Question

Are there more rational or irrational numbers?

I am thinking irrational, because the product of a rational and irrational number is irrational. Is this enough of a reason to prove it?

Answer 1

There are many many more irrationals than rationals. In fact, there are so many more that if you threw a dart at the number line, the chance of hitting a rational is zero.

The proof is relatively simple, and involves the idea of countability. You can count the rationals, but not the irrationals.