well, title says it pretty much … were there real life usages of systems different then 10 in history ?
how did e.g. the dozen come to happen?
Binary and hexadecimal has made great inroads in popularity in the computer age.
The best-known is probably the Mayan base-20 system. It’s actually a little more complicated than that: It’s also sort of base 5, and partly base 18, but it’s definitely not base 10.
Another famous case is that of Sumerians and Babylonians use of base-60, from which we still retain 60 s to the minute, 60 min to the hour, and probably 24 hours to the day (base sixty has lots of sub-bases, including 12).
And don’t forget clams
Popular wisdom says it’s a feature of being able to divide into 2,3,4,6 portions.
Note the British old money system - 12 pence to the shilling, 20 shillings to the pound (so 240 pence per pound). So a pound could be divided by 2,3,4,5,6,8,10,12,15,16,20, etc… Add in ha’penny coinage could be divided even further.
In the days before everyone was conversant with long division and decimal fractions, or had a calculator app on their phone, “one for you, one for me, one for you, one for me…” was a common method and so divisibility by assorted numbers was an asset.
Do Roman numerals count?
(No, the pun was not inTENded!)
Ancient Sumerians counted on their phalanx, with the thumb as pointer, so 4x3 = 12 (“une douzaine”) as base. And 12x12 =144 ( which is still called "une grosse " in France.
Dozen is from old French doze which is from Latin duodecum (duo ‘two’ plus decum ‘10’)
Lots of languages use a mix of base ten and base 20. In northern Ghana, Waali has unique words for 1 to 10, then 11 to 19 are formed from them. Twenty has its own unique word. Higher numbers are created by mixing 10’s and 20’s. So fifty three would be 20 by 2 and 10 and 3. (lejar-ayi aneng pienataa, if anyone is interested.)
FIfity by itself is a funny exception, as it’s rendered as double twenty and five (Phong lejen-anu)
French is kinda the same. 80 is “4 20s”, and 90 is “4 20 10”.
And both dozen and gross have the same meaning in English.
What I know about the old Babylonian base 60 was that it was really a combination of bases 6 and 10. I no longer recall the details, but we talked about in a history of math course I taught a few times.
Now we use every day a system in which the first place uses a variable base that can be any number from 28 to 31, depending on the value of the higher places.
And can be divided even farther with farthings…
And let’s not forget about quarter farthings.
Quarter farthing - Wikipedia
Then there’s guineas: the currency in which a gentleman is supposed to pay his tailor (but unfortunately, often does not)…
Wasn’t there something about base 3 being inherently more efficient because e (2.718281828…) is closer to 3 than to 2?
Are there any number systems or calculation processes that use a non-integer base?
All kinds of weird bases are/have been used in various languages (6, 8, 12, 20, 32, 60, …) but over 60% are decimal, and especially anything other than 10 or 20 is not really “popular”. Map:
https://wals.info/feature/131A
You can do that, and it might be useful for some practical applications, but not so much for counting sheep.
E.g.,
Yes, see Wikipedia on radix economy. It has to do with how efficiently numbers can be stored. Among real numbers, e (Euler’s number) has the lowest (hence best) average radix economy, equal to Euler’s number itself (about 2.72). But non-integer bases are not practical in digital computing. Among integers, ternary (base 3) is the best by this measure, with an average radix economy of about 2.73, closely followed by binary (base 2) and quaternary (base 4) both at about 2.89. Bases higher than 3 have progressively worse average radix economy. In general the average radix economy of base n is n/ln (n).
Ternary computers have been built but all things considered, binary is more practical.
Not all French though…
Bah, this idea of using the same base for every digit in a number is so old fashioned.
How delightful! Thanks!!