Would there be any advantages in having a numerical system other than base ten ?

Factual Questions
1

I’m curious.

Obviously, the reason why our numerical system is base 10 is because we have 10 digits on our hands. If we had evolved with four digits on each hand, like the Simpsons, (or do they only have three ??.. can’t remember) then presumably we would have developed a numerical system with base eight.

Now, I am aware that the Babylonians, supposedly, used a base 60 system (yeah right, they got their kids to memorize a 60 times table … I don’t think so ) but my question is this …

Would there be any significant advantage to using a duodecimal based system …ie where the number twelve was the base instead of the number ten … obviously twelve is divisible by more factors than ten is …2,3,4,6 as opposed to only 2 and 5 …but, would that be a significant advantage in any way, and if so why ?

The other more radical thought that occurred to me is that it might actually have been advantageous to have developed a numerical system based on a prime number such as eleven or thirteen … I have no idea whatsoever why this might or might not have been advantageous, but I do know that there are some clever dudes on this forum who will elucidate me … (and in return, I will help them with any woodworking queries they may have )…

2

There are plenty of real world (computer engineering) applications for bases 2 (binary), 8 (octal), and 16 (hexadecimal).

3

Aside from base 2, those are really only used by humans working with the computers. Eight and 16 are convenient because they are powers of 2, meaning binary numbers can be represented with a consistent number of digits. (Four bits always equals exactly one hexadecimal digit or two octal digits; not true for base 10.)

Anyway, here’s a previous thread about this topic.

4

And another earlier thread:

is Base 12 mathmatics easier

There is some convenience in having a base with a lot of factors, in particular a factor of 3 as well as 2. The most commonly used fractions are 1/2, 1/3 and 1/4. In base 6 or 12, 1/3 is a single place decimal, rather than a repeating decimal. In base 12, 1/4 comes out to be a single place decimal instead of two. Of course, you would lose 1/5 coming out even, which would seem to be an equitable trade.

5

Our brains don’t seem well equipped to deal with larger bases (imagine trying to memorize a base 60 multiplication table!).

Both base 8 and 12 have some things going for them, as mentioned above. They also happen to be a good size, neither too large or too small. Maybe base 16 is close enough, too, but it’s probably pushing the limits for what we should expect kids to learn for mental arithmetic.

6

I see where you are coming from, but I would really love to see a practical example of a real life situation where things would be totally simplified **or ** complicated by the adoption of a different number base … (real life …not computer based hexadecimal or octal) …I am asking how life would be different if we had adopted a different base than ten from the gitgo …

7

Literate/numerate Babylonian bureaucrats/scribes, who made up a very small proportion of the Babylonian population, did use a base-60 numeral system for specialized scientific and accounting applications. That’s where our present-day base-60 units for time and angle measurement came from.

They didn’t have to memorize all 1800 values of a 60x60 multiplication table, of course. They knew how to partition, say, 49 into 40+9 and add up the appropriate results from the 40-times-table and the 9-times-table. Moreover, they had lots of such tables written down on clay tablets so they could look up the results of operations instead of memorizing them, just as modern humans until the recent advent of easily available electronic calculators looked up values of, say, trig functions or log functions in reference tables.

Naturally, ordinary Babylonians didn’t learn any such advanced place-value arithmetic, since they got by just fine without it. They used various everyday systems of metrological units, some decimal and some not, just as our everyday household arithmetic can handle non-decimal conversions like “two cups to a pint” and “twelve inches to a foot” and “four quarts to a gallon”.

8

The Maya used a base-20 (vigesimal) system for several centuries during the heyday of their civilization.

9

A base-twelve system would be more convenient for impromptu dividing stuff up… that’s one area, maybe the only one, where the old systems of measurement had an advantage. I think we missed a great opportunity by not basing the metric system on dozens. If we’d gone to base-twelve math at the same time, no-one would have been able to resist.

10

If we went to a pure binary number system, it might be possible to get today’s kids to memorize their times tables.

Back in the day, when even reading/writing were not universally known by the masses, they had scribes: Paid professional readers/writers to do their claywork for them. Businessmen hired professional scribes to do their books. I believe the highest echelons of the scrivener trade were those who knew their arithmetic too, who could read/write/reckon and compute the right taxes to pay. The common folk, even businessmen or at least tradesmen, probably couldn’t to that themselves.

And how is this different from today? Today, reading/writing/arithmetic are nearly-universally taught in industrialized countries. Yet the complexities of accounting and law are still beyond most of us. To this day, we have the paid professional scribes – accountants and lawyers – who are hired and paid by companies and businessmen to do their reckoning and lawing for them.

Le plus ca change, le plus c’est la meme chose.

(And, the cynical among us might suggest, that arcane base-60 system of the Babylonians was specifically designed by the scribes to make it impossible for anyone else to get by without hiring them!)

