# Nakafa Framework: LLM

URL: https://nakafa.com/en/subject/high-school/11/mathematics/complex-number/conjugate-complex-numbers
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/complex-number/conjugate-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: "Complex Number Conjugate",
  description: "Find complex number conjugates by changing imaginary signs. Explore geometric reflections, properties, and why z×z̄ produces real numbers.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/01/2025",
  subject: "Complex Number",
};

## What is a Complex Number Conjugate?

Every complex number <InlineMath math="z = x + iy" /> has a "pair" called the **conjugate**. The conjugate of <InlineMath math="z" /> is written with the symbol <InlineMath math="\bar{z}" />.

Getting the conjugate is very easy: **just change the sign of the imaginary part**.

## Formal Definition

If <InlineMath math="z = x + iy" /> is a complex number, with <InlineMath math="x" /> as the real part and <InlineMath math="y" /> as the imaginary part, then its conjugate is:

<BlockMath math="\bar{z} = x - iy" />

This means the real part (<InlineMath math="x" />) stays the same, while the sign of the imaginary part (<InlineMath math="y" />) is flipped (positive becomes negative, negative becomes positive).

## Examples of Finding the Conjugate

Let's look at some examples:

1.  **If** <InlineMath math="z = 2 + i" />

    Here, <InlineMath math="x=2" /> and <InlineMath math="y=1" />.

    Then its conjugate is <InlineMath math="\bar{z} = 2 - i" />. (The sign of the imaginary part <InlineMath math="+1" /> becomes <InlineMath math="-1" />)

2.  **If** <InlineMath math="z = 2" />

    We can write <InlineMath math="z = 2 + 0i" />. Here, <InlineMath math="x=2" /> and <InlineMath math="y=0" />.

    Then its conjugate is <InlineMath math="\bar{z} = 2 - 0i = 2" />. (The imaginary part is 0, its sign doesn't change)

    The conjugate of a real number is the real number itself.

3.  **If** <InlineMath math="z = 3 - 2i" />

    Here, <InlineMath math="x=3" /> and <InlineMath math="y=-2" />.

    Then its conjugate is <InlineMath math="\bar{z} = 3 - (-2i) = 3 + 2i" />. (The sign of the imaginary part <InlineMath math="-2" /> becomes <InlineMath math="+2" />)

4.  **If** <InlineMath math="z = 3i" />

    We can write <InlineMath math="z = 0 + 3i" />. Here, <InlineMath math="x=0" /> and <InlineMath math="y=3" />.

    Then its conjugate is <InlineMath math="\bar{z} = 0 - 3i = -3i" />. (The sign of the imaginary part <InlineMath math="+3" /> becomes <InlineMath math="-3" />)

    The conjugate of a purely imaginary number is its negative.

## Visualization of the Conjugate

Geometrically, the conjugate <InlineMath math="\bar{z}" /> is the **reflection** of <InlineMath math="z" /> across the **real axis (X-axis)** in the complex plane.

<LineEquation
  title={
    <>
      Visualization of <InlineMath math="z = 3+2i" /> and its Conjugate{" "}
      <InlineMath math="\bar{z} = 3-2i" />
    </>
  }
  description={
    <>
      Notice how <InlineMath math="z" /> and <InlineMath math="\bar{z}" /> are
      like reflections across the real axis.
    </>
  }
  cameraPosition={[0, 0, 10]}
  showZAxis={false}
  data={[
    {
      points: [
        { x: 0, y: 0, z: 0 },
        { x: 3, y: 2, z: 0 },
      ],
      color: getColor("SKY"),
      labels: [{ text: "z = 3+2i", at: 1, offset: [0.5, 0.5, 0] }],
      cone: { position: "end" },
    },
    {
      points: [
        { x: 0, y: 0, z: 0 },
        { x: 3, y: -2, z: 0 },
      ],
      color: getColor("LIME"),
      labels: [{ text: "z̄ = 3-2i", at: 1, offset: [0.5, -0.5, 0] }],
      cone: { position: "end" },
    },
    // Real axis line as mirror (optional)
    {
      points: [
        { x: -5, y: 0, z: 0 },
        { x: 5, y: 0, z: 0 },
      ],
      color: getColor("AMBER"),
    },
  ]}
/>

## Complex Number Congruence

Is it possible for a complex number <InlineMath math="z=x+iy" /> to be equal to its conjugate <InlineMath math="\bar{z}=x-iy" />? If so, what is the condition?

