Exercises: The Real Numbers (1.6)

Exercises: Simplify Expressions with Square Roots

Instructions: For questions 1-12, simplify.

1. [latex]\sqrt{36}[/latex]

Solution

[latex]6[/latex]

2. [latex]\sqrt4[/latex]
3. [latex]\sqrt{64}[/latex]
Solution

[latex]8[/latex]

4. [latex]\sqrt{169}[/latex]

5. [latex]\sqrt9[/latex]
Solution

[latex]3[/latex]

6. [latex]\sqrt{16}[/latex]

7. [latex]\sqrt{100}[/latex]

Solution

[latex]10[/latex]

8. [latex]\sqrt{144}[/latex]

9. [latex]-\sqrt4[/latex]

Solution

[latex]-2[/latex]

10. [latex]-\sqrt{100}[/latex]

11. [latex]-\sqrt1[/latex]
Solution

[latex]-1[/latex]

12. [latex]-\sqrt{121}[/latex]

Exercises: Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers

Instructions: For questions 13-16, write as the ratio of two integers.

13.

a. [latex]5[/latex]
b. [latex]3.19[/latex]

Solution

a. [latex]\frac51[/latex]
b. [latex]\frac{319}{100}[/latex]

14.

a. [latex]8[/latex]
b. [latex]1.61[/latex]

15.

a. [latex]-12[/latex]
b. [latex]9.279[/latex]

Solution

a. [latex]\frac{-12}{1}[/latex]
b. [latex]\frac{9297}{1000}[/latex]

16.

a. [latex]-16[/latex]
b. [latex]4.399[/latex]

Exercises: Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers

Instructions: For questions 17-20, list the

a. rational numbers
b. irrational numbers

17. [latex]0.75,0.22\overline{3},1.39174...[/latex]
Solution

a. [latex]0.75,0.22\overline{3}[/latex]
b. [latex]1.39174...[/latex]

18. [latex]0.36,0.94729...,2.52\overline{8}[/latex]

19. [latex]0.4\overline{5},1.919293...,3.59[/latex]

Solution

a. [latex]0.4\overline{5},3.59[/latex]
b. [latex]1.919293...[/latex]

20. [latex]0.1\overline{3},0.42982...,1.875[/latex]

Exercises: Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers

Instructions: For questions 21-24, identify whether each number is rational or irrational.

21.

a. [latex]\sqrt{25}[/latex]
b. [latex]\sqrt{30}[/latex]

Solution

a. rational
b. irrational

22.

a. [latex]\sqrt{44}[/latex]
b. [latex]\sqrt{49}[/latex]

23.

a. [latex]\sqrt{164}[/latex]
b. [latex]\sqrt{169}[/latex]

Solution

a. irrational
b. rational

24.

a. [latex]\sqrt{225}[/latex]
b. [latex]\sqrt{216}[/latex]

Exercises: Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers

Instructions: For questions 25-28, identify whether each number is a real number or not a real number.

25.

a. [latex]-\sqrt{81}[/latex]
b. [latex]\sqrt{-121}[/latex]

Solution

a. real number
b. not a real number

26.

a. [latex]-\sqrt{64}[/latex]
b. [latex]\sqrt{-9}[/latex]

27.

a. [latex]\sqrt{-36}[/latex]
b. [latex]-\sqrt{144}[/latex]

Solution

a. not a real number
b. real number

28.

a. [latex]\sqrt{-49}[/latex]
b. [latex]-\sqrt{144}[/latex]

Exercises: Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers

Instructions: For each set of numbers in questions 29-32, list the

a. whole numbers
b. integers
c. rational numbers
d. irrational numbers
e. real numbers

29. [latex]-8,0,1.95286...,\frac{12}{5},\sqrt{36},9[/latex]
Solution

a. [latex]0,\sqrt{36},9[/latex]
b. [latex]-8,0,\sqrt{36},9[/latex]
c. [latex]-8,0,\frac{12}{5},\sqrt{36},9[/latex]
d. [latex]1.95286...[/latex]
e. [latex]-8,0,1.95286...,\frac{12}{5},\sqrt{36},9[/latex]

30. [latex]-9,-3\frac49,-\sqrt9,0.40\overline{9},\frac{11}{6},7[/latex]

31. [latex]-\sqrt{100},-7,-\frac83,-1,0.77,3\frac14[/latex]

Solution

a. none
b. [latex]-\sqrt{100},-7,-1[/latex]
c. [latex]-\sqrt{100},-7,-\frac83,-1,0.77,3\frac14[/latex]
d. none
e. [latex]-\sqrt{100},-7,-\frac83,-1,0.77,3\frac14[/latex]

32. [latex]-6,-\frac52,0,0.\overline{714285},2\frac15,\sqrt{14}[/latex]

Exercises: Locate Fractions on the Number Line

Instructions: For questions 33-40, locate the numbers on a number line.

