Is it possible to represent every huge number in abbreviated form?

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?

Answer

No. The problem is very simple: there are more huge numbers than abbreviated forms. Suppose that I allow you to use the numerals 0-9, the English alphabet, and spaces to describe whatever numbers you want; that’s still only $37$ characters. There are only $37^{140}$ expressions you can write down using these characters which is $140$ characters or shorter (the length of a tweet!), so there are only $37^{140}$ numbers that you could conceivably tweet to someone else. In particular, since $\log_{10} 37^{140} = 219.5$, this implies that there is at least one number with $220$ digits which is impossible to tweet to someone else without using something other than numerals and English.

All right, I’ll be even more generous: I’ll let you use all $127$ ASCII characters. That means there are $127^{140}$ possible tweets, some of which mathematically describe some kind of number, and since $\log_{10} 127^{140} = 294.5$, there is at least one number with $295$ digits which is impossible to tweet to someone else (again, assuming that Twitter uses ASCII). I could give you the entire Unicode alphabet and you still won’t be able to tweet, for example, the kind of prime numbers that public key cryptosystems use.

These kinds of questions are studied in computer science and information theory; you will find the word Kolmogorov complexity tossed around, as well as the word entropy.

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