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1.8 The Real Numbers

Square roots, sorting numbers into rational, irrational, and real, and locating fractions and decimals on the number line.

Elementary Algebra 2e, 1.8 The Real Numbers

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58 exercises

1.8 Exercise 1

Simplify: ⓐ 25\sqrt{25} ⓑ 121.\sqrt{121} .

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1.8 Exercise 2

Simplify: ⓐ 36\sqrt{36} ⓑ 169.\sqrt{169} .

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1.8 Exercise 3

Simplify: ⓐ 16\sqrt{16} ⓑ 196.\sqrt{196} .

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1.8 Exercise 4

Simplify: ⓐ −9− \sqrt{9} ⓑ −144.− \sqrt{144} .

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1.8 Exercise 5

Simplify: ⓐ −4− \sqrt{4} ⓑ −225.− \sqrt{225} .

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1.8 Exercise 6

Simplify: ⓐ −81− \sqrt{81} ⓑ −100.− \sqrt{100} .

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1.8 Exercise 7

Write as the ratio of two integers: ⓐ −27−27 ⓑ 7.31.

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1.8 Exercise 8

Write as the ratio of two integers: ⓐ −24−24 ⓑ 3.57.

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1.8 Exercise 9

Write as the ratio of two integers: ⓐ −19−19 ⓑ 8.41.

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1.8 Exercise 10

Given the numbers 0.583–,0.47,3.605551275...0.58 \overset{–}{3} , 0.47 , 3.605551275 . . . list the ⓐ rational numbers ⓑ irrational numbers.

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1.8 Exercise 11

For the given numbers list the ⓐ rational numbers ⓑ irrational numbers: 0.29,0.816–,2.515115111….0.29 , 0.81 \overset{–}{6} , 2.515115111 \ldots .

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1.8 Exercise 12

For the given numbers list the ⓐ rational numbers ⓑ irrational numbers: 2.63–,0.125,0.418302…2.6 \overset{–}{3} , 0.125 , 0.418302 \ldots

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1.8 Exercise 13

For each number given, identify whether it is rational or irrational: ⓐ 36\sqrt{36} ⓑ 44.\sqrt{44} .

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1.8 Exercise 14

For each number given, identify whether it is rational or irrational: ⓐ 81\sqrt{81} ⓑ 17.\sqrt{17} .

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1.8 Exercise 15

For each number given, identify whether it is rational or irrational: ⓐ 116\sqrt{116} ⓑ 121.\sqrt{121} .

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1.8 Exercise 16

For each number given, identify whether it is a real number or not a real number: ⓐ −169\sqrt{−169} ⓑ −64.− \sqrt{64} .

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1.8 Exercise 17

For each number given, identify whether it is a real number or not a real number: ⓐ −196\sqrt{−196} ⓑ −81.− \sqrt{81} .

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1.8 Exercise 18

For each number given, identify whether it is a real number or not a real number: ⓐ −49− \sqrt{49} ⓑ −121.\sqrt{−121} .

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1.8 Exercise 19

Given the numbers −7,145,8,5,5.9,−64,−7 , \frac{14}{5} , 8 , \sqrt{5} , 5.9 , − \sqrt{64} , list the ⓐ whole numbers ⓑ integers ⓒ rational numbers ⓓ irrational numbers ⓔ real numbers.

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1.8 Exercise 20

For the given numbers, list the ⓐ whole numbers ⓑ integers ⓒ rational numbers ⓓ irrational numbers ⓔ real numbers: −3,−2,0.3–,95,4,49.−3 , − \sqrt{2} , 0. \overset{–}{3} , \frac{9}{5} , 4 , \sqrt{49} .

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1.8 Exercise 21

For the given numbers, list the ⓐ whole numbers ⓑ integers ⓒ rational numbers ⓓ irrational numbers ⓔ real numbers: −25,−38,−1,6,121,2.041975…− \sqrt{25} , - \frac{3}{8} , −1 , 6 , \sqrt{121} , 2.041975 \ldots

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1.8 Exercise 26

Order each of the following pairs of numbers, using < or >:

ⓐ −13___−1- \frac{1}{3} ___ −1 ⓑ −112___−2−1 \frac{1}{2} ___ −2 ⓒ −23___−13- \frac{2}{3} ___ - \frac{1}{3} ⓓ −3___−73.−3 ___ - \frac{7}{3} .

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1.8 Exercise 27

Order each of the following pairs of numbers, using < or >:

ⓐ −1___−23−1 ___ - \frac{2}{3} ⓑ −214___−2−2 \frac{1}{4} ___ −2 ⓒ −35___−45- \frac{3}{5} ___ - \frac{4}{5} ⓓ −4___−103.−4 ___ - \frac{10}{3} .

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1.8 Exercise 34

Order 0.64___0.60.64 ___ 0.6 using << or >.> .

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1.8 Exercise 35

Order each of the following pairs of numbers, using <or>:0.42___0.4.< \text{or} > : 0.42 ___ 0.4 .

