Section 8.8Use the Complex Number System

Evaluate the Square Root of a Negative Number

Whenever we have a situation where we have a square root of a negative number we say there is no real number that equals that square root. For example, to simplify \(\sqrt{-1},\) we are looking for a real number x so that x² = –1. Since all real numbers squared are positive numbers, there is no real number that equals –1 when squared.

Mathematicians have often expanded their numbers systems as needed. They added 0 to the counting numbers to get the whole numbers. When they needed negative balances, they added negative numbers to get the integers. When they needed the idea of parts of a whole they added fractions and got the rational numbers. Adding the irrational numbers allowed numbers like \(\sqrt{5}.\) All of these together gave us the real numbers and so far in your study of mathematics, that has been sufficient.

But now we will expand the real numbers to include the square roots of negative numbers. We start by defining the imaginary unit \(i\) as the number whose square is –1.

We will use the imaginary unit to simplify the square roots of negative numbers.

We will use this definition in the next example. Be careful that it is clear that the i is not under the radical. Sometimes you will see this written as \(\sqrt{\text{-}b}=i\sqrt{b}\) to emphasize the i is not under the radical. But the \(\sqrt{\text{-}b}=\sqrt{b}\,i\) is considered standard form.

Now that we are familiar with the imaginary number i, we can expand our concept of the number system to include imaginary numbers. The complex number system includes the real numbers and the imaginary numbers. A complex number is of the form a + bi, where a, b are real numbers. We call a the real part and b the imaginary part.

Complex Number

A complex number is of the form a + bi, where a and b are real numbers.

A complex number is in standard form when written as \(a+bi,\) where a and b are real numbers.

If \(b=0,\) then \(a+bi\) becomes \(a+0·i=a,\) and is a real number.

If \(b\ne 0,\) then \(a+bi\) is an imaginary number.

If \(a=0,\) then \(a+bi\) becomes \(0+bi=bi,\) and is called a pure imaginary number.

We summarize this here.

The standard form of a complex number is \(a+bi,\) so this explains why the preferred form is \(\sqrt{\text{-}b}=\sqrt{b}i\) when \(b>0.\)

The diagram helps us visualize the complex number system. It is made up of both the real numbers and the imaginary numbers.

Add or Subtract Complex Numbers

We are now ready to perform the operations of addition, subtraction, multiplication and division on the complex numbers—just as we did with the real numbers.

Adding and subtracting complex numbers is much like adding or subtracting like terms. We add or subtract the real parts and then add or subtract the imaginary parts. Our final result should be in standard form.

Remember to add both the real parts and the imaginary parts in this next example.

Multiply Complex Numbers

Multiplying complex numbers is also much like multiplying expressions with coefficients and variables. There is only one special case we need to consider. We will look at that after we practice in the next two examples.

In the next example, we multiply the binomials using the Distributive Property or FOIL.

In the next example, we could use FOIL or the Product of Binomial Squares Pattern.

Example 6

Multiply: \({(3+2i)}^{2}\)

Apply the binomial squares pattern \({(a+b)}^{2}={a}^{2}+2ab+{b}^{2},\) then simplify \({i}^{2}.\)

Show solution

Table 10

Use the Product of Binomial Squares Pattern, \({(a+b)}^{2}={a}^{2}+2ab+{b}^{2}.\)
Simplify.
Simplify \({i}^{2}.\)
Simplify.

Since the square root of a negative number is not a real number, when we have the square roots of two negative numbers, we cannot use the Product Property for Radicals. In order to multiply square roots of negative numbers we should first write them as complex numbers, using \(\sqrt{\text{-}b}=\sqrt{b}i.\) This is one place students tend to make errors, so be careful when you see multiplying with a negative square root.

In the next example, each binomial has a square root of a negative number. Before multiplying, each square root of a negative number must be written as a complex number.

