# Reciprocal of a complex number

Calculator and formulas for reciprocal (inverse) of a complex number

## Reciprocal value calculator

This function calculates the reciprocal of a complex number.

To perform the calculation, enter the value of the complex number whose reciprocal is to be calculated. Then click the button 'Calculate'.

Reciprocal value calculator

 Input Complex number + i Decimal places 0 1 2 3 4 6 8 10 Results Reciprocal

## Formula for reciprocal value

In the following description, $$z$$ stands for the complex number. $$x$$ stands for the real value $$Re$$ and $$y$$ for the imaginary value $$Im$$.

$$z = Re(x)+Im(y)$$
$$\displaystyle\frac{1}{z}=\frac{x}{x^2+y^2}+\frac{-y}{x^2+y^2}$$

Example

$$z = Re(3)+Im(5)$$
$$\displaystyle\frac{1}{z}=\frac{3}{3^2+5^2}+\frac{-5}{3^2+5^2}$$
$$\displaystyle\frac{1}{z}=\frac{3}{9+25}+\frac{-5}{9+25}$$
$$\displaystyle\frac{1}{z}=\frac{3}{34}+\frac{-5}{34}$$
$$\displaystyle\frac{1}{z}= 0.088-0.147i$$