Complex Numbers - Basic Definitions
by M. Bourne
Let's first consider what we learned before in Quadratic Equations and Equations of Higher Degree, so we can better understand where complex numbers are coming from.
Quadratic Equations
On this page:
Roots of cubic equations
Imaginary Numbers
Powers of j
Complex Numbers
Equivalent complex numbers
Forms
Examples of quadratic equations:
- 2x2 + 3x − 5 = 0
- x2 − x − 6 = 0
- x2 = 4
The roots of an equation are the x-values that make it "work" We can find the roots of a quadratic equation either by using the quadratic formula or by factoring.
We can have 3 situations when solving quadratic equations.
Case 1: Two roots
Example: 2x2 + 3x − 5 = 0
We proceed to solve this equation using the quadratic formula as we did earlier:
We find 2 roots. The graph of the quadratic equation y = 2x2 + 3x − 5 cuts the x-axis at x = -2.5 and x = 1, as expected, showing our 2 roots:
More examples of quadratic equations with 2 roots:
x2 = 4 has 2 solutions, x = -2 and x = 2.
x2 − x − 6 = 0 has 2 solutions, x = -2 and x = 3.
2x2 + 13x − 7 = 0 has 2 solutions, x = -7 and x = ½.
Case 2: One Root
Example: 4x2 − 12x + 9 = 0
Notice what happens when we use the quadratic formula this time. Under the square root we get 144 − 144 = 0.
So it means we only have one root. We can also say that this is a repeated root, since we are expecting 2 roots.
On the graph of y = 4x2 −12x + 9, we can see that the graph cuts the x-axis in one place only, at x = 1.5.
Case 3: No Real Roots
Example: x2 −4x + 20 = 0
This example gives us a problem. Under the square root, we get √(-64), and we have been told repeatedly by our teachers that we cannot have the square root of a negative number. Can we find such a root?
Summary
A quadratic equation has degree 2 (the highest power of x is 2) and we can have either 2 real roots, one real repeated root or something that involves the square root of a negative number.
Cubic Equations
Cubic equations are polynomials which have degree 3 (this highest power of x is 3).
In the case of a cubic equation, we expect (up to) 3 real solutions:
Example 1: x3 − 2x2 − 5x + 6 = 0 has solutions x = -2, 1 and 3.
Example 2: If x3 = 8, we know the solution x = 2, but we expect 2 other solutions. What are they?
Imaginary Numbers
To allow for these "hidden roots", around the year 1800, the concept of
√(-1)
was proposed and is now accepted as an extension of the real number system. The symbol used is
j = √(-1)
and j is called an imaginary number.
Why Not i for Imaginary Numbers?
Many textbooks use i as the symbol for imaginary numbers. We use j, because the main application of imaginary numbers is in electricity and electronics, so there is less confusion with i (which is used for current).
Your calculator or computer algebra system will probably use i.
Powers of j
You may need to look at this reminder example about multiplying square roots before you go any further.
- Reminder
-
Reminder Example about Repeated Multiplying of Square Roots
If x = √10, then
x2 = (√10)2 = 10
[This because square and square root are inverse processes.]
Let's multiply our previous answer by √10.
x3 = (√10)3 = (√10)2√10 = 10√10
The next step in the pattern is:
x4 = (√10)4 = 10 (√10)2 = 10 × 10 = 100
[Once again, we are multiplying our previous answer by √10.]
The next step in the pattern is:
x5 = (√10)5 = 100√10
The next step in the pattern is:
x6 = (√10)6 = 1000
What we are doing next in Powers of j works in the same way.
Recall:
(√a)2 = a, for any value of a.
and
j = √(-1)
Using these, we can derive the following:
j2 = (√-1)2 = -1
Multiplying by j again gives us:
j3 = j2(j) = -j
Continuing the process gives us:
j4 = j3(j) = -j(j) = -(-1) = 1
j5 = j4(j) = 1 × j = j
j6 = j5(j) = j × j = -1 etc
Examples using j
1. Express in terms of j: √(-16)
- Answer
-
2. Express in terms of j: √(-100)
- Answer
-
3. Express in terms of j: √(-7)
- Answer
-
Care: Do not write this as
, which means 
4. Express in terms of j: √(-2)√(-18)
- Answer
-
5. Express in terms of j: √(-2 × -18)
(NOT the same as Number 4! - Note the difference.)
- Answer
-
Complex Numbers
Complex numbers have a real part and an imaginary part.
Examples:
(1) 5 + 6j
Real part: 5, Imaginary part: 6j
(2) -3 + 7j
Real part: -3, Imaginary part: 7j
NOTE: We can write the complex number 2 + 5j as 2 + j5.
There is no difference in meaning.
Solving Equations with Complex Numbers
We now return to our problem from above. We didn't know then what to do with √(-64). Now we can write the solution using complex numbers, as follows:
Equivalent Complex Numbers
Two complex numbers x + yj and a + bj are equivalent if:
The real parts are equal (x = a), and
The imaginary parts are equal (y = b).
Example:
Given that 3 + 2j = a + bj, then
a = 3 and b = 2.
Exercises
1. Express in terms of j:
- Answer
-
2. Simplify:
(a) 
- Answer
-
(b) 
- Answer
-
3. j2 − j6
- Answer
-
(-1) − (-1) = 0
4. 
- Answer
-
-2 + 1 = -1
Forms of Complex Numbers
We can write complex numbers in 3 different ways:
| Example | ||
|---|---|---|
| Rectangular form: | x + yj | 5 + 6j |
| Polar form: | r(cos θ + j sin θ) | 8(cos24° + j sin 24°) |
| Exponential form: | rejθ | 6e2.5j |
We will meet polar form and exponential form later in this chapter, but first, let's see how to perform basic operations with complex numbers.
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