Definition:Complex Number

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 the square root of $-1$, i.e. $\sqrt {-1}$.

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

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

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:


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

Imaginary Unit
The entity $i$ as defined here is called the imaginary unit.

Equivalence of Definitions
The two definitions as given above are equivalent.

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

Real Part
The real part of a complex number $a + i b$ is the coefficient $a$.

The real part of a complex number $z$ is often denoted $\Re \left({z}\right)$ or $\operatorname{Re} \left({z}\right)$ or $\operatorname{re} \left({z}\right)$.

Imaginary Part
The imaginary part of a complex number $a + i b$ is the coefficient $b$ (note: not $i b$).

The imaginary part of a complex number $z$ is often denoted $\Im \left({z}\right)$ or $\operatorname{Im} \left({z}\right)$ or $\operatorname{im} \left({z}\right)$.

Wholly Real
The complex number $z = a + i b$ is called wholly real or completely real, or entirely real, etc. iff $b = 0$.

Wholly Imaginary
The complex number $z = a + i b$ is called wholly imaginary or completely imaginary, or entirely imaginary, etc. iff $a = 0$.

Notation
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 usually gets 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.

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 $\R^2$:


 * ComplexPlane.png

See Argand diagram.

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 $y$-axis.

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

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 $\left \langle {0, 0} \right \rangle$.

From Euler's Theorem we have that $e^{i \theta} = \cos \theta + i \sin \theta$, so we can also write $z$ in the form:
 * $z = r e^{i \theta}$