# Is there an irrational number arbitrarily close to another irrational number?

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!

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.

• 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. – ANANDA PADHMANABHAN S Jan 21 '16 at 5:25
• Irrationals can differ in how well they can be approximated by rationals, check this out: mathworld.wolfram.com/IrrationalityMeasure.html – Dan Brumleve Jan 21 '16 at 5:29