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5.3

5.3. The Rational Numbers. 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. Fractions. Fractions are numbers such as: The numerator is the number above the fraction line.

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5.3

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  1. 5.3 The Rational Numbers

  2. 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. 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. 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. 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. 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. 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. Example • Convert to an improper fraction.

  9. 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. Example • Convert to a mixed number. • The mixed number is

  11. 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. Multiplication of Fractions • Division of Fractions

  13. Evaluate the following. a) b) Example: Multiplying Fractions

  14. Evaluate the following. a) b) Example: Dividing Fractions

  15. Addition and Subtraction of Fractions

  16. Add: Subtract: Example: Add or Subtract Fractions

  17. Fundamental Law of Rational Numbers • If a, b, and c are integers, with b 0, c  0, then

  18. Example: • Evaluate: • Solution:

  19. 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

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