We first looked at conjugate pairs when we studied polynomials. We said that a pair of binomials that each have the same first term and the same last term, but one is a sum and one is a difference is called a conjugate pair and is of the form \((a-b),(a+b).\)

A complex conjugate pair is very similar. For a complex number of the form \(a+bi,\) its conjugate is \(a-bi.\) Notice they have the same first term and the same last term, but one is a sum and one is a difference.

We will multiply a complex conjugate pair in the next example.

From our study of polynomials, we know the product of conjugates is always of the form \((a-b)(a+b)={a}^{2}-{b}^{2}.\) The result is called a difference of squares. We can multiply a complex conjugate pair using this pattern.

The last example we used FOIL. Now we will use the Product of Conjugates Pattern.

Notice this is the same result we found in Example 9.

When we multiply complex conjugates, the product of the last terms will always have an \({i}^{2}\) which simplifies to \(-1.\)

\[\begin{array}{l}(a-bi)(a+bi) \\ {a}^{2}-{(bi)}^{2} \\ {a}^{2}-{b}^{2}{i}^{2} \\ {a}^{2}-{b}^{2}(-1) \\ {a}^{2}+{b}^{2}\end{array}\]

This leads us to the Product of Complex Conjugates Pattern: \((a-bi)(a+bi)={a}^{2}+{b}^{2}\)

Example 10

Multiply using the Product of Complex Conjugates Pattern: \((8-2i)(8+2i).\)

Apply the pattern \((a-bi)(a+bi)={a}^{2}+{b}^{2}\) directly.

Show solution

Table 14

Use the Product of Complex Conjugates Pattern, \((a-bi)(a+bi)={a}^{2}+{b}^{2}.\)
Simplify the squares.
Add.

Divide Complex Numbers

Dividing complex numbers is much like rationalizing a denominator. We want our result to be in standard form with no imaginary numbers in the denominator.

Example 11How to Divide Complex Numbers

Divide: \(\frac{4+3i}{3-4i}.\)

Multiply the numerator and denominator by the complex conjugate of the denominator.

Show solution

We summarize the steps here.

Be careful as you find the conjugate of the denominator.

Simplify Powers of i

The powers of \(i\) make an interesting pattern that will help us simplify higher powers of i. Let’s evaluate the powers of \(i\) to see the pattern.

\[\begin{array}{llllllllll}{i}^{1} & & & \,{i}^{2} & & & \,{i}^{3} & & & \,{i}^{4} \\ i & & & \,-1 & & & \,{i}^{2}·i & & & \,{i}^{2}·{i}^{2} \\ & & & & & & \,-1·i & & & \,(-1)(-1) \\ & & & & & & \,-i & & & \,1 \\ \\ \\ {i}^{5} & & & \,{i}^{6} & & & \,{i}^{7} & & & \,{i}^{8} \\ {i}^{4}·i & & & \,{i}^{4}·{i}^{2} & & & \,{i}^{4}·{i}^{3} & & & \,{i}^{4}·{i}^{4} \\ 1·i & & & \,1·{i}^{2} & & & \,1·{i}^{3} & & & \,1·1 \\ i & & & \,{i}^{2} & & & \,{i}^{3} & & & \,1 \\ & & & \,-1 & & & \,-i\end{array}\]

We summarize this now.

\[\begin{array}{llllllll}{i}^{1} & = & i & & & \,{i}^{5} & = & i \\ {i}^{2} & = & -1 & & & \,{i}^{6} & = & -1 \\ {i}^{3} & = & \text{-}i & & & \,{i}^{7} & = & \text{-}i \\ {i}^{4} & = & 1 & & & \,{i}^{8} & = & 1\end{array}\]

If we continued, the pattern would keep repeating in blocks of four. We can use this pattern to help us simplify powers of i. Since i⁴ = 1, we rewrite each power, iⁿ, as a product using i⁴ to a power and another power of i.

We rewrite it in the form \({i}^{n}={({i}^{4})}^{q}·{i}^{r},\) where the exponent, q, is the quotient of n divided by 4 and the exponent, r, is the remainder from this division. For example, to simplify i⁵⁷, we divide 57 by 4 and we get 14 with a remainder of 1. In other words, \(57=4·14+1.\) So we write \({i}^{57}={({1}^{4})}^{14}·{i}^{1}\) and then simplify from there.

