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8.5 Polar form of complex numbers

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

13

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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.
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Questions & Answers

for the "hiking" mix, there are 1,000 pieces in the mix, containing 390.8 g of fat, and 165 g of protein. if there is the same amount of almonds as cashews, how many of each item is in the trail mix?
ADNAN Reply
linear speed of an object
Melissa Reply
an object is traveling around a circle with a radius of 13 meters .if in 20 seconds a central angle of 1/7 Radian is swept out what are the linear and angular speed of the object
Melissa
test
Matrix
how to find domain
Mohamed Reply
like this: (2)/(2-x) the aim is to see what will not be compatible with this rational expression. If x= 0 then the fraction is undefined since we cannot divide by zero. Therefore, the domain consist of all real numbers except 2.
Dan
define the term of domain
Moha
if a>0 then the graph is concave
Angel Reply
if a<0 then the graph is concave blank
Angel
what's a domain
Kamogelo Reply
The set of all values you can use as input into a function su h that the output each time will be defined, meaningful and real.
Spiro
how fast can i understand functions without much difficulty
Joe Reply
what is inequalities
Nathaniel
functions can be understood without a lot of difficulty. Observe the following: f(2) 2x - x 2(2)-2= 2 now observe this: (2,f(2)) ( 2, -2) 2(-x)+2 = -2 -4+2=-2
Dan
what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
158.5 This number can be developed by using algebra and logarithms. Begin by moving log(2) to the right hand side of the equation like this: t/100 log(2)= log(3) step 1: divide each side by log(2) t/100=1.58496250072 step 2: multiply each side by 100 to isolate t. t=158.49
Dan
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
how to find x: 12x = 144 notice how 12 is being multiplied by x. Therefore division is needed to isolate x and whatever we do to one side of the equation we must do to the other. That develops this: x= 144/12 divide 144 by 12 to get x. addition: 12+x= 14 subtract 12 by each side. x =2
Dan
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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