Polar Form of Complex Numbers

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Polar Form of Complex

in science

We first encountered complex numbers in Complex Numbers In this section, we willfocus on the mechanics of working with complex numbers: translation of complexnumbers from polar form to rectangular form and vice versa, interpretation of complexnumbers in the scheme of applications, and application of De Moivre’s Theorem

Plotting Complex Numbers in the Complex Plane

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

How To

Given a complex number a + bi, plot it in the complex plane.

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

axis.

2 Plot the point in the complex plane by moving a units in the horizontal direction and b units in the vertical direction.

Plotting a Complex Number in the Complex Plane

Plot the complex number 2 − 3i in the complex plane.

From the origin, move two units in the positive horizontal direction and three units inthe negative vertical direction See[link].

Try It

Plot the point 1 + 5i in the complex plane.

Finding the Absolute Value of a Complex Number

The first step toward working with a complex number in polar form is to find theabsolute value The absolute value of a complex number is the same as its magnitude,

or|z| It measures the distance from the origin to a point in the plane For example, the

graph of z = 2 + 4i, in[link], shows|z|

A General Note

Absolute Value of a Complex Number

Given z = x + yi, a complex number, the absolute value of z is defined as

|z|= √x2+ y2

It is the distance from the origin to the point(x, y)

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 theorigin,(0, 0)

Finding the Absolute Value of a Complex Number with a Radical

Find the absolute value of z =√5 − i.

Using the formula, we have

Finding the Absolute Value of a Complex Number

Given z = 3 − 4i, find|z|

Using the formula, we have

Writing Complex Numbers in Polar Form

The polar form of a complex number expresses a number in terms of an angle θ and

its distance from the origin r Given a complex number in rectangular form expressed

as z = x + yi, we use the same conversion formulas as we do to write the number in

trigonometric form:

x = rcos θ

y = rsin θ

r = √x2+ y2

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(x, y) The modulus, then, is the same as r,

the radius in polar form We use θ to indicate the angle of direction (just as with polarcoordinates) Substituting, we have

z = x + yi

z = rcos θ +(rsin θ)i

z = r(cos θ + isin θ)

a general note label

Polar Form of a Complex Number

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

x = rcos θ

y = rsin θ

r = √x2+ y2

Making a direct substitution, we have

Expressing a Complex Number Using Polar Coordinates

Express the complex number 4i using polar coordinates.

On the complex plane, the number z = 4i is the same as z = 0 + 4i Writing it in polar form, we have to calculate r first.

r = √x2+ y2

r = √02+ 42

r = √16

r = 4

Next, we look at x If x = rcos θ, and x = 0, then θ = π2 In polar coordinates, the

complex number z = 0 + 4i can be written as z = 4(cos(π

2)+ isin(π

2) )or 4cis( π

2) See[link]

Finding the Polar Form of a Complex Number

Find the polar form of − 4 + 4i.

First, find the value of r.

4 ).Try It

Write z =√3 + i in polar form.

z = 2(cos(π

6)+ isin(π

6) )

Converting a Complex Number from Polar to Rectangular Form

Converting a complex number from polar form to rectangular form is a matter ofevaluating what is given and using the distributive property In other words, given

z = r(cos θ + isin θ), first evaluate the trigonometric functions cos θ and sin θ Then,

multiply through by r.

Converting from Polar to Rectangular Form

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

The rectangular form of the given point in complex form is 6√3 + 6i.

Finding the Rectangular Form of a Complex Number

Find the rectangular form of the complex number given r = 13 and tan θ = 125

If tan θ = 125, and tan θ = y x , we first determine r = √x2+ y2=√122+ 52= 13 We thenfind cos θ = x r and sin θ = y r

Finding Products of Complex Numbers in Polar Form

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

A General Note

Products of Complex Numbers in Polar Form

If z1= r1(cos θ1+ isin θ1) and z2= r2(cos θ2+ isin θ2), then the product of thesenumbers is given as:

z1z2 = r1r2[cos(θ1+ θ2)+ isin(θ1+ θ2) ]

z1z2 = r1r2cis(θ1+ θ2)

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 of z1z2, given z1= 4(cos(80°) + isin(80°)) and

z2= 2(cos(145°) + isin(145°)).

