Neighbourhood of an irrational

Could someone point me towards any proof for whether or not the neighbourhood around an irrational number is rational? Would the rational numbers being a dense subset of the real numbers cause there to be a rational arbitrarily close to any irrational?

Thanks!

Re: Neighbourhood of an irrational

Quote:

Originally Posted by

**svanbeek** Could someone point me towards any proof for whether or not the neighborhood around an irrational number is rational? Would the rational numbers being a dense subset of the real numbers cause there to be a rational arbitrarily close to any irrational?

Neighborhoods are neither rational nor irrational.

Do you mean that any neighborhood of an irrational contains a rational?

The answer to that is yes. The rationals are dense in the reals.