Complex Division¶

Definition¶

For any two complex numbers $z$ and $w$ (where $w \neq 0$$0$), the quotient of complex numbers $z/w$ is defined as $z/w = z \cdot w^{-1} = \dfrac{z\overline{w}}{|w|^2}.$

Interpretation¶

Dividing by $w$ means multiplying by its reciprocal $w^{-1}$, the number with $w\,w^{-1}=1$$1$. In practice one multiplies numerator and denominator by the conjugate $\overline w$, which turns the denominator into the real number $w\overline w=|w|^2$$|w{|}^2$.

In the figure $z=3\mathrm{e}^{\mathrm{i}4\pi/9}$ has length $3$ and angle $80^\circ$, and $w=1.5\mathrm{e}^{\mathrm{i}5\pi/18}$ has length $1.5$ and angle $50^\circ$. The quotient has length $3/1.5=2$ and angle $80^\circ-50^\circ=30^\circ$$30^{\circ }$, so $z/w=2\mathrm{e}^{\mathrm{i}\pi/6}$$2{\mathrm{e}}^{\mathrm{i}\pi /6}$: moduli divide and arguments subtract modulo $2\pi$, the reverse of complex multiplication.

The condition $w\neq0$ is needed just as for real numbers: $0\cdot v=0\neq1$ for every $v$, so $0$ has no reciprocal. Every other complex number has one, which is what makes $\mathbb{C}$ a field.

Key Results¶

• $|z/w|=|z|/|w|$$|z|/|w|$.
• $\overline{z/w}=\overline z /\overline w$$\overline{z}/\overline{w}$.

Examples¶

• $(1+\mathrm{i})/(2-\mathrm{i})=\dfrac{(1+\mathrm{i})(2+\mathrm{i})}{(2-\mathrm{i})(2+\mathrm{i})}=\dfrac{1+3\mathrm{i}}{5}$$\frac{1+3\mathrm{i}}{5}$.

Prerequisites¶

• Complex Reciprocal
• Complex Multiplication
• Complex Number

Related Concepts¶

• Complex Conjugate

Sources¶

• Math II: Linear Algebra · Week 1

Aliases: division of complex numbers, quotient of complex numbers Symbols: complex numbers, pi

Last updated: 20 October 2026 03:02