# Nakafa Framework: LLM

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

Output docs content for large language models.

---

export const metadata = {
  title: "Properties of Addition of Complex Numbers",
  description: "Explore algebraic properties of complex addition: commutative, associative, identity, inverse. Master scalar multiplication and prove expressions.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/01/2025",
  subject: "Complex Number",
};

## Operation Basics

Addition and scalar multiplication operations on complex numbers have interesting properties, similar to those of real numbers. These properties help us in performing calculations.

Let <InlineMath math="z_1" />, <InlineMath math="z_2" />, and <InlineMath math="z_3" /> be any complex numbers, and let <InlineMath math="c" /> and <InlineMath math="d" /> be any scalars (real numbers).

## Addition Properties

### Commutativity

The order of addition does not matter; the result remains the same.

<BlockMath math="z_1 + z_2 = z_2 + z_1" />

Example: <InlineMath math="(2+i) + (1-3i) = (1-3i) + (2+i) = 3-2i" />

### Addition Associativity

When adding three complex numbers, the grouping of the addition does not affect the result.

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

### Identity Element

There exists a complex number <InlineMath math="0 = 0 + 0i" /> (zero) such that when added to any complex number <InlineMath math="z_1" />, the result is <InlineMath math="z_1" /> itself.

<BlockMath math="z_1 + 0 = z_1" />

### Inverse Element

Every complex number <InlineMath math="z_1 = x + iy" /> has an additive inverse (opposite), denoted by <InlineMath math="-z_1 = -x - iy" />, such that their sum is the zero element (0).

<BlockMath math="z_1 + (-z_1) = 0" />

Example:

If <InlineMath math="z_1 = 5-2i" />, then <InlineMath math="-z_1 = -5+2i" />.

Then <InlineMath math="(5-2i) + (-5+2i) = (5-5) + i(-2+2) = 0 + 0i = 0" />.

## Scalar Multiplication

### Multiplication Associativity

The grouping of scalar multiplication does not affect the result.

<BlockMath math="c(dz_1) = (cd)z_1" />

### Scalar Distributivity

A scalar can be distributed over the addition of scalars.

<BlockMath math="(c + d)z_1 = cz_1 + dz_1" />

### Complex Distributivity

A scalar can be distributed over the addition of complex numbers.

<BlockMath math="c(z_1 + z_2) = cz_1 + cz_2" />

### Scalar Identity

Multiplying a complex number by the scalar 1 does not change the complex number.

<BlockMath math="1 z_1 = z_1" />

### Zero Scalar

Multiplying a complex number by the scalar 0 results in the complex number zero.

<BlockMath math="0 z_1 = 0" />

## Property Applications

These properties can be used to simplify or prove expressions involving complex numbers.

### Application Example

Show that for any complex number <InlineMath math="z" />, <InlineMath math="4z + (-4)z = 0" /> holds.

**Solution:**

We can use the distributivity of scalar over scalar addition (property f) and the multiplication by zero scalar property (property i).

<div className="flex flex-col gap-4">
  <BlockMath math="4z + (-4)z = (4 + (-4))z \quad \text{(Distributive Property)}" />
  <BlockMath math="= (0)z \quad \text{(Scalar addition)}" />
  <BlockMath math="= 0 \quad \text{(Multiplication by Zero Scalar Property)}" />
</div>

Thus, it is proven that <InlineMath math="4z + (-4)z = 0" />.

## Exercise

Using the properties above, prove that <InlineMath math="3z - \frac{1}{2}(2z) = 2z" /> for any complex number <InlineMath math="z" />.

### Answer Key

<div className="flex flex-col gap-4">
  <BlockMath math="3z - \frac{1}{2}(2z) = 3z + (-\frac{1}{2})(2z) \quad \text{(Definition of subtraction)}" />
  <BlockMath math="= 3z + ((-\frac{1}{2}) \times 2)z \quad \text{(Associativity of Scalar Multiplication)}" />
  <BlockMath math="= 3z + (-1)z \quad \text{(Scalar multiplication)}" />
  <BlockMath math="= (3 + (-1))z \quad \text{(Distributive Property)}" />
  <BlockMath math="= (2)z \quad \text{(Scalar addition)}" />
  <BlockMath math="= 2z" />
</div>

Thus, it is proven that <InlineMath math="3z - \frac{1}{2}(2z) = 2z" />.