I considered deleting the post, but this seems more cowardly than just admitting I was wrong. But TIL something!
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I don't see what you are trying to say. You can also match 200 $1 bills with each $100 bill. The correspondence does not need to be one-to-one.
You're right that we don't need to, but mathematicians can use this method to prove that two infinite sets are the same size. This is how we know that the infinite set of whole numbers is the same size as the infinite set of integers. We can also prove that the set of real numbers is larger than the set of whole numbers.
I'm not quite sure how else to explain it, so I'll link a Numberphile video where they do the demonstration on paper: https://www.youtube.com/watch?v=elvOZm0d4H0&t=19s . Here you can see why it's useful to try to establish this 1-1 correspondence. If you can't do so, then the size of the two infinite sets are not equal.
The video shows that rational numbers (aka fractions) are countable (or listable). Did you mean real numbers?
Good catch, I'll edit that sentence