1. Thinking Mathematically
The Rational Numbers

2. The Rational Numbers
The set of rational numbers is the set of all numbers which can be expressed in the form a/b, where a and b are integers and b is not equal to 0. The integer a is called the numerator and the integer b is called the denominator.
Note that every integer is a rational number. For example 17 can be written 17/1.

3. The Fundamental Principle of Rational Numbers
If is a rational number and c is any number other than 0,
The rational numbers and are called
equivalent fractions

4. Lowest Terms
A rational number can be reduced to its lowest terms by dividing the numerator and denominator by the greatest common divisor.
Example: Reduce 24/40 to lowest terms.
The greatest common divisor of 24 and 40 is 2 x 2 x 2 = 8.
(24/8) =
(40/8) 40 24 5 = 3

5. Converting a Positive Mixed Number to an Improper Fraction
Multiply the denominator of the rational number by the integer and add the numerator to this product.
Place the sum in step 1 over the denominator in the mixed number.

6. Converting a Positive Improper Fraction to a Mixed Number
Divide the denominator into the numerator. Record the quotient and the remainder.
Write the mixed number using the following form:

7. Rational Numbers and Decimals
Any rational number can be expressed as a decimal. The resulting decimal will either terminate, or it will have a digit that repeats or a block of digits that repeat.
Examples: 5 3
= = .60
The “line” over the zero indicates that it is to be repeated infinitely often. 3 1
= = .3

8. Expressing a Repeating Decimal as a Quotient of Integers
Step 1 Let n equal the repeating decimal.
Step 2 Multiply both sides of the equation in step 1 by 10 if one digit repeats, by 100 if two digits repeat, by 1000 if three digits repeat, and so on.
Step 3 Subtract the equation in step 1 from the equation in step 2.
Step 4 Divide both sides of the equation in step 3 by an appropriate number and solve for n.

9. Multiplying Rational Numbers
The product of two rational numbers is the product of their numerators divided by the product of their denominators.
Example: 5 3 x 7 4 =
(5 x 7)
(3 x 4) 35 12

10. Dividing Rational Numbers
The quotient of two rational numbers is the product of the first number and the reciprocal of the second number.
If a/b and c/d are rational numbers, and c/d is not 0, then

11. Adding and Subtracting Rational Numbers with Identical Denominators
The sum or difference of two rational numbers with identical denominators is the sum or difference of their numerators over the common denominator. If a/b and c/d are rational numbers, then

12. Density of Rational Numbers
If r and t represent rational numbers, with r<t, then there is a rational number s such that s is between r and t.
r < s < t.

13. Thinking Mathematically
The Rational Numbers