# Definition:Complex Number/Definition 2

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## 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 number.

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

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

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

## Notation

The **set of complex numbers** is usually denoted $\C$.

Variants on $\C$ are often seen, for example $\mathbf C$, $\CC$ and $\mathfrak C$, or even just $C$.

When $a$ and $b$ are symbols representing variables or constants, the form $a + i b$ is usually (but not universally) seen.

Similarly, when $a$ and $b$ are actual numbers, for example $3$ and $4$, it is usually (but not universally) written $3 + 4 i$.

## Also see

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

## Sources

- 1957: E.G. Phillips:
*Functions of a Complex Variable*(8th ed.) ... (previous) ... (next): Chapter $\text I$: Functions of a Complex Variable: $\S 1$. Complex Numbers - 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): Chapter $\text{II}$: Groups: 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 - 1990: H.A. Priestley:
*Introduction to Complex Analysis*(revised ed.) ... (previous) ... (next): $1$ The complex plane: Complex numbers $\S 1.1$ Complex numbers and their representation - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**complex number**