Complex Numbers in Polar Form

Modulus of a Complex Number

The modulus of a complex number is the length of the vector determined by the origin of its coordinates and affix. It is denoted by |z|.

Modulus of a Complex Number

Modulus of a Complex Number

Modulus of a Complex Number

Argument of a Complex Number

The argument of a complex number is the angle that forms the vector with the real axis. It is denoted by arg(z).

Argument of a Complex Number

Expression of a Complex Number in Polar Form

z = rα

|Z| = r r is the modulus.

arg(z) = ArgumentArgument is the argument.

Examples

Express in polar form:

Complex Number

Complex Number Operations

Complex Number Operations

z = 260º


Complex Number

Complex Number Operations

Complex Number Operations

z = 2120º


Complex Number

Complex Number Operations

Argument

z = 2240º


Complex Number

Complex Number Operations

Argument

z = 2300º


Z = 2

Complex Number Operations

Argument

z = 2


Z = −2

Complex Number Operations

Argument

z = 2180º


Z = 2i

Complex Number Operations

Argument

z = 290º


Z = −2i

Complex Number Operations

Argument

z = 2270º


Trigonometric form

rα = r (cos α + i sin α)

Examples

z = 2120º

z = 2 · (cos 120º + i sin 120º)

Complex Number Operations

Complex Number Operations

Complex Number Operations


z =1 = 1

z =1180º = −1

z =190º = i

z =1270º = −i