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Chapter 1 Review of Real Numbers

Symbols and Sets of Numbers 1.2 Symbols and Sets of Numbers

Set of Numbers Natural numbers – {1, 2, 3, 4, 5, 6 . . .} Whole numbers – {0, 1, 2, 3, 4 . . .} Integers – {. . . –3, -2, -1, 0, 1, 2, 3 . . .} Rational numbers – the set of all numbers that can be expressed as a quotient of integers, with denominator  0. Irrational numbers – the set of all numbers that can NOT be expressed as a quotient of integers. Real numbers – the set of all rational and irrational numbers combined.

Equality and Inequality Symbols Meaning a = b a  b a < b a > b a  b a  b a is equal to b. a is not equal to b. a is less than b. a is greater than b. a is less then or equal to b. a is greater than or equal to b.

A number line is a line on which each point is associated with a number. 2 – 2 1 3 4 5 – 1 – 3 – 4 – 5 – 4.8 1.5 Negative numbers Positive numbers

Example Insert <, >, or = in the space between each pair of numbers to make each statement true. a. 3 4 b. 8 2 Solution a. 3 4 3 < 4 since 3 is to the left of 4 on the number line. b. 8 2 8 > 2 since 8 is to the right of 2 on the number line.

Example Tell whether each statement is true or false. a. 3 ≤ 3 b. 25 ≥ 0 Solution a. 3 ≤ 3 True, since 3 = 3. b. 25 ≥ 0 True, since 25 > 0.

Example Translate each sentence into a mathematical statement. a. 12 is greater than or equal to seven. b. Nine is less than twelve. c. Five is not equal to six. Solution 12 ≥ 7 9 < 12 5 ≠ 6

Example Given the set list the numbers in this set that belong to the set of: a. Natural numbers b. Whole numbers c. Integers d. Rational numbers e. Irrational numbers f. Real numbers Solution a. The natural numbers are 13 and 115. b. The whole numbers are 0, 13, and 115. c. The integers are 4, 3, 0, 13, and 115. d. The rational numbers are 4, 3, 2.5, 0, 1/4, 13, and 115. e. The irrational number is f. The real numbers are all the numbers in the given set.

Order Property for Real Numbers For any two real numbers a and b, a is less than b if a is to the left of b on the number line. a < b means a is to the left of b on a number line. a > b means a is to the right of b on a number line.

Example Insert <, >, or = in the space between each pair of numbers to make each statement true. a. 3 2 b. 8 10 Solution a. 3 2 3 < 2 since 3 is to the left of 2 on the number line. b. 8 10 8 > 10 since 8 is to the right of 10 on the number line.

The absolute value of a real number a, denoted by |a|, is the distance between a and 0 on the number line. | – 4| = 4 |5| = 5 Symbol for absolute value Distance of 4 Distance of 5 2 – 2 1 3 4 5 – 1 – 3 – 4 – 5

Example Find each absolute value. a. |4| b. c. |3.4| d. |8| Solution a. 4 since 4 is located 4 units from 0 on the number line. b. since 2/3 is 2/3 units from 0 on the number line. c. |3.4| = 3.4 d. |8| = 8

In mathematics, we can use counterexamples to prove that a statement is false. A counterexample is a specific example of the falsity of a statement.

Example Tell whether each statement is true or false. If false, give a counterexample. a. The absolute value of a negative number is always a positive number. b. 2 is a natural number. Solution a. True, by definition of absolute value. b. False, by definition of natural numbers.