chapter 4 the real number system -...

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4.1 Thinking About NumbersCounting Numbers, Whole Numbers,Integers, Rational and Irrational Numbers ● p. 211

4.2 Real NumbersProperties of the Real Number System ● p. 217

4.3 Man-Made NumbersImaginary Numbers and Complex Numbers ● p. 223

4.4 The Complete Number SystemOperations with Complex Numbers ● p. 229

The abacus, or counting frame, has been used since ancient times to perform basic numericaloperations such as addition, subtraction, multiplication, and division. In some countries, accountants still calculate on an abacus instead of an electronic calculator. You will learn aboutthe history of numbers.

4C HA PT E R

The Real Number System

Chapter 4 ● The Real Number System 209

210 Chapter 4 ● The Real Number System

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Lesson 4.1 ● Counting Numbers, Whole Numbers, Integers, Rational and Irrational Numbers 211

4

ObjectivesIn this lesson, you will

● Define the following sets of numbers:● rational numbers● irrational numbers● real numbers

● Define closure.● Determine which sets are closed

under the operations of addition, subtraction, multiplication, and division.

Key Terms● closure● rational numbers● irrational numbers● real numbers

4.1 Thinking About NumbersCounting Numbers, Whole Numbers,Integers, Rational and Irrational Numbers

Problem 1 A Short History of NumbersAt some time in mankind’s past, it became necessary for people to count theirpossessions, and a rudimentary number system was born. The first system probablyconsisted of making pictures of the items being counted. Later, symbols in the formof tally marks were used to “count” the number of objects that a person saw orowned. This set of numbers, the first set that you learned, is called the set of naturalnumbers, or more commonly, the set of counting numbers.

Set of Counting Numbers: {1, 2, 3, 4, . . . }

Once people were able to count their possessions, they soon developed the needto determine the total number of possessions when two or more groups were combined. Initially, they probably just put the two groups of possessions togetherand counted the resulting group, and then did the same thing with their symbols.But eventually, combining groups led to the development of the first operation,addition, which provided rules for calculating the sum of two or more countingnumbers.

Addition’s inverse operation, subtraction, developed in a similar way: as people lostpossessions, they initially took away pictures or tally marks and recounted theirtotal. Eventually, they established the rules for subtraction of two counting numbers.

The need for the ability to “solve” problems gave rise to mathematics. Next camethe need to do repeated additions, from which came multiplication, and repeatedsubtractions, from which came division. The set of counting numbers and these fouroperations served mankind for many millennia, but there were some problems.


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1. When you add or multiply any counting number with any other counting number,is your answer a counting number? Explain.

2. When you subtract any counting number from any other counting number, is youranswer a counting number? Explain.

3. When you divide any counting number by any other counting number, is youranswer a counting number? Explain.

4. When you can perform an operation on a set of numbers and the answer isalways in the same set, the set is said “to be closed” or “to have closure.” Underwhich operations is the set of counting numbers closed? Explain.

Problem 2 ZeroThe next major innovation was the discovery or invention of zero, 0, a number thatrepresented the lack of any quantity. When zero is added to the counting numbers,the resulting set is the whole numbers.

Set of Whole Numbers: {0, 1, 2, 3, 4, . . . }

Although having a number that represented the lack of a quantity was a majoradvance, its utility really became important when zero was paired with the conceptof place value. If you have ever tried to add, subtract, multiply, or divide usingRoman numerals, the lack of place value and zero becomes very apparent. (Note:See www.carnegielearning.com/Operations/RomanNumerals for examples.)

1. Under which operations is the set of whole numbers closed? Explain.

2. What numbers must be added to the whole numbers in order for the new set tobe closed under subtraction? Explain.

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3. This new set of numbers includes all whole numbers and all negative numbers.Write the name of the set on the line, and write the numbers that are in this setin the braces.

Name of set: __________ { }

At some point, people confronted the problem of having to divide one thing—apiece of food, a herd of animals, a collection of tools—among more than one person. From this dilemma came the rational numbers.

The set of rational numbers consists of all numbers that can be written as wherea and b are integers.

4. Under which operations are the rational numbers closed? Explain.

Finally, humankind realized there are some numbers that are not rational numbers.You know some of them, including p, . Although we often use a fractionto approximate these numbers, they cannot truly be written in the form of a rationalnumber . A more thorough discussion of why these numbers cannot be written as

fractions can be found at www.carnegielearning.com/rationalnumbers.

