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Since early childhood we are used to work with the decimal numbers. Living in the world, where everything is written is terms of decimals people feel most convenient, when performing the calcuations in this system, wherears when dealing with binary, hex, and any other kind of data we usually at first transform and process them as decimals.

However, for purely mathematical point of view there seems nothing to be special about the decimal system. $$10 = 2 * 5$$ The product of the first and the third prime number. Other base, like the binary, looks more natural, writing everything in terms of sum of powers of the smallest prime number.

However, writing the numbers with this base is not very convenient, due to the fact, that even moderately small numbers, like $73$ would require $7$ digits to be written.

The decimal system in this sense, is a compromise, that one doesn't have to remember a lot of symbols, and the numbers, appearing in most real-life applications can be written rather compactly. From the point of view of humans, the historical reason, why this system became dominant is most likely due to the number of fingers on both hands (for normal people). And the hands were often used for representing small numbers.

My question is - do there exist some other properties, or remarkable features, which distinguish the decimal system?

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    $\begingroup$ We can use fingers for counting $\endgroup$ Commented Dec 18, 2020 at 19:55
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    $\begingroup$ And that's why we use the word "digit" in math and to describe a finger. $\endgroup$ Commented Dec 18, 2020 at 20:06
  • $\begingroup$ Since the beginning of the age of computers it has become obvious that base 8 or base 16 would be better. However we have to live with reality. $\endgroup$ Commented Dec 18, 2020 at 20:14

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If you're a fan of divisibility tests and wanted a good base, you'd want it divisible by lots of small primes. You wouldn't want to waste size by using a prime more than once. But even the product of the first $3$ primes is $30,$ which is a bit large. We can barely get school children to learn the $55$ multiplication facts in base $10.$ (Even though a third of them involve $0$ or $1$.) I'd hate to think of the size of the flashcard deck if we went base $30.$

So let's just pick two of the primes. Use base $6$, $10$ or $15$. In each case, think about how you'd test for divisibility of the missing prime. In base $15,$ divisibility by $2$ is not obvious, and that seems a serious flaw. In base $6,$ if you want to test for divisibility by $5$, you'd have the "sum of digits" trick that works for $9$ in base $10$, so that might not be a bad choice. (And unlike base $10$, you get an easy test for divisibility by $7$.)

Oh, but in base $10$, checking for $3$ gets you $9$ for free, so it wins the "2 primes" contest. Base 10 gives quick divisibility tests for everything up to $12$, except for $7$. Hard to beat.

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  • $\begingroup$ How does $9$ help in the primes contest? $\endgroup$ Commented Dec 18, 2020 at 23:21
  • $\begingroup$ @NoahSchweber It's just a bonus. The more I think about it, base 6 is a stiff competitor. $\endgroup$
    – B. Goddard
    Commented Dec 19, 2020 at 0:17

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