De Moivre’s Theorem and Euler’s Formula

Imagine you need to compute $(1+i)^{20}$. Multiplying $1+i$ by itself twenty times is a nightmare of bookkeeping. But there is a viewpoint from which the answer falls out in a single line. That viewpoint is the polar form of complex numbers, and the two results that unlock it are De Moivre’s theorem and Euler’s formula. Together they reveal something startling: rotation, exponential growth, and the trigonometric functions are all the same idea wearing different costumes. This page will teach you why that is true, not just how to use it.

Learning Objectives

• Write complex numbers in polar form and multiply them by combining moduli and adding arguments.
• State and apply De Moivre’s theorem to compute integer powers of complex numbers.
• Find all $n$ complex $n$-th roots of a number and see why they sit evenly on a circle.
• Understand Euler’s formula $e^{i\theta}=\cos\theta+i\sin\theta$ and where it comes from.
• Interpret Euler’s identity $e^{i\pi}+1=0$ and use the exponential form to prove trig identities effortlessly.

Quick Answer

A complex number can be written as $z=r(\cos\theta+i\sin\theta)$, where $r$ is its distance from the origin (modulus) and $\theta$ is its angle from the positive real axis (argument). De Moivre’s theorem says that raising $z$ to a power multiplies the angle: $z^n=r^n(\cos n\theta+i\sin n\theta)$. The same rule run backwards gives $n$ equally-spaced $n$-th roots. Euler’s formula, $e^{i\theta}=\cos\theta+i\sin\theta$, explains why angles add under multiplication — because exponents add. Setting $\theta=\pi$ produces Euler’s identity $e^{i\pi}+1=0$, linking $e$, $i$, $\pi$, $1$, and $0$ in one equation.

Where It Came From

The story begins with a practical frustration. By the early 1700s mathematicians could add, subtract, multiply, and divide complex numbers, but taking high powers and roots was brutal. Abraham de Moivre (1667–1754), a French Huguenot who fled religious persecution to London and earned his living tutoring and computing odds for gamblers and insurers, noticed a pattern while working on problems in probability and the roots of equations. Around 1707, and more explicitly by 1722, he saw that the relationship $(\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta$ turned the tedious repeated multiplication of complex numbers into simple multiplication of angles. The need was concrete: to solve equations like $x^n=1$ and to expand $\cos n\theta$ in terms of $\cos\theta$, which appeared throughout the era’s work on trigonometric series.

But why it worked was still mysterious. That answer came from Leonhard Euler (1707–1783), the most prolific mathematician in history. Euler was studying infinite series — the power series for $e^x$, $\sin x$, and $\cos x$. Around 1740–1748 he made a daring move: substitute an imaginary number $i\theta$ into the exponential series. When he did, the terms sorted themselves perfectly into the cosine series and $i$ times the sine series. Out popped $e^{i\theta}=\cos\theta+i\sin\theta$. Suddenly De Moivre’s theorem was no longer a lucky pattern — it was just the ordinary law of exponents $(e^{i\theta})^n=e^{in\theta}$ in disguise. Euler had unified trigonometry with the exponential function, revealing that the sines and cosines describing oscillation and the exponentials describing growth are two faces of one object. The special case $e^{i\pi}+1=0$ has been voted the most beautiful equation in mathematics ever since.

Polar Form and Why Multiplication Adds Angles

Every complex number $z=x+iy$ can be located as a point in the plane. Instead of coordinates $x$ and $y$, use its modulus $r=|z|=\sqrt{x^2+y^2}$ and its argument $\theta=\arg z$, the angle satisfying $\tan\theta=y/x$ (chosen in the correct quadrant). Then

$x=r\cos\theta,\qquad y=r\sin\theta,\qquad z=r(\cos\theta+i\sin\theta).$

The magic appears when you multiply two numbers. Let $z_1=r_1(\cos\alpha+i\sin\alpha)$ and $z_2=r_2(\cos\beta+i\sin\beta)$. Multiplying out and using the angle-sum identities for sine and cosine:

$z_1 z_2=r_1 r_2\big(\cos(\alpha+\beta)+i\sin(\alpha+\beta)\big).$

So moduli multiply and arguments add. Multiplication by a complex number is a rotation combined with a scaling. This single fact is the engine behind everything below.

Worked example. Convert $z=1+i$ to polar form. Its modulus is $r=\sqrt{1^2+1^2}=\sqrt{2}$, and since it lies in the first quadrant with equal parts, $\theta=45^\circ=\pi/4$. Thus $1+i=\sqrt{2}\left(\cos\dfrac{\pi}{4}+i\sin\dfrac{\pi}{4}\right)$.

De Moivre’s Theorem: Powers Made Easy

If multiplying adds angles, then multiplying $z$ by itself $n$ times adds the angle $n$ times and multiplies the modulus $n$ times. That is De Moivre’s theorem:

$\big[r(\cos\theta+i\sin\theta)\big]^n=r^n\big(\cos n\theta+i\sin n\theta\big).$

For the special case $r=1$ it reads $(\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta$. It holds for every integer $n$ (positive, negative, or zero).

Worked example — the hard power made trivial. Compute $(1+i)^{20}$. From above, $1+i=\sqrt{2}\,(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4})$. Applying De Moivre’s theorem:

$(1+i)^{20}=(\sqrt{2})^{20}\left(\cos\frac{20\pi}{4}+i\sin\frac{20\pi}{4}\right).$

Now $(\sqrt{2})^{20}=2^{10}=1024$, and $\frac{20\pi}{4}=5\pi$. Since $\cos 5\pi=-1$ and $\sin 5\pi=0$:

$(1+i)^{20}=1024(-1+0i)=-1024.$

Twenty multiplications collapsed to one clean line.

Worked example — a trig identity for free. Take $n=2$: $(\cos\theta+i\sin\theta)^2=\cos 2\theta+i\sin 2\theta$. Expand the left side: $\cos^2\theta-\sin^2\theta+2i\sin\theta\cos\theta$. Matching real and imaginary parts gives the double-angle formulas $\cos 2\theta=\cos^2\theta-\sin^2\theta$ and $\sin 2\theta=2\sin\theta\cos\theta$ instantly. De Moivre’s theorem is a factory for such identities.

Roots of Complex Numbers

Running De Moivre’s theorem backwards finds roots. To solve $w^n=z$ where $z=r(\cos\theta+i\sin\theta)$, note that the angle $\theta$ is only defined up to full turns, so we should really write the argument as $\theta+2\pi k$ for any integer $k$. The $n$-th roots are

$w_k=r^{1/n}\left(\cos\frac{\theta+2\pi k}{n}+i\sin\frac{\theta+2\pi k}{n}\right),\qquad k=0,1,\dots,n-1.$

There are exactly $n$ distinct roots. They all share the modulus $r^{1/n}$, so they lie on a circle, and their angles are spaced $2\pi/n$ apart — the vertices of a regular $n$-gon.

Worked example — the cube roots of unity. Solve $w^3=1$. Here $r=1$ and $\theta=0$. The roots are

$w_k=\cos\frac{2\pi k}{3}+i\sin\frac{2\pi k}{3},\quad k=0,1,2.$

• $k=0:\ \cos 0+i\sin 0=1.$
• $k=1:\ \cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3}=-\tfrac{1}{2}+i\tfrac{\sqrt{3}}{2}.$
• $k=2:\ \cos\frac{4\pi}{3}+i\sin\frac{4\pi}{3}=-\tfrac{1}{2}-i\tfrac{\sqrt{3}}{2}.$

These three points form an equilateral triangle inscribed in the unit circle. Check: cubing $-\frac{1}{2}+i\frac{\sqrt{3}}{2}$ (which has modulus $1$ and angle $120^\circ$) triples the angle to $360^\circ$, landing back at $1$. Correct.

Euler’s Formula and the Exponential Form

Euler discovered that the whole apparatus above is just the exponential function evaluated at imaginary arguments:

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

Here is the heart of his reasoning. The power series are $e^{x}=\sum \frac{x^n}{n!}$, $\cos x=\sum \frac{(-1)^k x^{2k}}{(2k)!}$, and $\sin x=\sum \frac{(-1)^k x^{2k+1}}{(2k+1)!}$. Substitute $x=i\theta$ into the exponential series. Because the powers of $i$ cycle as $i,-1,-i,1,\dots$, the even-power terms become the cosine series and the odd-power terms become $i$ times the sine series. The identity follows.

