With the fractions we have just converted, we saw that each one reached an exact-value equivalent decimal number after some finite number of digits. The world of arithmetic would be very nice if this were true for all fractions, but our next example will show this not to be. We now tackle the fraction 1/3. For the 1/10 position, we look up the 1/3 equivalents, 2/6, 3/9, and 4/12. Doing as before, we find that 3/10 is closest-under 3/9, and we have 3 for the first digit. Subtracting 3/9 – 3/10 gives us 3/90 as remainder for the 1/100 position. 3/100 is closest-under 3/90, and we have 3 for the second digit. Continuing, the remainder after subtracting 3/90 – 3/100 is 3/900. 3/1000 is the closest-under, and the third digit gets another 3. At this point, it is easy to suspect that this process may fall into an endless loop that forever continues to generate 3’s for every successive digit. If you have the patience and curiosity to perform the exercise, you will find that the suspicion is justified.