1. Copyright © 2005 Pearson Education, Inc. 5.3 The Rational Numbers

2. Slide 5-2 Copyright © 2005 Pearson Education, Inc. The Rational Numbers The set of rational numbers, denoted by Q, is the set of all numbers of the form p/q, where p and q are integers and q  0.

3. Slide 5-3 Copyright © 2005 Pearson Education, Inc. Fractions Fractions are numbers such as: The numerator is the number above the fraction line. The denominator is the number below the fraction line.

4. Slide 5-4 Copyright © 2005 Pearson Education, Inc. Reducing Fractions In order to reduce a fraction, we divide both the numerator and denominator by the greatest common divisor. Example: Reduce to its lowest terms. Solution:

5. Slide 5-5 Copyright © 2005 Pearson Education, Inc. Mixed Numbers A mixed number consists of an integer and a fraction. For example, 3 ½ is a mixed number. 3 ½ is read “three and one half” and means “3 + ½”.

6. Slide 5-6 Copyright © 2005 Pearson Education, Inc. Improper Fractions Rational numbers greater than 1 or less than -1 that are not integers may be written as mixed numbers, or as improper fractions. An improper fraction is a fraction whose numerator is greater than its denominator. An example of an improper fraction is 12/5.

7. Slide 5-7 Copyright © 2005 Pearson Education, Inc. Converting a Positive Mixed Number to an Improper Fraction Multiply the denominator of the fraction in the mixed number by the integer preceding it. Add the product obtained in step 1 to the numerator of the fraction in the mixed number. This sum is the numerator of the improper fraction we are seeking. The denominator of the improper fraction we are seeking is the same as the denominator of the fraction in the mixed

8. Slide 5-8 Copyright © 2005 Pearson Education, Inc. Example Convert to an improper fraction.

9. Slide 5-9 Copyright © 2005 Pearson Education, Inc. Converting a Positive Improper Fraction to a Mixed Number Divide the numerator by the denominator. Identify the quotient and the remainder. The quotient obtained in step 1 is the integer part of the mixed number. The remainder is the numerator of the fraction in the mixed number. The denominator in the fraction of the mixed number will be the same as the denominator in the original fraction.

10. Slide 5-10 Copyright © 2005 Pearson Education, Inc. Convert to a mixed number. The mixed number is Example

11. Slide 5-11 Copyright © 2005 Pearson Education, Inc. Terminating or Repeating Decimal Numbers Every rational number when expressed as a decimal number will be either a terminating or repeating decimal number. Examples of terminating decimal numbers 0.7, 2.85, 0.000045 Examples of repeating decimal numbers 0.44444… which may be written

12. Slide 5-12 Copyright © 2005 Pearson Education, Inc. Division of Fractions Multiplication of Fractions

13. Slide 5-13 Copyright © 2005 Pearson Education, Inc. Example: Multiplying Fractions Evaluate the following. a) b)

14. Slide 5-14 Copyright © 2005 Pearson Education, Inc. Example: Dividing Fractions Evaluate the following. a) b)

15. Slide 5-15 Copyright © 2005 Pearson Education, Inc. Addition and Subtraction of Fractions

16. Slide 5-16 Copyright © 2005 Pearson Education, Inc. Example: Add or Subtract Fractions Add: Subtract:

17. Slide 5-17 Copyright © 2005 Pearson Education, Inc. Fundamental Law of Rational Numbers If a, b, and c are integers, with b  0, c  0, then

18. Slide 5-18 Copyright © 2005 Pearson Education, Inc. Example: Evaluate: Solution:

19. Slide 5-19 Copyright © 2005 Pearson Education, Inc. Next Steps Read Examples 1-6, 11-15 Work Problems in text on 13-45, odds; 101-107, all Do Online homework corresponding to this section Do Online quiz for Secs. 5.1-5.3