Not all fractions lead to simple decimal representations. For example, consider the fraction 31. To compute it, we seek groups of three in the following picture.
We see three groups of 3 leaving one behind.
Unexploding gives another ten dots to examine.
We find another three groups of 3 leaving one behind.
And so on. We are caught in an infinitely repeating cycle.
This puts us in a philosophically interesting position. As human beings we cannot conduct this, or any, activity for an infinite amount of time. But it seems very tempting to write 31=0.33333... with the ellipsis representing the instruction “keep going with this pattern forever.” In our minds we can almost imagine what this means. But as a practical human being it is beyond our abilities: one cannot actually write down those infinitely many 3s represented by the ellipses.
Nonetheless, many people choose not to contemplate what an infinite statement like this means and just carry on to say that some decimals are infinitely long and not be worried by it. In which case, the fraction 31 is one of those fractions whose decimal expansion goes on forever.
Notation: Many people make use of a vinculum (a horizontal bar) to represent infinitely long repeating decimals. For example, 0.3 means “repeat the 3 forever”
and 0.38142 means “repeat the group 142 forever” after the beginning “38” hiccup:
As another (complicated) example, here is the work that converts the fraction 76 to an infinitely long repeating decimal. Make sure to understand the steps one line to the next.
Do you see, with this 6 in the final right-most box that we have returned to the very beginning of the problem? This means that we shall simply repeat the work we have done and obtain the same sequence 857142 of answers, and then again, and then again. We have