# Definition:Complex Number

## 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 $-1$, that is, $i = \sqrt {-1}$.

### Formal 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:

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

### 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 $\tuple {a, b} \in \C$, where:

- $\tuple {a, b} \tuple {c, d} = \tuple {a c - d \overline b, \overline a d + c b}$
- $\overline {\tuple {a, b} } = \tuple {\overline a, -b}$

where:

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

and

- $\overline {\tuple {a, b} }$ is the conjugation operation on $\C$.

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

- $\tuple {a, b} \tuple {c, d} = \tuple {a c - d b, a d + c b}$
- $\overline {\tuple {a, b} } = \tuple {a, -b}$

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

## 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 $3 + 4 i$.

## 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 on $\mathsf{Pr} \infty \mathsf{fWiki}$ by $\map \Re z$ or $\mathop \Re z$.

### 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 on $\mathsf{Pr} \infty \mathsf{fWiki}$ by $\map \Im z$ or $\Im z$.

### 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** if and only if $b = 0$.

### Wholly Imaginary

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

## 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 $\R^2$:

This representation is known as the **complex plane**.

### Real Axis

Complex numbers of the form $\tuple {x, 0}$, being wholly real, appear as points on the $x$-axis.

### Imaginary Axis

Complex numbers of the form $\tuple {0, y}$, being wholly imaginary, appear as points on the points on the $y$-axis.

This line is known as the **imaginary axis**.

## Polar Form

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

\(\ds r\) | \(=\) | \(\ds \cmod z = \sqrt {x^2 + y^2}\) | the modulus of $z$, and | |||||||||||

\(\ds \theta\) | \(=\) | \(\ds \arg z\) | the argument of $z$ (the angle which $z$ yields with the real line) |

where $x, y \in \R$.

From the definition of $\arg z$:

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

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

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:

- $z = x + i y = r \paren {\cos \theta + i \sin \theta}$

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

The number $z = 0 + 0 i$ is defined as $\polar {0, 0}$.

## Also known as

Some soruces refer to a **complex number** as an **imaginary number**, but it is too easy to confuse this with a **wholly imaginary number**.

Hence this usage is not endorsed by $\mathsf{Pr} \infty \mathsf{fWiki}$.

## 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**.

## Historical Note

The concept of a **complex number** originated in the $16$th century as during the course of developing the solution to the general cubic equation.

In his *Artis Magnae, Sive de Regulis Algebraicis* of $1545$, Gerolamo Cardano considered the simultaneous equations:

- $\begin {cases} x + y & = 10 \\ x y & = 40 \end {cases}$

and obtained the solution:

- $\begin {cases} x & = 5 + \sqrt {-15} \\y & = 5 - \sqrt {-15} \end {cases}$

He made no attempt to interpret the meaning of the square root of a negative number, dismissing it with the comment:

*So progresses arithmetic subtlely, the end result of which ... is as refined as it is useless.*

On the other hand, he applied what is now known as Cardano's Formula to obtain a solution to:

- $x^3 = 15 x + 4$

which leads to the expression:

- $x = \sqrt [3] {2 + \sqrt {-121} } + \sqrt [3] {2 - \sqrt {-121} }$

whereas the "obvious" answer is $x = 4$.

Rafael Bombelli responded by treating $\sqrt {-121}$ in the same way as conventional numbers, showing that:

- $\paren {2 \pm \sqrt {-1} }^3 = 2 \pm \sqrt {-121}$

from which we obtain:

- $x = \paren {2 + \sqrt {-1} } + \paren {2 - \sqrt {-1} } = 4$

René Descartes, in his *La Géométrie* of $1637$, distinguished between "real numbers" and "imaginary numbers", concluding that if the latter occurred during the solution of a problem, it was in fact insoluble.

This view was endorsed by Isaac Newton.

However, by the $18$th century, complex numbers had gained acceptance.

## Sources

- 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory - 1968: Ian D. Macdonald:
*The Theory of Groups*... (previous) ... (next): Appendix: Elementary set and number theory - 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $2$: Some examples of rings: Ring Example $4$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): Notation and Terminology - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 2$: Introductory remarks on sets: $\text{(b)}$ - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: The Complex Number System - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.26$: Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem - 1990: H.A. Priestley:
*Introduction to Complex Analysis*(revised ed.) ... (previous) ... (next): Notation and terminology - 1994: H.E. Rose:
*A Course in Number Theory*(2nd ed.) ... (previous) ... (next): Preface to first edition: Prerequisites - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): Appendix $\text A$: Sets and Functions: $\text{A}.1$: Sets - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**number**:**2.** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**number**:**2.** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**complex number**