system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: If our current numbering system is more efficient than Roman numerals, is there another system better than the one we have now? Have we reached the best efficiency?
gpt
In order to answer this, it's worth reviewing *why* the decimal system is better than roman numerals:

- In principle, it can handle numbers of unlimited size, and it does so without adding extra symbols.
- The magnitude of the number is proportional to its length. All numbers of similar magnitude use the same number of digits.
- It can handle fractional numbers reasonably well.
- There is always one unambiguous way to write a finite decimal number.
- It generally requires fewer symbols than roman numerals.
- It's easy to do ~~arithmetic~~ addition, subtraction, multiplication, and comparison on decimal numbers.

That said, it has some limitations:

- Very large numbers can be unwieldy. Extremely large numbers are impossible (the universe contains a finite amount of paper).
- Most fractions can't be exactly represented, even common fractions like one-third.
- Division is inconvenient.
- Converting between fractions and decimals (or decimal approximations) generally requires doing division.

There are other notation systems that perform better in some circumstances. And we use them!

Fractions:

- Can exactly represent any rational number.
- Are easy to multiply and divide.
- Aren't quite as convenient to add, subtract, or compare (you have to convert them to a common base).
- Are not unique; two different fractions can represent the same number.

Exponential notation:

- Can represent much larger or smaller numbers than ordinary decimal notation.
- …but only if you're willing to accept a bit of imprecision.
- Easy to multiply, divide, and compare.

There's binary notation, which maps each digit onto a single bit. There's hexadecimal notation, which is more compact than binary. There are various arrow-based notations for representing extremely large numbers. There are esoteric mathematical notations like continued fraction expansions and p-adic numbers.

We have no shortage of notations, and we do make up new ones when necessary.

EDIT: Clarified that division is not easy with decimals (though it's easy with fractions).