Ah, that clears up the mystery. It makes sense as far as it goes; if it's completely indivisible by the base we should expect to always have a remainder, and with an infinite series of numbers a certain percentage repeat. That explains things like thirds (and, by extension, sixths and ninths) but not the peculiar case of requiring no less than six digits to express the reciprocal of 7. Nor Wikipedias startling (even to me) assertion that 142857 is the ONLY decimal cyclic number unless one allows leading zeroes (and I see no reason why one should. ) It doesn't help matters much that the square of 22 is 484 while the square of 7 is 49, making that rational approximation of irrational pi very close to the square root of 10. In fact, we might say that 10=((10+10/10)^2+7-10/10^2)/7^2. *shoots self*