# Nakafa Framework: LLM

URL: https://nakafa.com/en/subject/high-school/11/mathematics/complex-number/inverse-complex-numbers
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/complex-number/inverse-complex-numbers/en.mdx

Output docs content for large language models.

---

import { LineEquation } from "@repo/design-system/components/contents/line-equation";
import { getColor } from "@repo/design-system/lib/color";

export const metadata = {
  title: "Inverse of Complex Numbers",
  description: "Calculate complex number inverses using conjugate and modulus formulas. Master z⁻¹ = z̄/|z|² for division and reciprocal operations with examples.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/01/2025",
  subject: "Complex Number",
};

## What is the Inverse of a Complex Number?

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

The defining characteristic of the multiplicative inverse is that if we multiply the complex number <InlineMath math="z" /> by its inverse <InlineMath math="z^{-1}" />, the result is **1** (the multiplicative identity element).

<BlockMath math="z \times z^{-1} = 1" />

## Finding the Inverse Formula

We already know from the material on [properties of complex number multiplication](/subject/high-school/11/mathematics/complex-number/properties-multiplication-complex-numbers#multiplicative-inverse) that for <InlineMath math="z = x + iy" />, its inverse is:

<BlockMath 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:

<BlockMath 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 (<InlineMath math="\bar{z} = x-iy" />) and the modulus squared (<InlineMath math="|z|^2 = x^2+y^2" />):

<BlockMath math="z^{-1} = \frac{\bar{z}}{|z|^2}" />

## Example Inverse Calculation

Let the complex number be <InlineMath math="z = 1 - i" />. Find its inverse!

**Solution:**

Here, <InlineMath math="x=1" /> and <InlineMath math="y=-1" />.

Using the first formula:

<div className="flex flex-col gap-4">
  <BlockMath math="x^2+y^2 = (1)^2 + (-1)^2 = 1 + 1 = 2" />
  <BlockMath math="z^{-1} = \frac{x}{x^2+y^2} - i\frac{y}{x^2+y^2}" />
  <BlockMath math="= \frac{1}{2} - i\frac{-1}{2}" />
  <BlockMath math="= \frac{1}{2} + \frac{1}{2}i" />
</div>

Using the conjugate and modulus formula:

<div className="flex flex-col gap-4">
  <BlockMath math="\bar{z} = 1 - (-1)i = 1+i" />
  <BlockMath math="|z|^2 = x^2+y^2 = 1^2 + (-1)^2 = 2" />
  <BlockMath math="z^{-1} = \frac{\bar{z}}{|z|^2} = \frac{1+i}{2} = \frac{1}{2} + \frac{1}{2}i" />
</div>

The result is the same, namely:

<div className="flex flex-col gap-4">
  <BlockMath math="z^{-1} = \frac{1}{2} + \frac{1}{2}i \text{ or } \left( \frac{1}{2}, \frac{1}{2} \right)" />
  <LineEquation
    title={
      <>
        Visualization of <InlineMath math="z" /> and{" "}
        <InlineMath math="z^{-1}" />
      </>
    }
    description={
      <>
        Visualization of <InlineMath math="z = 1-i" /> and its inverse{" "}
        <InlineMath math="z^{-1} = \frac{1}{2} + \frac{1}{2}i" />. 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" },
      },
    ]}
  />
</div>

## Exercise

Given the complex numbers <InlineMath math="z_1 = 1-i" /> and <InlineMath math="z_2 = 2+3i" />. Find the inverse of <InlineMath math="z_1 + z_2" />.

### Answer Key

Step 1: Find <InlineMath math="z_1 + z_2" />.

<BlockMath math="z = z_1 + z_2 = (1-i) + (2+3i) = (1+2) + (-1+3)i = 3+2i" />

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

<div className="flex flex-col gap-4">
  <BlockMath math="\bar{z} = 3-2i" />
  <BlockMath math="|z|^2 = x^2+y^2 = 3^2 + 2^2 = 9 + 4 = 13" />
  <BlockMath math="z^{-1} = \frac{3-2i}{13} = \frac{3}{13} - \frac{2}{13}i" />
</div>

So, the inverse of <InlineMath math="z_1 + z_2" /> is <InlineMath math="\frac{3}{13} - \frac{2}{13}i" />.