Inverse of Complex Numbers

What is the Inverse of a Complex Number?

Every non-zero complex number $z = x + iy$ has a "reciprocal" friend called the multiplicative inverse (or just inverse), which we write as $z^{-1}$ or $1/z$.

The defining characteristic of the multiplicative inverse is that if we multiply the complex number $z$ by its inverse $z^{-1}$, the result is 1 (the multiplicative identity element).

Finding the Inverse Formula

We already know from the material on properties of complex number multiplication that for $z = x + iy$, its inverse is:

This formula can also be written as an ordered pair:

Remember also the other often useful form, using the conjugate ($\bar{z} = x-iy$) and the modulus squared ($|z|^2 = x^2+y^2$):

Example Inverse Calculation

Let the complex number be $z = 1 - i$. Find its inverse!

Solution:

Here, $x=1$ and $y=-1$.

Using the first formula:

Using the conjugate and modulus formula:

The result is the same, namely:

Exercise

Given the complex numbers $z_1 = 1-i$ and $z_2 = 2+3i$. Find the inverse of $z_1 + z_2$.

Answer Key

Step 1: Find $z_1 + z_2$.

Step 2: Find the inverse of $z = 3+2i$. Here $x=3$ and $y=2$. We use the formula $z^{-1} = \frac{\bar{z}}{|z|^2}$.

So, the inverse of $z_1 + z_2$ is $\frac{3}{13} - \frac{2}{13}i$.