Can you prove that there are more irrational numbers then there are rational.?

Question

I know that there are more irrational numbers then there are rational but how do you prove it.

Answer 1

Cantor's diagonal argument shows that the set of real numbers is uncountable. The set of rational numbers is countable.

If you start with the real numbers, and take away the rational numbers, you are left with the irrational numbers. A uncountable set, minus a countable sub-set, is still uncountable.

Answer 2

Suppose that I have 2 irrational numbers, unrelated (that is that the difference between them is not a rational number). Now, I can add each of those two irrational numbers to EACH rational number in existance, getting twice as many irrational numbers as there are rational ones as a result.
That would work for me as a starting point...

Answer 3

Technically there are more, obviously, but when going into infinty the amount does not really matter. What you want is that for every single rational number, there are an infinite number of irrational using the constant i.

Answer 4

The set of irrationals is bigger than the set of rationals.

Answer 5

Huh?? How do you know that there are more irrational numbers then there are rational?

Answer 6

Actually they are both infinite and one infinite thing can't be more than another.

Similarly, you can't say there are more points on a 5 cm segment than there are on a 2 cm segment, or that there are more rational numbers from 1 to 100 than there are from 1 to 5.

Answer 7

http://www.public.iastate.edu/~joefish/proof.doc


it's weird, but there are as many rational numbers as irrational. there are as many natural numbers as rational numbers. there are as many even numbers as there are odd numbers. there are as many prime numbers as there are natural numbers. there are as many numbers divisible by 2 as there are divisible by pi. freaky stuff, huh?

http://mathforum.org/library/drmath/view/51914.html