That mankind adopted the decimal system is a physiological accident. Those who see the hand of Providence in everything will have to admit that Providence is a poor mathematician. For outside its physiological merit the decimal base has little to commend itself. Almost any other base, with the possible exception of nine, would have done as well and probably better.

Indeed, if the choice of a base were left to a group of experts, we should probably witness a conflict between the practical man, who would insist on a base with the greatest number of divisors, such as twelve, and the mathematician, who would want a prime number, such as seven or eleven, for a base.

As a matter of fact, late in the eighteenth century the great naturalist Buffon proposed that the duodecimal system (base 12) be universally adopted. He pointed to the fact that 12 has 4 divisors, while 10 has only two, and maintained that throughout the ages this inadequacy of our decimal system had been so keenly felt that, in spite of ten being the universal base, most measures had 12 secondary units.

On the other hand the great mathematician Lagrange claimed that a prime base is far more advantageous. He pointed to the fact that with a prime base every systematic fraction would be irreducible and would therefore represent the number in a unique way. In our present numeration, for instance, the decimal fraction .36 stands really for many fractions: 36/100, 18/50, and 9/25 .... Such an ambiguity would be considered lessened if a prime base, such as eleven, were adopted.

But whether the enlightened group to whom we would entrust the selection of the base decided on a prime or a composite base, we may rest assured that the number ten would not even be considered, for it is neither prime nor has it a sufficient number of divisors.

In our own age, when calculating devices have largely supplanted mental arithmetic, nobody would take either proposal seriously. The advantages gained are so slight, and the tradition of counting by tens so firm, that the challenge seems ridiculous. From the standpoint of the history of culture a change of base, even if practicable, would be highly undesirable. As long as man counts by tens, his ten fingers will remind him of the human origin of this most important phase of his mental life. So may the decimal system stand as a living monument to the proposition: Man is the measure of all

which more or less sums the whole thing up

this copyright protected material taken from

Number Sense in Humans

* By Barry Mazur, Tobias Dantzig, Joseph Mazur.
* Sample Chapter is provided courtesy of Pi Press.
* Date: May 12, 2005.

at URL: http://www.informit.com/articles/article.asp?p=376880&rl=1

[reference at bottom of webpage]

posted without permission - without credit to source

That is the case in every number base. 0.555555... in base eleven means 1/2, but also 2/4, 3/6, 4/8, 5/A, ... You cannot avoid that, nor would it be desirable.

Forget the ten fingers. As all good Dozenalists- and Indians- know, man has twelve finger segments on each hand.

What could be even remotely advantageous about reducing the set of representable fractions to a minimum? The fractions 36/100, 18/50 and 9/25 are merely different ways of looking at the same quantity. Changing the base to a prime number won't change the fact there is an infinite set of fractions corresponding to every rational number; instead, it will only increase the number of fractions that lie outside its representational range, which is not something I would consider precisely as an advantage.

0.36 in base 7 = 28/49 = 56/98 = 84/147 = 112/196 = 140/245 = 168/294 = etc. ad infinitum
0.36 in base 10 = 9/25 = 18/50 = 27/75 = 36/100 = 45/125 = 54/150 = etc. ad infinitum
0.36 in base 12 = 7/24 = 14/48 = 21/72 = 28/96 = 35/120 = 42/144 = etc. ad infinitum
0.36 in base 13 = 45/169 = 90/338 = 135/507 = 180/676 = 225/845 = 270/1014 = etc. ad infinitum

Instead of a prime base, a much better choice would be to use the factorized representation of the number (which is already unique and irreducible), multiplied by either 1 or -1 to represent the sign, or by 0 to represent the number zero. This system not only does not restrict the set of representable rationals to a minimum, but makes it possible to represent every rational in a unique and relatively easy way; e.g., 0.36 = 9/25 = +1 · 3^2 · 5^-2 = {1;0,2,-2} (well, potentially, there would be many ways to represent zero, but if standardized as {0;0,0,0...}, this would leave plenty of possible representations unused, ready to be assigned for special purposes such as coding infinity, NaN and the like).

