Complex Numbers:The Polar Form

Complex Plane:

A complex number z can be viewed as a point or a position vector in a two-dimensional Cartesian coordinate system called the complex plane.

Polar Form:

Alternatively, the complex number z can be specified by polar coordinates. The polar coordinates are r = |z| ≥ 0, called the absolute value or modulus, and φ = arg(z), called the argument of z.

Conversion from the polar form to the Cartesian form:

Notation of The Polar Form:

The notation of the polar form as is called trigonometric form. The notation cis φ is sometimes used as an abbreviation for cos φ + i sin φ. Using Euler's formula it can also be written as:

which is called exponential form.

Multiplication, division, and exponentiation in the polar form:

Multiplication, division, and exponentiation of complex numbers are much easier to perform in the polar form than in the Cartesian form. For example:

De Moivre's Formula: