# Definition:Complex Number/Definition 2

## Definition

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

### Complex Addition

Let $\left({x_1, y_1}\right)$ and $\left({x_2, y_2}\right)$ be complex numbers.

Then $\left({x_1, y_1}\right) + \left({x_2, y_2}\right)$ is defined as:

- $\left({x_1, y_1}\right) + \left({x_2, y_2}\right):= \left({x_1 + x_2, y_1 + y_2}\right)$

### Complex Multiplication

Let $\left({x_1, y_1}\right)$ and $\left({x_2, y_2}\right)$ be complex numbers.

Then $\left({x_1, y_1}\right) \left({x_2, y_2}\right)$ is defined as:

- $\left({x_1, y_1}\right) \left({x_2, y_2}\right) := \left({x_1 x_2 - y_1 y_2, x_1 y_2 + y_1 x_2}\right)$

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 - 1964: Murray R. Spiegel:
*Theory and Problems of Complex Variables*... (previous) ... (next): $1$: Complex Numbers: Axiomatic Foundations of the Complex Number System - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): $\text{II}$: Subgroups