Example 14

Simplify: \({i}^{86}.\)

Divide the exponent by 4 and rewrite \({i}^{86}\) as \({({i}^{4})}^{q}·{i}^{r}\) using the quotient and remainder.

Show solution

Table 17

	\({i}^{86}\)
Divide 86 by 4 and rewrite \({i}^{86}\) in the \({i}^{n}={({i}^{4})}^{q}·{i}^{r}\) form.	\({({1}^{4})}^{21}·{i}^{2}\)
Simplify.	\({(1)}^{21}·(-1)\)
Simplify.	\(-1\)

Key Concepts

• Square Root of a Negative Number
  • If b is a positive real number, then \(\sqrt{\text{-}b}=\sqrt{b}i\)
    

    Table 18

    	\(a+bi\)
    \(b=0\)	\(\begin{array}{l} \\ a+0·i \\ \\ \,a\end{array}\)	Real number
    \(b\ne 0\)	\(a+bi\)	Imaginary number
    \(a=0\)	\(\begin{array}{l}0+bi \\ \\ \\ \,bi\end{array}\)	Pure imaginary number

  • A complex number is in standard form when written as a + bi, where a, b are real numbers.
    
  See Example 1.
• Product of Complex Conjugates
  • If a, b are real numbers, then
    \((a-bi)(a+bi)={a}^{2}+{b}^{2}\)
  See Example 9 and Example 10.
• How to Divide Complex Numbers
  • Write both the numerator and denominator in standard form.
  • Multiply the numerator and denominator by the complex conjugate of the denominator.
  • Simplify and write the result in standard form.
  See Example 11, Example 12, and Example 13.

Section Exercises

Practice Makes Perfect

Evaluate the Square Root of a Negative Number

In the following exercises, write each expression in terms of i and simplify if possible.

4

ⓐ \(\sqrt{-16}\) ⓑ \(\sqrt{-11}\)
ⓒ \(\sqrt{-8}\)

Show answer

ⓐ \(4i\) ⓑ \(\sqrt{11}i\) ⓒ \(2\sqrt{2}i\)

5

ⓐ \(\sqrt{-121}\) ⓑ \(\sqrt{-1}\) ⓒ \(\sqrt{-20}\)

6

ⓐ \(\sqrt{-100}\) ⓑ \(\sqrt{-13}\) ⓒ \(\sqrt{-45}\)

Show answer

ⓐ \(10i\) ⓑ \(\sqrt{13}i\) ⓒ \(3\sqrt{5}i\)

7

ⓐ \(\sqrt{-49}\) ⓑ \(\sqrt{-15}\) ⓒ \(\sqrt{-75}\)

Add or Subtract Complex Numbers In the following exercises, add or subtract.

8

\(\sqrt{-75}+\sqrt{-48}\)

Show answer

\(9\sqrt{3}i\)

9

\(\sqrt{-12}+\sqrt{-75}\)

10

\(\sqrt{-50}+\sqrt{-18}\)

Show answer

\(8\sqrt{2}i\)

11

\(\sqrt{-72}+\sqrt{-8}\)

12

\((1+3i)+(7+4i)\)

Show answer

\(8+7i\)

13

\((6+2i)+(3-4i)\)

14

\((8-i)+(6+3i)\)

Show answer

\(14+2i\)

15

\((7-4i)+(-2-6i)\)

16

\((1-4i)-(3-6i)\)

Show answer

\(-2+2i\)

17

\((8-4i)-(3+7i)\)

18

\((6+i)-(-2-4i)\)

Show answer

\(8+5i\)

19

\((-2+5i)-(-5+6i)\)

20

\((5-\sqrt{-36})+(2-\sqrt{-49})\)

Show answer

\(7-13i\)

21

\((-3+\sqrt{-64})+(5-\sqrt{-16})\)

22

\((-7-\sqrt{-50})-(-32-\sqrt{-18})\)

Show answer

\(25-2\sqrt{2}i\)

23

\((-5+\sqrt{-27})-(-4-\sqrt{-48})\)

Multiply Complex Numbers

In the following exercises, multiply.

