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URL: https://nakafa.com/en/subjects/mathematics/complex-number/properties-modulus-complex-numbers
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Learn modulus laws: |z₁×z₂|=|z₁|×|z₂|, triangle inequality, |z|²=z×z̄. Simplify calculations using properties instead of complex arithmetic.

---

## Properties of Modulus Operations

Let $$z_1$$ and $$z_2$$ be complex numbers.

Visible text: Let and be complex numbers.

### Modulus of a Number, its Negative, and its Conjugate

The modulus of a complex number is equal to the modulus of its negative, and also equal to the modulus of its conjugate.

```math
|z_1| = |-z_1| = |\bar{z_1}|
```

**Explanation:**

Recall that if $$z_1 = x + iy$$, then $$-z_1 = -x - iy$$ and $$\bar{z_1} = x - iy$$.

Visible text: Recall that if , then and .

- $$|z_1| = \sqrt{x^2 + y^2}$$
- $$|-z_1| = \sqrt{(-x)^2 + (-y)^2} = \sqrt{x^2 + y^2}$$
- $$|\bar{z_1}| = \sqrt{x^2 + (-y)^2} = \sqrt{x^2 + y^2}$$

Visible text: - 
- 
-

All three yield the same value.

### Modulus of Difference

The modulus of the difference of two complex numbers is the same if the order is reversed.

```math
|z_1 - z_2| = |z_2 - z_1|
```

**Explanation:**

This is a direct consequence of the first property. We know $$z_1 - z_2 = -(z_2 - z_1)$$. Then:

Visible text: This is a direct consequence of the first property. We know . Then:

```math
|z_1 - z_2| = |-(z_2 - z_1)| = |z_2 - z_1|
```

### Square of Modulus

The square of the modulus of a complex number is equal to the complex number multiplied by its conjugate.

```math
|z_1|^2 = z_1 \times \bar{z_1}
```

**Explanation:**

If $$z_1 = x + iy$$, then $$\bar{z_1} = x - iy$$.

Visible text: If , then .

```math
z_1 \times \bar{z_1} = (x + iy)(x - iy) = x^2 - (iy)^2 = x^2 - i^2y^2 = x^2 - (-1)y^2 = x^2 + y^2
```

We also know that $$|z_1| = \sqrt{x^2 + y^2}$$, so $$|z_1|^2 = (\sqrt{x^2 + y^2})^2 = x^2 + y^2$$.

Visible text: We also know that , so .

Thus, both sides are equal.

### Modulus of Product

The modulus of the product of two complex numbers is equal to the product of their individual moduli.

```math
|z_1 \times z_2| = |z_1| \times |z_2|
```

### Modulus of Quotient

The modulus of the quotient of two complex numbers is equal to the quotient of their individual moduli (provided the denominator is non-zero).

```math
\left| \frac{z_1}{z_2} \right| = \frac{|z_1|}{|z_2|}, \quad \text{for } z_2 \neq 0
```

### Triangle Inequality

The modulus of the sum of two complex numbers is less than or equal to the sum of their individual moduli.

```math
|z_1 + z_2| \leq |z_1| + |z_2|
```

**Explanation:**

Geometrically, if we consider $$z_1$$, $$z_2$$, and $$z_1 + z_2$$ as sides of a triangle on the complex plane, this property states that the length of one side ($$|z_1 + z_2|$$) cannot be greater than the sum of the lengths of the other two sides ($$|z_1| + |z_2|$$).

Visible text: Geometrically, if we consider , , and as sides of a triangle on the complex plane, this property states that the length of one side () cannot be greater than the sum of the lengths of the other two sides ().

## Using Modulus Properties

Suppose we are given the complex number $$z = \frac{1 - 2i}{3 + 4i}$$. Find $$|z|$$!

Visible text: Suppose we are given the complex number . Find !

**Solution:**

We can view $$z = \frac{z_1}{z_2}$$ with $$z_1 = 1 - 2i$$ and $$z_2 = 3 + 4i$$.

Visible text: We can view with and .

Using the **Modulus of Quotient** property:

```math
|z| = \left| \frac{1 - 2i}{3 + 4i} \right| = \frac{|1 - 2i|}{|3 + 4i|}
```

Now we calculate the moduli of $$z_1$$ and $$z_2$$:

Visible text: Now we calculate the moduli of and :

Component: MathContainer
Children:

```math
|z_1| = |1 - 2i| = \sqrt{1^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5}
```

```math
|z_2| = |3 + 4i| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
```

Therefore,

```math
|z| = \frac{\sqrt{5}}{5}
```

This is much easier than first multiplying by the conjugate of the denominator and then calculating the modulus.

## Exercise

1.  If $$z_1 = 6 + 8i$$ and $$z_2 = 3 - i$$, calculate $$|z_1 \times z_2|$$ using the modulus properties.
2.  If $$z = 5 - 12i$$, prove that $$|z|^2 = z \times \bar{z}$$.

Visible text: 1. If and , calculate using the modulus properties.
2. If , prove that .

### Answer Key

1.  We use the property $$|z_1 \times z_2| = |z_1| \times |z_2|$$.

    Calculate each modulus:

    <MathContainer>
      
    
    ```math
    |z_1| = |6 + 8i| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10
    ```

      
    
    ```math
    |z_2| = |3 - i| = \sqrt{3^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10}
    ```

    </MathContainer>

    Then:

    
    
    ```math
    |z_1 \times z_2| = |z_1| \times |z_2| = 10 \times \sqrt{10} = 10\sqrt{10}
    ```

2.  Given $$z = 5 - 12i$$.

    Calculate the left side ($$|z|^2$$):

    <MathContainer>
      
    
    ```math
    |z| = |5 - 12i| = \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13
    ```

      
    
    ```math
    |z|^2 = 13^2 = 169
    ```

    </MathContainer>

    Calculate the right side ($$z \times \bar{z}$$):

    The conjugate of $$z$$ is $$\bar{z} = 5 + 12i$$.

    
    
    ```math
    z \times \bar{z} = (5 - 12i)(5 + 12i) = 5^2 - (12i)^2 = 25 - 144i^2 = 25 - 144(-1) = 25 + 144 = 169
    ```

    Since the left side ($$169$$) equals the right side ($$169$$), the statement $$|z|^2 = z \times \bar{z}$$ is proven.

Visible text: 1. We use the property .

 Calculate each modulus:

 <MathContainer>
 
 

 
 

 </MathContainer>

 Then:

 
 

2. Given .

 Calculate the left side ():

 <MathContainer>
 
 

 
 

 </MathContainer>

 Calculate the right side ():

 The conjugate of is .

 
 

 Since the left side () equals the right side (), the statement is proven.