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Mathematics LibreTexts

8.3 Polar Form of Complex Numbers

From previous classes, you may have encountered “imaginary numbers” – the square roots of negative numbers – and, more generally, complex numbers which are the sum of a real number and an imaginary number.  While these are useful for expressing the solutions to quadratic equations, they have much richer applications in electrical engineering, signal analysis, and other fields. Most of these more advanced applications rely on properties that arise from looking at complex numbers from the perspective of polar coordinates.

We will begin with a review of the definition of complex numbers.

 

Imaginary Number i

The most basic complex number is i, defined to be File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image002.gif, commonly called an imaginary number.  Any real multiple of i is also an imaginary number.

 

 

Example 1

Simplify File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image004.gif.

 

We can separate File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image004.gif as File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image007.gif.  We can take the square root of 9, and write the square root of -1 as i

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image004.gif=File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image009.gif

 

A complex number is the sum of a real number and an imaginary number.

 

Complex Number

A complex number is a number File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image011.gif, where a and b are real numbers

is the real part of the complex number

b  is the imaginary part of the complex number

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image002.gif

Plotting a complex number

We can plot real numbers on a number line.  For example, if we wanted to show the number 3, we plot a point:

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image013.jpg

 

To plot a complex number like File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image015.gif, we need more than just a number line since there are two components to the number.  To plot this number, we need two number lines, crossed to form a complex plane. 

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image016.gif

 

Complex Plane

In the complex plane, the horizontal axis is the real axis and the vertical axis is the imaginary axis.

 

 

 

Example 2

 

imaginary

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image018.jpgPlot the number File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image015.gif on the complex plane.

 

 

real

The real part of this number is 3, and the imaginary part is -4.  To plot this, we draw a point 3 units to the right of the origin in the horizontal direction and 4 units down in the vertical direction.

 

Because this is analogous to the Cartesian coordinate system for plotting points, we can think about plotting our complex number File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image011.gif as if we were plotting the point (a, b) in Cartesian coordinates.  Sometimes people write complex numbers as File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image020.gif to highlight this relation.

Arithmetic on Complex Numbers

Before we dive into the more complicated uses of complex numbers, let’s make sure we remember the basic arithmetic involved.  To add or subtract complex numbers, we simply add the like terms, combining the real parts and combining the imaginary parts.

 

Example 3

Add File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image015.gif and File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image022.gif.

 

Adding File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image024.gif, we add the real parts and the imaginary parts

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image026.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image028.gif

 

 

Try it Now

1. Subtract File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image022.gif from File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image015.gif.

 

 

We can also multiply and divide complex numbers.

 

Example 4

Multiply:  File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image030.gif.

 

To multiply the complex number by a real number, we simply distribute as we would when multiplying polynomials.

 

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image030.gif

=File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image033.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image035.gif

 

 

Example 5

Divide File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image037.gif.

 

To divide two complex numbers, we have to devise a way to write this as a complex number with a real part and an imaginary part. 

 

We start this process by eliminating the complex number in the denominator.  To do this, we multiply the numerator and denominator by a special complex number so that the result in the denominator is a real number.  The number we need to multiply by is called the complex conjugate, in which the sign of the imaginary part is changed.  Here, 4+i  is the complex conjugate of 4–i.  Of course, obeying our algebraic rules, we must multiply by 4+i  on both the top and bottom.

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image039.gif

 

To multiply two complex numbers, we expand the product as we would with polynomials (the process commonly called FOIL – “first outer inner last”).  In the numerator:

 

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image041.gif                                    Expand

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image043.gif                            Since File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image002.gif, File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image045.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image047.gif                        Simplify

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image049.gif

 

Following the same process to multiply the denominator

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image051.gif                                      Expand

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image053.gif                              Since File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image002.gif, File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image045.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image055.gif

=17

 

Combining this we get File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image057.gif

Try it Now

2.  Multiply File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image015.gif and File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image059.gif.

 

With the interpretation of complex numbers as points in a plane, which can be related to the Cartesian coordinate system, you might be starting to guess our next step – to refer to this point not by its horizontal and vertical components, but using its polar location, given by the distance from the origin and an angle.

Polar Form of Complex Numbers

Remember, because the complex plane is analogous to the Cartesian plane that we can think of a complex number File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image020.gif as analogous to the Cartesian point (x, y) and recall how we converted from (x, y) to polar (r, θ) coordinates in the last section.

