Products, de Moivre’s formula, and Euler form

Content updated: October 22, 2026

Multiplication of complex numbers multiplies their moduli and adds their angles. Induction from the product formula proves de Moivre’s formula; Euler’s formula provides concise exponential notation with angles measured in radians.

Learning goals: prove the general properties before applying them; retain nonzero conditions for arguments and distinguish two similar complex numbers.

Prerequisites: polar form, complex multiplication, and trigonometric addition formulas.

On this page: Definition · Method · Examples · Practice · Further learning

1. Modulus, conjugation, and the scope of arguments

For $z=a+bi$, we have $|\overline z|=|z|$ because the sums of squares agree. If $z\ne0$ has an argument $\alpha$, its conjugate has an argument $-\alpha$ by reflection across the real axis. Angles are understood modulo $2\pi$.

Do not assign an argument to 0. Argument formulas for products and powers apply only when the relevant numbers are nonzero. With principal values in an interval of length $2\pi$, an angle sum may require adjustment by a multiple of $2\pi$; do not assume unrestricted equality as real numbers for all principal values.

2. From products to de Moivre’s formula and Euler form

Let $z_j=r_j(\cos\alpha_j+i\sin\alpha_j)$ with $r_j>0$. Expanding the product and using $i^2=-1$ and the two addition formulas yields

Since $r_1r_2>0$, the product has modulus $r_1r_2$ and an argument $\alpha_1+\alpha_2$. The identity $|z_1z_2|=|z_1||z_2|$ also holds when either number is 0, as both sides then vanish. An algebraic check is the identity

Induction by repeated multiplication establishes the following for every integer $n\ge1$: if $z=r(\cos\alpha+i\sin\alpha)\ne0$, then

The base case $n=1$ holds; an additional factor $z$ multiplying the formula at $n$ adds the angles to $(n+1)\alpha$ and multiplies the modulus by $r$. This is de Moivre’s formula. Hence $|z^n|=|z|^n$; the modulus identity also holds for $z=0$, although no angle is assigned in that case.

We use Euler’s standard formula without proving it here: for $\theta\in\mathbb R$ measured in radians, $e^{i\theta}=\cos\theta+i\sin\theta$. Thus $z=re^{i\alpha}$, and de Moivre’s formula becomes $z^n=r^ne^{in\alpha}$. Do not write $e^{i30}$ for an angle of $30^\circ$; use $e^{i\pi/6}$. The de Moivre formula above follows from multiplication, without treating Euler’s formula as a proof of that same product formula.

3. Four worked examples

Example 1. Conjugating 1 + √3i

For $z=1+\sqrt3i$ and $\overline z=1-\sqrt3i$, both moduli are 2. Choose arguments $\pi/3$ and $-\pi/3$, corresponding to $60^\circ$ and $-60^\circ$. Their respective polar forms use those angle signs and represent reflections across the real axis.

Example 2. Multiplying two polar forms

Let $z_1=\sqrt3-i$ and $z_2=1+\sqrt3i$. Both have modulus 2; choose angles $-\pi/6$ and $\pi/3$. The product has modulus 4 and angle $\pi/6$, so

Expanding $(\sqrt3-i)(1+\sqrt3i)$ produces the same result. Both checks identify the same complex number in both components.

Example 3. Three powers of √3 + i

Set $u=\sqrt3+i=2[\cos(\pi/6)+i\sin(\pi/6)]$. By de Moivre’s formula,

Example 4. Distinguishing two Euler forms

For the same $u=\sqrt3+i$, we have $u=2e^{i\pi/6}$; exponential calculation of $u^2,u^3,u^4$ gives the three results in Example 3. If instead $v=1+\sqrt3i$ is intended, then $v=2e^{i\pi/3}$ has a different angle. In that case

The numbers $u,v$ have the same modulus 2 but different arguments; do not switch the complex number between the problem and its calculation. The number $u^4$ has real part $-8$, consistent with its quadrant-II angle $2\pi/3$.

4. Practice

1. Find the modulus and an argument of $(1+i)(1-i)$.
2. Calculate $(1+i)^4$ by de Moivre’s formula or expansion.
3. If $z\ne0$ has an argument $\alpha$, what is an argument of $z^3$? Does the statement apply to $z=0$?

5. Common mistakes

Do not treat argument as a unique angle before fixing a branch. Do not assign an angle to 0. Use radians in Euler exponents. Follow the same complex number throughout the calculation; equal moduli do not imply equal arguments.

6. Further learning

Complex numbers (English lesson) · Representing complex numbers (English lesson) · Roots of complex numbers (English lesson) · Quadratic equations over the complex numbers (English lesson) · Trigonometric identities (English lesson).