Trivial/pointless math puzzle I'm too stupid to solve

[Michael12345]

Pretend you own a printing press and you want to be able to represent any arbitrarily large natural number. You also want to store the fewest possible number of characters in your collection, just to save space. What base, if any, would yield the largest ratio of numbers you can represent to numbers of characters you have to store?

I can't figure it out. Please mock the question and don't forget to tell me all the ways I have missed the point or shamed my country's public education system. I've trained myself to turn your abuse into cuts and cigarette burns on my skin and holes in my landlord's walls.

 

[Joffan]

This is just using place notation right? No sneaky alternative depictions of numbers.

Optimizing for the number of characters, you want to divide up your storeroom into n (different digits, the base) x m (copies of each digit) in order to maximize n^m (approximately). Or equivalently maximize m log n ;)

Feel free to try this out for smallish numbers. Assume you can store 20 characters or something like that.