Wikia

See also the Wikipedia article:

Complex number

A complex number is any number of the form

$a + b i$ ,

where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Definition
Set of all complex numbers $\mathbb{C} = \left\{a + bi\mid a,b\in\mathbb R\right\}$ , where $i^2 = -1$ .

The set of complex numbers, denoted C or $\mathbb C$ , is a field under the operations of addition (+) and multiplication ( $\cdot$ ) defined as follows:

$(a_1 + b_1 i) + (a_2 + b_2 i) = (a_1 + a_2) + (b_1 + b_2) i$

$(a_1 + b_1 i) \cdot (a_2 + b_2 i) = (a_1 a_2 - b_1 b_2) + (a_1 b_2 + a_2 b_1) i$

See the article Complex numbers as a field for more details. In particular, this means that these complex-number operations are commutative, associative and distributive in the same way as their real-number counterparts.

Note that the final expression above is the result of multiplying out the left hand side in the usual way (see, for example, FOIL) and simplifying using the fact that $i^2=-1$ (see Imaginary unit).

It is easily seen that all real numbers are also complex numbers, since

$a = a + 0 i$ .

All purely imaginary numbers are also complex, since

$b \cdot i = 0 + b i$ .

The complex numbers can also be thought of as a vector space over the real numbers, with basis vectors 1 (one, the real unit) and $i$ (the imaginary unit). In this case, a complex number may be written as:

$a \cdot 1 + b \cdot i$

Regardless of the interpretation used, $a$ is also known as the real part of the complex number and $b$ the imaginary part; or where $a$ is a purely real number and $bi$ is a purely imaginary number.

Number lines and rectangular form Edit

If the real numbers have a real number line, and the imaginary numbers have their own number line, these two number lines can be interpreted as being perpendicular to one another.

These perpendicular lines form axes in a Cartesian coordinate system where all complex numbers lie somewhere on the plane.

In this fashion, the real value of magnitude 'a' forms the x-coordinate and the imaginary value, of magnitude 'b', forms the y-coordinate. This way, all complex numbers exist somewhere on the complex number plane at coordinate loci $(a, b)$ , which equates to a simple numeric value of $a+bi$ .

Polar form Edit

Since all coordinates on a rectangular coordinate plane can be interpreted using the polar coordinate system, all complex numbers can also be interpreted in terms of a polar coordinate set $(r,\theta)$ and using the trigonometric based function cis.

In this way:

$a + bi = r\cdot\mathrm{cis}\;\theta$

Where:

$a = r\cdot\cos\theta$

$b = r\cdot\sin\theta$

$\theta = \arctan(b/a)$

$r = \sqrt{a^2 + b^2}$

Matrix representations Edit

In a more constructive approach, complex numbers can also be represented as $2 \times 2$ matrices of the form $\left[\begin{array}{cc} a & -b\\ b & a\end{array}\right]$ , where $a$ and $b$ are real numbers.

Indeed, as a vector space over the real numbers, the following subspace can be shown to be a field, and moreover, isomorphic to $\mathbb C$ :

$\mathrm{span}\left\{\left[\begin{array}{cc} 1 & 0\\ 0 & 1\end{array}\right], \left[\begin{array}{cc} 0 & -1\\ 1 & 0\end{array}\right]\right\}$

Where the isomorphism is given by $\phi\left(a+bi\right)=a\left[\begin{array}{cc} 1 & 0\\ 0 & 1\end{array}\right]+b\left[\begin{array}{cc} 0 & -1\\ 1 & 0\end{array}\right]$