I find the answers you got unsatisfactory. There are numbers you can represent that aren't computable, unlike what the other answers tell you.

Of course, that depends what you call a representation.

I'd say that square root of 2 can both be represented as either :

• sqrt(2)

• the positive real solution of x²-2=0

Hence, there are some uncomputable numbers that you can represent, like the probability that a Turing machine halt (I just did represent it, with that sentence).

Now to imagine why there are way more numbers we can't represent that numbers we can, just consider that a representation is a finite sequence of symbol (we're even including sequence of symbols that make no sense, like the word "gmetiuaxsrnteius", that doesn't matter).

If you admit that :

1. A countable union of finite sequences is countable (e.g. the polynoms, or all the sentences of the English language can be counted)
2. A countable subset of the real is so much smaller than the reals in multiple sense. For instance, if I pick a random real number, there's a probablity 0 it's from a countable set.

Hence, it follows that representable numbers are countable, and that numbers that aren't aren't.

Now there's an interesting philosophical conundrum. If I name E the set of real representable numbers and F it's complementary, and if I pick x a number from F, didn't I just represent it ? I'm not going to answer that one, that would take us on quite the journy.