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20 Polar Form of Complex Numbers

Learning Objectives

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 German mathematician Leopold Kronecker 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 investigation and struggle by mathematicians such as Pythagoras, Descartes, De Moivre, Euler, Gauss, and others. Complex numbers answered questions that for centuries had puzzled the greatest minds in science.

We first 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.

Plotting Complex Numbers in the Complex Plane

Plotting a complex numberis similar to plotting a real number, except that the horizontal axis represents the real part of the number,and the vertical axis represents the imaginary part of the number,

How To

Given a complex numberplot it in the complex plane.

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

Plot the point in the complex plane by movingunits in the horizontal direction andunits in the vertical direction.

Plotting a Complex Number in the Complex Plane

Plot the complex number in the complex plane.

Show Solution

From the origin, move two units in the positive horizontal direction and three units in the negative vertical direction. See (Figure).

Try It

Plot the pointin the complex plane.

Show Solution

Finding the Absolute Value of a Complex Number

The first step toward working with a complex number in polar form is to find the absolute value. The absolute value of a complex number is the same as its magnitude, orIt measures the distance from the origin to a point in the plane. For example, the graph ofin (Figure), shows

Absolute Value of a Complex Number

Givena complex number, the absolute value ofis defined as

It is the distance from the origin to the point

Notice that the absolute value of a real number gives the distance of the number from 0, while the absolute value of a complex number gives the distance of the number from the origin,

Finding the Absolute Value of a Complex Number with a Radical

The polar form of a complex number expresses a number in terms of an angleand its distance from the originGiven a complex number in rectangular form expressed aswe use the same conversion formulas as we do to write the number in trigonometric form:

We use the term modulus to represent the absolute value of a complex number, or the distance from the origin to the pointThe modulus, then, is the same asthe radius in polar form. We useto indicate the angle of direction (just as with polar coordinates). Substituting, we have

Polar Form of a Complex Number

Writing a complex number in polar form involves the following conversion formulas:

Making a direct substitution, we have

whereis the modulus and is the argument. We often use the abbreviationto represent

Expressing a Complex Number Using Polar Coordinates

Express the complex numberusing polar coordinates.

Show Solution

On the complex plane, the numberis the same asWriting it in polar form, we have to calculatefirst.

Next, we look atIfandthenIn polar coordinates, the complex numbercan be written asorSee (Figure).

Try It

Express as in polar form.

Show Solution

Finding the Polar Form of a Complex Number

Find the polar form of

Show Solution

First, find the value of

Find the angleusing the formula:

Thus, the solution is

Try It

Writein polar form.

Show Solution

Converting a Complex Number from Polar to Rectangular Form

Converting a complex number from polar form to rectangular form is a matter of evaluating what is given and using the distributive property. In other words, givenfirst evaluate the trigonometric functionsandThen, multiply through by

Converting from Polar to Rectangular Form

Convert the polar form of the given complex number to rectangular form:

Show Solution

We begin by evaluating the trigonometric expressions.

After substitution, the complex number is

We apply the distributive property:

The rectangular form of the given point in complex form is

Finding the Rectangular Form of a Complex Number

Find the rectangular form of the complex number givenand

Show Solution

Ifandwe first determine We then findand

The rectangular form of the given number in complex form is

Try It

Convert the complex number to rectangular form:

Show Solution

Finding Products of Complex Numbers in Polar Form

Now that we can convert complex numbers to polar form we will learn how to perform operations on complex numbers in polar form. For the rest of this section, we will work with formulas developed by French mathematician Abraham de Moivre (1667-1754). These formulas have made working with products, quotients, powers, and roots of complex numbers much simpler than they appear. The rules are based on multiplying the moduli and adding the arguments.

Products of Complex Numbers in Polar Form

Ifand then the product of these numbers is given as:

Notice that the product calls for multiplying the moduli and adding the angles.

Finding the Product of Two Complex Numbers in Polar Form

Find the product ofgivenand

Show Solution

Follow the formula

Finding Quotients of Complex Numbers in Polar Form

The quotient of two complex numbers in polar form is the quotient of the two moduli and the difference of the two arguments.

Quotients of Complex Numbers in Polar Form

Ifand then the quotient of these numbers is

Notice that the moduli are divided, and the angles are subtracted.

How To

Given two complex numbers in polar form, find the quotient.

Divide

Find

Substitute the results into the formula:Replacewithand replacewith

Calculate the new trigonometric expressions and multiply through by

Finding the Quotient of Two Complex Numbers

Find the quotient ofand

Show Solution

Using the formula, we have

Try It

Find the product and the quotient ofand

Show Solution

Finding Powers of Complex Numbers in Polar Form

Finding powers of complex numbers is greatly simplified using De Moivre’s Theorem. It states that, for a positive integeris found by raising the modulus to thepower and multiplying the argument byIt is the standard method used in modern mathematics.