The set of irrational numbers consists of all numbers that cannot be written as where a and b are integers.

The set of real numbers is the set produced by combining the set of rationalnumbers with the set of irrational numbers.

5. Under which of the operations is the set of real numbers closed?

Irrational numbers can be represented by a symbol, such as p, or by using

radicals or fractional exponents, such as . While irrational numbers are

often approximated by a fraction or a decimal, these numbers have no exact numerical representation. In fact, p has been calculated to over a million places, andthe calculation of p is often used as a gauge of the computation speed of certaincomputers. Here is the value of p to 400 decimal places.

�2,�3, 523

ab

ab

�2, and�3

ab

4


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p � 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094 . . .

All irrational numbers have an infinite number of non-repeating decimal places. Anyinfinite decimal that repeats single digits or blocks of digits can be written as afraction and is therefore a rational number. For instance, we know that

To convert a repeating decimal to a fraction, first set the decimal representationequal to a variable:

Multiply both sides of this equation by the multiple of 10 that has the same number ofzeros as there are repeated digits (in 0. , multiply by 10, but in 0. , multiply by 100):

Subtract the first equation from the second, and solve for x:

Determine the fractional representation of each of the following repeating decimals.

6. 0.22222 . . .

7. 0.27

x �3199

9999

x �3199

99x � 31 x � 0.313131 . . .

100x � 31.313131 . . .

100x � 31.313131 . . .

313

x � 0.313131 . . .

13� 0.333333 . . .

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8. 4.121212 . . .

9.

10.

11.

12. 0.

Be prepared to share your work with another pair, group, or the entire class.

1234

9.99999 . . .

7.76

5.512512 . . .

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Lesson 4.2 ● Properties of the Real Number System 217

4Problem 1 Operations on Real Numbers

Addition and multiplication are said to be well-defined operations over the real number system and have the following properties:

1. Addition and multiplication are commutative:

a. �a, b � �, a � b � b � a

This is read as: “For all (�) numbers a and b that are elements of the set ofreal numbers (� �), a plus b equals b plus a.”

Using the same notation, write the commutative property for multiplication.

b. Are subtraction and division commutative? If not, give a counterexample.

ObjectiveIn this lesson, you will

● Identify the following properties of thereal number system:● commutative● associative● distributive● additive identity● multiplicative identity● additive inverse● multiplicative inverse

Key Terms● commutative● associative● distributive● additive identity● multiplicative identity● additive inverse● multiplicative inverse

4.2 Real NumbersProperties of the Real Number System


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2. Addition and multiplication are associative:

a. �a, b, and c � �, (a � b) � c � a � (b � c)

This is read as:

b. Using the same notation, write the associative property for multiplication.

c. Are subtraction and division associative? If not, give a counterexample.

Multiplication is distributive over addition:

3. �a, b and c � �, a(b � c) � ab � ac

This is read as:

4. Is multiplication distributive over subtraction? If so, write the property of multiplication over subtraction using the same notation as used in Question 3. If not, give a counterexample.

5. Is division distributive over addition? Over subtraction? If so, write each propertyusing the same notation as used in Question 3. If not, give a counterexample foreach.

6. There is a number that when added to any real number a, the sum is equal to a. Thisnumber is called the additive identity. What number is the additive identity? Explain.

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7. Similarly, there is a number called the multiplicative identity that when multipliedby any real number a, the product is equal to a. What number is the multiplicativeidentity? Explain.

8. When a number is added to its additive inverse, the sum is the additive identity.For any real number a, what is its additive inverse? Explain.

9. When a number is multiplied by its multiplicative inverse, the product is themultiplicative identity. For any real number a, what is its multiplicative inverse?Explain.

Problem 2 Simplifying ExpressionsEach of the following expressions has been simplified one step at a time. Next toeach step, identify the property, transformation, or simplification used in the step.

1.

2.

x � 25

(7x � 6x) � (4 � 21)

7x � 6x � 4 � 21

7x � 4 � 6x � 21

7x � 4 � 3(2x � 7)

5x � 39

5x � (4 � 35)

5x � 4 � 35

4 � (5x � 35)

4 � 5(x � 7)


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Each of the following equations has been solved one step at a time. Next to eachstep, identify the property used in the step.

