Chapter 02 – Section 04 Rational Numbers

2.4.1 DEFINITION OF A RATIONAL NUMBER 2-4 RATIONAL NUMBERS OBJECTIVES To compare and order rational numbers, and to find a number between two rational numbers. rational number – any number that can be written as a fraction A rational number is a number that can be expressed in the form , where a and b are integers and b is not equal to 0. 2.4.1 DEFINITION OF A RATIONAL NUMBER Rational numbers are any number that can be written as a fraction. Examples of rational numbers: Rational Numbers Form

2-4 RATIONAL NUMBERS Rational numbers can be graphed on a number line the same way as integers. Examples of the graphs of rational numbers. Rational Numbers Form -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8

2-4 RATIONAL NUMBERS We use the inequality signs, < and > to compare rational numbers. You might remember the shark (or PacMan or alligator) always eats the larger number. The set {-5, -2, 11/2, 3.5} is graphed below. -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 You can say the following about these numbers: A. The graph of -2 is to the left of the graph of 3.5. B. The graph of 11/2 is to the right of the graph of -5. -2 < 3.5 11/2 > -5

2.4.2 COMPARING NUMBERS ON THE NUMBER LINE 2-4 RATIONAL NUMBERS If a and b represent any numbers and the graph of a is to the left of the graph of b, then a < b. If the graph of a is to the right of the graph of b, then a > b. 2.4.2 COMPARING NUMBERS ON THE NUMBER LINE The larger number is always to the right of the smaller number. For any two numbers a and b, exactly one of the following sentences is true. a < b a = b a > b 2.4.3 COMPARISON PROPERTY A number is either smaller than, greater than, or equal to any other number.

2-4 RATIONAL NUMBERS Here is a chart of symbols you need to know. Symbol Meaning < is less than > is greater than = is equal to ≠ is not equal to ≤ is less than or equal to ≥ is greater than or equal to Notice these statements have verbs – mathematical sentences to follow.

2-4 RATIONAL NUMBERS EXAMPLE 1α: Replace each _?_ with <, >, or = to make each sentence true. a. -75 _?_ 13 b. -14 _?_ -22 + 9 c. 3/8 _?_ -7/8 Since any negative number is always less than any positive number, the true sentence is: -75 < 13. Simplify the right-hand side. -14 _?_ -22 + 9 -14 _?_ -13 Since -14 is less than -13, the true sentence is: -14 < -22 + 9. Since 3 is greater than -7, the true sentence is: 3/8 > -7/8.

2-4 RATIONAL NUMBERS EXAMPLE 1β: Replace each _?_ with <, >, or = to make each sentence true. a. -11 _?_ -20 + 9 b. -15 + 6 _?_ 0 c. 5 _?_ 8 + (-12)

2.4.4 COMPARISON PROPERTY FOR RATIONAL NUMBERS Example 1C had fractions in it. The problem was easy because the denominators were the same. If you have a similar problem that has different denominators, you can use cross products to compare the two fractions. For any rational numbers and , with b > 0 and d > 0: 1. if then ad < bc, and 2. if ad < bc, then . 2.4.4 COMPARISON PROPERTY FOR RATIONAL NUMBERS

2-4 RATIONAL NUMBERS EXAMPLE 2α: Replace each _?_ with <, >, or = to make each sentence true. A. _?_ B. _?_ 7(15) _?_ 13(4) 9(7) _?_ 8(8) 105 > 52 63 < 64 EXAMPLE 2β: Replace each _?_ with <, >, or = to make each sentence true. A. _?_ B. _?_

2-4 RATIONAL NUMBERS Every rational number can be expressed as a terminating or repeating decimal. EXAMPLE 3α: Use a calculator to write the fractions , , and as decimals. Then write the fractions in order from least to greatest. MORE HERE ON RE-WRITE = 0.375 This is a terminating decimal. = 0.8 This is also a terminating decimal. = 0.166666… or This is a repeating decimal. Ordering the decimals, you get: , 0.375, 0.8. So, the fractions in order would be , , .

2-4 RATIONAL NUMBERS EXAMPLE 3β: Use a calculator to write the fractions as decimals. Then write the fractions in order from least to greatest.

2.4.5 DENSITY PROPERTY FOR RATIONAL NUMBERS You can always find the midpoint of any distance. Eventually you will get down to the atomic level and be working with miniscule lengths. However, you can always find a point mid-way between any two other points - no matter how small the gap. When you are asked to find a point in-between two numbers, always calculate the midpoint – its easier for me to grade. Between every pair of distinct rational numbers, there are infinitely many rational numbers. 2.4.5 DENSITY PROPERTY FOR RATIONAL NUMBERS

2-4 RATIONAL NUMBERS EXAMPLE 4α: Find a rational number between and . Since we can choose which one, we will just find the midpoint. To do this, we take the average of the two points - add them and divide by 2. or -0.3875 EXAMPLE 4β: Find a rational number between and .

2-4 RATIONAL NUMBERS HOMEWORK PAGE 97 #15 – 33 odd plus #39