# Why don't we use base 6 or 11?

Another question on this site asks why we have chosen our number system to be decimal base 10. There are others asking basically the same thing as well.

I'm not really satisfied with any of the answers, because most of the answers given seem to suggest that base 10 was chosen because we have $10$ fingers. However, this would seem to me to imply that we should be using decimal base $11$. Supposing we use the scheme of calling decimal 10 "A", then on our fingers we would could count $1, 2, 3, 4, 5$ on the first hand, and then $6, 7, 8, 9, A,$ on the second. Only then would we be out of fingers and need to roll over to 10 which would be decimal $11$. Likewise, a similar argument could be made for base 6 counting on only one hand, as there are five digits before one runs out and needs to roll over to $10$, in this case for the decimal value 6.

For base 6 the argument could be made that the thumb is not counted, and thus base 5 is more natural, but the fact remains that we don't use base 5 either, we use base 10, and not counting thumbs on either hand would result in us using base 9, not decimal base 10, so I feel like this argument does not hold water either.

An alternate explanation, that base 10 is an abbreviation of base 60 seems slightly more likely, but base 60 seems rather unweildy to being with, which leads me to the question, why don't we simply use base 11, as our 10 fingers seem most suited to it? As far as I am aware no culture has ever widely used it.

• Historical considerations aside, every positional number system needs a digit (corresponding to) $0$. Base $10$ uses $10$ digits, and we have $10$ fingers (and the etymology of "digit" goes back to finger). – Marc van Leeuwen Sep 21 '13 at 5:08
• 36 is base 6 is 100..now try to divide it by 5 ..you will yourself understand why decimal ..! – arnab Nov 22 '13 at 21:13
• There's a near-extinct language (Native American, I think) which uses base 8. Apparently, they counted on the spaces between their fingers. (Bonus binary compatibility!) There's a Numberphile video on this. – Akiva Weinberger Aug 19 '15 at 20:43
• @arnab : Just like Dadam said: 36/6 in base-10 is the same as 100/10 in base 6. By choosing the right numbers you can defend any base. – Oded Breiner Nov 26 '15 at 11:06

The idea of a "base", and even the idea implicit in it of a consistent positional numbering system, is a relatively modern one. For that matter, even "base 10", in the sense of decimal numerals — the Hindu–Arabic numeral system we all use today — is relatively modern; witness the fact that much of Europe didn't begin to use it until well into the second millenium. The idea that we can count in any base is even newer.

Your argument might make sense if humanity started with the idea in mind of using a base-$b$ representation for some $b$, and then looked to their fingers to decide what the base $b$ should be. This of course is not what happened, nor even is it imaginable in any culture. The concrete precedes the abstract.

Instead, what we see historically is counting on one's fingers, and thus counting by fives or by tens, not counting in base 10. If you're counting on two hands, when you reach $10$ you literally run out of fingers to count, and that is where you have to leave a mental (or physical) note to yourself that you're done with one round of counting, and begin counting anew. (Similarly $5$, if using one hand.)

We can see traces of this "counting by $5$" or "counting by $10$" in early systems like Roman numerals: note that $8$ is represented as "VIII", denoting one count of five (done with one hand's worth of counting), and starting again, reaching up to $3$ in the process. Similarly, the representations "XX" and "XXX" show that they were being thought of as "two tens" and "three tens", rather than as "in base $10$, three in the tens place and zero in the units place" — the idea of base $10$ is not actually present. Thus "XXXVIII" literally denotes the process of counting: "three tens (three double-hands), a five (one hand) and three fingers". (There are even traces of counting by $20$s; consider the English "score" and French number names).

It was only after centuries of already counting by $10$s, conducting transactions with $10$-based words for numbers and so on, that the base-$10$ representation arose, as a representation system for the same numbers that everyone was already accustomed to thinking of in tens, and (very slowly) spread across the world.

