# Complex numbers

A complex number is a number made up of a real and imaginary component. A complex number can be written in the form

z = a + bi

where a and b are real numbers, i represents the imaginary unit (in some disciplines, j is used instead), and z is the complex number. When referencing complex numbers, the real part can be denoted as Re(z). The imaginary part is denoted as Im(z). In reference to the general equation for a complex number, Re(z) = a and Im(z) = b.

Technically, all real and imaginary numbers are complex numbers, even if they don't have an explicit real or imaginary component. In such cases, the missing component is just 0, which is considered both real and imaginary. In z = 2 + 0i for example,

Re(z) = 2

Im(z) = 0

Note that the i is left out of Im(z), since Im(z) is the imaginary component not the imaginary number as a whole.

Complex numbers can be represented in the complex plane as the vector formed by the pair of numbers, (a, b), as shown in the figure below.

## Adding and subtracting complex numbers

To add (or subtract) complex numbers, add the real and imaginary components separately. Generally,

(a + bi) + (c + di) = (a + c) + (b + d)i

Examples

1. (5 - 6i) + (7 + 3i):

(5 - 6i) + (7 + 3i) = (5 + 7) + (-6 + 3)i = 12 - 3i

2. (5 - 6i) - (7 + 3i):

(5 - 6i) - (7 + 3i) = (5 - 6i) - 7 - 3i = (5 - 7) - 6i - 3i = -2 - 9i

When performing subtraction, just distribute the negative as you would with real numbers.

## Multiplying complex numbers

The FOIL method of binomial expansion can be used to multiply complex numbers; essentially, multiply each term in each complex number by each of the terms in the other complex number, as shown in the figure below.

Example

Solve (5 - 6i)(7 + 3i).

(5 - 6i)(7 + 3i) = | 5×7 + 5×3i - 6i×7 - 6i×3i |

= | 35 + 15i - 42i - 18i^{2} |

= | 35 - 27i - 18(-1) |

= | 53 - 27i |

## Dividing complex numbers

Dividing complex numbers involves using the complex conjugate to simplify the expression. Given a division problem of the form:

multiply both complex numbers by the complex conjugate of the denominator:

This results in a real number in the denominator, which makes simplifying the expression simpler, because any complex number multiplied by its complex conjugate results in a real number:

(c + di)(c - di) = c^{2} - (di)^{2} = c^{2} + d^{2}

Example

Solve .