TITLE: Is it correct to say that $\mathbb{R}$ has fewer elements than $\mathbb{C}$ if both are infinite?
QUESTION [8 upvotes]: My math teacher said that. I disagreed, but he said that I was wrong. But I'm not convinced - is it really right? Please notice that I'm not talking about $\mathbb{R}$ $⊂$ $\mathbb{C}$,  but $\mathbb{R}$ $<$ $\mathbb{C}$.

REPLY [2 votes]: There are two senses of size, which I think any mathematician has thought in terms of (here, both strict comparisons):

Subsets: $A \subset B$: $A$ is a smaller set than $B$ in that $B$ contains elements in $A$, and more.
Cardinality: $A \prec B$: for some bijection $f$, $f[A] \subset B$

In this context, both $\mathbb{R} \subset \mathbb{C}$ and $\mathbb{R} \prec \mathbb{C}$.  One way to see the second fact is that $\mathbb{C} = \mathbb{R}\times\mathbb{R}$ as sets and that the product (or sum, or max) of two infinite cardinalities is the maximum of those two values.
Anyways, I think it would be hard to find a mathematician who hasn't thought in terms of 'bigger', 'smaller', and 'size' in both the comparative senses of set membership (1) and set cardinality (2).  The notion of subset is obviously useful, while the notion of cardinality shows up everywhere when you don't care about the properties of the elements of the set, you just care about what you can construct out of a set.  Blind to any properties the elements of a set may have, the only thing left over is cardinality.  
I think the only trouble here is not the mathematics, but semantics and terminology, which may be a point of contention.  For example, I've known a few people who object to any association between 'size' in the intuitive sense and the mathematical definition of cardinality.  I haven't seen this happen within the mathematical community.  Perhaps it's because mathematicians are wired differently or that we often find the exotic objects in the mathematical world not more unnatural than the exotic species in the biological world... and that in fact the mathematical universe wouldn't work right without them.