Can the Set of Rational Numbers Be Counted? And the Irrational Numbers?

[dabdobber]

I need help with this math problem:

Show that the set of rational numbers, Q, is countable.

and

Show that the set of irrational numbers is uncountable.

 

[micromass]

What do you know about cardinality already? Are there any theorems you've seen that might be useful?
What did you try already??

 

[dabdobber]

well as far as the first one goes I know that the set of integers Z is a subset of Q and that Z is countable. now I make B as the set of all non-integral rational numbers.

if B is countable, then Q is countable because a union of two countable sets is countable.
that's my approach, but I'm not much of an example person.

and for the second one I'm thinking that I can show the union of the reals IR with the irrational, thus making them uncountable as well?

 

[lanedance]

1) try and set up a 1:1 map between Q and Z
2) show that no such map exists for R-Q