1. Section 5.3 The Rational Numbers

2. What You Will Learn Rational Numbers
Multiplying and Dividing Fractions
Adding and Subtracting Fractions

3. 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:

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

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

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

7. 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”.

8. 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

9. 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.

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

11. 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.

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

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

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

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

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

17. 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 … or and … or

18. Example 5: Terminating Decimal Numbers
Show that the following rational numbers can be expressed as terminating decimal numbers.
= 0.6 =
= –0.65

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

20. 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:

21. Converting Decimal Numbers to Fractions

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

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

24. Example 11: Multiplying Fractions
Evaluate

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

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

27. Example 12: Dividing Fractions
Evaluate

28. 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.

29. Example 13: Adding and Subtracting Fractions
Evaluate

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

31. 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.

32. Example 14: Subtracting Fractions with Unlike Denominators
Evaluate