Number of Digits in Base k Expansion of Fractions

base

10

numerator

4

maximum denominator

5

Some fractions are terminating decimals, like

1/8=0.125

; some are periodic, with repeating digits in cycles, like

1/7=0.142857

; and some are eventually periodic, like

5/6=0.83333

. The plot shows the number of digits in the base

k

expansion of fractions with denominators up to and including

q

, with the numerator varying from 1 to

q

.