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"The digits of every rational number repeat after some finite number of digits, so the "period" of every rational number is finite. However, there is no upper
by ttiitg 13y ago
"The digits of every rational number repeat after some finite number
of digits, so the "period" of every rational number is finite.
However, there is no upper bound on the period of rational numbers,
i.e., the periods are all finite, but there is no largest period.
Thus, in a manner of speaking, the least common multiple of this set
of strictly finite things is infinite."
Got lost here, what is the LCM of this set; which set?
- wetmore 13y agoThe set of lengths of periods of decimal expansions. It's an infinite set of finite numbers, with no bound.
- thaumasiotes 13y agothis set (the set of periods of decimal expansions of rationals) seems to be oddly defined to no purpose. Consider the set of positive integers. It has the same property described (in fact, it has all the same properties, since it's the same set). Why go to all the trouble of defining Z+ as "the set of periods of decimal expansions of rationals", which is harder to parse?