Rational Numbers

A rational number is a number that can be written as a ratio of two integers. The decimal of a rational number will either repeat or terminate. There is a way to tell in advance whether a rational number’s decimal representation will repeat or terminate. When trying to find a pattern in the relationship between rational numbers and their decimals, it is best to start with a list. A random list of rational numbers and their decimal values was made in order to find a pattern. The list included ½, 5/6, 44/10, 3/7, 23/36, 89/53, 3/50, and 4/31. These ratios were 0.5, 0.8333…, 4.4, 0.42857…, 0.63888…, 1.67924…, 0.06, and 0.12903… in decimal form. When analyzing these rational numbers, they were placed into categories of repeating and terminating decimals. In the terminating decimal category, the numbers ½, 44/10, and 3/50 had something in common. It was determined that each of the denominators were multiples of two. To solidly identify a pattern, more examples were used. Rational numbers such as 2/25, 6/8, and 9/64 helped to identify a definite pattern. What all the denominators of terminating decimals shared was the fact that their prime factors included only 2’s and 5’s. When 2, 10, 50, 25, 8, and 64 were prime factorized the only prime factors were 2’s and 5’s. When the denominators of the repeating decimals were prime factorized, there were other prime numbers although some 2’s and 5’s were still included. These observations helped lead to the conclusion that there is a way to tell in advance whether a rational number’s decimal representation will be repeating or terminating. If the denominator’s prime factors are only 2’s and 5’s, then the decimal will terminate. If not, it will be a repeating