Coming to grips with decimal expansions

Writing a fraction as 31/12 does not always give a clear sense of its value. To get a clearer picture, we can use its decimal expansion, found by dividing the numerator by the denominator.

In our example, this gives:

From a certain point onward, the remainder is always 4, so the digit 3 appears in every new decimal place of the quotient. Thus, the fraction 31/12 has the infinite decimal expansion 2.5833333…, in which the digit 3 repeats indefinitely. More rigorously, we denote this expansion by $2,58\bar{3}2,58\backslash overline\{3\}$ to indicate the periodic repetition of the digit 3. In other words, we have $31\mathrm{/}12=2,58\bar{3}\mathrm{.}31/12\; =\; 2,58\backslash overline\{3\}.$ Now try another fraction, 75/34. Dividing 75 by 34 begins with the quotient

2.2058823529411764705882352941176470588235294117647…