Many entry-level real analysis courses will cover cardinality after introducing sets and functions. There's also a short article on Wikipedia. [0]

Your intuition about there being more rational numbers might be based on viewing the rationals as a proper superset of the natural numbers. You might similarly consider that there are more natural numbers than even natural numbers. However, by "renaming" every even number, and that shouldn't change how many there are, to half its value, we obtain the natural numbers. Formally, there exists a bijection between N and 2N, as between N and Q, and this is what mathematicians mean when they say that sets have the same size, or cardinality.

[0]: https://en.wikipedia.org/wiki/Cardinality

To add, for hutzlibu and others like them: as soon as infinite sets (or sequences. See https://plus.maths.org/content/when-things-get-weird-infinit...) are involved math gets counter-intuitive.

In this case, at most one of these two a priori quite reasonable statements can be true:

- if set B is a strict superset of A, B is larger than A.

- if you can map the times in A to the items in B in 1:1 fashion, A and B have the same size.

Giving up the first is deemed less problematic than giving up the second (probably because that means giving up comparing sizes of infinite sets at all, but I’m not familiar enough with that to be sure about that)

  for a in range(1, Infinity):
      for b in range(a):
          print(a-b, "/" , b+1)

Will print out:

      1/1, 2/1, 1/2, 3/1, 2/2, 1/3, 4/1, 3/2, 2/3, 1/4, ...

Hence, we've just mapped all rationals (with duplicates) to a single linear list. Since any linear list can always be mapped to the naturals (1, 2, 3, ...), they have the same cardinality.

QED

The proof is when you can create a bijective function between the sets then they have the same cardinality. The “zip” function can do this between integers and rationals.

You can count rational numbers and everything countable is the size of infinity as natural numbers.

Google ”counting rational numbers” and ”different sizes of infinity” to learn more.