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URL: https://nakafa.com/en/subjects/mathematics/complex-number/properties-multiplication-complex-numbers
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Learn commutative, associative, distributive properties and multiplicative inverse of complex numbers with worked examples and algebraic proofs.

---

## Properties of Multiplication Operation

Just like arithmetic operations on real numbers, the multiplication operation on complex numbers also has several important properties. Let $$z_1, z_2,$$ and $$z_3$$ be any complex numbers.

Visible text: Just like arithmetic operations on real numbers, the multiplication operation on complex numbers also has several important properties. Let and be any complex numbers.

### Commutative Property

The commutative property means that the order in the multiplication of two complex numbers does not affect the result.

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

**Example:**

Let $$z_1 = 1+2i$$ and $$z_2 = 3-i$$.

Visible text: Let and .

Component: MathContainer
Children:

```math
z_1 \times z_2 = (1+2i)(3-i) = 1(3-i) + 2i(3-i) = 3-i+6i-2i^2 = 3+5i-2(-1) = 5+5i
```

```math
z_2 \times z_1 = (3-i)(1+2i) = 3(1+2i) - i(1+2i) = 3+6i-i-2i^2 = 3+5i-2(-1) = 5+5i
```

The results are proven to be the same.

### Associative Property

The associative property states that when multiplying three or more complex numbers, the grouping of the multiplication does not change the result.

```math
(z_1 \times z_2) \times z_3 = z_1 \times (z_2 \times z_3)
```

**Example:**

Let $$z_1 = i$$, $$z_2 = 2$$, and $$z_3 = 3-i$$.

Visible text: Let , , and .

Component: MathContainer
Children:

```math
(z_1 \times z_2) \times z_3 = (i \times 2) \times (3-i) = 2i(3-i) = 6i - 2i^2 = 6i - 2(-1) = 2+6i
```

```math
z_1 \times (z_2 \times z_3) = i \times (2 \times (3-i)) = i \times (6-2i) = 6i - 2i^2 = 6i - 2(-1) = 2+6i
```

The results are proven to be the same.

### Multiplicative Identity

The complex number $$1 = 1+0i$$ is the identity element for multiplication. This means that any complex number multiplied by $$1$$ results in the complex number itself.

Visible text: The complex number is the identity element for multiplication. This means that any complex number multiplied by results in the complex number itself.

```math
z \times 1 = z = 1 \times z
```

**Example:**

Let $$z = 4-7i$$.

Visible text: Let .

Component: MathContainer
Children:

```math
z \times 1 = (4-7i)(1+0i) = 4(1) - (-7)(0) + i(4(0) + (-7)(1)) = 4 - 7i = z
```

```math
1 \times z = (1+0i)(4-7i) = 1(4) - (0)(-7) + i(1(-7) + 0(4)) = 4 - 7i = z
```

### Distributive Property of Multiplication over Addition

This property connects the operations of multiplication and addition of complex numbers.

```math
z_1 \times (z_2 + z_3) = (z_1 \times z_2) + (z_1 \times z_3)
```

**Example:**

Let $$z_1=2$$, $$z_2 = 1+i$$, $$z_3 = 3-2i$$.

Visible text: Let , , .

**Left side:** $$z_1 \times (z_2 + z_3)$$

Visible text: **Left side:**

Component: MathContainer
Children:

```math
z_1 \times (z_2 + z_3) = 2 \times ((1+i) + (3-2i))
```

```math
= 2 \times (1+3 + i-2i)
```

```math
= 2 \times (4-i)
```

```math
= 8-2i
```

**Right side:** $$(z_1 \times z_2) + (z_1 \times z_3)$$

Visible text: **Right side:**

Component: MathContainer
Children:

```math
(z_1 \times z_2) + (z_1 \times z_3) = (2(1+i)) + (2(3-2i))
```

```math
= (2+2i) + (6-4i)
```

```math
= (2+6) + (2i-4i)
```

```math
= 8-2i
```

The results are proven to be the same.

## Example Proof Using Properties

We can prove several algebraic identities using these properties. Let's prove that $$(z_1 + z_2)^2 = z_1^2 + 2z_1z_2 + z_2^2$$ for any $$z_1, z_2$$.

Visible text: We can prove several algebraic identities using these properties. Let's prove that for any .

Component: MathContainer
Children:

```math
(z_1 + z_2)^2 = (z_1 + z_2)(z_1 + z_2) \quad \text{(Definition of square)}
```

```math
= z_1(z_1 + z_2) + z_2(z_1 + z_2) \quad \text{(Distributive Property)}
```

```math
= (z_1z_1 + z_1z_2) + (z_2z_1 + z_2z_2) \quad \text{(Distributive Property)}
```

```math
= z_1^2 + z_1z_2 + z_2z_1 + z_2^2 \quad \text{(Definition of square)}
```

```math
= z_1^2 + z_1z_2 + z_1z_2 + z_2^2 \quad \text{(Commutative Property: } z_2z_1 = z_1z_2)
```

```math
= z_1^2 + 2z_1z_2 + z_2^2 \quad \text{(Combining like terms)}
```

## Multiplicative Inverse

Every non-zero complex number $$z = x + iy \neq 0$$ has a multiplicative inverse, denoted as $$z^{-1}$$ or $$1/z$$, such that $$z \times z^{-1} = 1$$.

Visible text: Every non-zero complex number has a multiplicative inverse, denoted as or , such that .

Let $$z^{-1} = u + iv$$. Then:

Visible text: Let . Then:

Component: MathContainer
Children:

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

```math
(x+iy)(u+iv) = 1 + 0i
```

```math
(xu - yv) + i(xv + uy) = 1 + 0i
```

Based on the equality of two complex numbers, we obtain the system of equations:

1.  $$xu - yv = 1$$
2.  $$xv + uy = 0$$

Visible text: 1. 
2.

By solving this system of equations (for example, by multiplying equation $$1$$ by $$x$$, equation $$2$$ by $$y$$, then adding them, and using the substitution method), we will get:

Visible text: By solving this system of equations (for example, by multiplying equation by , equation by , then adding them, and using the substitution method), we will get:

Component: MathContainer
Children:

```math
u = \frac{x}{x^2+y^2}
```

```math
v = -\frac{y}{x^2+y^2}
```

So, the multiplicative inverse of $$z = x+iy$$ is:

Visible text: So, the multiplicative inverse of is:

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

Note that $$x^2+y^2 = |z|^2$$ and $$x-iy = \bar{z}$$. Thus the inverse formula can also be written as:

Visible text: Note that and . Thus the inverse formula can also be written as:

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