Section 2.3 Rational Numbers A rational number is a number that may be written in the form a b for any integer a and any nonzero integer b. Why is division by zero undefined? For example, we know that 3 4 = 12 and so 12 = 3. We get one and only one 4 number for this quotient. We also know that any number, n 0= 0 so 0 0 = n. But remember, n was any number, not a unique one! This is one example as to why division by 0 is undefined! The rational numbers are ratios of whole numbers. The formal symbol for the rational numbers is. Integers are may also be called rational numbers since any integer n may n be written as = n. 1 Note: a a a = = b b b Reducing Fractions To reduce a fraction, we must find its greatest common factor, gcf, and divide it out. The resulting fraction is in simplest form (lowest terms). If finding the gcf is too much work, start by dividing out a common factor (you may use divisibility rules here), the fraction will become simpler. Continue in this manner until you see that its gcf is 1. Example 1: Reduce 240 to lowest terms. 360 Section 2.3 Rational Numbers 1
Mixed Numbers and Improper Fractions The form b a c is called a mixed numeral. The form a b where a > b is called an improper fraction. We can convert from a mixed numeral to an improper fraction and vice versa. Mixed numeral to an improper fraction: b a = ac + b c c Improper fraction to a mixed numeral: simply divide Example 2: Convert the following mixed numbers to improper fractions. a. 9 3 b. 10 5 1 7 Example 3: Convert the following improper fractions to mixed numbers. a. 4 9 b. 135 4 Section 2.3 Rational Numbers 2
Every rational number when expressed as a decimal number will be either a terminating or a repeating decimal number. Example of terminating decimal is 3 = 0.12 since upon dividing, the division ended. 25 Example of repeating decimal is 1 = 0.3333 = 0.3 since upon dividing, the division 3 did not end and it repeated. Converting Decimal Numbers to Fractions First recall that 1 0.1 10 =, 1 0.01 100 =, 1 0.001 100 =, etc. Example 4: Convert the following terminating decimal numbers to a quotient of integers. a. 0.6 b. 0.0622 c. 1.02 Section 2.3 Rational Numbers 3
Converting Repeating Decimal Numbers to a Fractions Example 5: Convert 0.35 to a quotient of integers. Example 6: Convert 12.142 to a quotient of integers. Section 2.3 Rational Numbers 4
Addition and Subtraction of Fractions In order to add or subtract fractions, we first need a common denominator. Once we write each fraction using the same denominator, simply add or subtract the numerators and keep the same denominator. Reduce the answer if possible. If any of the numbers are mixed numerals, first convert them to an improper fraction then add/subtract as described above. Example 7: Perform the indicated operation. 8 3 a. + 2 13 13 b. 5 3 6 8 c. 2 1 1 2 4 5 Section 2.3 Rational Numbers 5
Multiplication of Fractions In order to multiply fractions, first make sure the fractions are in simplest form. Then multiply the numerators together and multiply the denominators together. If any of the numbers are mixed numerals, first convert them to an improper fraction then multiply as described above. Division of Fractions The reciprocal of a fraction, a b is the fraction b a. In order to divide fractions, change the division to multiplication and find the reciprocal of the second fraction (the one that is dividing). Then follow the rules for multiplying fractions. Example 8: Evaluate the following. a. 3 22 2 15 b. 2 3 1 3 7 c. 7 2 22 2 18 Section 2.3 Rational Numbers 6