How can we predict whether a certain rational number recurs or terminates in a base system other than 10?

Question

So for example how could I know whether 1/6 recurred or terminated in say base 8??? can this be defined in a rule???

Answer 1

Set up an equation B^x = n where B is the base and n is the number.

If the solution is irrational then so is the number and the decimal will recur.

Ex. 1/6 as base 8

8^x = 1/6

(2^3)^x = 6^(-1)

2^(3x) = 6^(-1)

takes log of both sides:

(3x) log 2 = - log 6

3x = - log 6/log2

x = -log 6 / (3 log 2)

severely irrational, decimal recurs