Euler’s identity is one of my favorite equations.

It contains five constants, each exactly once: the additive identity $0$, the multiplicative identity $1$, the circle constant $\pi$, the base of the natural exponential $e$, and the imaginary unit $i = \sqrt{-1}$. They are joined by one addition and one exponentiation, with no other symbols.

The identity is the $\theta = \pi$ case of Euler’s formula

For a real exponent, $e^{x}$ is a positive real number. For a purely imaginary exponent, $e^{i\theta}$ has modulus $1$ and argument $\theta$, so it is the point on the unit circle at angle $\theta$. At $\theta = \pi$ that point is $-1$, which is the identity. This post derives Euler’s formula three ways, geometrically, as a limit, and from the power series, then reads off what each constant contributes.

Multiplying by i is a quarter turn

Put a complex number $a + bi$ at the point $(a, b)$ in the plane. Addition is then ordinary vector addition. Multiplication has a geometric description too.

Write each number in terms of its modulus $r$ (its distance from the origin) and its argument $\phi$ (its angle from the positive real axis), so $z = r(\cos\phi + i\sin\phi)$. Multiplying two such numbers and using the angle-addition formulas for sine and cosine gives

So multiplying two complex numbers multiplies their lengths and adds their angles.

Now take the special case of multiplying by $i$. Its modulus is $1$ and its argument is a quarter turn, $90^\circ$. So multiplying any point by $i$ leaves its length unchanged and rotates it a quarter turn counterclockwise. Check it on the powers of $i$: starting from $1$ at $(1, 0)$, one multiplication gives $i$ at $(0, 1)$, the next gives $i^2 = -1$ at $(-1, 0)$, then $i^3 = -i$ at $(0, -1)$, then $i^4 = 1$, home again after four quarter turns. The imaginary unit is the $90^\circ$ rotation of the plane, written as a number.

The complex exponential traces the unit circle

Define $z(\theta) = e^{i\theta}$ and follow the point as $\theta$ increases.

The exponential is the function that is its own derivative. With the constant $i$ sitting in the exponent, the chain rule gives

Read that as a statement about motion: the velocity $z'(\theta)$ is the position $z(\theta)$ multiplied by $i$, that is, the position turned a quarter turn. Two things follow.

• The velocity is always perpendicular to the position. A point whose velocity stays perpendicular to its own radius never moves closer to or farther from the origin; only its angle changes. It travels on a circle.
• The speed equals the radius. Since $|i| = 1$, we have $|z'| = |z|$. Starting from $z(0) = e^0 = 1$, the radius is $1$ and stays $1$, and the speed is $1$.

So $e^{i\theta}$ travels around the unit circle at unit speed. After parameter $\theta$ it has covered arc length $\theta$, which on the unit circle is an angle of $\theta$ radians (angle measured so that a full turn is $2\pi$). Its coordinates are therefore $(\cos\theta, \sin\theta)$, which is Euler’s formula read straight off the motion:

Drag $\theta$ below and watch $e^{i\theta}$ move around the circle, its projections on the two axes tracing cosine and sine. The red velocity arrow stays exactly a quarter turn ahead of the radius. At $\theta = \pi$ the point sits at $-1$:

The compound-interest limit

A second derivation comes from continuous compounding.

Money at annual rate $r$, compounded $n$ times, multiplies by $(1 + r/n)^n$ over the year, and as the compounding gets finer this tends to $e^r$. Each of the $n$ steps multiplies by $1 + r/n$, a number just larger than $1$, so the balance grows.

Now make the rate imaginary, $r = i\theta$, so each step multiplies by $1 + i\theta/n$. That factor has modulus $\sqrt{1 + \theta^2/n^2}$, barely more than $1$, and argument $\arctan(\theta/n) \approx \theta/n$. Each step barely stretches, but it does turn, by about $\theta/n$. Over $n$ steps the turning accumulates to $\theta$ while the modulus tends to $1$, since $(1 + \theta^2/n^2)^{n/2} \to 1$. In the limit,

a point on the unit circle at angle $\theta$.

The widget draws the partial products $1,\ (1 + i\theta/n),\ (1 + i\theta/n)^2,\ \dots$ as a chain of segments. Because each factor stretches a little, the chain bulges just outside the circle and spirals; as $n$ grows the stretch per step shrinks and the corners settle onto the red arc, the open blue endpoint closing in on the true $e^{i\theta}$ in red. Slide $n$ up and watch the polygon flatten onto the circle:

The readout total turn is $n\arctan(\theta/n)$, the angle actually accumulated by the $n$ steps; it climbs to $\theta$ as $n$ grows, while the modulus $(1 + \theta^2/n^2)^{n/2}$ falls back to $1$.

The series check

For a derivation with no picture at all, expand. The exponential series $e^{x} = \sum_{k \ge 0} x^k/k!$ converges for every complex $x$, and absolutely, so we may substitute $x = i\theta$ and regroup the terms:

The powers of $i$ run through $1, i, -1, -i$ with period four. The even powers $k = 2m$ give $i^{2m} = (-1)^m$ and land on the real axis; the odd powers $k = 2m+1$ give $i^{2m+1} = (-1)^m i$ and land on the imaginary axis. Splitting the sum along that parity,

The two halves are exactly the Taylor series of cosine and sine. Same conclusion as the geometry, now as an algebraic identity. (Absolute convergence is what licenses the regrouping.)

