The title speaks for itself. I guess the question should be narrowed down to irrational numbers, since the rest are all rationals. I'm leaning towards no, although that's just intuition speaking. Note that I'm not asking if we can find a closed form expression for a given number; just if it always exists.

Depending on how you define 'closed form', transcendental numbers do not have a closed form.

Depending on how you define 'closed form', transcendental numbers do not have a closed form.

That's not very depending .

Yes it is, some conventions allow 'closed form' to include all kinds of messy expression.

I was making fun of the syntax of your statement ("do not" instead of "may not"), not its content. Might still be correct English -I'm just a stupid non-native speaker- but I sure found it funny.

 

Perhaps as something more constructive (humorless mob!): For any representation with a finite number of symbols, say ASCII, the answer is probably "no".

Oh I see, you're correct in that case.

Note that I'm not asking if we can find a closed form expression for a given number; just if it always exists.

No. There are non-computable numbers. In fact, almost all real numbers are not computable.