# Exercises: The Real Numbers (1.6)

## Exercises: Simplify Expressions with Square Roots

Instructions: For questions 1-12, simplify.

1. $\sqrt{36}$

Solution

$6$

2. $\sqrt4$

3. $\sqrt{64}$
Solution

$8$

4. $\sqrt{169}$

5. $\sqrt9$
Solution

$3$

6. $\sqrt{16}$

7. $\sqrt{100}$

Solution

$10$

8. $\sqrt{144}$

9. $-\sqrt4$

Solution

$-2$

10. $-\sqrt{100}$

11. $-\sqrt1$
Solution

$-1$

12. $-\sqrt{121}$

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

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

13.

a. $5$
b. $3.19$

Solution

a. $\frac51$
b. $\frac{319}{100}$

14.

a. $8$
b. $1.61$

15.

a. $-12$
b. $9.279$

Solution

a. $\frac{-12}{1}$
b. $\frac{9297}{1000}$

16.

a. $-16$
b. $4.399$

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

Instructions: For questions 17-20, list the

a. rational numbers
b. irrational numbers

17. $0.75,0.22\overline{3},1.39174...$
Solution

a. $0.75,0.22\overline{3}$
b. $1.39174...$

18. $0.36,0.94729...,2.52\overline{8}$

19. $0.4\overline{5},1.919293...,3.59$

Solution

a. $0.4\overline{5},3.59$
b. $1.919293...$

20. $0.1\overline{3},0.42982...,1.875$

## 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. $\sqrt{25}$
b. $\sqrt{30}$

Solution

a. rational
b. irrational

22.

a. $\sqrt{44}$
b. $\sqrt{49}$

23.

a. $\sqrt{164}$
b. $\sqrt{169}$

Solution

a. irrational
b. rational

24.

a. $\sqrt{225}$
b. $\sqrt{216}$

## 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. $-\sqrt{81}$
b. $\sqrt{-121}$

Solution

a. real number
b. not a real number

26.

a. $-\sqrt{64}$
b. $\sqrt{-9}$

27.

a. $\sqrt{-36}$
b. $-\sqrt{144}$

Solution

a. not a real number
b. real number

28.

a. $\sqrt{-49}$
b. $-\sqrt{144}$

## 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. $-8,0,1.95286...,\frac{12}{5},\sqrt{36},9$
Solution

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

30. $-9,-3\frac49,-\sqrt9,0.40\overline{9},\frac{11}{6},7$

31. $-\sqrt{100},-7,-\frac83,-1,0.77,3\frac14$

Solution

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

32. $-6,-\frac52,0,0.\overline{714285},2\frac15,\sqrt{14}$

## Exercises: Locate Fractions on the Number Line

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

33. $\frac35,\frac85,\frac{10}{3}$
Solution

34. $\frac14,\frac95,\frac{11}{3}$

35. $\frac{3}{10},\frac72,\frac{11}{6},4$

Solution

36. $\frac{7}{10},\frac52,\frac{13}{8},3$

37. $\frac25,-\frac25$

Solution

38. $\frac34,-\frac34$

39. $\frac34,-\frac34,1\frac23,-1\frac23,\frac52,-\frac52$

Solution

40. $\frac25,-\frac25,1\frac34,-1\frac34,\frac83,-\frac83$

## Exercises: Order Pairs of Numbers

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

41. $-1\;\underline{\;\;\;\;}\;-\frac14$
Solution

$<$

42. $-1\;\underline{\;\;\;\;}\;-\frac13$

43. $-2\frac12\;\underline{\;\;\;\;}\;-3$

Solution

$>$

44. $-1\frac34\;\underline{\;\;\;\;}\;-2$

45. $-\frac{5}{12}\;\underline{\;\;\;\;}\;-\frac{7}{12}$

Solution

$>$

46. $-\frac{9}{10}\;\underline{\;\;\;\;}\;-\frac{3}{10}$

47. $-3\;\underline{\;\;\;\;}\;-\frac{13}{5}$

Solution

$<$

48. $-4\;\underline{\;\;\;\;}\;-\frac{23}{6}$

## Exercises: Locate Decimals on the Number Line

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

49. $0.8$
Solution

50. $-0.9$

51. $-1.6$

Solution

52. $3.1$

## Exercises: Order Pairs of Numbers

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

53. $0.37\;\underline{\;\;\;\;}\;0.63$
Solution

$<$

54. $0.86\;\underline{\;\;\;\;}\;0.69$

55. $0.91\;\underline{\;\;\;\;}\;0.901$

Solution

$>$

56. $0.415\;\underline{\;\;\;\;}\;0.41$

57. $-0.5\;\underline{\;\;\;\;}\;-0.3$

Solution

$<$

58. $-0.1\;\underline{\;\;\;\;}\;-0.4$

59. $-0.62\;\underline{\;\;\;\;}\;-0.619$

Solution

$<$

60. $-7.31\;\underline{\;\;\;\;}\;-7.3$

## 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 $147$ people. Each bus holds $44$ 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. $4$ buses

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