Is binary more ideal than decimal?

Question

We only chose the decimal system because we have 10 fingers. Binary is the most basic positional numbering system, so would it make sense to say that it would be the most ideal system? Is it better than hexadecimal, base-12 base-10 and base-8 since those seem arbitrary? Would it be considered the most "wholesome" positional system?

Answer 1

Mathematically, possibly the most "pure" numbering system is the system consisting of zero and the successor function $S$. In this system, instead of writing the first several non-negative integers as $0, 1,2,3,4,5,6$ and so forth, we write

$$ 0,$$ $$ S(0),$$ $$ S(S(0)),$$ $$ S(S(S(0))),$$ $$ S(S(S(S(0)))),$$ $$ S(S(S(S(S(0))))),$$ and so forth. It's cumbersome, but it is related very closely to the fundamental concepts of arithmetic; at least, it is much closer to the fundamentals than binary is, in my opinion.

As a practical matter, most digital electronic computers use binary arithmetic for most internal calculations. That's because "on" and "off" are relatively simple concepts to get modern electronics to keep track of accurately. There were attempts to use more than two possible voltage levels as "digits" inside computers, which would have allowed computers to do their internal calculations truly in a base higher than two, but these efforts were not successful enough. I think there may still be some computers (or at least electronic hand calculators) that still use binary-coded decimal, which means they're basically working in base ten but some of the details (such as determining that $2 + 3 = 5$) are worked out in binary. The only reason I can see for this is to accommodate the fact that the humans using these computers want to see base-ten numbers going in and base-ten numbers coming out. (The humans may also particularly want to see "rounding" done in a base-ten fashion for some applications.)

As someone who works professionally with computers and occasionally has to deal with binary numeric representations directly, I find that as the numbers get larger in size it becomes very difficult to work directly with the binary representation, simply because the numbers have too many digits to keep track of (and certainly far more digits than I care to have to copy). Octal and hexadecimal are much more convenient. If we ever give up on decimal notation, I think it's much more likely that human society will switch to octal or hexadecimal for most numbers than that we will every make binary our usual numbering system.

Regardless of the numbering system, each number is what it is, and most mathematical properties (other than obvious ones such as number of digits or having the digit $6$) are not affected. For example, a number that is prime when written in base ten is a prime number in any system of writing the integers.