Actually, I DID phrase that poorly, because of course there's no unique "closest number" even over the rationals.
I'm trying to remember the nature of that proof I read, and it's escaping me. I'm going to _try_, but this hasn't been peer reviewed... But by definition the decimal expansion of an irrational has an infinite number of nonzero digits, while a rational can have a finite number (ignoring repeating decimals, of course). This means every irrational whose expansion starts at a given digit is greater than a single well-defined rational whose expansion simply _is_ that single digit. So the race can go on, but the rationals have won as soon as you've started writing the first nonzero digit of your irrational. That's not a good formal proof :). On Fri, Mar 29, 2019 at 2:54 PM Don Guinn <[email protected]> wrote: > > Just curious. Can you site a reference on this comment: > > Not particularly relevant; only a countable number of irrationals can > appear as the limit of computations. And some pairs of rationals have > no irrationals between them (for example, 0 and the nearest rational > to it, which is of the form 1/x, cannot have any irrationals between > them, even though (1/x and 1/(x-1)) has an countable infinity of > rationals between _them_). > > What are some other rational pairs which don't have any irrationals between > them? > > I can see that given an irrational number close to zero one can generate a > rational that is closer to zero. But then one can find an irrational > smaller than that rational. Continue forever. The method of showing that > there are as many counting numbers as rationals is by setting up a > one-to-one correspondence. But the only reasoning I can think of for no > irrationals to come between 0 and 1/x sounds a lot like saying one cannot > reach the finish line because you have to go half-way to the finish first. > > What the rational (pun intended) behind this statement? > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
