<< Chapter < Page Chapter >> Page >
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     a + b i 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, b i .

Given a complex number a + b i , 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 3 i 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 of 2-3i in the complex plane (2 along the real axis, -3 along the imaginary axis).
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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

Plot of 1+5i in the complex plane (1 along the real axis, 5 along the imaginary axis).
Got questions? Get instant answers now!

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 , or | z | . It measures the distance from the origin to a point in the plane. For example, the graph of z = 2 + 4 i , in [link] , shows | z | .

Plot of 2+4i in the complex plane and its magnitude, |z| = rad 20.

Absolute value of a complex number

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

| z | = x 2 + y 2

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 the origin, ( 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

| z | = x 2 + y 2 | z | = ( 5 ) 2 + ( 1 ) 2 | z | = 5 + 1 | z | = 6

See [link] .

Plot of z=(rad5 - i) in the complex plane and its magnitude rad6.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the absolute value of the complex number z = 12 5 i .

13

Got questions? Get instant answers now!

Finding the absolute value of a complex number

Given z = 3 4 i , find | z | .

Using the formula, we have

| z | = x 2 + y 2 | z | = ( 3 ) 2 + ( 4 ) 2 | z | = 9 + 16 | z | = 25 | z | = 5

The absolute value z is 5. See [link] .

Plot of (3-4i) in the complex plane and its magnitude |z| =5.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 4

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask