All the fractions we've looked at so far were terminating decimals, and their denominators were all powers of 2 and/or 5. The fractions in this section have other factors in their denominators, and as a result they will not have terminating decimal representations.
As you can see in the division problem below, the decimal expansion of 1/3 does not fit the pattern we've observed so far in this session:
Since the remainder of this division problem is never 0, this decimal does not end, and the digit 3 repeats infinitely. For decimals of this type, we can examine the period of the decimal, or the number of digits that appear before the digit string begins repeating itself. In the decimal expansion of 1/3, only the digit 3 repeats, and so the period is one.
To indicate that 3 is a repeating digit, we write a bar over it, like this:
The fraction 1/7 converts to 0.142857142857.... In this case, the repeating part is 142857, and its period is six. We write it like this:
The repetend is the digit or group of digits that repeats infinitely in a repeating decimal. For example, in the repeating decimal 0.3333..., the repetend is 3 and, as we've just seen, the period is one; in 0.142857142857..., the repetend is 142857, and the period is six.