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OpenStax-CNX module: m49408

1

Polar Form of Complex Numbers

OpenStax College

This work is produced by OpenStax-CNX and licensed under the

Creative Commons Attribution License 4.0

Abstract

In this section, you will:

Plot complex numbers in the complex plane.

Find the absolute value of a complex number

Write complex numbers in polar form.

Convert a complex number from polar to rectangular form.

Find products of complex numbers in polar form.

Find quotients of complex numbers in polar form.

Find powers of complex numbers in polar form.

Find roots of complex numbers in polar form.

God made the integers; all else is the work of man. This rather famous quote by nineteenth-century

Kronecker

German mathematician Leopold

sets the stage for this section on the polar form of a complex

number.

Complex numbers were invented by people and represent over a thousand years of continuous

Pythagoras

Descartes

Euler

Gauss

investigation and struggle by mathematicians such as

,

, De Moivre,

,

,

and others.

Complex numbers answered questions that for centuries had puzzled the greatest minds in

science.

We rst encountered complex numbers in Complex Numbers.

In this section, we will focus on the

mechanics of working with complex numbers: translation of complex numbers from polar form to rectangular

form and vice versa, interpretation of complex numbers in the scheme of applications, and application of De

Moivre's Theorem.

1 Plotting Complex Numbers in the Complex Plane

complex number

a + bi

Plotting a

is similar to plotting a real number, except that the horizontal axis

a,

bi.

represents the real part of the number,

and the vertical axis represents the imaginary part of the number,

Given a complex number

plot it in the complex plane.

a + bi,

how to feature:

1.Label the horizontal axis as the real axis and the vertical axis as the imaginary axis.

a

b

2.Plot the point in the complex plane by moving

units in the horizontal direction and

units

in the vertical direction.

Example 1

Plotting a Complex Number in the Complex Plane

complex plane

2

3i

Plot the complex number

in the

.

Version 1.2: Jul 8, 2014 10:00 am +0000

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Parent category: Education