Definition:Complex Number

Informal Definition
A complex number is a number in the form $$a + b \imath$$ or $$a + \imath b$$ where:
 * $$a$$ and $$b$$ are real numbers;
 * $$\imath$$ is the square root of $-1$, i.e. $$\sqrt {-1}$$.

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

Formal Definition
A complex number is an ordered pair $$\left({x, y}\right)$$ where $$x, y \in \mathbb{R}$$ 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:

$$\left({x_1, y_1}\right) + \left({x_2, y_2}\right) \ \stackrel {\mathbf {def}} {=\!=} \ \left({x_1 + x_2, y_1 + y_2}\right)$$

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:

$$\left({x_1, y_1}\right) \left({x_2, y_2}\right) \ \stackrel {\mathbf {def}} {=\!=} \ \left({x_1 x_2 - y_1 y_2, x_1 y_2 + y_1 x_2}\right)$$

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

The $$a + \imath b$$ notation 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 $$a + \imath b$$ is the coefficient $$a$$.

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

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

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

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

Wholly Imaginary
The complex number $$z = a + \imath b$$ 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 $$a + \imath b$$ is usually seen.

When $$a$$ and $$b$$ are actual numbers, for example 3 and 4, it usually gets written $$3 + 4 \imath$$.

The symbol $$\imath$$ can also be seen as $$i$$.

When mathematics is applied to engineering, in particular electrical and electronic engineering, the symbol $$\jmath$$ or $$j$$ 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 $$x + \imath y$$ on the Real Number Plane $$\mathbb{R}^2$$:



This representation is also known as an Argand Diagram or a Gauss Plane, but as it is difficult to establish exactly who had precedence over the concept of plotting complex numbers on a plane, the more neutral term complex plane is usually preferred nowadays.

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
The polar form of a complex number $$x + \imath y$$ is written $$\left \langle {r, \theta} \right \rangle$$, where:
 * $$x = r \cos \theta$$;
 * $$y = r \sin \theta$$;

and $$\theta$$ is measured in radians.

Thus $$x + \imath y$$ can be expressed $$r \left({\cos \theta + \imath \sin \theta}\right)$$.

The value $$r$$ is the modulus of $$x + \imath y$$:

$$\left|{x + \imath y}\right| = \sqrt {x^2 + y^2} = \sqrt {r^2 \left({\cos^2 \theta + \sin^2 \theta}\right)} = r$$.