5.3

1. 5.3 The Rational Numbers

2. Objectives • Define the rational numbers. • Reduce rational numbers. • Convert between mixed numbers and improper fractions. • Express rational numbers as decimals. • Express decimals in the form a / b. • Multiply and divide rational numbers. • Add and subtract rational numbers. • Use the order of operations agreement with rational numbers. • Apply the density property of rational numbers. • Solve problems involving rational numbers.

3. Defining the Rational Numbers • The set of rational numbers is the set of all numbers which can be expressed in the form , where a and b are integers and b is not equal to 0. • The integer a is called the numerator. • The integer b is called the denominator. The following are examples of rational numbers: ¼, ½, ¾, 5, 0

4. Example 1: Reducing a Rational Number Reduce to lowest terms.

5. Mixed Numbers and Improper Fractions • A mixed number consists of the sum of an integer and a rational number, expressed without the use of an addition sign. Example: • An improper fraction is a rational number whose numerator is greater than its denominator. Example: 19 is larger than 5

6. Converting a Positive Mixed Number to an Improper Fraction • Multiply the denominator of the rational number by the integer and add the numerator to this product. • Place the sum in step 1 over the denominator of the mixed number.

7. Example 2: Converting a Positive Mixed Number to an Improper Fraction Example: Convert to an improper fraction.

8. Converting a Positive Improper Fraction to a Mixed Number • Divide the denominator into the numerator. Record the quotient and the remainder. • Write the mixed number using the following form:

9. Example 3: Converting from an Improper Fraction to a Mixed Number Convert to a mixed number. Solution: Step 1 Divide the denominator into the numerator. Step 2 Write the mixed number using Thus,

10. Rational Numbers and Decimals • Any rational number can be expressed as a decimal by dividing the denominator into the numerator.

11. Example 4: Expressing Rational Numbers as Decimals Express each rational number as a decimal. a. b.

12. Expressing Decimals as a Quotient of Two Integers • Terminating decimals can be expressed with denominators of 10, 100, 1000, 10,000, and so on. • Using the chart, the digits to the right of the decimal point are the numerator of the rational number.

13. Example 5: Expressing Decimals as a Quotient of Two Integers Express each terminating decimal as a quotient of integers: a. 0.7 b. 0.49 c. 0.048

14. Example 6: Expressing a Repeating Decimal in a/b Form Express as a quotient of integers. Solution: Step1 Let n equal the repeating decimal such that n = , or 0.6666… Step 2 If there is one repeating digit, multiply both sides of the equation in step 1 by 10. n = 0.66666… 10n = 10(0.66666…) 10n = 6.66666… Multiplying by 10 moves the decimal point one place to the right.

15. Example 6: Expressing a Repeating Decimal in a/b Form (continued) Step 3 Subtract the equation in step 1 from the equation in step 2. Step 4 Divide both sides of the equation in step 3 by the number in front of n and solve for n. We solve 9n = 6 for n: Thus, .

17. Multiplying Rational Numbers • The product of two rational numbers is the product of their numerators divided by the product of their denominators. • If and are rational numbers, then

18. Example 8b: Multiplying Rational Numbers Multiply. If possible, reduce the product to its lowest terms:

19. Dividing Rational Numbers • The quotient of two rational numbers is a product of the first number and the reciprocal of the second number. If and are rational numbers, then

20. Example 9b: Dividing Rational Numbers Divide. If possible, reduce the quotient to its lowest terms:

21. Adding and Subtracting Rational Numbers with Identical Denominators The sum or difference of two rational numbers with identical denominators is the sum or difference of their numerators over the common denominator. If and are rational numbers, then and

22. Example 10: Adding & Subtracting Rational Numbers with Identical Denominators Perform the indicated operations: a. b. c. Solution:

23. Adding and Subtracting Rational NumbersUnlike Denominators • If the rational numbers to be added or subtracted have different denominators, we use the least common multiple of their denominators to rewrite the rational numbers. • The least common multiple of their denominators is called the least common denominator or LCD.

24. Example 11: Adding Rational NumbersUnlike Denominators Find the sum of Solution: Find the least common multiple of 4 and 6 so that the denominators will be identical. LCM of 4 and 6 is 12. Hence, 12 is the LCD. We multiply the first rational number by 3/3 and the second one by 2/2 to obtain 12 in the denominator for each number. Notice, we have 12 in the denominator for each number. Add numerators and put this sum over the least common denominator.

25. Density of Rational Numbers If r and t represent rational numbers, with r < t, then there is a rational number s such that s is between r and t: r < s < t.

26. Example 14: Illustrating the Density Property Find a rational number halfway between ½ and ¾. Solution: First add ½ and ¾. Next, divide this sum by 2. The number is halfway between ½ and ¾. Thus,