Complex Reciprocal¶

Definition¶

The reciprocal of the complex number $z\neq 0$ is defined as $z^{-1} = |z|^{-2} \overline{z}$$|z{|}^{-2}\overline{z}$.

Interpretation¶

The reciprocal $z^{-1}$ is the number that gives $1$ when multiplied by $z$. Since the product of $z$ with its conjugate is $z\bar z=|z|^2$$|z{|}^2$, a positive real number for $z\neq0$$0$, dividing $\bar z$ by the square of the modulus does this: $z\cdot\bar z/|z|^2=1$$1$.

In the figure $z$ has length $2$ and angle $40^\circ$; its conjugate $\bar z$ (dashed) has the same length and the angle $-40^\circ$, and dividing by $|z|^2=4$ shrinks it along the same ray to length $\tfrac{1}{2}$. In general $1/z$ has the direction of $\bar z$ and modulus $1/|z|$, so points outside the unit circle are sent inside it and nonzero points inside are sent outside.

The condition $z\neq0$ is needed because $0\cdot w=0\neq1$ for every $w$. Every other complex number has a reciprocal, which is what makes $\mathbb{C}$ a field, and dividing by $w$ means multiplying by $w^{-1}$ (complex division). Reciprocals are not taken part by part: $1/(1+\mathrm{i})$ is $\tfrac{1}{2}(1-\mathrm{i})$$\mathrm{i})$, not $1+1/\mathrm{i}$$1/\mathrm{i}$.

Key Results¶

• We have $|z^{-1}|=1/|z|$ and $\arg(z^{-1})=-\arg z$ modulo $2\pi$. For the principal argument, $\operatorname{Arg}(z^{-1})=-\operatorname{Arg}z$ unless $z$ is a negative real number, in which case both equal $\pi$.

Examples¶

• The reciprocal of $z = 1 + 2\mathrm{i}$ is $\displaystyle z^{-1} = \frac{1}{5} - \frac{2}{5}\mathrm{i}$$\frac{2}{5}\mathrm{i}$.

Prerequisites¶

• Complex Conjugate
• Complex Absolute Value
• Complex Number

Related Concepts¶

• Complex Multiplication

Sources¶

• Math II: Linear Algebra · Week 1

Aliases: reciprocal of a complex number Symbols: complex numbers, pi

Last updated: 20 October 2026 03:02