# Nakafa Framework: LLM

URL: /en/subject/high-school/11/mathematics/complex-number/multiplication-complex-numbers
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/complex-number/multiplication-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: "Multiplication of Complex Numbers",
  description: "Multiply complex numbers using distributive property and formula (x₁x₂-y₁y₂)+i(x₁y₂+x₂y₁). Master binomial expansion with i²=-1 examples.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/01/2025",
  subject: "Complex Number",
};

## 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 <InlineMath math="z_1 = x_1 + iy_1" /> by <InlineMath math="z_2 = x_2 + iy_2" />.

<div className="flex flex-col gap-4">
  <BlockMath math="z_1 \times z_2 = (x_1 + iy_1)(x_2 + iy_2)" />
  <BlockMath math="= x_1(x_2 + iy_2) + iy_1(x_2 + iy_2)" />
  <BlockMath math="= (x_1x_2 + ix_1y_2) + (ix_2y_1 + i^2y_1y_2)" />
</div>

Remember that <InlineMath math="i^2 = -1" />, so we can substitute:

<div className="flex flex-col gap-4">
  <BlockMath math="= (x_1x_2 + ix_1y_2) + (ix_2y_1 + (-1)y_1y_2)" />
  <BlockMath math="= x_1x_2 + ix_1y_2 + ix_2y_1 - y_1y_2" />
</div>

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

<div className="flex flex-col gap-4">
  <BlockMath math="= (x_1x_2 - y_1y_2) + (ix_1y_2 + ix_2y_1)" />
  <BlockMath math="= (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)" />
</div>

So, the general formula for complex number multiplication is:

<BlockMath math="z_1 \times z_2 = (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)" />

## Calculation Example

Let <InlineMath math="z_1 = 2+i" /> and <InlineMath math="z_2 = 1-2i" />. Find <InlineMath math="z_1 \times z_2" />.

**Solution:**

Using the distributive property:

<div className="flex flex-col gap-4">
  <BlockMath math="z_1 \times z_2 = (2+i)(1-2i)" />
  <BlockMath math="= 2(1-2i) + i(1-2i)" />
  <BlockMath math="= (2 - 4i) + (i - 2i^2)" />
  <BlockMath math="= (2 - 4i) + (i - 2(-1)) \quad \text{(since } i^2 = -1)" />
  <BlockMath math="= (2 - 4i) + (i + 2)" />
  <BlockMath math="= (2+2) + (-4i + i)" />
  <BlockMath math="= 4 - 3i" />
</div>

Or using the general formula with <InlineMath math="x_1 = 2, y_1 = 1, x_2 = 1, y_2 = -2" />:

<div className="flex flex-col gap-4">
  <BlockMath math="z_1 \times z_2 = (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)" />
  <BlockMath math="= (2)(1) - (1)(-2) + i((2)(-2) + (1)(1))" />
  <BlockMath math="= (2 - (-2)) + i(-4 + 1)" />
  <BlockMath math="= (2+2) + i(-3)" />
  <BlockMath math="= 4 - 3i" />
</div>

The result is the same!

<LineEquation
  title="Visualization of Complex Number Multiplication"
  description={
    <>
      Visualization of <InlineMath math="z_1 = 2+i" />,{" "}
      <InlineMath math="z_2 = 1-2i" />, and their product{" "}
      <InlineMath math="z_1 \times z_2 = 4-3i" />.
    </>
  }
  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 <InlineMath math="z_1 = 1+i" /> and <InlineMath math="z_2 = \frac{1}{2} - 2i" />. Find <InlineMath math="z_1 \times z_2" />.

### Answer Key

Using the distributive property:

<div className="flex flex-col gap-4">
  <BlockMath math="z_1 \times z_2 = (1+i)(\frac{1}{2} - 2i)" />
  <BlockMath math="= 1(\frac{1}{2} - 2i) + i(\frac{1}{2} - 2i)" />
  <BlockMath math="= (\frac{1}{2} - 2i) + (\frac{1}{2}i - 2i^2)" />
  <BlockMath math="= (\frac{1}{2} - 2i) + (\frac{1}{2}i - 2(-1))" />
  <BlockMath math="= (\frac{1}{2} - 2i) + (\frac{1}{2}i + 2)" />
  <BlockMath math="= (\frac{1}{2} + 2) + (-2i + \frac{1}{2}i)" />
  <BlockMath math="= (\frac{1}{2} + \frac{4}{2}) + (-\frac{4}{2}i + \frac{1}{2}i)" />
  <BlockMath math="= \frac{5}{2} - \frac{3}{2}i" />
</div>

Using the general formula with <InlineMath math="x_1 = 1, y_1 = 1, x_2 = \frac{1}{2}, y_2 = -2" />:

<div className="flex flex-col gap-4">
  <BlockMath math="z_1 \times z_2 = (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)" />
  <BlockMath math="= ((1)(\frac{1}{2}) - (1)(-2)) + i((1)(-2) + (\frac{1}{2})(1))" />
  <BlockMath math="= (\frac{1}{2} - (-2)) + i(-2 + \frac{1}{2})" />
  <BlockMath math="= (\frac{1}{2} + 2) + i(-\frac{4}{2} + \frac{1}{2})" />
  <BlockMath math="= (\frac{1}{2} + \frac{4}{2}) + i(-\frac{3}{2})" />
  <BlockMath math="= \frac{5}{2} - \frac{3}{2}i" />
</div>