3.

x � 2

x � 1 � 2

x(20) � 120

� 40 � 120

20x � 120

� 40 � 120

20x � 40

20x � 0 � 40

20x � (�18 � 18) � (22 � 18)

20x � 18 � 18 � 22 � 18

20x � 18 � 22

20x � (�24 � 6) � 22

20x � 24 � 6 � 22

4(5x � 6) � 6 � 22

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4.


x � �7

1 � x � �7

( 12

� 2 ) x � �7

12

� 2x �12

� (�14)

2x � �14

2x � 0 � 14

2x � (x � x) � 14

3x � x � x � x � 14

3x � x � 14

3x � 0 � x � 14

3x � (12 � 12) � x � (�2 � 12)

3x � 12 � 12 � x � 2 � 12

3x � 12 � x � 2

(6x � 3x) � 12 � x � 2

6x � 3x � 12 � x � 2

6x � 12 � 3x � x � 2

6x � 12 � 3x �(4x � 8)

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Lesson 4.3 ● Imaginary Numbers and Complex Numbers 223

4

Problem 1 Exponentiation and Real NumbersAs we have seen, the real numbers are closed under the operations of addition,subtraction, multiplication, and division. Now we’ll explore whether the set of realnumbers is closed under another important operation, exponentiation.

1. Are the real numbers closed for all integer exponents? Explain why or why not. Ifnot, give a counterexample.

2. Are the real numbers closed for all real-number exponents? Explain why or whynot. If not, give a counterexample.

3. For each of following, simplify the power when possible.

a.

b.

c. (�27)23

10052

823

ObjectivesIn this lesson, you will:

● Identify as i.● Identify the powers of i.● Identify where a is a real number.● Define the set of imaginary numbers.● Define the set of complex numbers.● Define the real term of a complex number.● Define the imaginary term of a complex

number.

��a

��1

Key Terms● exponentiation● rational exponents● imaginary numbers● complex numbers● real term of a complex number● imaginary term of a complex

number

4.3 Man-Made NumbersImaginary Numbers and Complex Numbers

RememberRational exponents are

defined as:

abc

c ab (c a)b


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d.

e.

f.

In order to have a number system that is closed under exponentiation, there mustbe some way to define . If a definition exists, then it is possible to calculate anyroot of any real number. So, let us define .

4. Calculate the following powers of i.

a.

b.

c.

d.

e.

f.

g.

h.

i.

j.

5. Describe how you can calculate the value of any integer power of i.

6. Using i, we can now calculate the square root of any negative real number.Calculate the following square roots.

a.

b.

c.

d.

e. ��25

��144

��64

��3

��4

i�13

i�2

i102

i8

i7

i6

i5

i4

i3

i2

��1 � i��1

(�4)32

10,00034

�6432

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7. We are also now able to solve some quadratic equations that we could not solvebefore. Solve the following quadratics by solving for x2 and taking the square rootof each side of the equation.

a. x2 � 4 � 0

b. x2 � 4 � 0

c. x2 � 196 � 0

d. x2 � 7 � 0

Problem 2 Imaginary and Complex NumbersThe set of imaginary numbers consists of all numbers that can be written in theform bi where b is a real number.

1. For each number, list all of the sets to which it belongs.

a. 3

b.

c. 5.45

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d.

e. 5i

2. Solve each of the following quadratic equations using the quadratic formula.Simplify your answer using imaginary numbers.

a. x 2 � 4x � 13 � 0

b. 2x 2 � 4x � 5 � 0

Even under the imaginary numbers, there are still values of real exponents andsolutions to quadratic equations that cannot be calculated. The solutions to thequadratic equations in Question 2 are members of a new set of numbers, thecomplex numbers.

The set of complex numbers consists of all the numbers of the form a � bi where a and b are real numbers and . The term a is called the real term of a complex number and bi is called the imaginary term of a complex number.

Decide if each of the following statements is true or false. Explain your answers.

3. All real numbers are complex numbers.

i ��1

�7

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4. All imaginary numbers are real numbers.

5. All integers are real numbers.

6. All rational numbers are complex numbers.

7. All counting numbers are imaginary numbers.

For each of the following complex numbers, identify the real term and theimaginary term.

8. 3 � 6i

9. �3i � 9

10.

11. �11i

12. 13 � 6 i

13.

