Complex Numbers

Complex numbers have a real and imaginary part.

Description

A complex number has a real part and an imaginary part. It can be expressed as a + bi where a and b are from the set of real numbers.

The defining characteristic of i is that i² = -1.

Coordinates

Complex numbers can be expressed as being in a 2-dimensional plane: given z = x + yi, z = (x,y).

It follows that they can also be expressed in polar coordinates. In this case, they are expressed either as z = r(cosθ + i sinθ) or z = r e^θi. This reveals a relation between complex numbers and rotation in a coordinate system.

To convert the above z into polar coordinates:

• r = |z| = √(x² + y²)

• θ = tan^-1(y/x)

To convert back:

• x = r cosθ

• y = r sinθ

Vectors

Complex numbers are sometimes expressed as a 2-dimensional vector with a real first member and an imaginary second member. In this case they are expressed as z = [a b] where z is in R² space.

Complex Conjugates

For a complex number as z = a + bi, there is a complex conjugate notated and evaluated as z̅ = a - bi.

Properties

The double conjugate of a complex number is the original complex number: a̿ = a.

Conjugation is distributive.

• If c = ab, then c̅ = a̅b̅.

• If c = a + b, then c̅ = a̅ + b̅.

• If c = a - b, then c̅ = a̅ - b̅.

• If d = exp(a), then d̅ = exp(a̅)

• If d = ln(a), then d̅ = ln(a̅)

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