However, even though division and exponentiation become almost trivial with this kind of factorized representation —e.g., (0.36^2) / 2 = ({1;0,2,-2}^2) / {1;1,0,0} = {1^2; 0×2, 2×2, -2×2}/{1;1,0,0} = {1;0,4,-4}/{1;1,0,0} = {1×1; 0-1, 4-0, -4-0} = {1;-1,4,-4} = +1 · 2^-1 · 3^4 · 5^-4 = 81/1250 = 0.0648—, addition gets rather tricky —e.g., 2+2 = {1;1}+{1;1} = {1;2} = +1 · 2^2 = 4, looks simple, but now try with 3+3 = {1;0,1}+{1;0,1} = ...???... = {1;1,1} = +1 · 2^1 · 3^1 = 6, or with 2+3 = {1;1,0,0} + {1;0,1,0} = ...!!!... = {1;0,0,1} = 5.

The set of representable numbers using this system can be expanded to include any irrational that is the nth root of a rational, if we allow the representation to consist of a set of rationals instead of a set of integers. For example, sqrt(9/8) = (3/4)·sqrt(2) = +1 · 2^-3/2 · 3^1 = {1; -3/2, 1} = {{1;0,0}; {-1;-1,1}, {1;0,0}}. A more complex example: (26/15)·5thrt(7)·8thrt(45/2) = +1 · 2^1 · 13^1 · 3^-1 · 5^-1 · 7^1/5 · (3^2 · 5^1 · 2^-1)^1/8 = +1 · 2^7/8 · 3^-3/4 · 5^-7/8 · 7^1/5 · 13^1 = {1; 7/8, -3/4, -7/8, 1/5, 0, 1} = {{1;0,0,0,0}; {1;-3,0,0,1}, {-1;-2,1,0,0}, {-1;-3,0,0,1}, {1;0,0,-1,0}, {0;0,0,0,0}, {1;0,0,0,0}}. This makes it possible to represent all of the rationals plus also some of the irrationals using merely a matrix of signed integers of finite size, but makes division potentially as tricky as addition.

In fact, counting decimally is about the least efficient way to use one's hands to count; that way, you can only count from 0 to 10. Quinary is three times as efficient, allowing you to count from 0 to 30 (quinary 110). Dozenal is still more efficient, allowing you to count from 0 to 156 (dozenal 110):



Then, using four fingers of each hand to code the hexadecimal digits in binary, you can reach up to 255. And using all ten fingers as binary digits, the limit soars up to 1023. Decimal loses this battle miserably.

You can also count decimally to 99 with both hands:
Count 0-5 as usual, for 6 remove the thumb, for 7 the index finger (thus having 3 fingers raised), and 9 shows only the little finger. Thus all 10 digits can be represented with only one hand.
With both hands you can count to 99, and I have experimented with showing the hundred-digit by your arm position (also possible for dozenal). But that's of course only playing around, nobody would actually count arm positions... :D

ok, well i can count to 150000 with the hairs on my head
but how is this useful in the age of pocket calculators and computers?

technology makes the whole issue of mental calculation rather redundant.

until your batteries go flat ...
You reject mental calculation? What's wrong with mental calculation?

photovoltaics are your friend! ;)

i surely do not reject mental calculation, i use it quite often
there's nothing at all wrong with mental calculation

everyone uses it at least sometimes,
not many people whip out a calculator at the supermarket cashier

what i mean to say is that for complex calculations
we have electronic computers to do our dirty work

just as robotics makes arduous repetitive labour unnecessary


surely you use your computer for more than just a hightech typewriter

I tot up the bill as I go along; it stops me spending more than I've got in my pocket.
And that goes back to the days when if you didn't have the money you couldn't buy the goods.
(There was hire-purchase, of course, but that was treated like a dirty word by many).

wait, are you saying they refused to sell you any of the items brought to the cashier
if you were 1 penny short?

No, if I didn't have enough money then I couldn't buy the item so I wouldn't have asked for it in the first place. We used to save up for the item; then when we had enough, we could buy it. For example, in the days when there were model cars - Dinky Toys - the average one (in the days when I was still buying toys ...) cost 3s6d and I had 6d a week pocket money; 3s6d = 7 x 6d, so I had to save up for seven weeks to get it. Money was tight - and don't forget I'm talking about sixty years now. Old habits die hard.