With this, polar form becomes the exponential form $z=re^{i\theta}$, and De Moivre’s theorem is simply the exponent rule:

$(re^{i\theta})^n=r^n e^{in\theta}.$

Worked example — division the easy way. Compute $\dfrac{e^{i\pi/2}}{e^{i\pi/6}}$. Subtract exponents: $e^{i(\pi/2-\pi/6)}=e^{i\pi/3}=\cos 60^\circ+i\sin 60^\circ=\tfrac{1}{2}+i\tfrac{\sqrt{3}}{2}$.

Euler’s identity. Set $\theta=\pi$: $e^{i\pi}=\cos\pi+i\sin\pi=-1+0i=-1$, so

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

Five fundamental constants — $e$, $i$, $\pi$, $1$, and $0$ — bound together by the three basic operations of addition, multiplication, and exponentiation. Geometrically it says: starting at $1$ and rotating a half-turn about the origin lands you at $-1$.

A useful pair of consequences, the inverse Euler formulas, come from adding and subtracting $e^{i\theta}$ and $e^{-i\theta}$:

$\cos\theta=\frac{e^{i\theta}+e^{-i\theta}}{2},\qquad \sin\theta=\frac{e^{i\theta}-e^{-i\theta}}{2i}.$

These turn hard trigonometric problems into easy algebra with exponentials.

Real-World Applications

• Electrical engineering (AC circuits). Alternating voltages and currents are modeled as $V e^{i\omega t}$. Impedance, phase lag, and resonance all become simple complex arithmetic instead of trigonometric slog. Engineers use $j$ instead of $i$ (since $i$ means current), but it is exactly Euler’s formula.
• Signal processing and the Fourier transform. Every signal is decomposed into rotating components $e^{i\omega t}$. MP3 compression, image formats like JPEG, MRI reconstruction, and Wi-Fi all run on this.
• Quantum mechanics. The wavefunction evolves by the factor $e^{-iEt/\hbar}$; physical probabilities and interference come directly from complex exponentials.
• Computer graphics and robotics. Rotations in the plane are multiplication by $e^{i\theta}$; the 3D generalization (quaternions) powers game engines and spacecraft attitude control.

Common Mistakes

1. Forgetting the $+2\pi k$ when taking roots. Misconception: $w^n=z$ has one answer. Why wrong: the argument of a complex number is defined only up to full turns, so there are $n$ distinct roots. Correction: always list $k=0,1,\dots,n-1$ and expect $n$ equally-spaced answers on a circle.

2. Putting the argument in the wrong quadrant. Misconception: $\theta=\arctan(y/x)$ always. Why wrong: $\arctan$ only returns angles in $(-90^\circ,90^\circ)$, so for $z=-1-i$ (third quadrant) it gives $45^\circ$ instead of the correct $225^\circ$. Correction: check the signs of $x$ and $y$ and adjust by $\pm\pi$.

3. Applying De Moivre’s theorem to a non-unit modulus without powering it. Misconception: $[r(\cos\theta+i\sin\theta)]^n=\cos n\theta+i\sin n\theta$. Why wrong: the modulus must also be raised. Correction: the result is $r^n(\cos n\theta+i\sin n\theta)$ — never drop the $r^n$.

Comparison and Connections

De Moivre’s theorem and Euler’s formula describe the same phenomenon at different levels. De Moivre is the rule for powers; Euler is the reason the rule holds.

Feature	De Moivre’s theorem	Euler’s formula
Statement	$(\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta$	$e^{i\theta}=\cos\theta+i\sin\theta$
Primary use	Powers and roots of complex numbers	Converting between exponential and trig form
Restriction on $n$	Integer $n$ (for the classic statement)	Holds for all real $\theta$
Deeper meaning	A pattern	Explains why the pattern exists

The polar form $r(\cos\theta+i\sin\theta)$ and the exponential form $re^{i\theta}$ are the same object — Euler’s formula is the bridge. Do not confuse Euler’s formula ($e^{i\theta}=\cos\theta+i\sin\theta$) with Euler’s identity ($e^{i\pi}+1=0$), which is just the formula at $\theta=\pi$.

