Wikipedia:Reference desk/Archives/Mathematics/2020 August 5

= August 5 =

Why we decided to use base 10 instead of base 11, base 6, base 5, base 25, base 30 or base 36?
Why we decided to use base 10 instead of one of the other fingers related bases (base 11, base 6, base 5, base 25, base 30 or base 36)? 2804:7F2:688:3746:DD8D:36F4:453C:DC0F (talk) 16:23, 5 August 2020 (UTC)


 * Two hand, ten fingers.jpg
 * See Decimal. [Makes you wonder what the Babylonians looked like. ;-)  Hmm. I see that article disputes my long held (though not particularly informed) belief that theirs was more of a "mixed-radix system of bases 10 and 6" than a true base-60 system.  I stand corrected.] -- ToE 16:59, 5 August 2020 (UTC)
 * And I see my response doesn't really answer your question as why that particular choice dominated over other finger or finger & toe related bases described in Decimal, included Yuki octal where "speakers count using the spaces between their fingers rather than the fingers themselves". -- ToE 17:08, 5 August 2020 (UTC)
 * According to the article, the difference between the Babylonian system and a mixed radix system is that the Babylonian used a different set of digits for the 10 place. Plus the digits themselves were more like tick-marks than true digits. They also didn't have a digit for 0; I gather they just left a space when one was required but that got confusing so they started using a filler character to make it clear that no digit was in that position. But other than those differences, which to me seem notational rather than conceptual, their system was a mixed radix positional system. By the same token you can think of an abacus as a mixed radix (alternating 5 and 2) system.


 * I think one factor that influenced the choice of 10 was that it lies in a Goldilocks zone of not too big and not too small. The simplest system is base 2, but a binary expression takes three times as much space to write and three times as much time speak as a decimal expression. It's perfect for electronic circuits though. A system with a much larger base (say 1000) would be more compact than base 10 but it would be too hard to memorize. It seems natural to compare this with various phonetic alphabets. We have 26 letters which (somewhat vaguely in English) correspond to different sounds. Other systems like Japanese Kana go syllable by syllable, but I think this would only be possible in a relatively sound-poor language like Japanese, otherwise it would be too hard to memorize. (Pictogram based writing systems like Chinese would seem to contradict this reasoning, but the Chinese spend years trying to master it.) Then there are mixed syllabaries (abugida) which combine consonant and vowel components similar to the way the Babylonians combined 10's and units into a single base 60 digit. The point is that whatever the system, it seems there must be a balance between conciseness and ease of use. A base of about 10 would seem to be the right size to maintain this balance when expressing numbers. --RDBury (talk) 22:25, 5 August 2020 (UTC)
 * Interestingly, the word finger is etymologically related to the word five; both are related to Proto-Indo-European *pénkʷe meaning "five". The German philologist Hans Kuhn has also sought to explain the Dutch word for the fifth finger, pink (whence English pinkie) as related (Hans Kuhn, "Anlautend p- im Germanischen". Zeitschrift für Mundartforschung 28, 1961, 1–31). --Lambiam 06:49, 6 August 2020 (UTC)