The half-turn from 1 to −1

Set $\theta = \pi$. The point $e^{i\theta}$ has walked halfway around the unit circle, from $1$ to $-1$:

Now each constant contributes something specific, and none can be changed:

• $e$ is the base of the exponentiation.
• $i$ sends the exponent to the imaginary axis, where $e^{i\theta}$ lies on the unit circle at angle $\theta$.
• $\pi$ is the length of a half-turn in radians, so $i\pi$ is precisely half a circle.
• $1$ is where the walk begins, and the quantity added back at the end.
• $0$ is where you land: start at $1$, rotate to $-1$, add $1$.

Change any one of them and the statement is false. The identity is a genuine relation among these five constants, not a coincidence of notation.

Applications

Euler’s formula is why complex numbers are the standard tool for periodic and rotational problems.

• Fourier analysis. A periodic signal is a sum of terms $c_k\, e^{ik\omega t}$, each a vector of length $|c_k|$ spinning at frequency $k\omega$. Adding spinning vectors reconstructs any waveform, and pulling the coefficients back out is what a Fourier transform does. The plane waves $e^{ipx/\hbar}$ in the uncertainty-principle post are the same objects carrying a particle’s momentum.
• Rotations in the plane. Multiplying a point by $e^{i\alpha}$ rotates it by $\alpha$, so one complex multiplication replaces a $2\times 2$ rotation matrix. Graphics, robotics, and navigation use this, and its three-dimensional cousin is the quaternion.
• AC circuits. An engineer writes an alternating voltage as $\mathrm{Re}\,(V e^{i\omega t})$ and tracks the complex amplitude $V$, called a phasor. The $90^\circ$ phase shifts of a capacitor and an inductor are just multiplications by $-i$ and $i$ (their impedances are $1/(i\omega C)$ and $i\omega L$), which replaces the calculus of sinusoids with algebra on complex numbers.
• Quantum mechanics. A state of definite energy $E$ evolves by the pure phase $e^{-iEt/\hbar}$: it rotates in the complex plane without changing magnitude, so total probability is conserved, and it is the relative phase between states that produces interference.

Three corollaries

• The roots of unity. The solutions of $z^n = 1$ are $z_k = e^{2\pi i k / n}$ for $k = 0, 1, \dots, n-1$: $n$ equally spaced points on the unit circle, the vertices of a regular $n$-gon with one corner at $1$. Multiplying two of them adds their angles, so they form a group under multiplication.
• de Moivre’s formula. Raising Euler’s formula to the $n$-th power gives $(\cos\theta + i\sin\theta)^n = e^{in\theta} = \cos n\theta + i\sin n\theta$. Expanding the left side with the binomial theorem and matching real and imaginary parts hands you $\cos n\theta$ and $\sin n\theta$ as polynomials in $\cos\theta$ and $\sin\theta$, the multiple-angle identities at no extra cost.
• $i$ to the $i$ is real. Since $i = e^{i\pi/2}$, we get $i^i = \bigl(e^{i\pi/2}\bigr)^i = e^{i^2 \pi/2} = e^{-\pi/2} \approx 0.2079$. An imaginary number raised to an imaginary power lands on the real line.

A little history

Euler’s formula predates Euler’s compact statement of it. Roger Cotes published the equivalent logarithmic relation $i\theta = \ln(\cos\theta + i\sin\theta)$ in 1714, though he did not exponentiate it. Abraham de Moivre had the $n$-th power identity that carries his name, in equivalent form, by the 1720s. Leonhard Euler stated $e^{i\theta} = \cos\theta + i\sin\theta$ in his 1748 Introductio in analysin infinitorum and put the exponential at the center of the subject. The compact $e^{i\pi} + 1 = 0$ is a later way of writing the $\theta = \pi$ case. When David Wells polled readers of The Mathematical Intelligencer in 1990 for the most beautiful theorem in mathematics, Euler’s identity finished first.

Common misconceptions

• $e^{i\pi}$ is not $e$ multiplied by itself $i\pi$ times. Doing something an imaginary number of times has no meaning. For complex $z$, the symbol $e^{z}$ is defined by the power series $\sum_k z^k/k!$, and equivalently by the limit and the differential equation used above; all three agree. Repeated multiplication is only the special case of a positive-integer exponent.
• The complex logarithm is multivalued, and so are complex powers. Because $e^{i\theta}$ returns to the same point every $2\pi$, the equation $e^{w} = z$ has infinitely many solutions differing by $2\pi i$. So $\log z$, and any power $z^a = e^{a\log z}$, come with a choice of branch; the principal value takes the angle in $(-\pi, \pi]$. This is the fine print behind $i^i$: the other branches give the further real values $e^{-(\pi/2 + 2\pi k)}$, and $e^{-\pi/2}$ is only the principal one.

References

• Leonhard Euler. Introductio in analysin infinitorum. Lausanne, 1748. States $e^{i\theta} = \cos\theta + i\sin\theta$ and builds analysis around the exponential.
• Roger Cotes. Logometria. Philosophical Transactions of the Royal Society, 29(338):5-45, 1714. The logarithmic form of the formula, before Euler.
• David Wells. Are these the most beautiful? The Mathematical Intelligencer, 12(3):37-41, 1990. The reader poll in which Euler’s identity finished first.
• Paul J. Nahin. Dr. Euler’s Fabulous Formula. Princeton University Press, 2006. A book-length tour of the formula and its uses.
• Tristan Needham. Visual Complex Analysis. Oxford University Press, 1997. Chapter 1 develops the rotational, geometric reading of complex multiplication and $e^{i\theta}$.