Definition:Complex Number

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

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.

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:
 * $z = r e^{i \theta}$

Also see

 * Equivalence of Definitions of Complex Number

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


 * Definition:Complex Addition
 * Definition:Complex Multiplication