1
$\begingroup$

I know that there is a rational number arbitrarily close to an irrational, due to the density of real number. But what about an irrational number? Thanks!

$\endgroup$
2
$\begingroup$

Yes, consider $\alpha + \frac{1}{n}$ where $\alpha$ is irrational and $n$ is an integer. $\alpha + \frac{1}{n}$ is also irrational and can be made arbitrarily close to $\alpha$ by choosing $n$ to be sufficiently large.

$\endgroup$
  • $\begingroup$ Thank you. Stupid question but can one define something like the number closest to an irrational? For example, the question "which number is closest to a given irrational number is rational or irrational?" makes no sense right? just making sure. $\endgroup$ – ANANDA PADHMANABHAN S Jan 21 '16 at 5:25
  • $\begingroup$ Irrationals can differ in how well they can be approximated by rationals, check this out: mathworld.wolfram.com/IrrationalityMeasure.html $\endgroup$ – Dan Brumleve Jan 21 '16 at 5:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.