In this section, you will:
“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 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,
**Given a complex number
plot it in the complex plane.**
units in the horizontal direction and
units in the vertical direction.
Plot the complex number
in the complex plane.
From the origin, move two units in the positive horizontal direction and three units in the negative vertical direction. See [link].
Plot the point
in the complex plane.
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, or
It measures the distance from the origin to a point in the plane. For example, the graph of
in [link], shows
Given
a complex number, the absolute value of
is 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,
Find the absolute value of
Find the absolute value of the complex number
13
Given
find
Given
find
The polar form of a complex number expresses a number in terms of an angle
and its distance from the origin
Given a complex number in rectangular form expressed as
we use the same conversion formulas as we do to write the number in trigonometric form:
We review these relationships in [link].
We use the term modulus to represent the absolute value of a complex number, or the distance from the origin to the point
The modulus, then, is the same as
the radius in polar form. We use
to indicate the angle of direction (just as with polar coordinates). Substituting, we have
Writing a complex number in polar form involves the following conversion formulas:
Making a direct substitution, we have
where
is the modulus and
is the argument. We often use the abbreviation
to represent
Express the complex number
using polar coordinates.
On the complex plane, the number
is the same as
Writing it in polar form, we have to calculate
first.
Next, we look at
If
and
then
In polar coordinates, the complex number
can be written as
or
See [link].
Express
as
in polar form.
Find the polar form of
First, find the value of
Find the angle
using the formula:
Thus, the solution is
Write
in polar 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, given
first evaluate the trigonometric functions
and
Then, multiply through by
Convert the polar form of the given complex number to rectangular form:
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
Find the rectangular form of the complex number given
and
If
and
we first determine
We then find
and
The rectangular form of the given number in complex form is
Convert the complex number to rectangular 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.
If
and
then the product of these numbers is given as:
Notice that the product calls for multiplying the moduli and adding the angles.
Find the product of
given
and
Follow the formula
The quotient of two complex numbers in polar form is the quotient of the two moduli and the difference of the two arguments.
If
and
then the quotient of these numbers is
Notice that the moduli are divided, and the angles are subtracted.
Given two complex numbers in polar form, find the quotient.
Replace
with
and replace
with
Find the quotient of
and
Using the formula, we have
Find the product and the quotient of
and
Finding powers of complex numbers is greatly simplified using De Moivre’s Theorem. It states that, for a positive integer
is found by raising the modulus to the
power and multiplying the argument by
It is the standard method used in modern mathematics.
If
is a complex number, then
where
is a positive integer.
Evaluate the expression
using De Moivre’s Theorem.
Since De Moivre’s Theorem applies to complex numbers written in polar form, we must first write
in polar form. Let us find
Then we find
Using the formula
gives
Use De Moivre’s Theorem to evaluate the expression.
To find the nth root of a complex number in polar form, we use the
Root 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 finding
roots of complex numbers in polar form.
To find the
root of a complex number in polar form, use the formula given as
where
We add
to
in order to obtain the periodic roots.
Evaluate the cube roots of
We have
There will be three roots:
When
we have
When
we have
When
we have
Remember to find the common denominator to simplify fractions in situations like this one. For
the angle simplification is
Find the four fourth roots of
Access these online resources for additional instruction and practice with polar forms of complex numbers.
are 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 [link].
and
Then,
raise
to the power
and multiply
by
See [link].
A complex number is
Explain each part.
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?
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?
It is used to simplify polar form when a number has been raised to a power.
For the following exercises, find the absolute value of the given complex number.
For the following exercises, write the complex number in polar form.
For the following exercises, convert the complex number from polar to rectangular form.
For the following exercises, find
in polar form.
For the following exercises, find
in polar form.
For the following exercises, find the powers of each complex number in polar form.
Find
when
Find
when
Find
when
Find
when
Find
when
Find
when
For the following exercises, evaluate each root.
Evaluate the cube root of
when
Evaluate the square root of
when
Evaluate the cube root of
when
Evaluate the square root of
when
Evaluate the cube root of
when
For the following exercises, plot the complex number in the complex plane.
For the following exercises, find all answers rounded to the nearest hundredth.
Use the rectangular to polar feature on the graphing calculator to change
to polar form.
Use the rectangular to polar feature on the graphing calculator to change
to polar form.
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 change
to rectangular form.
Use the polar to rectangular feature on the graphing calculator to change
to rectangular form.
Use the polar to rectangular feature on the graphing calculator to change
to rectangular form.
power or nth roots of a complex number; states that, for a positive integer
is found by raising the modulus to the
power and multiplying the angles by
also called the amplitude
and its distance from the origin
can be found by using conversion formulas
and
You can also download for free at http://cnx.org/contents/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1
Attribution: