Section 5 3 The Rational Numbers Copyright 2013

Section 5. 3 The Rational Numbers Copyright 2013, 2010, 2007, Pearson, Education, Inc.

What You Will Learn Rational Numbers Multiplying and Dividing Fractions Adding and Subtracting Fractions 5. 3 -2 Copyright 2013, 2010, 2007, 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. The following are examples of rational numbers: 5. 3 -3 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Fractions are numbers such as: The numerator is the number above the fraction line. The denominator is the number below the fraction line. 5. 3 -4 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Reducing Fractions To reduce a fraction to its lowest terms, divide both the numerator and denominator by the greatest common divisor. 5. 3 -5 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 1: Reducing a Fraction to Lowest Terms Reduce to lowest terms. Solution GCD of 54 and 90 is 18 5. 3 -6 Copyright 2013, 2010, 2007, 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”. 5. 3 -7 Copyright 2013, 2010, 2007, 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. 5. 3 -8 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Converting a Positive Mixed Number to an Improper Fraction 1. Multiply the denominator of the fraction in the mixed number by the integer preceding it. 2. 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 number. 5. 3 -9 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 2: Converting Mixed Numbers to Improper Fractions Convert the following mixed numbers to improper fractions. 5. 3 -10 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Converting a Positive Improper Fraction to a Mixed Number 1. Divide the numerator by the denominator. Identify the quotient and the remainder. 2. 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. 5. 3 -11 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 3: From Improper Fraction to Mixed Number Convert the following improper fraction to a mixed number. Solution 5. 3 -12 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 3: From Improper Fraction to Mixed Number Solution The mixed number is 5. 3 -13 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 3: From Improper Fraction to Mixed Number Convert the following improper fraction to a mixed number. Solution 5. 3 -14 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 3: From Improper Fraction to Mixed Number Solution The mixed number is 5. 3 -15 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Terminating or Repeating Decimal Numbers Every rational number when expressed as a decimal number will be either a terminating or a repeating decimal number. 5. 3 -16 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Terminating or Repeating Decimal Numbers Examples of terminating decimal numbers are 0. 5, 0. 75, 4. 65 Examples of repeating decimal numbers 0. 333… which may be written 0. 2323… or and 8. 13456456… or 5. 3 -17 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 5: Terminating Decimal Numbers Show that the following rational numbers can be expressed as terminating decimal numbers. = 0. 6 = 1. 4375 = – 0. 65 5. 3 -18 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 6: Repeating Decimal Numbers Show that the following rational numbers can be expressed as repeating decimal numbers. 5. 3 -19 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Converting Decimal Numbers to Fractions We can convert a terminating or repeating decimal number into a quotient of integers. The explanation of the procedure will refer to the positional values to the right of the decimal point, as illustrated here: 5. 3 -20 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Converting Decimal Numbers to Fractions 5. 3 -21 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 10: Converting a Repeating Decimal Number to a Fraction Convert integers. 5. 3 -22 to a quotient of Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Multiplication of Fractions The product of two fractions is found by multiplying the numerators together and multiplying the denominators together. 5. 3 -23 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 11: Multiplying Fractions Evaluate 5. 3 -24 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Reciprocal The reciprocal of any number is 1 divided by that number. The product of a number and its reciprocal must equal 1. 5. 3 -25 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Division of Fractions To find the quotient of two fractions, multiply the first fraction by the reciprocal of the second fraction. 5. 3 -26 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 12: Dividing Fractions Evaluate 5. 3 -27 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Addition and Subtraction of Fractions To add or subtract two fractions with a common denominator, we add or subtract their numerators and retain the common denominator. 5. 3 -28 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 13: Adding and Subtracting Fractions Evaluate 5. 3 -29 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Fundamental Law of Rational Numbers If a, b, and c are integers, with b ≠ 0, and c ≠ 0, then and 5. 3 -30 are equivalent fractions. Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Adding or Subtracting Fractions with Unlike Denominators When adding or subtracting two fractions with unlike denominators, first rewrite each fraction with a common denominator. Then add or subtract the fractions. 5. 3 -31 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 14: Subtracting Fractions with Unlike Denominators Evaluate 5. 3 -32 Copyright 2013, 2010, 2007, Pearson, Education, Inc.