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 , commonly called an imaginary number. Any real multiple of i is also an imaginary number.
We can separate as . We can take the square root of 9, and write the square root of -1 as i.
A complex number is the sum of a real number and an imaginary number.
A complex number is a number , where a and b are real numbers
a is the real part of the complex number
b is the imaginary part of the complex number
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:
To plot a complex number like , 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.
In the complex plane, the horizontal axis is the real axis and the vertical axis is the imaginary axis.
Because this is analogous to the Cartesian coordinate system for plotting points, we can think about plotting our complex number as if we were plotting the point (a, b) in Cartesian coordinates. Sometimes people write complex numbers as 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.
Add and .
Adding , we add the real parts and the imaginary parts
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1. Subtract from .
We can also multiply and divide complex numbers.
To multiply the complex number by a real number, we simply distribute as we would when multiplying polynomials.
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.
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:
Following the same process to multiply the denominator
Combining this we get
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2. Multiply and .
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 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.
Bringing in all of our old rules we remember the following:
With this in mind, we can write .
Express the complex number using polar coordinates.
On the complex plane, the number 4i is a distance of 4 from the origin at an angle of , so
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 , where Euler’s Formula holds:
Similar to plotting a point in the polar coordinate system we need r and to find the polar form of a complex number.
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 .
Plugging r = 7 and θ = π back into Euler’s formula, we have:
Find the polar form of .
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.
To find θ, we can use
This is one of known cosine values, and since the point is in the second quadrant, we can conclude that .
The polar form of this complex number is .
Note we could have used instead to find the angle, so long as we remember to check the quadrant.
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3. Write in polar form.
Write in complex form.
Evaluate the trig functions
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:
- Convert to polar form
- Raise to the power, using exponent rules to simplify
- Convert back to a + bi form, if needed
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 . Using this,
Write the complex number in polar form
Utilize the exponent rule
On the second factor, use the rule
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.
Since is coterminal with , we can use our special angle knowledge to evaluate the sine and cosine.
We have found that .
The result of the process can be summarized by DeMoivre’s Theorem.
If , then for any integer n,
We omit the proof, but note we can easily verify it holds in one case using Example 10:
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 : .
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 . This gives the polar form: .
Use the polar form
= Use exponent rules to simplify
Rewrite using Euler’s formula if desired
Evaluate the sine and cosine
Using the polar form, we were able to find a square root of a complex number.
Alternatively, using DeMoivre’s Theorem we can write and simplify
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4. Write in polar form.
You may remember that equations like have two solutions, 2 and -2 in this case, though the square root 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 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.
Find all complex solutions to .
Since we are trying to solve , we can solve for x as . 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 . Taking the 1/3 power of this gives the real solution:
However, since the angle 2π is coterminal with the angle of 0, we could also represent the number 8 as . Taking the 1/3 power of this gives a first complex solution:
To find the third root, we use the angle of 4π, which is also coterminal with an angle of 0.
Altogether, we found all three complex solutions to ,
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
3. in polar form is
Section 8.3 Exercises
Simplify each expression to a single complex number.
1. 2. 3.
4. 5. 6.
Simplify each expression to a single complex number.
25. 26. 27. 28.
Rewrite each complex number from polar form into form.
29. 30. 31. 32.
Rewrite each complex number into polar form.
35. 36. 37. 38.
39. 40. 41. 42.
43. 44. 45. 46.
47. 48. 49. 50.
Compute each of the following, leaving the result in polar form.
51. 52. 53.
54. 55. 56.
Compute each of the following, simplifying the result into form.
59. 60. 61.
62. 63. 64.
Solve each of the following equations for all complex solutions.
65. 66. 67. 68.
- David Lippman (Pierce College)
- Melonie Rasmussen (Pierce College)