# Nakafa Framework: LLM

URL: https://nakafa.com/en/subject/high-school/11/mathematics/complex-number/properties-multiplication-complex-numbers
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/complex-number/properties-multiplication-complex-numbers/en.mdx

Output docs content for large language models.

---

export const metadata = {
  title: "Properties of Multiplication of Complex Numbers",
  description: "Master commutative, associative, distributive properties and multiplicative inverse of complex numbers with step-by-step examples and algebraic proofs.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/01/2025",
  subject: "Complex Number",
};

## Properties of Multiplication Operation

Just like arithmetic operations on real numbers, the multiplication operation on complex numbers also has several important properties. Let <InlineMath math="z_1, z_2," /> and <InlineMath math="z_3" /> 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.

<BlockMath math="z_1 \times z_2 = z_2 \times z_1" />

**Example:**

Let <InlineMath math="z_1 = 1+2i" /> and <InlineMath math="z_2 = 3-i" />.

<MathContainer>
  <BlockMath 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" />
  <BlockMath 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" />
</MathContainer>

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.

<BlockMath math="(z_1 \times z_2) \times z_3 = z_1 \times (z_2 \times z_3)" />

**Example:**

Let <InlineMath math="z_1 = i" />, <InlineMath math="z_2 = 2" />, and <InlineMath math="z_3 = 3-i" />.

<MathContainer>
  <BlockMath 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" />
  <BlockMath 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" />
</MathContainer>

The results are proven to be the same.

### Multiplicative Identity

The complex number <InlineMath math="1 = 1+0i" /> is the identity element for multiplication. This means that any complex number multiplied by 1 results in the complex number itself.

<BlockMath math="z \times 1 = z = 1 \times z" />

**Example:**

Let <InlineMath math="z = 4-7i" />.

<MathContainer>
  <BlockMath math="z \times 1 = (4-7i)(1+0i) = 4(1) - (-7)(0) + i(4(0) + (-7)(1)) = 4 - 7i = z" />
  <BlockMath math="1 \times z = (1+0i)(4-7i) = 1(4) - (0)(-7) + i(1(-7) + 0(4)) = 4 - 7i = z" />
</MathContainer>

### Distributive Property of Multiplication over Addition

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

<BlockMath math="z_1 \times (z_2 + z_3) = (z_1 \times z_2) + (z_1 \times z_3)" />

**Example:**

Let <InlineMath math="z_1=2" />, <InlineMath math="z_2 = 1+i" />, <InlineMath math="z_3 = 3-2i" />.

**Left side:** <InlineMath math="z_1 \times (z_2 + z_3)" />

<MathContainer>
  <BlockMath math="z_1 \times (z_2 + z_3) = 2 \times ((1+i) + (3-2i))" />
  <BlockMath math="= 2 \times (1+3 + i-2i)" />
  <BlockMath math="= 2 \times (4-i)" />
  <BlockMath math="= 8-2i" />
</MathContainer>

**Right side:** <InlineMath math="(z_1 \times z_2) + (z_1 \times z_3)" />

<MathContainer>
  <BlockMath math="(z_1 \times z_2) + (z_1 \times z_3) = (2(1+i)) + (2(3-2i))" />
  <BlockMath math="= (2+2i) + (6-4i)" />
  <BlockMath math="= (2+6) + (2i-4i)" />
  <BlockMath math="= 8-2i" />
</MathContainer>

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 <InlineMath math="(z_1 + z_2)^2 = z_1^2 + 2z_1z_2 + z_2^2" /> for any <InlineMath math="z_1, z_2" />.

<MathContainer>
  <BlockMath math="(z_1 + z_2)^2 = (z_1 + z_2)(z_1 + z_2) \quad \text{(Definition of square)}" />
  <BlockMath math="= z_1(z_1 + z_2) + z_2(z_1 + z_2) \quad \text{(Distributive Property)}" />
  <BlockMath math="= (z_1z_1 + z_1z_2) + (z_2z_1 + z_2z_2) \quad \text{(Distributive Property)}" />
  <BlockMath math="= z_1^2 + z_1z_2 + z_2z_1 + z_2^2 \quad \text{(Definition of square)}" />
  <BlockMath math="= z_1^2 + z_1z_2 + z_1z_2 + z_2^2 \quad \text{(Commutative Property: } z_2z_1 = z_1z_2)" />
  <BlockMath math="= z_1^2 + 2z_1z_2 + z_2^2 \quad \text{(Combining like terms)}" />
</MathContainer>

## Multiplicative Inverse

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

Let <InlineMath math="z^{-1} = u + iv" />. Then:

<MathContainer>
  <BlockMath math="z \times z^{-1} = 1" />
  <BlockMath math="(x+iy)(u+iv) = 1 + 0i" />
  <BlockMath math="(xu - yv) + i(xv + uy) = 1 + 0i" />
</MathContainer>

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

1.  <InlineMath math="xu - yv = 1" />
2.  <InlineMath math="xv + uy = 0" />

By solving this system of equations (for example, by multiplying equation 1 by <InlineMath math="x" />, equation 2 by <InlineMath math="y" />, then adding them, and using the substitution method), we will get:

<MathContainer>
  <BlockMath math="u = \frac{x}{x^2+y^2}" />
  <BlockMath math="v = -\frac{y}{x^2+y^2}" />
</MathContainer>

So, the multiplicative inverse of <InlineMath math="z = x+iy" /> is:

<BlockMath math="z^{-1} = \frac{x}{x^2+y^2} - i\frac{y}{x^2+y^2}" />

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

<BlockMath math="z^{-1} = \frac{\bar{z}}{|z|^2}" />