1-3 Real Numbers and the Number Line

Quick Review

The rational numbers and irrational numbers form the set of real numbers.

A rational number is any number that you can write as a b , eh over b , comma  where a and b are integers and b ± 0 . b plus minus 0 .  The rational numbers include all positive and negative integers, as well as fractions, mixed numbers, and terminating and repeating decimals.

Irrational numbers cannot be represented as the quotient of two integers. They include the square roots of all positive integers that are not perfect squares.

Example

Is the number rational or irrational?

1. − 5.422 negative , 5.422  rational
2. 7 square root of 7  irrational

Exercises

Tell whether each number is rational or irrational.

32. π
33. − 1 2 negative , 1 half
34. 2 3 square root of 2 thirds end root
35. 0. 57 ¯ 0. , 57 bar

Estimate each square root. Round to the nearest integer.

36. 99 square root of 99
37. 48 square root of 48
38. 30 square root of 30

Name the subset(s) of the real numbers to which each number belongs.

39. − 17 negative 17
40. 13 62 13 over 62
41. 94 square root of 94
42. 100 square root of 100
43. 4.288
44. 1 2 3 1 , and 2 thirds

Order the numbers in each exercise from least to greatest.

45. − 1 2 3 , negative 1 , and 2 thirds , comma  1.6, − 1 4 5 negative 1 , and 4 fifths
46. 7 9 , − 0.8 , 3 7 ninths , comma , minus 0.8 comma square root of 3

1-4 Properties of Real Numbers

Quick Review

You can use properties such as the ones below to simplify and evaluate expressions.

Example

Use an identity property to simplify − 7 a b a negative . fraction 7 eh b , over eh end fraction

− 7 a b a = − 7 b ⋅ a a = − 7 b ⋅ 1 = − 7 b negative . fraction 7 eh b , over eh end fraction . equals negative 7 b dot , eh over eh , equals negative 7 b dot 1 equals negative 7 b

Exercises

Simplify each expression. Justify each step.

47. − 8 + 9 w + ( − 23 ) negative 8 plus 9 w plus open negative 23 close
48. 6 5 · ( − 10 · 8 ) 6 fifths , middle dot open negative 10 middle dot 8 close
49. ( 4 3 · 0 ) · ( − 20 ) open , 4 thirds , middle dot 0 close middle dot open negative 20 close
50. 53 + ( − 12 ) + ( − 4 t ) 53 plus open negative 12 close plus open negative 4 t close
51. 6 + 3 9 fraction 6 plus 3 , over 9 end fraction

Tell whether the expressions in each pair are equivalent.

52. ( 5 − 2 ) c  and  c · 3 open 5 minus 2 close c , and , c middle dot 3
53. 41 + z + 9 and 41 · z · 9 41 middle dot z middle dot 9
54. 81 x y 3 x fraction 81 x y , over 3 x end fraction  and 9xy
55. 11 t ( 5 + 7 − 11 ) fraction 11 t , over open . 5 plus 7 minus 11 . close end fraction  and t

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