Frankly, I can't believe that you think technology renders mental artihmetic virtually useless.

that's quite understandable

i don't believe mental arithmatic is useless
as i said, i use it all the time, as most people do

im only saying that for more complex calculations
we have machines which can do the job much more reliably
and billions of times faster

Counting on fingers is still useful when you want to keep track of large numbers you count.
Occasionally, I count double-steps on my way from university home to keep my mind busy, so I count just one-two-three-...-eleven-twelve and keep track of dozens and grosses with my two hands.

how would you define large number?

counting by 1s on fingers is limited to 10 (unless you include bones, then it's 28)

if you want to count higher, then you would need to remove your shoes


how would subtract 0.00000007162534765243 from 0.0065412653578236478268
on your fingers for example?

or count to 765 876 543 098 123 770 000 433


why would you want to torture yourself trying?
when you can take out a pocket calculator and do it instantly

it's like swimming from England to Canada when one can fly by jet


most people these days (myself included) use their all-purpose mobile as a calculator, + watch, calendar, memo pad, alarm clock...
in some cases as digital camera, webcam, voice recorder...

who is ever without their mobile?
Swedish people would sooner leave home without their head, than forget their mobile

mobiles are very handy for making calculations whilst out on the town,
it's like having a little computer in your pocket

But as has been pointed out already you can count using your thumb and count up to twelve using the joints of your fingers. If you use both hands then you can count the dozens on the other hand and get up to a gross.

Come on, now, do be sensible; when exactly would anyone want to do this? No-one's talking about doing huge calculations on their fingers.

I can't remember the last time I actually needed a calculator when out in town, though I occasionally carry one. I have a mobile for the occasional text message but rarely use it for phone calls; I only carry it in case of an emergency.

Watching people in town using their mobiles for phone calls I'm surprised there isn't a "mobile user's elbow" to go with "Golfer's elbow" and "housemaid's knee"; and for "texting" there'll be a "texter's thumb" one of these days!

exactly, no one would

Ebbe suggested that...

and when i think of large numbers
i think 5.678 x 10^15

or 20 decimal place fractions

i don't see 12 or 28 as particularly large numbers

im willing to bet many teenagers get "texter's thumb" ;-)

some people use a handsfree to prevent "mobile elbow"
but then they seem like schizoid lunatics arguing with "the voices" within

Just like the ones who use the phone booth; they wave their hands about, make faces and generally behave as if the person they're talking to can actually see them.

yes, and even better - they often nod or shake their head rather than replying yes or no

<_<

Yeah!! People using hands-free kits RALLY, and I mean really, freak me out! You'l be standing there by 'em, and all of a sudden their arms will start flailing and they'll beginning yeklling excitedly.....


....it is worrying, because there actually ARE quite a few care i nthe community types near where I live.

especially when they're not actually using a mobile ...

it's the voices again

See The Machine Stops by E.M.Forster and that was written in 1909!!!

Lagrange did NOT claim that a prime base is far more advantageous than a non-prime base. That whole article is complete rubbish. There is not a serious, respected mathematician who believes that it would be more advantageous to use a prime number base.

The decimal system only stands as a living monument to THIS proposition:

Most men are complete morons who blindly follow thoughtless fools. The decimal system is only a measure of mans stupidity. It’s shameful how we put future generations through such crap just because the rest of us become comfortable with such rot.

Let’s get the fundamentals right. Imagine if you could measure how much time humans have wasted by using the base 10 rather than something more sensible … and how many errors could have been prevented. I’ll bet that if you could measure time wasted and errors made per capita it would be astounding. Problem with this world is that there are not enough people with the imagination of those posting on this message board … way too many mindless drones out there ... people who can’t appreciate how much better the world could be.

And that … “well, it doesn’t matter now because we all have calculators on our watches” argument does not appeal to me at all. Humans are making minor calculations in their heads all the time. And with a more sensible base, an even larger range of calculation would become minor. People, who presently run to their calculator because they have to deal with the decimal system, might actually work more things out in their heads if the base was more logical … and that mental exercise would be a good thing for the human population.

Werll, calculation on calculators would also be easier in base 12, because 1/3s and 1/16s will be easier to do on a calculator. Today, I think most people don't know right away that .375 is equal to 3/8, and certainly not that .4375 is equal to 7/16; However, shoelace's point is, in my opinion, correct as well.