Lesson 13-1: Terminating and Nonterminating Decimals

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.

Property

When a decimal is repeating, convention dictates using bar notation (a vinculum) to indicate the exact block of digits that repeats infinitely. The horizontal line must be drawn strictly over the repeating sequence, starting from the first instance of the repetition.

Examples

• The fraction 7/9 converts to 0.777... Because the digit 7 repeats infinitely, it is written as 0.7 with a bar over the 7.
• The fraction 5/6 converts to 0.8333... The digit 8 does not repeat, but the 3 does, so it is written as 0.83 with a bar over the 3 only.
• The fraction 8/11 converts to 0.727272... The block '72' repeats, so it is written as 0.72 with a bar over both the 7 and the 2.

Explanation

Writing out an endless string of numbers is impossible, so mathematicians use a shortcut. The bar acts as a shorthand command that says "repeat these specific digits forever." It is crucial to place the bar accurately; placing it over too many or too few digits changes the mathematical value of the number completely.