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URL: https://nakafa.com/en/subjects/mathematics/complex-number/conjugate-complex-numbers
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Find complex number conjugates by changing imaginary signs. Explore geometric reflections, properties, and why z×z̄ produces real numbers.

---

## What is a Complex Number Conjugate?

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

Visible text: Every complex number has a "pair" called the **conjugate**. The conjugate of is written with the symbol .

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

## Formal Definition

If $$z = x + iy$$ is a complex number, with $$x$$ as the real part and $$y$$ as the imaginary part, then its conjugate is:

Visible text: If is a complex number, with as the real part and as the imaginary part, then its conjugate is:

```math
\bar{z} = x - iy
```

This means the real part ($$x$$) stays the same, while the sign of the imaginary part ($$y$$) is flipped (positive becomes negative, negative becomes positive).

Visible text: This means the real part () stays the same, while the sign of the imaginary part () is flipped (positive becomes negative, negative becomes positive).

## Examples of Finding the Conjugate

Let's look at some examples:

1.  **If** $$z = 2 + i$$

    Here, $$x=2$$ and $$y=1$$.

    Then its conjugate is $$\bar{z} = 2 - i$$. (The sign of the imaginary part $$+1$$ becomes $$-1$$)

2.  **If** $$z = 2$$

    We can write $$z = 2 + 0i$$. Here, $$x=2$$ and $$y=0$$.

    Then its conjugate is $$\bar{z} = 2 - 0i = 2$$. (The imaginary part is $$0$$, its sign doesn't change)

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

3.  **If** $$z = 3 - 2i$$

    Here, $$x=3$$ and $$y=-2$$.

    Then its conjugate is $$\bar{z} = 3 - (-2i) = 3 + 2i$$. (The sign of the imaginary part $$-2$$ becomes $$+2$$)

4.  **If** $$z = 3i$$

    We can write $$z = 0 + 3i$$. Here, $$x=0$$ and $$y=3$$.

    Then its conjugate is $$\bar{z} = 0 - 3i = -3i$$. (The sign of the imaginary part $$+3$$ becomes $$-3$$)

    The conjugate of a purely imaginary number is its negative.

Visible text: 1. **If** 

 Here, and .

 Then its conjugate is . (The sign of the imaginary part becomes )

2. **If** 

 We can write . Here, and .

 Then its conjugate is . (The imaginary part is , its sign doesn't change)

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

3. **If** 

 Here, and .

 Then its conjugate is . (The sign of the imaginary part becomes )

4. **If** 

 We can write . Here, and .

 Then its conjugate is . (The sign of the imaginary part becomes )

 The conjugate of a purely imaginary number is its negative.

## Visualization of the Conjugate

Geometrically, the conjugate $$\bar{z}$$ is the **reflection** of $$z$$ across the **real axis** ($$x$$-axis) in the complex plane.

Visible text: Geometrically, the conjugate is the **reflection** of across the **real axis** (-axis) in the complex plane.

Component: LineEquation
Props:
- title: Visualization of $$z = 3+2i$$ and its Conjugate{" "}
$$\bar{z} = 3-2i$$
  Visible text: Visualization of and its Conjugate{" "}
- description: Notice how $$z$$ and $$\bar{z}$$ are
like reflections across the real axis.
  Visible text: Notice how and 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)
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points: [
{ x: -5, y: 0, z: 0 },
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]

## Complex Number Congruence

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

Visible text: Is it possible for a complex number to be equal to its conjugate ? If so, what is the condition?

**Answer:**

Yes, it's possible. For $$z = \bar{z}$$, then:

Visible text: Yes, it's possible. For , then:

```math
x+iy = x-iy
```

This can only happen if $$iy = -iy$$, which means $$2iy = 0$$.

Visible text: This can only happen if , which means .

Since $$i \neq 0$$, it must be that $$y=0$$.

Visible text: Since , it must be that .

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 $$z, z_1,$$ and $$z_2$$ be any complex numbers.

Visible text: The conjugate operation has several interesting properties that are useful in calculations. Let and 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.

Component: MathContainer
Children:

```math
\overline{z_1 + z_2} = \bar{z_1} + \bar{z_2}
```

```math
\overline{z_1 - z_2} = \bar{z_1} - \bar{z_2}
```

### Product and Quotient

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

Component: MathContainer
Children:

```math
\overline{z_1 \times z_2} = \bar{z_1} \times \bar{z_2}
```

```math
\overline{\left( \frac{z_1}{z_2} \right)} = \frac{\bar{z_1}}{\bar{z_2}}, \quad \text{for } z_2 \neq 0
```

### Inverse

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

```math
\overline{z^{-1}} = (\bar{z})^{-1}
```

### Double Conjugate

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

```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:

Component: MathContainer
Children:

```math
z + \bar{z} = 2 \text{Re}(z)
```

```math
z - \bar{z} = 2i \text{Im}(z)
```

### Multiplication by Conjugate

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

```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. $$2+i^2$$
2. $$1+\frac{1}{i}$$
3. $$1+2i$$

Visible text: 1. 
2. 
3.

### Answer Key

1. First, simplify the complex number:

   $$z = 2+i^2 = 2+(-1) = 1$$. Since $$z=1$$ is
   a real number ($$1+0i$$
   ),

   its conjugate is $$\bar{z} = 1$$.

2. Simplify first:

   Remember that

   
   
   ```math
   \frac{1}{i} = \frac{1}{i} \times \frac{-i}{-i} = \frac{-i}{-i^2} = \frac{-i}{-(-1)} = \frac{-i}{1} = -i
   ```

   So, $$z = 1 + \frac{1}{i} = 1 - i$$.

   Its conjugate is $$\bar{z} = 1 - (-i) = 1 + i$$.

3. $$z = 1+2i$$.

   Directly use the definition: $$\bar{z} = 1 - 2i$$.

Visible text: 1. First, simplify the complex number:

 . Since is
 a real number (
 ),

 its conjugate is .

2. Simplify first:

 Remember that

 
 

 So, .

 Its conjugate is .

3. .

 Directly use the definition: .