Fractions with Terminating and Non-Terminating Decimal Representations

Another representation of rational numbers aside from fractions is the decimal form.  Every fraction has a decimal representation:

\frac{1}{2} = 0.5, \frac{1}{5} = 0.2, \frac{2}{3} = 0.666 \cdots and \frac{1}{11} = 0.0909 \cdots.

Notice that some of the fractions above are terminating, while the others are repeating decimals. The fractions \frac{1}{2}  and \frac{1}{5} have only one decimal place, while \frac{2}{3} and \frac{1}{11} have infinitely many (Can you see why?). Now, given a  fraction, can we determine if it’s a terminating or non-terminating decimal without dividing?

Delving Deeper

First let us examine the characteristics of terminating decimals, say 0.125. The easiest way to convert this decimal into fraction is by dividing a whole number by a power of 10: » Read more