14. �i


�3 ��6i

p

p

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Lesson 4.4 ● Operations with Complex Numbers 229

4

Problem 1 Adding, Subtracting, andMultiplying Complex Numbers

To add and subtract complex numbers, combine the real terms and then combinethe imaginary terms to form an answer that consists of two terms.

1. For each of the following pairs of complex numbers, calculate the sum and thedifference.

a. 5 � 3i, 6 � 5i

b. �12 � 4i, 11 � 7i

c. �8i, 11 � 17i

d. 12 � 1.3i, 2.6 � 7.6i

Multiplying complex numbers is similar to multiplying binomials. Use the distributiveproperty twice so that each term of the first complex number is multiplied by eachterm of the second complex number.

ObjectivesIn this lesson, you will

● Add, subtract, multiply, and dividecomplex numbers.

● Determine the conjugate of acomplex number.

● Calculate powers and roots ofcomplex numbers.

Key Terms● conjugate of a complex number● power of a complex number● root of a complex number

4.4 The Complete NumberSystemOperations with Complex Numbers


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2. For each of the pairs of complex numbers in Question 1, calculate the product.

a.

b.

c.

d.

Problem 2 Dividing Complex NumbersDivision of complex numbers requires using the conjugate of a complex number,thus changing the divisor into a real number. The conjugate of a complex numbera � bi is a � bi.

1. For each of the following complex numbers, write its conjugate.

a. 7 � i

b. �5 � 3i

c. 12 � 11i

d. �4i

e. 9 � 7i

f. a � bi

2. For each complex number and its conjugate in Question 1, calculate the product.

a.

b.

c.

d.

e.

f.

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3. In each case, what happened when you multiplied the complex number (a � bi )by its conjugate (a � bi )? Explain.

Using this fact, we can simplify the division of a complex number by multiplyingboth the divisor and the dividend by the conjugate of the divisor, thus changing thedivisor into a real number. For example:

4. Calculate the following quotients.

a.

b.

c.

d.20 � 5i2 � 4i

5 � 2i1 � i

3 � 4i2 � 3i

2 � i3 � 2i

3 � 2i4 � 3i

�625

�1725

i

�6 � 17i

25

�12 � 17i � 6

16 � 9

�12 � 9i � 8i � 6i2

16 � 12i � 12i � 9i2

3 � 2i4 � 3i

�3 � 2i4 � 3i

�4 � 3i4 � 3i


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Problem 3 Calculating Powers and Roots of Complex Numbers

Calculating a whole number power of a complex number can be accomplished byrepeated multiplication, but this process is very time consuming.

1. Calculate the indicated power of each of the following complex numbers.

a. (2 � 3i )2

b. (�1 � 2i )3

c. (�1 � 3i )3

Calculating the square root of a complex number can be accomplished usingstraightforward algebraic techniques, but solving for higher roots or fractional rootsrequires the use of the mathematics that we will cover later. When solving for asquare root of a complex number, we first set the square root equal to a generalform of a complex number, then square both sides. By combining the real terms andcombining the imaginary terms, we are able to form two equations and solve usingsubstitution. For example:

Solve for a: a �

Then substitute for a in the first equation:

b2 ��16 � �400

8��16 � 20

8� �

92

, 12

b2 ��16 � �256 � 144

8

4b4 � 16b2 � 9 � 0

16b2 � 9 � 4b4

4 �9

4b2 � b2

4 � ( 32b )

2

� b2

32b

32b

4 � a2 � b2 and 3i � 2abi

4 � 3i � (a2 � b2) � 2abi

4 � 3i � a2 � 2abi � b2i2

�4 � 3i � a � bi

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Because b by definition is a real number, set the positive term equal to b2:

Check:

� 4 � 3i

�184�

124

i �24

i2

(�3�22

��22

i )2

� (�3�22 )

2

� 2 (�3�22 ) (��2

2 ) i � (��22

i )2

� 4 � 3i

�184�

124

i �24

i2

( 3�22

��22

i )2

� ( 3�22 )

2

� 2 ( 3�22 ) ( �2

2 ) i � ( �22

i )2

�4 � 3i �3�2

2�

�22

i, �3�2

2�

�22

i

a �3

2 (��22 )

�3

��2� �

3�22

b � ��12� �

�22

b2 �12

chapter 4 the real number system -...

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