Lesson 1: Rational Numbers and Decimal Expansions

Property

A rational number is one that can be expressed as a quotient (or ratio) of two integers, where the denominator is not zero. The decimal representation of a rational number has one of two forms.

1. The decimal representation terminates, or ends.

2. The decimal representation repeats a pattern.

Property

A repeating decimal is a decimal in which the last digit or group of digits repeats endlessly. A bar is placed over the repeating digit or digits.

Examples

• To convert $\frac{2}{3}$ to a decimal, dividing 2 by 3 gives $0.666...$. The digit 6 repeats endlessly, so we write it as $0.\overline{6}$.

• For the fraction $\frac{5}{11}$, dividing 5 by 11 gives $0.454545...$. The block of digits 45 repeats, so we write it as $0.\overline{45}$.

Property

To convert a rational number a/b (where a and b are integers) to a decimal, divide the numerator by the denominator using long division. The resulting decimal will either be a terminating decimal (if the division eventually results in a remainder of 0) or a repeating decimal (if a non-zero remainder repeats, creating an infinitely repeating sequence of digits in the quotient).

Examples

• Terminating: To convert 5/8 to a decimal, calculate 5 divided by 8. The division ends with a remainder of 0, resulting in the terminating decimal 0.625.
• Repeating: To convert 2/11 to a decimal, calculate 2 divided by 11. The remainders 2 and 9 alternate endlessly, resulting in the repeating decimal 0.181818...

Explanation

Think of a fraction as a division problem waiting to be solved. When you perform long division, you are looking at the leftovers (remainders). If you eventually have no leftovers, the decimal stops cleanly. If you start seeing the same leftovers over and over, you are caught in a mathematical loop, which means your decimal will repeat that pattern forever.