# Definition:Complex Number

## Contents

## Definition

### Informal Definition

A **complex number** is a number in the form $a + b i$ or $a + i b$ where:

- $a$ and $b$ are real numbers
- $i$ is a square root of $-1$, i.e. $\sqrt {-1}$.

### Formal 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 $\C$.

### Construction from Cayley-Dickson Construction

The **complex numbers** can be defined by the Cayley-Dickson construction from the set of real numbers $\R$.

From Real Numbers form Algebra, $\R$ forms a nicely normed $*$-algebra.

Let $a, b \in \R$.

Then $\left({a, b}\right) \in \C$, where:

- $\left({a, b}\right) \left({c, d}\right) = \left({a c - d \overline b, \overline a d + c b}\right)$
- $\overline {\left({a, b}\right)} = \left({\overline a, -b}\right)$

where:

- $\overline a$ is the conjugate of $a$

and

- $\overline {\left({a, b}\right)}$ is the conjugation operation on $\C$.

From Real Numbers form Algebra, $\overline a = a$ and so the above translate into:

- $\left({a, b}\right) \left({c, d}\right) = \left({a c - d b, a d + c b}\right)$
- $\overline {\left({a, b}\right)} = \left({a, -b}\right)$

It is clear by direct comparison with the formal definition that this construction genuinely does generate the **complex numbers**.

## Real and Imaginary Parts

### Real Part

Let $z = a + i b$ be a **complex number**.

The **real part** of $z$ is the coefficient $a$.

The real part of a complex number $z$ is usually denoted:

- $\Re \left({z}\right)$
- $\operatorname{Re} \left({z}\right)$
- $\operatorname{re} \left({z}\right)$

or a similar variant.

### Imaginary Part

Let $z = a + i b$ be a **complex number**.

The **imaginary part** of $z$ is the coefficient $b$ (**note:** not $i b$).

The imaginary part of a complex number $z$ is usually denoted:

- $\Im \left({z}\right)$
- $\operatorname{Im} \left({z}\right)$
- $\operatorname{im} \left({z}\right)$

or a similar variant.

### Imaginary Unit

The entity $i := 0 + 1 i$ is known as the **imaginary unit**.

### Wholly Real

A **complex number** $z = a + i b$ is **wholly real** iff $b = 0$.

### Wholly Imaginary

A **complex number** $z = a + i b$ is **wholly imaginary** iff $a = 0$.

## Complex Plane

Because a **complex number** can be expressed as an ordered pair, we can plot the number $x + i y$ on the real number plane $\R^2$:

### Real Axis

**Complex numbers** of the form $\left({x, 0}\right)$, being wholly real, appear as points on the $x$-axis.

### Imaginary Axis

**Complex numbers** of the form $\left({0, y}\right)$, being wholly imaginary, appear as points on the points on the $y$-axis.

## Polar Form

For any **complex number** $z = x + i y \ne 0$, let:

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle r\) | \(=\) | \(\displaystyle \) | \(\displaystyle \left\vert z \right\vert = \sqrt {x^2 + y^2}\) | \(\displaystyle \) | \(\displaystyle \) | the modulus of $z$, and | ||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \theta\) | \(=\) | \(\displaystyle \) | \(\displaystyle \arg \left({z}\right)\) | \(\displaystyle \) | \(\displaystyle \) | the argument of $z$ (the angle which $z$ yields with the real line) |

where $x, y \in \R$.

From the definition of $\arg \left({z}\right)$:

- $(1): \quad \dfrac x r = \cos \theta$

- $(2): \quad \dfrac y r = \sin \theta$

which implies that:

- $x = r \cos \theta$
- $y = r \sin \theta$

which in turn means that any number $z = x + i y \ne 0$ can be written as:

- $z = x + i y = r \left({\cos \theta + i \sin \theta}\right)$

The pair $\left \langle {r, \theta} \right \rangle$ is called the **polar form** of the **complex number** $z \ne 0$.

The number $z = 0 + 0i$ is defined as $\left \langle {0, 0} \right \rangle$.

## Also denoted as

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

When $a$ and $b$ are symbols representing variables or constants, the form $a + i b$ is usually seen.

When $a$ and $b$ are actual numbers, for example $3$ and $4$, it is usually written $3 + 4 i$.

When mathematics is applied to engineering, in particular electrical and electronic engineering, the symbol $j$ is usually used, as $i$ is the standard symbol used to denote the flow of electric current, and to use it also for $\sqrt {-1}$ would cause untold confusion.

## Also see

The $a + i b$ notation usually proves more convenient; the ordered pair version is generally used only for the formal definition as given above.

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

## Sources

- Murray R. Spiegel:
*Theory and Problems of Complex Variables*(1964)... (previous)... (next): $1$: Complex Numbers: The Complex Number System - Seth Warner:
*Modern Algebra*(1965)... (previous)... (next): $\S 1$ - George McCarty:
*Topology: An Introduction with Application to Topological Groups*(1967)... (previous)... (next): Introduction: Special Symbols - Ian D. Macdonald:
*The Theory of Groups*(1968)... (previous)... (next): Appendix: Elementary set and number theory - B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*(1970)... (previous)... (next): $\S 1.2$: Some examples of rings: Ring Example $4$ - W.A. Sutherland:
*Introduction to Metric and Topological Spaces*(1975)... (previous)... (next): Notation and Terminology - Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*(1978)... (previous)... (next): $\S 2 \ \text{(b)}$ - H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*(1996)... (previous)... (next): Appendix $\text{A}.1$: Sets