For that matter, an alien race that uses base 2 for their everyday routine number system might find our ten symbols (and hundred-element multiplication table) inconveniently large to learn.
The Soviet Union, back when it was the Soviet Union, built some ternary (base 3) computers. They mostly didn’t explode either.
Some African languages use a combo of base 20 and base 10.
When I was a kid, we learned the table up to 12x12. Some earlier generations learned up to 15x15. Base 12 would not be a problem,
Yeah, but their football players would have seven digits on their uniforms.
Actually, we learn less than 100 rules. We know what to do with 0 and 1 without memorizing all the pairs, and commutativity means that we only really have to learn the diagonal and one of the two off diagonal triangles for the other 8. I make it that you only need (n-2)*(n-3)/2 + (n-2) “flash cards” committed to memory for base n. But, yeah, it’s order n^2.
The base 60 systems I have seen do not have 60 symbols but use a mixture of base 10 and 6. On the other hand for hexadecimals we really do use 16 different symbols. I have never seen the details about base 20 systems.
We can also have a variable base system in which the first position is base 12, the second is base 20, and the third just devolves into base 10, although the first two positions do use base 10 for notation. It is called pounds, shillings and pence. Even worse is a system that we use daily in which the first position can be in base 28, 29, 30, or 31 depending on the second position and, sometimes, a function of the third position depending on its remainder upon division by 4, 100, and 400.
I wonder if the question is about (what we perceive to be) the remarkable convenience of operating in base 10 - dividing or multiplying by 10 is a simple case of moving the decimal point.
But it would be the same if we worked in any other base - for example, suppose we had 12 fingers and our number line went like this:
1
2
3
4
5
6
7
8
9
A
B
10
11
12
etc
in base 12, 10 times 10 is still 100 - it’s just that ‘10’ is how you write the equivalent of decimal 12, and ‘100’ is how you write the equivalent of decimal 144
The same tricks work - they just work in 12s instead of 10s
By which formula, base 2 needs 0 “flash cards”, an infinite improvement.
The Mayan one was fairly elegant, needing only three distinct symbols: The 20 digits were constructed in a manner reminiscent of Roman numerals. The digit for 1 consisted of a single dot:
1 = .
And so it continued up to 4:
4 = …
(the digits are written one atop another, so “one twenty and one one” would be : , not …)
For 5, you replace the dots with a line:
5 = ____
And then you start adding more dots and lines, so that the digit for 19 has three lines and four dots:
....
19 = ____
____
____
The digit for 0, meanwhile, resembled an Easter egg (I think it was actually supposed to be a seashell), something like (|/)
The above posts have clearly established that any base system can consistently be used for representing numbers / doing math.
The real question, for me, is the ease of learning to do math with numbers : addition, subtraction, multiplication, division, fractions, decimals, …
Is there scientific research or scientific consensus as to which base system is the easiest for kids to learn math ? If so, please provide a cite.
It’s most apparent in the upper numbers in French French, which was likely influenced by Gaullish:
60: “soixante” - ultimately derived from Proto-Indo-European for “six-ten”
70: “soixante-dix” - sixty-ten
80: “quatre-vingt” - four-twenty
90: “quatre-vingt-dix” - four-twenty-ten
But in Swiss French, not influenced by Gaullish and geographically closer to Latin areas, the upper numbers are “septante” (70), “octante” (80), and “nonante” (90).
It’s believed that the Roman numerals are derived from hand gestures in the marketplace. |, ||, |||, and |||| represented one finger, two fingers, three fingers, and four. The “V” for five was likely the whole hand held up, with the fingers together and the thumb separate. “V|” was one hand held up as five and the other with one finger, and so on to x, which was likely the hands crossed at the wrists. Quick, easy to understand in a noisy market, and transcended dialects and languages. Later got assimilated using the letters i, v and x.
But computer programmers (especially back when they worked mainly in assembly language) used either octal (base 8) or hexadecimal (base 16) to represent numbers in their computer programs. Basically for human convenience – it’s too hard for humans to recognize/distinguish a long string of only 2 symbols (0 & 1).
“Vingt” in French is believed to be a fossil of a 20 base system.
The first nineteen numbers in French are clearly decimal:
From one to ten: un, deux, trois, quatre, cinq, six, sept, huit, neuf et dix.
From eleven to nineteen, the numbers are based on the first ten numbers, with some variations: onze, douze, treize, quatorze, quinze, seize, dix-sept, dix-huit et dix-neuf.
But then vingt, which is not tied to the earlier numbers at all.
But then the next four decades are again clearly derived from the first numbers: trente (30), quarante (40), cinquante (50) and soixante (60).
'Vingt" is the odd number here, not tied to the word for “two” (“deux”) unlike the other decade numbers.
Whatever base system is easiest for kids to learn arithmetic in, it is not Roman numerals, since that is not positional notation at all.
And I never made any such claim.
“Vingt” is not some French invention; cf Latin viginti (cf triginta), Sanskrit visati. (Wiktionary says the root word means “two-ten”, despite your statement it is not thus tied.)
Didn’t think you did; the post above yours was wondering about which base might be easiest, but I think those differences are overshadowed by the ease of using positional notation in the first place, versus not doing so. Not to mention when it comes to fractions…
Basically, French numbers display a standard simplification. The more used a word is and for a long time, the shorter and simpler it is. So numbers up to 16 are single words. the higher teens are compound. Similarly, up to 60 is simple, a variant of the integers 3 to 6; 70, 80, 90 - were used a lot less in everyday life way back when, so complex constructs worked just fine. It suggests that the number system was frozen into the language earlier than in English, where we don’t have as much construction.
(Similarly, our “half” has its own word, while more complex fractions are derived from the whole numbers.)
Notable and fun, India still uses a more complex system for large numbers lakh (100,000) and crore (10,000,000). I noted in the papers there it is very commonly used for big numbers - and in the government and big business propensity to spend large sums, these numbers appear in the news quite commonly to describe situations like budgets and expensive items. (It helps that their Rupee is worth about 2 cents, so the numbers have a head start).
Its radix would be the duodecimal point, i.e. one or two duodecimal places…
Yup. But interestingly enough, the Babylonian base-60 system also arose out of counting fingers ! Or, rather, phalanges - specifically you counted the phalanges of your non-thumb fingers of one hand (1 to 12), then held up one finger of the other hand every time you ran out of phalanges (1 to 5).
As for numbering in French, don’t bother trying to make sense of it as French was clearly drunk at the time. Sixty-ten-eight, Sixty-ten-nine, Four-twenties whoooo ! collapses
The Waali language of northern Ghana uses a mix of base 10 and base 20. The numbers 1 - 10 are unique. 11 - 19 is composed of ten + one. ten + two, etc…
Then there is a unique number for 20 (lejare). 21-30 is tweny + one, twenty + two, etc… 31-39 is twenty + ten + one, etc,… 40 is twenty with a two suffix lejarayii
There are also unique numbers for 100 koo and 200 bora
My favorite though is the number 50, which was not twenty*two + ten as you would expect, but rather Kpong lejananu i.e. double twenty-five