 

imaginary

Bringing in all of our old rules we remember the following:

 

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image061.gifFile:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image063.gif                 File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image065.gif

 

real

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image067.gif                  File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image069.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image071.gif                  File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image073.gif

 

With this in mind, we can write File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image075.gif.

 

Example 6

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image077.gifGraphsExpress the complex number File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image081.gif using polar coordinates.

 

On the complex plane, the number 4i is a distance of 4 from the origin at an angle of File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image083.gif, so File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image085.gif

 

Note that the real part of this complex number is 0.

In the 18th century, Leonhard Euler demonstrated a relationship between exponential and trigonometric functions that allows the use of complex numbers to greatly simplify some trigonometric calculations.  While the proof is beyond the scope of this class, you will likely see it in a later calculus class.

Polar Form of a Complex Number and Euler’s Formula

The polar form of a complex number is  File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image087.gif, where Euler’s Formula holds:

 

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image089.gif

 

Similar to plotting a point in the polar coordinate system we need r and File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image091.gif to find the polar form of a complex number.

 

 

Example 7

Find the polar form of the complex number -7.

 

Treating this is a complex number, we can consider the unsimplified version -7+0i.

 

Plotted in the complex plane, the number -7 is on the negative horizontal axis, a distance of 7 from the origin at an angle of π from the positive horizontal axis. 

 

The polar form of the number -7 is File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image093.gif.

 

Plugging r = 7 and θ = π back into Euler’s formula, we have:

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image095.gif as desired.

 

 

Example 8

Find the polar form of File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image097.gif.

 

On the complex plane, this complex number would correspond to the point (-4, 4) on a Cartesian plane.  We can find the distance r and angle θ as we did in the last section.

 

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image099.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image101.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image103.gif

 

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image106.gifTo find θ, we can use File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image063.gif  

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image109.gif

This is one of known cosine values, and since the point is in the second quadrant, we can conclude that File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image111.gif.

The polar form of this complex number is File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image113.gif.

Note we could have used File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image071.gif instead to find the angle, so long as we remember to check the quadrant.

 

                                    

Try it Now

3.  Write File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image116.gif in polar form.

 

 

Example 9

Write File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image118.gif in complex File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image120.gif form.

 

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image122.gif                                   Evaluate the trig functions

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image124.gif                                                        Simplify

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image126.gif

 

The polar form of a complex number provides a powerful way to compute powers and roots of complex numbers by using exponent rules you learned in algebra.  To compute a power of a complex number, we:

  1. Convert to polar form
  2. Raise to the power, using exponent rules to simplify
  3. Convert back to a + bi form, if needed

 

 

Example 10

Evaluate File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image128.gif.

 

While we could multiply this number by itself five times, that would be very tedious.  To compute this more efficiently, we can utilize the polar form of the complex number.  In an earlier example, we found that File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image130.gif.  Using this,

 

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image128.gif                             Write the complex number in polar form

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image132.gif                       Utilize the exponent rule File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image134.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image136.gif                  On the second factor, use the rule File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image138.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image140.gif                      Simplify

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image142.gif            

 

At this point, we have found the power as a complex number in polar form.  If we want the answer in standard a + bi form, we can utilize Euler’s formula.

 

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image144.gif

 

Since File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image146.gif is coterminal with File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image083.gif, we can use our special angle knowledge to evaluate the sine and cosine.

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image149.gifFile:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image151.gif

 

We have found that File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image153.gif.

 

 

The result of the process can be summarized by DeMoivre’s Theorem.

 

 

DeMoivre’s Theorem

If File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image155.gif, then for any integer n, File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image157.gif

 

 

We omit the proof, but note we can easily verify it holds in one case using Example 10:

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image159.gif

 

Example 11

Evaluate File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image161.gif.

 

To evaluate the square root of a complex number, we can first note that the square root is the same as having an exponent of File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image163.gifFile:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image165.gif.

 

To evaluate the power, we first write the complex number in polar form.  Since 9i has no real part, we know that this value would be plotted along the vertical axis, a distance of 9 from the origin at an angle of File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image083.gif.  This gives the polar form:  File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image167.gif.

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image165.gif                                    Use the polar form

=File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image169.gif                           Use exponent rules to simplify

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image171.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image173.gif                            Simplify

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image175.gif                                   Rewrite using Euler’s formula if desired

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image177.gif        Evaluate the sine and cosine

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image179.gif

 

Using the polar form, we were able to find a square root of a complex number.

