1. © 2010 Pearson Prentice Hall. All rights reserved. CHAPTER 5 Number Theory and the Real Number System

2. © 2010 Pearson Prentice Hall. All rights reserved. 2 5.3 The Rational Numbers

3. © 2010 Pearson Prentice Hall. All rights reserved. 3 Objectives 1.Define the rational numbers. 2.Reduce rational numbers. 3.Convert between mixed numbers and improper fractions. 4.Express rational numbers as decimals. 5.Express decimals in the form a / b. 6.Multiply and divide rational numbers. 7.Add and subtract rational numbers. 8.Use the order of operations agreement with rational numbers. 9.Apply the density property of rational numbers. 10.Solve problems involving rational numbers.

4. © 2010 Pearson Prentice Hall. All rights reserved. 4 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

5. © 2010 Pearson Prentice Hall. All rights reserved. 5 Reducing a Rational Number If is a rational number and c is any number other than 0, The rational numbers and are called equivalent fractions. To reduce a rational number to its lowest terms, divide both the numerator and denominator by their greatest common divisor.

6. © 2010 Pearson Prentice Hall. All rights reserved. 6 Reduce to lowest terms. Solution: Begin by finding the greatest common divisor of 130 and 455. Thus, 130 = 2 · 5 · 13, and 455 = 5 · 7 · 13. The greatest common divisor is 5 · 13 or 65. Example 1: Reducing a Rational Number

7. © 2010 Pearson Prentice Hall. All rights reserved. 7 Divide the numerator and the denominator of the given rational number by 5 · 13 or 65. There are no common divisors of 2 and 7 other than 1. Thus, the rational number is in its lowest terms. Example 1: Reducing a Rational Number (continued)

8. © 2010 Pearson Prentice Hall. All rights reserved. 8 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

9. © 2010 Pearson Prentice Hall. All rights reserved. 9 1.Multiply the denominator of the rational number by the integer and add the numerator to this product. 2.Place the sum in step 1 over the denominator of the mixed number. Converting a Positive Mixed Number to an Improper Fraction

10. © 2010 Pearson Prentice Hall. All rights reserved. 10 Example: Convert to an improper fraction. Solution: Example 2: Converting a Positive Mixed Number to an Improper Fraction

11. © 2010 Pearson Prentice Hall. All rights reserved. 11 1.Divide the denominator into the numerator. Record the quotient and the remainder. 2.Write the mixed number using the following form: Converting a Positive Improper Fraction to a Mixed Number

12. © 2010 Pearson Prentice Hall. All rights reserved. 12 Convert to a mixed number. Solution: Step 1 Divide the denominator into the numerator. Step 2 Write the mixed number using Thus, Example 3: Converting from an Improper Fraction to a Mixed Number

13. © 2010 Pearson Prentice Hall. All rights reserved. 13 Rational Numbers and Decimals Any rational number can be expressed as a decimal by dividing the denominator into the numerator.

14. © 2010 Pearson Prentice Hall. All rights reserved. 14 Example 4: Expressing Rational Numbers as Decimals Express each rational number as a decimal. a.b. Solution: In each case, divide the denominator into the numerator.

15. © 2010 Pearson Prentice Hall. All rights reserved. 15 a. b. Notice the decimal stops. This is called a terminating decimal. Notice the digits 63 repeat over and over indefinitely. This is called a repeating decimal. Example 4: Expressing Rational Numbers as Decimals (continued)

16. © 2010 Pearson Prentice Hall. All rights reserved. 16 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.

17. © 2010 Pearson Prentice Hall. All rights reserved. 17 Example 5: Expressing Decimals as a Quotient of Two Integers Express each terminating decimal as a quotient of integers: a. 0.7b. 0.49 c. 0.048 Solution: a.0.7 = because the 7 is in the tenths position.

18. © 2010 Pearson Prentice Hall. All rights reserved. 18 b.0.49 = because the digit on the right, 9, is in the hundredths position. c. 0.048 = because the digit on the right, 8, is in the thousandths position. Reducing to lowest terms, Example 5: Expressing Terminating Decimals in a/b form (continued)

19. © 2010 Pearson Prentice Hall. All rights reserved. 19 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.

20. © 2010 Pearson Prentice Hall. All rights reserved. 20 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: Example 6: Expressing a Repeating Decimal in a/b Form (continued) Thus,.

21. © 2010 Pearson Prentice Hall. All rights reserved. Repeating Decimal to Rational __ Express 0.63 as a rational number. Solution n = 0.6363… 10n = 6.6363… 10n = 6.6363… - n = 0.6363… -------------------- 9n = 6 n = 2/3 21

22. © 2010 Pearson Prentice Hall. All rights reserved. 22 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

23. © 2010 Pearson Prentice Hall. All rights reserved. 23 Example 8b: Multiplying Rational Numbers Multiply. If possible, reduce the product to its lowest terms: Multiply across.Simplify to lowest terms.

24. © 2010 Pearson Prentice Hall. All rights reserved. 24 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

25. © 2010 Pearson Prentice Hall. All rights reserved. 25 Example 9b: Dividing Rational Numbers Divide. If possible, reduce the quotient to its lowest terms: Change to multiplication by using the reciprocal. Multiply across.

26. © 2010 Pearson Prentice Hall. All rights reserved. 26 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

27. © 2010 Pearson Prentice Hall. All rights reserved. 27 Perform the indicated operations: a. b. c. Solution: a. b. c. Example 10: Adding & Subtracting Rational Numbers with Identical Denominators

28. © 2010 Pearson Prentice Hall. All rights reserved. 28 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. Adding and Subtracting Rational Numbers Unlike Denominators

29. © 2010 Pearson Prentice Hall. All rights reserved. 29 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. Example 11: Adding Rational Numbers Unlike Denominators 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.

30. © 2010 Pearson Prentice Hall. All rights reserved. 30 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.

31. © 2010 Pearson Prentice Hall. All rights reserved. 31 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,

32. © 2010 Pearson Prentice Hall. All rights reserved. 32 Problem Solving with Rational Numbers A common application of rational numbers involves preparing food for a different number of servings than what the recipe gives. The amount of each ingredient can be found as follows:

33. © 2010 Pearson Prentice Hall. All rights reserved. 33 A chocolate-chip recipe for five dozen cookies requires ¾ cup of sugar. If you want to make eight dozen cookies, how much sugar is needed? Solution: Example 15: Changing the Size of a Recipe

34. © 2010 Pearson Prentice Hall. All rights reserved. 34 The amount of sugar, in cups, needed is determined by multiplying the rational numbers: Thus, cups of sugar is needed. Example 15: Changing the Size of a Recipe (continued)

35. © 2010 Pearson Prentice Hall. All rights reserved. Proportion 35 x 60” 240” 960” Find the height x of the tree, when a 60”-man casts a shadow 240” long.