• Aside: even with the "backward-compatible" base $10$, the positional ("place-value") system took several centuries between 1202, the year of being introduced in Europe by Leonardo of‌ Pisa (Fibonacci), and it being generally adopted in Europe. One can only imagine what would have happened if someone had tried to introduce a base-$11$ representation, in which a number like LXXIII, conceptualized by everyone as "a fifty, two tens, and three", would have to be represented by an entirely unrelated-looking "67". – ShreevatsaR Aug 11 '13 at 17:05
• What you're saying is true in respect to a given culture. However, positional systems are not new if you look at the world as a whole. Sumerians were using their base 60/10 system circa 23rd century BC. Just as you said, they had been using an additive system with these numbers for at least a millennium. – GregRos Aug 11 '13 at 17:51
• @GregRos: Yes, my "relatively modern" is to be loosely interpreted: e.g. it works to take it to mean "much more recent than counting itself". :-) BTW, even the Sumerian/Babylonian numerals seem to have used counting by tens, to represent the 59 different "digits" they needed (only 59 because they muddled through without a 0): en.wikipedia.org/wiki/File:Babylonian_numerals.svg It is said that the Yuki and Pamean peoples (both in the Americas) use octal because they look at the spaces between the fingers, but even there they're counting by number of "things", not one more (like base-11). – ShreevatsaR Aug 11 '13 at 18:07
• Yup, that's why I said base 60/10. More so; before they used cuneiform, they used a different number system in which the numbers 10 and 60 featured prominently. Their number names also betray the importance of the number 10. – GregRos Aug 11 '13 at 18:33

Let's consider your base 6 proposal. You motivate $10_6$ to be the first number that you can't count on one hand. To be consistent, $20_6$ should be the first number that you can't count on two hands. But it isn't! Instead:

• $10_6$ is $1$ more than the number you can count on $1$ hand
• $20_6$ is $2$ more than the number you can count on $2$ hands
• $30_6$ is $3$ more than the number you can count on $3$ hands

What a strange pattern. Wouldn't it be better if $d$ more than the number you can count on $d$ hands were written $dd$? Well, that's what we get in base 5:

• $11_5$ is $1$ more than the number you can count on $1$ hand
• $22_5$ is $2$ more than the number you can count on $2$ hands
• $33_5$ is $3$ more than the number you can count on $3$ hands

And there's an even nicer pattern for numbers of the form $d0$:

• $10_5$ is the number you can count on $1$ hand
• $20_5$ is the number you can count on $2$ hands
• $30_5$ is the number you can count on $3$ hands

The same arguments apply to 10 versus 11.

• It took me a minute to understand your assertion that "20 base 6 is 2 more than the number you can count on 2 hands", because in my mind I was picturing being able to count to 55 base 6 using two hands, with the left hand being the 10's (base 6) position and the right hand being the 1's position. – Michael Aug 6 '13 at 18:55

Well, you're clearly aware that different cultures have used different bases, and indeed developed sophisticated mathematics using them. Further, most of the mathematical advances in the West over the last 500 years (say) would seem to have nothing to do with what base we use in ordinary life.

So I'm not quite sure where the "why" question comes in. I agree that claims that base 10 is "natural", or is more efficient than some other base, seem dubious. So what are we seeking to explain here, exactly?

If it's a simple causal explanation that's wanted -- "How did it come to be that we use base 10" -- then the answer is, I'm willing to bet, that it's an accident of history.

In other words: isn't this a bit like asking why we have 26 letters in the alphabet?

