It’s larger than the rational numbers in the sense that it is a strict superset. Cardinality is what a lot of people reach for when they are talking about “larger” or “smaller”, but there are lots of other useful concepts which we can translate to “larger” and “smaller”.
So when someone says “larger” or “smaller”, your first step might be to try and translate that relationship into a more precise mathematical concept, like cardinality or measure.
Casual terminology also leads to weird discussions. Like when someone asks whether some function is “close” to another, and these functions are defined in terms of vector spaces. Unfortunately, “closeness” does not necessarily exist in a vector space. So the answer may be that the question does not make sense.
> “closeness” does not necessarily exist in a vector space.
The asker will give a definition. For example, two vectors are close if sqrt of dot product of difference of the two vectors is smaller than some number delta.
There is a 1-to-1 mapping between the sets, so by definition that means they have the same cardinality.
Let me give a more common example. Consider these two sets: N, the set of non-negative integers, and Z, the set of all integers. Clearly, everything in N is also in Z, and then some. But N and Z still have the same cardinality, because there are 1-to-1 mappings between the two sets. Here is one example of such a mapping:
..etc. The formula for this mapping would be floor(n/2)*(-1)^(n%2). Clearly everything in the left set has exactly one corresponding item in the right set and vice versa, so they must be the same "size", even though the right set contains every item in the left set and then some.
So when someone says “larger” or “smaller”, your first step might be to try and translate that relationship into a more precise mathematical concept, like cardinality or measure.
Casual terminology also leads to weird discussions. Like when someone asks whether some function is “close” to another, and these functions are defined in terms of vector spaces. Unfortunately, “closeness” does not necessarily exist in a vector space. So the answer may be that the question does not make sense.