3 ms·
> You sort of can't get arbitrarily close. Is there an obvious concrete example? I've never spent proper time studying Real Analysis, and I confess I have the
by jpmattia 7y ago
> You sort of can't get arbitrarily close.
Is there an obvious concrete example? I've never spent proper time studying Real Analysis, and I confess I have the same intuition as OP: That you could use rationals to approximate irrationals to arbitrary precision.
- throw_away_1979 7y agoNo, because that's not true, see e.g. https://en.wikipedia.org/wiki/Dirichlet%27s_approximation_theorem https://en.wikipedia.org/wiki/Dirichlet%27s_approximation_th... TL;DR: Every real number can be approximated by rationals. GP was being careless with language.
- SamReidHughes 7y agoThe operative words there are "sort of". You can in fact get arbitrarily close. Just as you can with other sets of numbers, like those the form m/10^n (decimal approximations), or the numbers tan(n) where n is an integer.
- gianduja 7y agoYou're right, you can. What's special about real numbers is that, if you have a sequence of reals for which the distance between two consecutive elements approaches zero, then there exists a real number that's the limit of the starting sequence. This isn't the case with rational numbers, eg. the sequence 1.4, 1.41, 1.414 ... (EDIT: these are increasing approximations of sqrt(2)) satisfies the hypothesis but there's no single rational number this sequence approaches. This property is called completeness. The real numbers are a complete topological space, whereas the rationals aren't.