TITLE: Is it possible to represent every huge number in abbreviated form?
QUESTION [88 upvotes]: Consider the following expression. 
$1631310734315390891207403279946696528907777175176794464896666909137684785971138$
$2649033004075188224$
This is a $98$ decimal digit number. 
This can be represented as $424^{37}$ which has just 5 digits.
or consider this number:
$1690735149233357049107817709433863585132662626950821430178417760728661153414605$
$2484795771275896190661372675631981127803129649521785142469703500691286617000071$
$8058938908895318046488014239482587502405094704563355293891175819575253800433524$
$5277559791129790156439596789351751130805731546751249418933225268643093524912185$
$5914917866181252548011072665616976069886958296149475308550144566145651839224313$
$3318400757678300223742779393224526956540729201436933362390428757552466287676706$
$382965998179063631507434507871764226500558776264$
This $200$ decimal digit number can be simply expressed as $\log_e 56$ when we discard first $6$ numbers and then consider first $200$ digits.
Now the question is, is it possible to represent any and every huge random number using very few characters as possible, theoretically.
...Also, is there any standard way to reduce it mathematically?

REPLY [5 votes]: As in other answers, at the very least a pigeon-hole principle shows that "most" numbers cannot be "described" in fewer characters than their decimal (or whatever-you-pick) expression... 
To my mind, the relevant developed body of ideas is "Solomonov-Chaitin-Kolmogorov" complexity, which is about descriptional (or program-length) complexity, rather than run-time "complexity".
This does remind one of the "smallest boring number" pseudo-paradox, which argues that the first non-interesting number has some interest because it is the first...
The bottom line is that "size" of numbers is not reliably comparable to "descriptional complexity", in general, although most large-ish numbers are also descriptionally complex, by pigeon-hole.
There is a book by Li and Vitanyi which is not only authoritative, but fascinatingly readable...