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image181.gif

Alternatively, using DeMoivre’s Theorem we can write File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image169.gifFile:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image183.gif and simplify

 

Try it Now

4.  Write File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image185.gif in polar form.

 

You may remember that equations like File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image187.gifhave two solutions, 2 and -2 in this case, though the square root File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image189.gif only gives one of those solutions.  Likewise, the square root we found in Example 11 is only one of two complex numbers whose square is 9i.  Similarly, the equation File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image191.gif would have three solutions where only one is given by the cube root.  In this case, however, only one of those solutions, z = 2, is a real value.  To find the others, we can use the fact that complex numbers have multiple representations in polar form.

 

 

Example 12

Find all complex solutions to File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image191.gif.

 

Since we are trying to solve File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image191.gif, we can solve for x as File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image194.gif.  Certainly one of these solutions is the basic cube root, giving z = 2.  To find others, we can turn to the polar representation of 8. 

 

Since 8 is a real number, is would sit in the complex plane on the horizontal axis at an angle of 0, giving the polar form File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image196.gif.  Taking the 1/3 power of this gives the real solution:

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image198.gif

 

However, since the angle 2π is coterminal with the angle of 0, we could also represent the number 8 as File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image200.gif.  Taking the 1/3 power of this gives a first complex solution:

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image202.gif

To find the third root, we use the angle of 4π, which is also coterminal with an angle of 0.

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image206.gifFile:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image208.gifAltogether, we found all three complex solutions to File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image191.gif,

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image210.gif

 

Graphed, these three numbers would be equally spaced on a circle about the origin at a radius of 2.

Important Topics of This Section

  • Complex numbers
  • Imaginary numbers
  • Plotting points in the complex coordinate system
  • Basic operations with complex numbers
  • Euler’s Formula
  • DeMoivre’s Theorem
  • Finding complex solutions to equations

 

Try it Now Answers

1. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image212.gif

2. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image214.gif

3. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image116.gif in polar form is File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image216.gif

4. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image218.gif

Section 8.3 Exercises

Simplify each expression to a single complex number.

1. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image002.gif                                    2. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image004.gif                                  3. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image006.gif             

4. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image008.gif                          5. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image010.gif                           6. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image012.gif

 

Simplify each expression to a single complex number.

7. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image014.gif                                         8. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image016.gif

9. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image018.gif                                        10. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image020.gif

11. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image022.gif                                               12. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image024.gif

13. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image026.gif                                                14. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image028.gif

15. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image030.gif                                           16. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image032.gif

17. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image034.gif                                         18. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image036.gif

19. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image038.gif                                                        20. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image040.gif

21. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image042.gif                                                     22. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image044.gif

23. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image046.gif                                                        24. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image048.gif

25. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image050.gif                          26. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image052.gif                          27. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image054.gif                         28. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image056.gif

 

Rewrite each complex number from polar form into File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image058.gif form.

29. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image060.gif                       30. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image062.gif                       31. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image064.gif                      32. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image066.gif                     

33. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image068.gif                     34. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image070.gif

 

Rewrite each complex number into polar File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image072.gif form.

35. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image074.gif                           36. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image076.gif                         37. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image078.gif                       38. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image080.gif             

39. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image082.gif                    40. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image084.gif                    41. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image086.gif                  42. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image088.gif     

43. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image090.gif                    44. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image092.gif                    45. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image094.gif                    46. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image096.gif

47. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image098.gif                  48. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image100.gif                  49. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image102.gif                      50. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image104.gif        

 

Compute each of the following, leaving the result in polar File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image072.gif form.

51. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image107.gif                  52. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image109.gif                53. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image111.gif                   

54. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image113.gif                              55. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image115.gif                            56. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image117.gif                            

57. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image119.gif                           58.File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image121.gif

 

Compute each of the following, simplifying the result into File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image058.gif form.

59. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image124.gif                            60. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image126.gif                            61. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image128.gif              

62. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image130.gif                           63. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image132.gif                             64. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image134.gif

 

Solve each of the following equations for all complex solutions.

65. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image136.gif                   66. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image138.gif                   67. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image140.gif                    68. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image142.gif

Contributors

  • David Lippman (Pierce College)
  • Melonie Rasmussen (Pierce College)