33. [latex]\frac35,\frac85,\frac{10}{3}[/latex]
Solution
There is a number line shown that runs from 0 to 6. From left to right, the numbers marked are 3/4, 8/5, and 10/3. The number 3/4 is slightly to the left of 1. The number 8/5 is halfway between 1 and 2. The number 10/3 is slightly to the right of 3.
Figure 1P.6.1

34. [latex]\frac14,\frac95,\frac{11}{3}[/latex]

35. [latex]\frac{3}{10},\frac72,\frac{11}{6},4[/latex]

Solution
There is a number line shown that runs from 0 to 6. From left to right, the numbers marked are 3/10, 11/6, 7/2, and 4. The number 3/10 is between negative 0 and 1 but slightly closer to 0. The number 11/6 is slightly to the left of 2. The number 4 lies right on the line for 4 on the number line.
Figure 1P.6.2

36. [latex]\frac{7}{10},\frac52,\frac{13}{8},3[/latex]

37. [latex]\frac25,-\frac25[/latex]

Solution
There is a number line shown that runs from negative 1 to 1, with increments of one-fifth. From left to right, the numbers marked are -2/5, and 2/5.
Figure 1P.6.3

38. [latex]\frac34,-\frac34[/latex]

39. [latex]\frac34,-\frac34,1\frac23,-1\frac23,\frac52,-\frac52[/latex]

Solution
There is a number line shown that runs from negative 4 to positive 4. From left to right, the numbers marked are negative 5/2, negative 1 and 2/3, negative 3/4, positive 3/4, positive 1 and 2/3, and positive 5/2. The number negative 5/2 is halfway between negative 3 and negative 2. The number negative 1 and 2/3 is between negative 2 and negative 1, but slightly closer to negative 2. The number negative 3/4 is slightly to the right of negative 1. The number positive 3/4 is slightly to the left of positive 1. The number positive 1 and 2/3 is between positive 1 and positive 2, but slightly closer to positive 2. The number positive 5/2 is halfway between positive 2 and positive 3.
Figure 1P.6.4

40. [latex]\frac25,-\frac25,1\frac34,-1\frac34,\frac83,-\frac83[/latex]

Exercises: Order Pairs of Numbers

Instructions: For questions 41-48, order each of the pairs of numbers, using < or >.

41. [latex]-1\;\underline{\;\;\;\;}\;-\frac14[/latex]
Solution

[latex]<[/latex]

42. [latex]-1\;\underline{\;\;\;\;}\;-\frac13[/latex]

43. [latex]-2\frac12\;\underline{\;\;\;\;}\;-3[/latex]

Solution

[latex]>[/latex]

44. [latex]-1\frac34\;\underline{\;\;\;\;}\;-2[/latex]

45. [latex]-\frac{5}{12}\;\underline{\;\;\;\;}\;-\frac{7}{12}[/latex]

Solution

[latex]>[/latex]

46. [latex]-\frac{9}{10}\;\underline{\;\;\;\;}\;-\frac{3}{10}[/latex]

47. [latex]-3\;\underline{\;\;\;\;}\;-\frac{13}{5}[/latex]

Solution

[latex]<[/latex]

48. [latex]-4\;\underline{\;\;\;\;}\;-\frac{23}{6}[/latex]

Exercises: Locate Decimals on the Number Line

Instructions: For questions 49-52, locate the number on the number line.

49. [latex]0.8[/latex]
Solution
There is a number line shown that runs from negative 4 to positive 4. The only point given is positive 0.8, which is slightly to the left of positive 1.
Figure 1P.6.5

50. [latex]-0.9[/latex]

51. [latex]-1.6[/latex]

Solution
There is a number line shown that runs from negative 4 to positive 4. The only point given is negative 1.6, which is between negative 2 and negative 1, but slightly closer to negative 2.
Figure 1P.6.6

52. [latex]3.1[/latex]

Exercises: Order Pairs of Numbers

Instructions: For questions 53-60, order each of the pairs of numbers, using < or >.

53. [latex]0.37\;\underline{\;\;\;\;}\;0.63[/latex]
Solution

[latex]<[/latex]

54. [latex]0.86\;\underline{\;\;\;\;}\;0.69[/latex]

55. [latex]0.91\;\underline{\;\;\;\;}\;0.901[/latex]

Solution

[latex]>[/latex]

56. [latex]0.415\;\underline{\;\;\;\;}\;0.41[/latex]

57. [latex]-0.5\;\underline{\;\;\;\;}\;-0.3[/latex]

Solution

[latex]<[/latex]

58. [latex]-0.1\;\underline{\;\;\;\;}\;-0.4[/latex]

59. [latex]-0.62\;\underline{\;\;\;\;}\;-0.619[/latex]

Solution

[latex]<[/latex]

60. [latex]-7.31\;\underline{\;\;\;\;}\;-7.3[/latex]

Exercises: Everyday Math

Instructions: For questions 61-62, answer the given everyday math word problems.

61. 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 [latex]147[/latex] people. Each bus holds [latex]44[/latex] people.

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

Solution

a. [latex]4[/latex] buses
b. Answers may vary
c. Answers may vary

62. Child Care. Serena wants to open a licensed child care center. Her state requires there be no more than [latex]12[/latex] children for each teacher. She would like her child care center to serve [latex]40[/latex] children.

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

Exercises: Writing Exercises

Instructions: For questions 63-64, answer the given writing exercises.

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

Solution

Answers may vary

64. Explain how the sets of numbers (counting, whole, integer, rational, irrationals, reals) are related to each other.

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Fanshawe Pre-Health Sciences Mathematics 1 Copyright © 2022 by Domenic Spilotro, MSc is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License, except where otherwise noted.

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