# Nakafa Learning Content

> For AI agents: use [llms.txt](https://nakafa.com/llms.txt) for the site index. Markdown versions are available by appending `.md` to content URLs or sending `Accept: text/markdown`.

URL: https://nakafa.com/en/subjects/mathematics/complex-number/inverse-complex-numbers
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/complex-number/inverse-complex-numbers/en.mdx

Calculate complex number inverses using conjugate and modulus formulas. Learn z⁻¹ = z̄/|z|² for division and reciprocal operations with examples.

---

## 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 $$1/z$$.

Visible text: Every **non-zero** complex number has a "reciprocal" friend called the **multiplicative inverse** (or just inverse), which we write as or .

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).

Visible text: The defining characteristic of the multiplicative inverse is that if we multiply the complex number by its inverse , the result is (the multiplicative identity element).

```math
z \times z^{-1} = 1
```

## Finding the Inverse Formula

We already know from the material on [properties of complex number multiplication](/en/subjects/mathematics/complex-number/properties-multiplication-complex-numbers#multiplicative-inverse) that for $$z = x + iy$$, its inverse is:

Visible text: We already know from the material on [properties of complex number multiplication](/en/subjects/mathematics/complex-number/properties-multiplication-complex-numbers#multiplicative-inverse) that for , its inverse is:

```math
z^{-1} = \frac{x}{x^2+y^2} - i\frac{y}{x^2+y^2}
```

This formula can also be written as an ordered pair:

```math
z^{-1} = \left( \frac{x}{x^2+y^2}, -\frac{y}{x^2+y^2} \right)
```

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

Visible text: Remember also the other often useful form, using the conjugate () and the modulus squared ():

```math
z^{-1} = \frac{\bar{z}}{|z|^2}
```

## Example Inverse Calculation

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

Visible text: Let the complex number be . Find its inverse!

**Solution:**

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

Visible text: Here, and .

Using the first formula:

Component: MathContainer
Children:

```math
x^2+y^2 = (1)^2 + (-1)^2 = 1 + 1 = 2
```

```math
z^{-1} = \frac{x}{x^2+y^2} - i\frac{y}{x^2+y^2}
```

```math
= \frac{1}{2} - i\frac{-1}{2}
```

```math
= \frac{1}{2} + \frac{1}{2}i
```

Using the conjugate and modulus formula:

Component: MathContainer
Children:

```math
\bar{z} = 1 - (-1)i = 1+i
```

```math
|z|^2 = x^2+y^2 = 1^2 + (-1)^2 = 2
```

```math
z^{-1} = \frac{\bar{z}}{|z|^2} = \frac{1+i}{2} = \frac{1}{2} + \frac{1}{2}i
```

The result is the same, namely:

Component: ContentStack
Children:

```math
z^{-1} = \frac{1}{2} + \frac{1}{2}i \text{ or} \left( \frac{1}{2}, \frac{1}{2} \right)
```

Component: LineEquation
Props:
- title: Visualization of $$z$$ and{" "}
$$z^{-1}$$
  Visible text: Visualization of and{" "}
- description: Visualization of $$z = 1-i$$ and its inverse{" "}
$$z^{-1} = \frac{1}{2} + \frac{1}{2}i$$. Notice their
positions relative to the origin.
  Visible text: Visualization of and its inverse{" "}
. Notice their
positions relative to the origin.
- cameraPosition: [0, 0, 8]
- showZAxis: false
- data: [
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 1, y: -1, z: 0 },
],
color: getColor("SKY"),
labels: [{ text: "z = 1-i", at: 1, offset: [0.5, -0.3, 0] }],
cone: { position: "end" },
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 0.5, y: 0.5, z: 0 },
],
color: getColor("LIME"),
labels: [
{
text: "z^{-1} = 1/2 + i/2",
at: 1,
offset: [1, 0.5, 0],
},
],
cone: { position: "end" },
},
]

## Exercise

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

Visible text: Given the complex numbers and . Find the inverse of .

### Answer Key

Step $$1$$: Find $$z_1 + z_2$$.

Visible text: Step : Find .

```math
z = z_1 + z_2 = (1-i) + (2+3i) = (1+2) + (-1+3)i = 3+2i
```

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

Visible text: Step : Find the inverse of .
Here and .
We use the formula .

Component: MathContainer
Children:

```math
\bar{z} = 3-2i
```

```math
|z|^2 = x^2+y^2 = 3^2 + 2^2 = 9 + 4 = 13
```

```math
z^{-1} = \frac{3-2i}{13} = \frac{3}{13} - \frac{2}{13}i
```

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

Visible text: So, the inverse of is .