Complex numbers
You have learned about the set of real numbers and should know that a real number is any number that can be graphed on a number line. Because there is no real number that is equal to 
, it is said that 
is not real.
Some will argue that for each real number there is a physical entity which can be matched to it—for example, a single pencil indicates a 1—but it is important for you to understand that the object is a pencil. The manner in which the item is quantified is a mental construct in the same way that all language is.
It was not until the sixteenth century that mathematicians began to pay greater attention to this number. Because was not real, the number was called imaginary and 
Powers of i
If 
, then 
. The rules of arithmetic have always stated that 
so i2 = −1. Multiply both sides of this equation by i to get i3 = −1(i) = −i. Do so again to find that i4 = 1. A rather surprising and interesting consequence of this is that the powers of i will repeat themselves. That is, i5 = i4i = i; i6 = i4i2 = −1; i7 = i4i3 = −i; i8 = i4i4 = 1. The rule is simple: when simplifying powers of i, divide the exponent by 4 and match the remainder with 0, 1, 2, or 3.
If i2 = −1, then what is the value of (4i)2? The rules of exponents tell you that (4i)2 = 16i2 = 16(−1) = −16.
If , what is the value of 
The rules for simplifying square roots tell you that 
The most difficult problem is 
The temptation is to multiply −16 and −25 to get 400 and conclude that the square root of 400 is 20. However, the must first be factored from each of the terms. 
Arithmetic of complex numbers
If the imaginary numbers cannot fit on a number line, where do they exist geometrically? The answer is to extend the number line to a number plane. Through the origin of the real number line, construct a new number line perpendicular to it. The numbers of this line are the imaginary numbers. i, or 1i, is one unit above the real number line while −i is one unit below. The numbers that appear in the quadrants of this number plane are the complex numbers. A point with coordinates (4, 3) in the number plane equals 4 + 3i. A number point with coordinates (−2, 5) equals −2 + 5i. The real numbers have an ordinate (second number in the ordered pair) of 0, while the imaginary numbers have an abscissa (first number in the ordered pair) of 0.
The sum of two complex numbers a + bi and c + di = (a + c) + (b + d)i. When you combine the like parts, real numbers are added to real numbers and imaginary numbers are added to imaginary numbers.
Multiplication of complex numbers is performed in the same way as the multiplication of binomials.
The division of complex numbers requires the use of conjugates. Recall that 
and 
are called conjugates and that 
Finally, two complex numbers, a + bi and c + di, are equal if and only if a = c and b = d.
Perform each operation. Write your answers in a + bi form.
1. (4 + 5i) + (9 − 3i)
2. (8 + 7i) − (−2 + 9i)
3. (2 − 3i)(4 + 5i)
4. 
5. 
6. 
7. 
8. Find the values of x and y if (3x − 2y) + (5x + 7y)i = 13 + i.
The discriminant and nature of the roots of a quadratic equation
With the inclusion of complex numbers, the set of solutions for equations is expanded. In the past, the solution to the equation x2 = −1 would have been “no real solution,” and now the answer is x = ±i.
It is possible to tell the type of answers an equation will generate (also known as the nature of the roots) prior to working through the entire problem. The term b2 − 4ac, the term within the square root when using the quadratic formula, is called the discriminant. The value of this number discriminates between real and complex solutions, equal and unequal solutions, and rational versus irrational solutions.
For questions 1–4, compute the discriminant and use this result to determine the nature of the roots of the given equation.
1. 9x2 − 24x + 16 = 0
2. 5x2 + 8x − 7 = 0
3. 10x2 + 9x + 4 = 0
4. 24x2 + 38x + 15 = 0
For questions 5 and 6, solve the quadratic equations. Write your answers in a + bi form when appropriate.
5. 16x2 − 16x + 13 = 0
6. 64x2 − 48x + 29 = 0
Sum and product of roots of a quadratic equation
The solutions to the quadratic equation ax2 + bx + c = 0 are 
The sum of these roots is 
while the product of these roots is 
For this problem, it appears to be easier to substitute the root for x and find the value of c. Is this true for the next problem also?
Notice that the two roots of the quadratic equation, 
are conjugates of each other. If the coefficients of the quadratic are integers, then irrational roots and complex roots will always occur in conjugate pairs. This makes writing a quadratic equation for a given set of roots much easier to do.
Solve questions 1–3 as indicated.
1. Determine the sum and product of the roots of the quadratic equation 2x2 + 7x − 3 = 0.
2. If −5 is a root of the equation 6x2 + 12x + c = 0, what is the other root?
3. Determine the value of c in the equation 4x2 − 9x + c = 0 if one root of the equation is 
For questions 4–6, write a quadratic equation with integral coefficients whose roots are given.
4. 
5. 
6. 