Practice Questions

Recall

State De Moivre’s theorem and write the exponential form of a complex number with modulus $r$ and argument $\theta$. Answer: $[r(\cos\theta+i\sin\theta)]^n=r^n(\cos n\theta+i\sin n\theta)$; exponential form is $z=re^{i\theta}$.

Understanding

Why does $w^n=1$ have exactly $n$ solutions rather than one? Guidance: the argument $0$ can be written as $2\pi k$ for any integer $k$; dividing by $n$ produces $n$ distinct angles $2\pi k/n$ before they start repeating, giving $n$ roots evenly spaced on the unit circle.

Application

Compute $(\sqrt{3}+i)^6$. Answer: Modulus $=\sqrt{3+1}=2$, argument $=30^\circ=\pi/6$. So $(\sqrt{3}+i)^6=2^6(\cos\pi+i\sin\pi)=64(-1)=-64$.

Analysis

Use the inverse Euler formulas to show $\cos^2\theta=\tfrac{1}{2}(1+\cos 2\theta)$. Guidance: write $\cos\theta=\frac{e^{i\theta}+e^{-i\theta}}{2}$, square it to get $\frac{e^{2i\theta}+2+e^{-2i\theta}}{4}=\frac{1}{4}(2\cos 2\theta+2)=\frac{1}{2}(1+\cos 2\theta)$.

FAQ

Is $i$ in the exponent "real"? What does raising $e$ to an imaginary power even mean? It is defined by the power series: $e^{i\theta}=\sum (i\theta)^n/n!$. That series converges to $\cos\theta+i\sin\theta$, so the definition is consistent and geometrically meaningful — it is a point on the unit circle at angle $\theta$.

Does De Moivre’s theorem work for fractional powers? The equation $\cos n\theta+i\sin n\theta$ is one of the values when $n$ is a fraction, but fractional powers are multi-valued (that is exactly the roots story). Use the full root formula with $k=0,\dots,n-1$ to get all of them.

Why do the roots always form a regular polygon? Because every root has the same modulus $r^{1/n}$ (same distance from origin) and consecutive arguments differ by $2\pi/n$ (equal angular spacing). Equal radius plus equal angle spacing is the definition of a regular polygon’s vertices.

What is the difference between the argument and the "principal argument"? The argument is any valid angle; the principal argument is the unique one in $(-\pi,\pi]$. When computing, keep track of which convention a problem expects.

How is Euler’s formula related to the unit circle? As $\theta$ increases, $e^{i\theta}$ traces the unit circle counterclockwise at unit speed. Its real part is $\cos\theta$ and its imaginary part is $\sin\theta$ — the circle is the definition of sine and cosine.

Quick Revision

• Polar form: $z=r(\cos\theta+i\sin\theta)$; exponential form: $z=re^{i\theta}$.
• Multiplication: multiply moduli, add arguments. Division: divide moduli, subtract arguments.
• De Moivre: $z^n=r^n(\cos n\theta+i\sin n\theta)$.
• $n$-th roots: modulus $r^{1/n}$, arguments $\dfrac{\theta+2\pi k}{n}$ for $k=0,\dots,n-1$; they form a regular $n$-gon.
• Euler’s formula: $e^{i\theta}=\cos\theta+i\sin\theta$. Identity: $e^{i\pi}+1=0$.
• Inverse Euler: $\cos\theta=\dfrac{e^{i\theta}+e^{-i\theta}}{2}$, $\sin\theta=\dfrac{e^{i\theta}-e^{-i\theta}}{2i}$.

Related Topics

Prerequisites

• Trigonometry overview
• Complex numbers and the trigonometric (polar) form
• Angle-sum and double-angle identities

Related Topics

• Trigonometric identities and their proofs
• The unit circle and radian measure

Next Topics

• Fourier series and signal analysis
• Complex analysis and functions of a complex variable
• Calculus