Q: Prove the ratio of integers to rationals is/isn't a 1:1 relationship?
Answer 1: Yes as integers and rationals go on infinitely so the ratio is infinity:infinity which simplifies to 1:1
[socratic question]Can this proof be used to show that the real numbers have a 1:1 relationship with the integers? If so, consider reading this
Now I have a feeling this is wrong, as these are mathsy people who did answer 1, and I'm a secondary school'er.
Your #1 is wrong, so whoever gave it to you probably shouldn't be trusted on this stuff. See, problem is, not all infinities are equal. See, there are infinities and then there are infinities
. The hotel paradox
is a good introduction to the idea that not all infinities are equal. Infinities don't cancel. You have to find other ways to get rid of them, usually involving some form of induction (prove it's true for n, then prove it's true for n+1 assuming it's true for n).
You're on the right track of breaking down the rationals to y/x, but you need to find an explicit formula that maps a given rational number to a unique integer (or natural number if you want to feel smart.) That is, find a way to give each rational number y/x a single integer n that uniquely defines it. Not every integer n needs a rational to go with it, though. It's okay to have holes. Also don't use the term "number" in your proofs, because it's necessarily ambiguous when you're talking about different kinds of numbers. Better just to say something's a rational or an integer.
It's not an easy proof. If you're interested in this stuff, you might pick up a book on Real Analysis. It's a grad level course, and the later stuff gets in to calculus, but the early stuff is all about building the notion of numbers from first principles, assuming as little as possible except a basic knowledge of proofs (proof by induction especially). The one I used for my class had sections about the history of civilizations and their concepts of what was or wasn't a number, which is also quite interesting.