Section 5.3 The Rational Numbers

What You Will Learn Upon completion of this section, you will be able to: Reduce fractions to lowest terms. Convert mixed numbers to improper fractions and vice versa. Express rational numbers as terminating or repeating decimal numbers. Convert decimal numbers to fractions. Multiply, divide, add and subtract fractions.

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:

Fractions Fractions are numbers such as: The numerator is the number above the fraction line. The denominator is the number below the fraction line.

Reducing Fractions To reduce a fraction to its lowest terms, divide both the numerator and denominator by the greatest common divisor.

Example 1: Reducing a Fraction to Lowest Terms Reduce to lowest terms. Solution GCD of 54 and 90 is 18

Mixed Numbers A mixed number consists of an integer and a fraction. For example, 2 ¾ is a mixed number. 2 ¾ is read “two and three quarters” and means “2 + ¾”.

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 .

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.

Example 2: Converting Mixed Numbers to Improper Fractions Convert the following mixed numbers to improper fractions.

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.

Example 3: From Improper Fraction to Mixed Number Convert the following improper fraction to a mixed number. Solution

Example 3: From Improper Fraction to Mixed Number Solution The mixed number is

Example 3: From Improper Fraction to Mixed Number Convert the following improper fraction to a mixed number. Solution

Example 3: From Improper Fraction to Mixed Number Solution The mixed number is

Terminating or Repeating Decimal Numbers Every rational number when expressed as a decimal number will be either a terminating or a repeating decimal number.

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

Example 5: Terminating Decimal Numbers Show that the following rational numbers can be expressed as terminating decimal numbers. = 0.8 = 2.425 = –0.84

Example 6: Repeating Decimal Numbers Show that the following rational numbers can be expressed as repeating decimal numbers.

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 on the next slide.

Converting Decimal Numbers to Fractions

Example 10: Converting a Repeating Decimal Number to a Fraction Convert to a quotient of integers.

Multiplication of Fractions The product of two fractions is found by multiplying the numerators together and multiplying the denominators together.

Example 11: Multiplying Fractions Evaluate

Reciprocal The reciprocal of any number is 1 divided by that number. The product of a number and its reciprocal must equal 1.

Division of Fractions To find the quotient of two fractions, multiply the first fraction by the reciprocal of the second fraction.

Example 12: Dividing Fractions Evaluate

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.

Example 13: Adding and Subtracting Fractions Evaluate

Fundamental Law of Rational Numbers If a, b, and c are integers, with b ≠ 0, and c ≠ 0, then and are equivalent fractions.

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.

Example 14: Subtracting Fractions with Unlike Denominators Evaluate