Show that the decimal expansion of a rational number must repeat from some point on. [In other words, if the fractional part of the number is 0.a1a2a3 ... , then an+k = an for some k > 0 and all n > some n0.]
Solution
We wish to find a/b. We may assume a < b. Carry out the traditional longhand calculation to find ai: 10a = a1b + r1 where 0≤r1<b, 10r1 = a2b + r2 where 0≤r2<b, ... ,10rn = an+1b + rn+1, where 0≤rn+1<b. If any ri = 0, then the calculation terminates and only finitely many ai are non-zero (so ai becomes periodic with period 1). If not, there are only b-1 possible values for ri. So we must get a repeat in the first b values, but ri completely determines all aj for j > i, so the sequence aj must become periodic with period ≤ b-1.
© John Scholes
jscholes@kalva.demon.co.uk
29 Oct 2003
Last corrected/updated 29 Oct 03