# Definition:Complex Number/Definition 2

## Contents

## Definition

A **complex number** is an ordered pair $\tuple {x, y}$ where $x, y \in \R$ are real numbers, on which the operations of addition and multiplication are defined as follows:

### Complex Addition

Let $\tuple {x_1, y_1}$ and $\tuple {x_2, y_2}$ be complex numbers.

Then $\tuple {x_1, y_1} + \tuple {x_2, y_2}$ is defined as:

- $\tuple {x_1, y_1} + \tuple {x_2, y_2}:= \tuple {x_1 + x_2, y_1 + y_2}$

### Complex Multiplication

Let $\tuple {x_1, y_1}$ and $\tuple {x_2, y_2}$ be complex numbers.

Then $\tuple {x_1, y_1} \tuple {x_2, y_2}$ is defined as:

- $\tuple {x_1, y_1} \tuple {x_2, y_2} := \tuple {x_1 x_2 - y_1 y_2, x_1 y_2 + y_1 x_2}$

### Scalar Product

Let $\tuple {x, y}$ be a complex numbers.

Let $m \in \R$ be a real number.

Then $m \tuple {x, y}$ is defined as:

- $m \tuple {x, y} := \tuple {m x, m y}$

The set of all complex numbers is denoted $\C$.

## Also see

- Results about
**complex numbers**can be found here.

## Sources

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 1.2$. The Algebraic Theory - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): $\text{II}$: Subgroups - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Axiomatic Foundations of the Complex Number System