De Moivre’s Theorem

Ifis a complex number, then

where
is a positive integer.

Evaluating an Expression Using De Moivre’s Theorem

Evaluate the expressionusing De Moivre’s Theorem.

Show Solution

Since De Moivre’s Theorem applies to complex numbers written in polar form, we must first writein polar form. Let us find

Then we findUsing the formulagives

Use De Moivre’s Theorem to evaluate the expression.

Finding Roots of Complex Numbers in Polar Form

To find the nth root of a complex number in polar form, we use theRoot Theorem or De Moivre’s Theorem and raise the complex number to a power with a rational exponent. There are several ways to represent a formula for findingroots of complex numbers in polar form.

The nth Root Theorem

To find theroot of a complex number in polar form, use the formula given as

whereWe add toin order to obtain the periodic roots.

Finding the nth Root of a Complex Number

Evaluate the cube roots of

Show Solution

We have

There will be three roots:Whenwe have

Whenwe have

When we have

Remember to find the common denominator to simplify fractions in situations like this one. Forthe angle simplification is

Try It

Find the four fourth roots of

Show Solution

Access these online resources for additional instruction and practice with polar forms of complex numbers.

Complex numbers in the formare plotted in the complex plane similar to the way rectangular coordinates are plotted in the rectangular plane. Label the x-axis as the real axis and the y-axis as the imaginary axis. See (Figure).

The absolute value of a complex number is the same as its magnitude. It is the distance from the origin to the point:See (Figure) and (Figure).

To write complex numbers in polar form, we use the formulasand Then,See (Figure) and (Figure).

To convert from polar form to rectangular form, first evaluate the trigonometric functions. Then, multiply through bySee (Figure) and (Figure).

To find the product of two complex numbers, multiply the two moduli and add the two angles. Evaluate the trigonometric functions, and multiply using the distributive property. See (Figure).

To find the quotient of two complex numbers in polar form, find the quotient of the two moduli and the difference of the two angles. See (Figure).

To find the power of a complex numberraise to the power and multiply by See (Figure).

Finding the roots of a complex number is the same as raising a complex number to a power, but using a rational exponent. See (Figure).

Section Exercises

Verbal

A complex number isExplain each part.

Show Solution

a is the real part, b is the imaginary part, and

What does the absolute value of a complex number represent?

How is a complex number converted to polar form?

Show Solution

Polar form converts the real and imaginary part of the complex number in polar form using and

How do we find the product of two complex numbers?

What is De Moivre’s Theorem and what is it used for?

Show Solution

It is used to simplify polar form when a number has been raised to a power.

Algebraic

For the following exercises, find the absolute value of the given complex number.

Show Solution

Show Solution

Show Solution

For the following exercises, write the complex number in polar form.

Show Solution

Show Solution

For the following exercises, convert the complex number from polar to rectangular form.

Show Solution

Show Solution

Show Solution

For the following exercises, findin polar form.

Show Solution

Show Solution

Show Solution

For the following exercises, findin polar form.

Show Solution

Show Solution

Show Solution

For the following exercises, find the powers of each complex number in polar form.

Findwhen

Show Solution

Findwhen

Findwhen

Show Solution

Findwhen

Findwhen

Show Solution

Findwhen

For the following exercises, evaluate each root.

Evaluate the cube root ofwhen

Show Solution

Evaluate the square root ofwhen

Evaluate the cube root ofwhen

Show Solution

Evaluate the square root ofwhen

Evaluate the cube root ofwhen

Show Solution

Graphical

For the following exercises, plot the complex number in the complex plane.

Show Solution

Show Solution

Show Solution

Show Solution

Show Solution

Technology

For the following exercises, find all answers rounded to the nearest hundredth.

Use the rectangular to polar feature on the graphing calculator to changeto polar form.

Use the rectangular to polar feature on the graphing calculator to change
to polar form.

Show Solution

Use the rectangular to polar feature on the graphing calculator to change
to polar form.

Use the polar to rectangular feature on the graphing calculator to changeto rectangular form.

Show Solution

Use the polar to rectangular feature on the graphing calculator to changeto rectangular form.

Use the polar to rectangular feature on the graphing calculator to changeto rectangular form.

Show Solution

Glossary

argument

the angle associated with a complex number; the angle between the line from the origin to the point and the positive real axis

De Moivre’s Theorem

formula used to find thepower or nth roots of a complex number; states that, for a positive integeris found by raising the modulus to thepower and multiplying the angles by

modulus

the absolute value of a complex number, or the distance from the origin to the pointalso called the amplitude

polar form of a complex number

a complex number expressed in terms of an angle and its distance from the origincan be found by using conversion formulasand