^{i\theta} = \left( 1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \cdots \right) + i\left( \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \cdots \right). \]

Those two brackets are exactly the Taylor series for \( \cos\theta \) and \( \sin\theta \). The real part is the cosine, the imaginary part is the sine, and the split is clean because the alternating signs of the trig series are the same signs the powers of i produce. Hence \( e^{i\theta} = \cos\theta + i\sin\theta \), and at \( \theta = \pi \) it collapses to \( e^{i\pi} = -1 \).

The differential-equation argument

If series feel like sleight of hand, define \( f(\theta) = \cos\theta + i\sin\theta \) and differentiate: \( f'(\theta) = -\sin\theta + i\cos\theta = i\,f(\theta) \), with \( f(0) = 1 \). The function \( g(\theta) = e^{i\theta} \) satisfies the same equation \( g' = i g \) with the same initial value. Since the initial-value problem \( y' = iy,\ y(0)=1 \) has a unique solution, the two functions agree everywhere. Equivalently, form \( h(\theta) = f(\theta)\,e^{-i\theta} \); then \( h' = 0 \) and \( h(0) = 1 \), so \( h \equiv 1 \) and \( f = e^{i\theta} \). The identity is the value at half a turn.

Roots of unity

Euler's formula turns the equation \( z^n = 1 \) into geometry. Its solutions are \( z_k = e^{2\pi i k / n} \) for \( k = 0, 1, \dots, n-1 \), the n points spaced evenly around the unit circle, the vertices of a regular n-gon. For \( n = 2 \) the two roots are \( 1 \) and \( e^{i\pi} = -1 \), so the identity is simply the statement that -1 is a square root of unity sitting half a lap around. These roots underpin the discrete Fourier transform and the fast algorithms built on it.

Phasors and Fourier analysis

The reason engineers and physicists lean on \( e^{i\theta} \) is that it packages an oscillation into a single rotating arrow, a phasor. A signal like \( A\cos(\omega t + \phi) \) is the real part of \( A\,e^{i(\omega t + \phi)} \), an arrow of length \( A \) spinning at angular frequency \( \omega \). Differentiation becomes multiplication by \( i\omega \), which is why linear circuits and oscillators are so much easier to handle in this language. Fourier analysis takes the step to the limit: any reasonable signal can be written as a sum, or an integral, of these pure rotations,

\[ f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} \hat{f}(\omega)\, e^{i\omega t}\, d\omega, \]

with each \( e^{i\omega t} \) a unit circle traced at its own rate. Euler's identity is the single grain of that idea, the case where the frequency and time conspire to put the arrow exactly on -1. Read that way, the most beautiful equation in mathematics is also the atom of how we decompose sound, light and every other signal into frequencies.