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10.5 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

how did you get 1640
Noor Reply
If auger is pair are the roots of equation x2+5x-3=0
Peter Reply
Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
MATTHEW Reply
420
Sharon
from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph
George
Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28
Salma
Devon is 32 32​​ years older than his son, Milan. The sum of both their ages is 54 54​. Using the variables d d​ and m m​ to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.
Aaron Reply
find product (-6m+6) ( 3m²+4m-3)
SIMRAN Reply
-42m²+60m-18
Salma
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-24m+3+3mÁ^2
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Bajemah
-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18
Salma
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Salma
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3x-12y=18
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A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg
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The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.
cameron Reply
Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.
mahnoor Reply
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12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00
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When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?
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Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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