Section 1
Terminating vs. Repeating 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.