24

\(4i(5-3i)\)

Show answer

\(12+20i\)

25

\(2i(-3+4i)\)

26

\(-6i(-3-2i)\)

Show answer

\(-12+18i\)

27

\(\text{-}i(6+5i)\)

28

\((4+3i)(-5+6i)\)

Show answer

\(-38++9i\)

29

\((-2-5i)(-4+3i)\)

30

\((-3+3i)(-2-7i)\)

Show answer

\(27+15i\)

31

\((-6-2i)(-3-5i)\)

In the following exercises, multiply using the Product of Binomial Squares Pattern.

32

\({(3+4i)}^{2}\)

Show answer

\(-7+24i\)

33

\({(-1+5i)}^{2}\)

34

\({(-2-3i)}^{2}\)

Show answer

\(-5+12i\)

35

\({(-6-5i)}^{2}\)

In the following exercises, multiply.

36

\(\sqrt{-25}·\sqrt{-36}\)

Show answer

\(-30\)

37

\(\sqrt{-4}·\sqrt{-16}\)

38

\(\sqrt{-9}·\sqrt{-100}\)

Show answer

\(-30\)

39

\(\sqrt{-64}·\sqrt{-9}\)

40

\((-2-\sqrt{-27})(4-\sqrt{-48})\)

Show answer

\(-44-4i\sqrt{3}\)

41

\((5-\sqrt{-12})(-3+\sqrt{-75})\)

42

\((2+\sqrt{-8})(-4+\sqrt{-18})\)

Show answer

\(-20-2\sqrt{2}i\)

43

\((5+\sqrt{-18})(-2-\sqrt{-50})\)

44

\((2-i)(2+i)\)

Show answer

5

45

\((4-5i)(4+5i)\)

46

\((7-2i)(7+2i)\)

Show answer

53

47

\((-3-8i)(-3+8i)\)

In the following exercises, multiply using the Product of Complex Conjugates Pattern.

48

\((7-i)(7+i)\)

Show answer

50

49

\((6-5i)(6+5i)\)

50

\((9-2i)(9+2i)\)

Show answer

85

51

\((-3-4i)(-3+4i)\)

Divide Complex Numbers

In the following exercises, divide.

52

\(\frac{3+4i}{4-3i}\)

Show answer

i

53

\(\frac{5-2i}{2+5i}\)

54

\(\frac{2+i}{3-4i}\)

Show answer

\(\frac{2}{25}+\frac{11}{25}i\)

55

\(\frac{3-2i}{6+i}\)

56

\(\frac{3}{2-3i}\)

Show answer

\(\frac{6}{13}+\frac{9}{13}i\)

57

\(\frac{2}{4-5i}\)

58

\(\frac{-4}{3-2i}\)

Show answer

\(-\frac{12}{13}-\frac{8}{13}i\)

59

\(\frac{-1}{3+2i}\)

60

\(\frac{1+4i}{3i}\)

Show answer

\(\frac{4}{3}-\frac{1}{3}i\)

61

\(\frac{4+3i}{7i}\)

62

\(\frac{-2-3i}{4i}\)

Show answer

\(-\frac{3}{4}+\frac{1}{2}i\)

63

\(\frac{-3-5i}{2i}\)

Simplify Powers of i

In the following exercises, simplify.

64

\({i}^{41}\)

Show answer

i

65

\({i}^{39}\)

66

\({i}^{66}\)

Show answer

\(-1\)

67

\({i}^{48}\)

68

\({i}^{128}\)

Show answer

1

69

\({i}^{162}\)

70

\({i}^{137}\)

Show answer

i

71

\({i}^{255}\)

Writing Exercises

72

Explain the relationship between real numbers and complex numbers.

Show answer

Answers will vary.

