Jan 19, 2021

There is no free lunch when you want to find an abstract system that will be capable of compressing all possible expressions, but there are free lunches in the practical day to day world where you do not need a 'perfect' system. Seems reasonable and yet, even that sensible statement is wrong. If you assume that unary notation (marks for each object to be counted) is the basis of arithmetic notation, then the introduction of an n-ary (or base n) system (n>1), like binary, proves that there is a way to effectively compress all possible numbers.

I should stop there and let it sink in. I mean, is there something wrong with that opinion? Did I make an obvious mistake?

SSD chips are actually analog systems (simple memory systems) that emulate digital characteristics. But then again so are all digital systems, (at least as far as I know). Is that wrong? So, assuming that this is correct so far, the question then becomes: is it possible for analog systems to emulate n-ary systems or combinations of n-ary systems in a way that could further compress and efficiently use any possible number? The one obvious mistake I made in that last statement is that no system is able to compress any possible number because all systems have some kind of finite range (or ranges) and therefore bounded in some way. And analog systems are analog relative to the powers of human discernment.

But, I believe the fundamental proposition is true: the n-ary system (where n>1) is an abstract compression of any number relative to the most natural expression of numbers which is unary. And computers are analog systems that emulate digital characteristics. Therefore, would analog systems that could efficiently emulate a range of n-ary (or base n) systems be able to obtain a greater range of compression methods than binary digital emulators?

The question for me is: Can combinations of n-ary system be used in efficient dynamic computations that can be used to find more efficient representations of particular numbers and computational methods? I guess I will work on that problem a little. I am pretty sure I could find effective ways to use combinations of binary and trinary representations in efficient computations.