Find All Solutions in Complex With Polar Coordinates
Further Applications of Trigonometry
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 number is 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,
Plotting a Complex Number in the Complex Plane
Plot the complex number in the complex plane.
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From the origin, move two units in the positive horizontal direction and three units in the negative vertical direction. See (Figure).
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Plot the pointin the complex plane.
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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
Find the absolute value of
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Using the formula, we have
See (Figure).
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Find the absolute value of the complex number
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Finding the Absolute Value of a Complex Number
Givenfind
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Using the formula, we have
The absolute valueis 5. See (Figure).
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Givenfind
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Writing Complex Numbers in Polar Form
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 review these relationships in (Figure).
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.
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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).
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Express as in polar form.
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Finding the Polar Form of a Complex Number
Find the polar form of
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First, find the value of
Find the angleusing the formula:
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Writein polar form.
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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:
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We begin by evaluating the trigonometric expressions.
After substitution, the complex number is
We apply the distributive property:
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Finding the Rectangular Form of a Complex Number
Find the rectangular form of the complex number givenand
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Ifandwe first determine We then findand
The rectangular form of the given number in complex form is
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Convert the complex number to rectangular form:
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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.
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Follow the formula
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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.
Finding the Quotient of Two Complex Numbers
Find the quotient ofand
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Using the formula, we have
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Find the product and the quotient ofand
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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.
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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.
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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
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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
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Find the four fourth roots of
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Key Concepts
Section Exercises
Verbal
A complex number isExplain each part.
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a is the real part, b is the imaginary part, and
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What does the absolute value of a complex number represent?
How is a complex number converted to polar form?
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Polar form converts the real and imaginary part of the complex number in polar form using and
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How do we find the product of two complex numbers?
What is De Moivre's Theorem and what is it used for?
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It is used to simplify polar form when a number has been raised to a power.
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Algebraic
For the following exercises, find the absolute value of the given complex number.
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For the following exercises, write the complex number in polar form.
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For the following exercises, convert the complex number from polar to rectangular form.
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For the following exercises, findin polar form.
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For the following exercises, findin polar form.
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For the following exercises, find the powers of each complex number in polar form.
Findwhen
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Findwhen
Findwhen
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Findwhen
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Findwhen
For the following exercises, evaluate each root.
Evaluate the cube root ofwhen
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Evaluate the square root ofwhen
Evaluate the cube root ofwhen
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Evaluate the square root ofwhen
Evaluate the cube root ofwhen
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Graphical
For the following exercises, plot the complex number in the complex plane.
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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.
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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.
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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.
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Find All Solutions in Complex With Polar Coordinates
Source: https://opentextbc.ca/algebratrigonometryopenstax/chapter/polar-form-of-complex-numbers/