• I don't think your last comparison is good. While the base $10$ is (almost?) exclusively used today in ordinary life, there are various alphabets with various numbers of letters. Just to name a few Slavic ones: Croatian/Serbian/Bosnian (30), Slovene (25), Russian (33),... There is a huge list of writing systems being used around the world, nothing like the number systems. (Upvote for the rest of your answer) – Vedran Šego Aug 11 '13 at 16:57
• @Vedran -- very true... I wish I could think of a better analogy but my poor brain's failing me. – helveticat Aug 11 '13 at 17:29
• It's a fact that the number 10 crops up all the time as part of number systems, and usually as a dominant part. Almost all other number systems use 20, 5, or 12. These are also related to finger counting (or toe counting). The names of numbers in many diverse cultures also betray the significance of finger counting, and the numbers 5 and 10. In contrast, I don't think a number system based on 7, 28, or 11 has ever been found. There is probably something about all these systems that makes them so common. It's not just a fluke. – GregRos Aug 11 '13 at 18:41
• @Greg -- I think tracing any base that's not coprime to 10 to finger-counting might be a stretch; it'd be a huge task to demonstrate all such bases originated from that. Not saying it couldn't feature in the history of a specific system, but the danger is that finger-counting becomes a "just-so" story that naturalises a contingent practice (base 10) that we happen to be used to. – helveticat Aug 12 '13 at 9:07
• The issue is not as theoretical as you seem to think, and it is not my off-hand remark that links such systems to finger counting. It is a large collection of evidence. You should look at that evidence yourself, instead of taking the armchair approach. – GregRos Aug 12 '13 at 13:03

I'm writing this answer because, I think you might not be satisfied with the answers given for now. In my opinion, your approach is also natural and probable. Also base $12$ is normal and natural. But, this things are like languages, not just logic. For example, we still uses $12$ based clocks. So, shortly, your approach is maybe more natural, more logical but this aspect of mathematics depends on occasions not just pure logic.

interesting question ! :)

I THINK THIS A POSITIONAL NUMBER SYSTEM .and that's of great advantage ..simple shifting the position of decimal . this is the most conventional answer ..! but somehow it seems , not that strong . we could have tried to construct a fractional system in other number systems as well !

but there are certain points i have made out :

1.if the base is more than 10 , say 20 . then if you want to use our 10 symbols (i.e. 0, 1..,9) it not possible for us to write numbers more than 9 and less than 20 or numbers more than 189 and less than 200 and numbers of similar form ) , so in that case for each unit either you will have to allot two places for digits (which will make it difficult to read a number) or you will have introduce more 10 symbols , which will make it more complex and we wont get any advantage. cause 20 is just a a multiple of 10 by 2 ..and 2 is a factor of 10 ( if the base would be 30 then , we could take it as an advantage that the problem in fixing the decimal we face while handling division by 3 could be simplified . but in that case just imagine how many symbols you will have to introduce !!! the base could be 12 . in that case dividing by three would not face any approximation problem..or if the base would be 14 then we would easily be able to divide by 7 ..but in both 12 and 14 base the problem in dividing by 5 arises. and in ancient days as our hand and toe consists of 5 fingers , i think divisibility ease for 5 was given preference ) so among systems with 10 or more than ten , the base 10 system , i.e. decimal system was given preference in different civilization , as a human mind always and everywhere think in the same direction and following our ancestors we set this as our ruling international number system mostly used ! 2.now , for bases less than 10 : there is no number which has three prime factors ! among numbers 10 or less than 10 , 10 has the highest number of one digit-ed prime factors ! and for that reason in decimal system , we can properly divide any number by most number of numbers compared to all the other systems with base less than 10 .!(except 6 . which was again given less priority because 10 is a multiple of 5 )

NOTE : some say , "you can order the numbers with specific space between in decimal system .. If we can't order the number with equal interval among them , then it wont be possible to get any number line,and whole number theory will possibly be collapse then . the way co-ordinate geometry ,..vector algebra work , where ordering is very important will collapse ..! e.g in octal system 63 =77 where 64 = 100 and 65 = 101 ..so , then what's the measure of unit countable increase ? .." but , that's not a proper argument in my opinion . there we will have to consider 77+1 = 100 , 7+1=10 , 777+1=1000...and so on ,..

added note : 11 is itself a prime number . so in dividing a number which is a fractional multiple of other primes , we will face great difficulty !

• 36 in base 6 is 100 . now try to divide it by 5 ..you will yourself understand why decimal system ! – arnab Nov 22 '13 at 21:12
• 36/6 in base-10 is the same as 100/10 in base 6. By choosing the right numbers you can defend any base. – Adam Trhon Dec 28 '13 at 10:18