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URL: https://nakafa.com/en/subjects/mathematics/complex-number/multiplication-complex-numbers
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Multiply complex numbers using distributive property and formula (x₁x₂-y₁y₂)+i(x₁y₂+x₂y₁). Learn binomial expansion with i²=-1 examples.

---

## Multiplying Two Complex Numbers

Multiplying two complex numbers is similar to multiplying two binomial algebraic expressions. We can use the distributive property of multiplication over addition.

Let's see how to multiply $$z_1 = x_1 + iy_1$$ by $$z_2 = x_2 + iy_2$$.

Visible text: Let's see how to multiply by .

Component: MathContainer
Children:

```math
z_1 \times z_2 = (x_1 + iy_1)(x_2 + iy_2)
```

```math
= x_1(x_2 + iy_2) + iy_1(x_2 + iy_2)
```

```math
= (x_1x_2 + ix_1y_2) + (ix_2y_1 + i^2y_1y_2)
```

Remember that $$i^2 = -1$$, so we can substitute:

Visible text: Remember that , so we can substitute:

Component: MathContainer
Children:

```math
= (x_1x_2 + ix_1y_2) + (ix_2y_1 + (-1)y_1y_2)
```

```math
= x_1x_2 + ix_1y_2 + ix_2y_1 - y_1y_2
```

Now, let's group the real and imaginary parts:

Component: MathContainer
Children:

```math
= (x_1x_2 - y_1y_2) + (ix_1y_2 + ix_2y_1)
```

```math
= (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)
```

So, the general formula for complex number multiplication is:

```math
z_1 \times z_2 = (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)
```

## Calculation Example

Let $$z_1 = 2+i$$ and $$z_2 = 1-2i$$. Find $$z_1 \times z_2$$.

Visible text: Let and . Find .

**Solution:**

Using the distributive property:

Component: MathContainer
Children:

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

```math
= 2(1-2i) + i(1-2i)
```

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

```math
= (2 - 4i) + (i - 2(-1)) \quad \text{(since } i^2 = -1)
```

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

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

```math
= 4 - 3i
```

Or using the general formula with $$x_1 = 2, y_1 = 1, x_2 = 1, y_2 = -2$$:

Visible text: Or using the general formula with :

Component: MathContainer
Children:

```math
z_1 \times z_2 = (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)
```

```math
= (2)(1) - (1)(-2) + i((2)(-2) + (1)(1))
```

```math
= (2 - (-2)) + i(-4 + 1)
```

```math
= (2+2) + i(-3)
```

```math
= 4 - 3i
```

The result is the same!

Component: LineEquation
Props:
- title: Visualization of Complex Number Multiplication
- description: Visualization of $$z_1 = 2+i$$,{" "}
$$z_2 = 1-2i$$, and their product{" "}
$$z_1 \times z_2 = 4-3i$$.
  Visible text: Visualization of ,{" "}
, and their product{" "}
.
- cameraPosition: [0, 0, 15]
- showZAxis: false
- data: [
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 2, y: 1, z: 0 },
],
color: getColor("SKY"),
labels: [{ text: "z₁ = 2+i", at: 1, offset: [0.5, 0.5, 0] }],
cone: { position: "end" },
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 1, y: -2, z: 0 },
],
color: getColor("EMERALD"),
labels: [{ text: "z₂ = 1-2i", at: 1, offset: [0.5, -0.5, 0] }],
cone: { position: "end" },
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 4, y: -3, z: 0 },
],
color: getColor("ROSE"),
labels: [{ text: "z₁ × z₂ = 4-3i", at: 1, offset: [0.5, -0.5, 0] }],
cone: { position: "end" },
},
]

## Exercise

Let $$z_1 = 1+i$$ and $$z_2 = \frac{1}{2} - 2i$$. Find $$z_1 \times z_2$$.

Visible text: Let and . Find .

### Answer Key

Using the distributive property:

Component: MathContainer
Children:

```math
z_1 \times z_2 = (1+i)(\frac{1}{2} - 2i)
```

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

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

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

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

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

```math
= (\frac{1}{2} + \frac{4}{2}) + (-\frac{4}{2}i + \frac{1}{2}i)
```

```math
= \frac{5}{2} - \frac{3}{2}i
```

Using the general formula with $$x_1 = 1, y_1 = 1, x_2 = \frac{1}{2}, y_2 = -2$$:

Visible text: Using the general formula with :

Component: MathContainer
Children:

```math
z_1 \times z_2 = (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)
```

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

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

```math
= (\frac{1}{2} + 2) + i(-\frac{4}{2} + \frac{1}{2})
```

```math
= (\frac{1}{2} + \frac{4}{2}) + i(-\frac{3}{2})
```

```math
= \frac{5}{2} - \frac{3}{2}i
```