**Answer:**

Yes, it's possible. For <InlineMath math="z = \bar{z}" />, then:

<BlockMath math="x+iy = x-iy" />

This can only happen if <InlineMath math="iy = -iy" />, which means <InlineMath math="2iy = 0" />.

Since <InlineMath math="i \neq 0" />, it must be that <InlineMath math="y=0" />.

So, a complex number is equal to its conjugate **if and only if its imaginary part is zero**, or in other words, **if the complex number is a real number**.

## Properties of Conjugate Operations

The conjugate operation has several interesting properties that are useful in calculations. Let <InlineMath math="z, z_1," /> and <InlineMath math="z_2" /> be any complex numbers.

### Sum and Difference

The conjugate of the sum (or difference) of two complex numbers is equal to the sum (or difference) of their conjugates.

<div className="flex flex-col gap-4">
  <BlockMath math="\overline{z_1 + z_2} = \bar{z_1} + \bar{z_2}" />
  <BlockMath math="\overline{z_1 - z_2} = \bar{z_1} - \bar{z_2}" />
</div>

### Product and Quotient

The conjugate of the product (or quotient) of two complex numbers is equal to the product (or quotient) of their conjugates.

<div className="flex flex-col gap-4">
  <BlockMath math="\overline{z_1 \times z_2} = \bar{z_1} \times \bar{z_2}" />
  <BlockMath math="\overline{\left( \frac{z_1}{z_2} \right)} = \frac{\bar{z_1}}{\bar{z_2}}, \quad \text{for } z_2 \neq 0" />
</div>

### Inverse

The conjugate of the inverse of a complex number is equal to the inverse of its conjugate.

<BlockMath math="\overline{z^{-1}} = (\bar{z})^{-1}" />

### Double Conjugate

Taking the conjugate twice returns the complex number to its original form.

<BlockMath math="\overline{(\bar{z})} = z" />

### Relationship with Real and Imaginary Parts

Adding and subtracting a complex number with its conjugate yields interesting relationships with its real and imaginary parts:

<div className="flex flex-col gap-4">
  <BlockMath math="z + \bar{z} = 2 \text{Re}(z)" />
  <BlockMath math="z - \bar{z} = 2i \text{Im}(z)" />
</div>

### Multiplication by Conjugate

Multiplying a complex number by its conjugate yields the square of its modulus (a non-negative real number).

<BlockMath math="z \times \bar{z} = (\text{Re}(z))^2 + (\text{Im}(z))^2 = |z|^2" />

## Exercise

Find the conjugate of each of the following complex numbers!

1. <InlineMath math="2+i^2" />
2. <InlineMath math="1+\frac{1}{i}" />
3. <InlineMath math="1+2i" />

### Answer Key

1. First, simplify the complex number:

   <InlineMath math="z = 2+i^2 = 2+(-1) = 1" />. Since <InlineMath math="z=1" /> is
   a real number (<InlineMath math="1+0i" />
   ),

   its conjugate is <InlineMath math="\bar{z} = 1" />.

2. Simplify first:

   Remember that

   <BlockMath math="\frac{1}{i} = \frac{1}{i} \times \frac{-i}{-i} = \frac{-i}{-i^2} = \frac{-i}{-(-1)} = \frac{-i}{1} = -i" />

   So, <InlineMath math="z = 1 + \frac{1}{i} = 1 - i" />.

   Its conjugate is <InlineMath math="\bar{z} = 1 - (-i) = 1 + i" />.

3. <InlineMath math="z = 1+2i" />.

   Directly use the definition: <InlineMath math="\bar{z} = 1 - 2i" />.