11

Huh. Gotta think about this for the bigger picture. How would that “Does .9999 = 1” thread go then?

Cite?

12

<nitpick> AFAIK, this phrase doesn’t need the “le”: Plus ça change, plus c’est la même chose.. We use “the” in the equivalent English phrase, but it is not needed in French. I think. (I know just enough French to get into trouble…)

I know programmers who think like this.

13

It’d still be the same, except instead of 0-9 you’d have additional (or fewer) digits; for example, hexadecimal uses the letters A through F to represent 10-15; for base-12, just add A and B, so it’d be “does 0.BBBBBBB… = 1?” (or 0.55555 in base-6).

14

It becomes a “Does .BBBB… = 1” thread.

15

Well, according to archaeological evidence, that’s probably not how it happened, AFAICT. It appears that as populations increased and urban civilization got more complex in fourth- and third-millennium Mesopotamia, the various naturally evolving different metrological systems started being a real hassle to convert back and forth.

So the bureaucrat-scribes (initially Sumerians, though the full place-value base-60 number system wasn’t completely developed until Babylonian times) ended up using a big base with a lot of convenient prime factors, namely the number 60, to put as many of the different unit systems as possible on a common basis.

That way, if, say, a guy from Larsa says that a basket is one-tenth of a bushel and a guy from Isin says a basket is one-twelfth of a bushel and a guy from Mari says a basket is one-fifteenth-of a bushel, you can just convert all the quantities to the same Official Governmental BigAssBushel unit containing 60 baskets, and you’ll get a standardized inventory in reasonably nice round numbers.

After all, the Mesopotamian bureaucrat-scribes before the establishment of a common base-60 numeral system were already a highly trained official elite who were essentially indispensable to the functioning of the city-state. They didn’t really need to make arithmetic incredibly more cumbrous for themselves just to enhance their job security.

16

Old Babylonian metrology, for example. Various larger units were defined as multiples of smaller units without any concern for consistency in the numbers used for the multiples, much less for making them all consistently decimal.

17

Here’s a Numberphile video on why Base 12 is easier. To drive the point home: high level math and physics is exactly the same in Base 10 or Base 12, the only time it would be easier is when it comes to multiplying and dividing in your head.

I won’t type up the Base-10 times table up to 12 because I assume you know it, but here’s the Base 12 times table. Remember that in the dozenal system 10 is replaced by “dec” (a Greek chi χ) and 11 is replaced by “el” (a Greek epsilon ε)



    1  2  3  4  5  6  7  8  9  χ  ε  10
1   1  2  3  4  5  6  7  8  9  χ  ε  10
2   2  4  6  8  χ  10 12 14 16 18 1χ 20
3   3  6  9  10 13 16 19 20 23 26 29 30
4   4  8  10 14 18 20 24 28 30 34 38 40
5   5  χ  13 18 21 26 2ε 34 39 42 47 50
6   6  10 16 20 26 30 36 40 46 50 56 60
7   7  12 19 24 2ε 36 41 48 53 5χ 65 70
8   8  14 20 28 34 40 48 54 60 68 74 80
9   9  16 23 30 39 46 53 60 69 76 83 90
χ   χ  18 26 34 42 50 5χ 68 76 84 92 χ0
ε   ε  1χ 29 38 47 56 65 74 83 92 χ1 ε0
10  10 20 30 40 50 60 70 80 90 χ0 ε0 100

Now, some people may be put off by how much harder 5 is to multiply by, but you exchange it for 3, 4, and 6. Even 8, χ, and ε aren’t that bad. Either way, the addition of more regular patterns compared to base-10 would make it easier to triple, quadruple, or hextuple things for kids, imo.

In addition, division becomes rather nice. 1/2 = .6, 1/3 = .4, 1/4 = .3, 1/6 = .2

Though I will mention something that numberphile didn’t, 1/χ and 1/5 become repeating in base 12 while 1/7 and 1/ε remain repeating, so it doesn’t really get rid of the “gnarly repeating decimal” problem like they say it does. That said, 1/3 is probably more a common operation than 1/5 is.

So there’s no magic scenario* you can give where it’s way easier or harder, it’s just one of those things where it has properties with multiplication and division that would make math slightly easier to do in your head and to learn as a child. I don’t think it’s really “worth it” to switch, it would probably cause more problems than it would solve.

  • The only modern scenario I can think of is inches->feet in America. 1ft = 1 * 10[1]1[/sup] in in base 12, and that’s not really a magic bullet.

  1. sup ↩︎

18

I look forward to algebra students having to use χ as a digit. 3χY? Or 3XY?

19

Naturally I think they’d stop using x as the default “unknown” letter. One would hope at least. (That or they’d write chi in some really fancy script to make it obvious).

20

Seeing this thread made me wonder whether the “Duodecimal Society of America” was still around. With a slight name change, they are: