> 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 assume you are talking about mathematical numbers rather than finite-precision floating point representations of same. In that case, the first claim (countable number of irrationals can appear as the limit of computations) is false. The second claim (some pairs of rationals have no irrationals between them) is false. Between any two (different) rational numbers there is an uncountable number of irrationals. In particular, for the rational pair 0 and 1%x, there is (for example) the irrational number %x*%:0.5. It is true that between two rationals there is a countable infinity of rationals. The last question (other rational pairs which don't have any irrationals between them) can be answered, there are no such rational pairs. 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