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1.8 Exercise 36

Order each of the following pairs of numbers, using <or>:0.18___0.1.< \text{or} > : 0.18 ___ 0.1 .

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1.8 Exercise 37

Order 0.83___0.8030.83 ___ 0.803 using << or >.> .

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1.8 Exercise 38

Order the following pair of numbers, using <or>:0.76___0.706.< \text{or} > : 0.76 ___ 0.706 .

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1.8 Exercise 39

Order the following pair of numbers, using <or>:0.305___0.35.< \text{or} > : 0.305 ___ 0.35 .

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1.8 Exercise 40

Use << or >> to order −0.1___−0.8.−0.1 ___ −0.8 .

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1.8 Exercise 41

Order the following pair of numbers, using < or >: −0.3___−0.5.−0.3 ___ −0.5 .

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1.8 Exercise 42

Order the following pair of numbers, using < or >: −0.6___−0.7.−0.6 ___ −0.7 .

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1.8 Exercise 43

Simplify.

36\sqrt{36}

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1.8 Exercise 45

Simplify.

64\sqrt{64}

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1.8 Exercise 47

Simplify.

9\sqrt{9}

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1.8 Exercise 49

Simplify.

100\sqrt{100}

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1.8 Exercise 51

Simplify.

−4− \sqrt{4}

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1.8 Exercise 53

Simplify.

−1− \sqrt{1}

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1.8 Exercise 55

Write as the ratio of two integers.

ⓐ 5 ⓑ 3.19

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1.8 Exercise 57

Write as the ratio of two integers.

ⓐ −12− 12 ⓑ 9.279

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1.8 Exercise 59

List the ⓐ rational numbers, ⓑ irrational numbers

0.75,0.223–,1.39174…0.75 , 0.22 \overset{–}{3} , 1.39174 \ldots

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1.8 Exercise 61

List the ⓐ rational numbers, ⓑ irrational numbers

0.45–,1.919293…,3.590.4 \overset{–}{5} , 1.919293 \ldots , 3.59

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1.8 Exercise 63

Identify whether each number is rational or irrational.

ⓐ 25\sqrt{25} ⓑ 30\sqrt{30}

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1.8 Exercise 65

Identify whether each number is rational or irrational.

ⓐ 164\sqrt{164} ⓑ 169\sqrt{169}

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1.8 Exercise 67

Identify whether each number is a real number or not a real number.

ⓐ −81− \sqrt{81} ⓑ −121\sqrt{−121}

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1.8 Exercise 69

Identify whether each number is a real number or not a real number.

ⓐ −36\sqrt{−36} ⓑ −144− \sqrt{144}

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1.8 Exercise 71

List the ⓐ whole numbers, ⓑ integers, ⓒ rational numbers, ⓓ irrational numbers, ⓔ real numbers for each set of numbers.

−8,0,1.95286…,125,36,9−8 , 0 , 1.95286 \ldots , \frac{12}{5} , \sqrt{36} , 9

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1.8 Exercise 73

List the ⓐ whole numbers, ⓑ integers, ⓒ rational numbers, ⓓ irrational numbers, ⓔ real numbers for each set of numbers.

−100,−7,−83,−1,0.77,314− \sqrt{100} , −7 , - \frac{8}{3} , −1 , 0.77 , 3 \frac{1}{4}

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1.8 Exercise 83

Order each of the pairs of numbers, using < or >.

−1___−14−1 ___ - \frac{1}{4}

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1.8 Exercise 85

Order each of the pairs of numbers, using < or >.

−212___−3−2 \frac{1}{2} ___ −3

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1.8 Exercise 87

Order each of the pairs of numbers, using < or >.

−512___−712- \frac{5}{12} ___ - \frac{7}{12}

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1.8 Exercise 89

Order each of the pairs of numbers, using < or >.

−3___−135−3 ___ - \frac{13}{5}

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1.8 Exercise 95

Order each pair of numbers, using < or >.

0.37___0.630.37 ___ 0.63

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1.8 Exercise 97

Order each pair of numbers, using < or >.

0.91___0.9010.91 ___ 0.901

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1.8 Exercise 99

Order each pair of numbers, using < or >.

−0.5___−0.3−0.5 ___ −0.3

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1.8 Exercise 101

Order each pair of numbers, using < or >.

−0.62___−0.619−0.62 ___ −0.619

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1.8 Exercise 103

Field trip All the 5th graders at Lincoln Elementary School will go on a field trip to the science museum. Counting all the children, teachers, and chaperones, there will be 147 people. Each bus holds 44 people.

ⓐ How many busses will be needed?
ⓑ Why must the answer be a whole number?
ⓒ Why shouldn’t you round the answer the usual way, by choosing the whole number closest to the exact answer?

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1.8 Exercise 105

In your own words, explain the difference between a rational number and an irrational number.

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Adapted from Elementary Algebra 2e © 2020 Rice University. Licensed under CC BY 4.0. Source: Elementary Algebra 2e. Imported at revision 77a933c.