73

Aniket multiplied as follows and he got the wrong answer. What is wrong with his reasoning?

\(\begin{array}{l}\sqrt{-7}·\sqrt{-7} \\ \sqrt{49} \\ 7\end{array}\)

74

Why is \(\sqrt{-64}=8i\) but \(\sqrt[3]{-64}=-4.\)

Show answer

Answers will vary.

75

Explain how dividing complex numbers is similar to rationalizing a denominator.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of \(1-10,\) how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

Chapter Review Exercises

Simplify Expressions with Roots

Simplify Expressions with Roots

In the following exercises, simplify.

Estimate and Approximate Roots

In the following exercises, estimate each root between two consecutive whole numbers.

In the following exercises, approximate each root and round to two decimal places.

Simplify Variable Expressions with Roots

In the following exercises, simplify using absolute values as necessary.

Simplify Radical Expressions

Use the Product Property to Simplify Radical Expressions

In the following exercises, use the Product Property to simplify radical expressions.

In the following exercises, simplify using absolute value signs as needed.

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

Simplify Rational Exponents

Simplify expressions with \({a}^{\frac{1}{n}}\)

In the following exercises, write as a radical expression.

In the following exercises, write with a rational exponent.

In the following exercises, simplify.

Simplify Expressions with \({a}^{\frac{m}{n}}\)

In the following exercises, write with a rational exponent.

In the following exercises, simplify.

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

Add, Subtract and Multiply Radical Expressions

Add and Subtract Radical Expressions

In the following exercises, simplify.

Multiply Radical Expressions

In the following exercises, simplify.

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

Divide Radical Expressions

Divide Square Roots

In the following exercises, simplify.

Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

Rationalize a Two Term Denominator

In the following exercises, simplify.

Solve Radical Equations

Solve Radical Equations

In the following exercises, solve.

Solve Radical Equations with Two Radicals

In the following exercises, solve.

Use Radicals in Applications

In the following exercises, solve. Round approximations to one decimal place.

Use Radicals in Functions

Evaluate a Radical Function

In the following exercises, evaluate each function.

Find the Domain of a Radical Function

In the following exercises, find the domain of the function and write the domain in interval notation.

Graph Radical Functions

In the following exercises, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

81

\(g(x)=2\sqrt{x}\)

Show answer

ⓐ domain: \([0,\infty )\)
ⓑ

ⓒ range: \([0,\infty )\)

83

\(f(x)=\sqrt[3]{x}+3\)

Show answer

ⓐ domain: \((\text{-}\infty ,\infty )\)
ⓑ

ⓒ range: \((\text{-}\infty ,\infty )\)

Use the Complex Number System

Evaluate the Square Root of a Negative Number

In the following exercises, write each expression in terms of i and simplify if possible.

Add or Subtract Complex Numbers

In the following exercises, add or subtract.

Multiply Complex Numbers

In the following exercises, multiply.

In the following exercises, multiply using the Product of Binomial Squares Pattern.

In the following exercises, multiply using the Product of Complex Conjugates Pattern.

Divide Complex Numbers

In the following exercises, divide.

Simplify Powers of i

In the following exercises, simplify.

Practice Test

In the following exercises, simplify using absolute values as necessary.

In the following exercises, simplify. Assume all variables are positive.

In the following exercises, solve.

In the following exercise, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

25

\(g(x)=\sqrt{x+2}\)

Show answer

ⓐ domain: \([-2,\infty )\)
ⓑ

ⓒ range: \([0,\infty )\)

Glossary

complex conjugate pair

A complex conjugate pair is of the form a + bi, a – bi.

complex number

A complex number is of the form a + bi, where a and b are real numbers. We call a the real part and b the imaginary part.

complex number system

The complex number system is made up of both the real numbers and the imaginary numbers.

imaginary unit

The imaginary unit \(i\) is the number whose square is –1. i² = –1 or \(i=\sqrt{-1}.\)

standard form

A complex number is in standard form when written as \(a+bi,\) where a, b are real numbers.