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 moduliand the difference of the two arguments

A General Note

Quotients of Complex Numbers in Polar Form

If z1= r1(cos θ1+ isin θ1) and z2= r2(cos θ2+ isin θ2), then the quotient of thesenumbers is

4 Calculate the new trigonometric expressions and multiply through by r.

Finding the Quotient of Two Complex Numbers

Find the quotient of z1 = 2(cos(213°) + isin(213°)) and z2 = 4(cos(33°) + isin(33°)).

Using the formula, we have

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 integer n, z n is found by raising the modulus to the nth power and multiplying the argument by n It is the standard method used in modern

where n is a positive integer.

Evaluating an Expression Using De Moivre’s Theorem

Evaluate the expression(1 + i)5using De Moivre’s Theorem.

Since De Moivre’s Theorem applies to complex numbers written in polar form, we mustfirst write(1 + i)in polar form Let us find r.

Use De Moivre’s Theorem to evaluate the expression

(a + bi) n = r n [cos(nθ) + isin(nθ)]

Finding Roots of Complex Numbers in Polar Form

To find the nth root of a complex number in polar form, we use the nth 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 nth roots of complex

numbers in polar form

A General Note label

The nth Root Theorem

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

Finding the nth Root of a Complex Number

Evaluate the cube roots of z = 8(cos(2π

Remember to find the common denominator to simplify fractions in situations like this

one For k = 1, the angle simplification is

Find the four fourth roots of 16(cos(120°) + isin(120°)).

• Complex numbers in the form a + bi 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]

• The absolute value of a complex number is the same as its magnitude It is thedistance from the origin to the point:|z|= √a2+ b2 See[link]and[link]

• To write complex numbers in polar form, we use the formulas

x = rcos θ, y = rsin θ, and r = √x2+ y2 Then, z = r(cos θ + isin θ) See[link]and[link]

• To convert from polar form to rectangular form, first evaluate the trigonometric

functions Then, multiply through by r See[link]and[link]

• To find the product of two complex numbers, multiply the two moduli and addthe two angles Evaluate the trigonometric functions, and multiply using thedistributive property See[link].

• To find the quotient of two complex numbers in polar form, find the quotient ofthe two moduli and the difference of the two angles See[link]

• To find the power of a complex number z n , raise r to the power n, and

multiply θ by n See[link]

• Finding the roots of a complex number is the same as raising a complex

number to a power, but using a rational exponent See[link]

Section Exercises

Verbal

A complex number is a + bi Explain each part.

a is the real part, b is the imaginary part, and i =√ − 1

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 x = rcosθ and y = rsinθ.

How do we find the product of two complex numbers?

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

z n = r n(cos(nθ)+ isin(nθ) )It is used to simplify polar form when a number has beenraised to a power

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

Find z3when z = 5cis(45°)

125cis(135°)

Find z4when z = 2cis(70°)

Find z2when z = 3cis(120°)

4 )

Find z3when z = 3cis(5π

3).For the following exercises, evaluate each root

Evaluate the cube root of z when z = 27cis(240°)

3cis(80°), 3cis(200°), 3cis(320°)

Evaluate the square root of z when z = 16cis(100°)

Evaluate the cube root of z when z = 32cis(2π

Evaluate the square root of z when z = 32cis(π)

Evaluate the cube root of z when z = 8cis(7π

4)

3 + 2i

2i

− 4

6 − 2i

− 2 + i

1 − 4i

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

Use the rectangular to polar feature on the graphing calculator to change 5 + 5i to polar

Use the polar to rectangular feature on the graphing calculator to change 2cis(45°)torectangular form.

Use the polar to rectangular feature on the graphing calculator to change 5cis(210°)torectangular form

− 4.33 − 2.50i