TITLE: The set of rationals has the same cardinality as the set of integers
QUESTION [6 upvotes]: The set of rationals $\mathbb{Q}$ has the same cardinality as
the set of integers $\mathbb{Z}$. True or false?
This was a question on an old exam for our class. The correct answer is true. However, I did some additional reading and came across Cantor's transfinite numbers. In the book I'm reading, it says that "there are more real numbers (which include rational and irrational numbers) than there are integers". So can it also be said that there are more rational numbers than integers? And so can we say that the above statement is false?

REPLY [3 votes]: Just because the integers are a proper subset of the rationals doesn't mean that the rationals have a higher cardinality than the integers.  Actually, there is a theorem that says that a set is infinite if and only if it has the same cardinality to a proper subset of itself (so your logic would only apply to a finite set).  
A set is countable, or has the same cardinality as the integers, if you can count the elements.  In other words, you can label each element by a unique positive integer.  We can see from the diagonals argument (see this image on Wikipedia for a good illustration) that this holds for that rational numbers.  Once you get the hang of it, you can see that a lot of sets that seem to be a lot bigger than the set of integers are in fact the same "size" (have the same cardinality) as the integers.  See Hilbert's Paradax of the Grand Hotel for a good example of this.