# why have we chosen our number system to be decimal (base 10)

After learning about the binary number system (only 2 symbols, i.e. 0 and 1), I just thought why did we adopt the decimal number system (10 symbols) after all?
I mean if you go to see, it's rather inefficient when compared to the octal (8 symbols) and the hexadecimal (16 symbols)?

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Because we have ten fingers. –  Ｊ. Ｍ. Nov 3 '10 at 6:34
My personal opinion is that duodecimal might have been a better number system than decimal, but sadly we don't have 12 "digits" on our hands. –  Ｊ. Ｍ. Nov 3 '10 at 6:42
+1 @J.M. ...why didn't you post this as an answer...i cannot accept a comment! @deb by inefficient, i mean that octals get 3 bits completely utilized to represent 8 symbols, but decimal is 10 symbols which requires 4 bits not fully utilized...i guess it sounds silly... –  Samrat Patil Nov 3 '10 at 6:55
@Samrat: That's just a byproduct of us choosing to make electronics binary, which wasn't relevant when our number system was invented. –  Cam Nov 3 '10 at 7:06
ObTomLehrer: Base-8 is just like Base-10 ... if you're missing two fingers. (from "New Math") –  Blue Nov 3 '10 at 11:28

Expanding on the comment by J.M., let me quote from the (highly recommended) book by Georges Ifrah The Universal History of Numbers (Wiley, 2000, pp. 21-22):

Traces of the anthropomorphic origin of counting systems can be found in many languages. In the Ali language (Central Africa), for example, "five" and "ten" are respectively moro and mbouna: moro is actually the word for "hand" and mbouna is a contraction of moro ("five") and bouna, meaning "two" (thus "ten"="two hands").

It is therefore very probable that the Indo-European, Semitic and Mongolian words for the first ten numbers derive from expressions related to finger-counting. But this is an unverifiable hypothesis, since the original meanings of the names of the numbers have been lost.

Ifrah then goes on to explain that

...the hand makes the two complementary aspects of integers entirely intuitive. It serves as an instrument permitting natural movement between cardinal and ordinal numbering. If you need to show that a set contains three, four, seven or ten elements, you raise or bend simultaneously three, four, seven or ten fingers, using your hand as cardinal mapping. If you want to count out the same things, then you bend or raise three, four, seven or ten fingers in succession, using the hand as an ordinal counting tool.

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I will give the linguistic note that "digit" is in fact a synonym for "finger", and in fact stems from the Latin digitus. –  Ｊ. Ｍ. Nov 3 '10 at 11:10
BEWARE Ifrah's book has received highly critical reviews by experts, so one should be wary of any historical claims. For example see this review part1, part2 by the eminent mathematical historian Joseph Dauben. –  Bill Dubuque Nov 3 '10 at 13:36
@Bill Dubuque: Thanks for the comment. I have not been aware of the controversy. –  Andrey Rekalo Nov 3 '10 at 13:58
I feel bad. The m and rn look rather similar and I almost read that "Traces of the anthropornorphic origin"... –  Drew Nov 24 '10 at 15:19
"BEWARE Ifrah's book has received highly critical reviews by experts" So what then would those experts have to say on the matter? –  mike4ty4 May 16 '13 at 5:38

I think the answer here might be, that the guys who thought base 10 was a good idea had the largest sticks.

If one trusts the wikipedia, the Babylonians had a base 60 system, which can still be felt today with this "60 minutes in an hour" nonesense, and a (related) base 12 system was widely in use too. There are still unique words for "eleven" and "twelve", as well as expressions as "a dozen". After all, you can count to twelve using a single hand.

Then, there was the base 1 latin system, and (wikipedia again) a base 20 system for the mayan, who obvsiously had no shoes, since they could count their toes, too.

Something as easy as "base 10 is natural for humans" does not explain it all. =)

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"why do we all use base 10?" ... but "we" didn't all use base 10. Very perceptive. –  a little don Nov 3 '10 at 12:23
Awesome! And I always wondered who came up with the 12 hours, 60 mins and 60 secs thingy.. Can you please post the wiki or whatever read up on this topic. –  Samrat Patil Nov 4 '10 at 6:22
@Sumrat: en.wikipedia.org/wiki/Numeral_system and related pages. –  Jens Nov 4 '10 at 7:13
@Jens you're pretty much right on; Europes use of place-valued base-10 numbers comes from the Arabic numbers, which were introduced by The Pope in ~1000AD. You don't get a bigger stick in Middle Ages Europe than the Church. (It also made accounting easier, which was why powerful people tended to like them and teach their kids how to use them) –  Michael Edenfield Jul 5 '12 at 15:10
"60 minutes in an hour" nonesense? You can easily divide hour in 2, 3, 4, 5 or 6 parts. OK, by 7, 8, 9 it's not working, as well as with 11, but 10 and 12 still works. –  РСТȢѸФХѾЦЧШЩЪЫЬѢѤЮѦѪѨѬѠѺѮѰѲѴ Jun 25 '13 at 19:07

Because it makes the metric system so much simpler :).

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Now that's just backwards. ;) –  Ｊ. Ｍ. Nov 3 '10 at 11:45
Haha. Sure, and, in particular, it makes the deciliter useful. –  user02138 Nov 3 '10 at 16:09

I don't believe you understand the notion of efficiency in terms of encoding. Informally speaking, you have to keep it mind there are two factors involved: (i) cost of having different symbols (in case of base 10 there as 10 different symbols, in case of base 16 there are 16 different symbols etc) and the length of the resulting string to encode a particular number.

When you consider both factors and apply some basic information theory to it, the answer may look a bit surprising: the most efficient encoding has a base $e$ (yes, that very $e = 2.718\dots$). Since we'd rather have some natural number as a base, the best we can get is base 3, and the next is base 2.

So, why, then, computers use base 2 (0 and 1) rather than base 3 (say, -1, 0, and 1)? The answer is that it is simple to design the circuits that distinguish between two (rather than three) states. (I do remember reading some of the earlier computers did use base 3, but I can't recall all the details.)

Now, with respect to octals and hexes, those are simply convenient ways to record the binary strings. If you did some machine-level debugging, you probably had a chance to read what's known as "hexadecimal dump" (contents of a memory). Surely it's easy to read than if it were written as binary dump. But what's lurking underneath of that is base 2.

(The answer on "why do we use base 10" has been answered elsewhere.)

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+1 nice to know. –  Samrat Patil Nov 8 '10 at 17:00
There was the russian Setun computer system among others. –  ogerard Apr 13 '11 at 14:59
There's a Wikipedia article on radix economy that gives the argument for base $e$ being the most efficient. –  Simon Nov 21 '11 at 4:25
@Simon: The argument used for getting $e$ as most efficient radix is entirely based on the totally unmotivated definition of radix economy using the value of the radix to multiply by. From the point of view of information theory it is obviously more natural to use the logarithm of the radix instead, in which case all radices come out equally efficient. –  Marc van Leeuwen Sep 6 '13 at 9:16
@Marc - Have you got a link to a longer discussion of this? Both the use of $\log(b)$ instead of $b$ and the final result of all radices being equal seem reasonable, but it would be nice to see more on it. –  Simon Sep 8 '13 at 10:30

It is believed that the decimal system evolved mainly due to anthropomorphic reasons (5 digits on each hand) and is thought to be a simplification of the Babylonian sexagesimal (base 60) counting method.

To make this analogy precise, note that the normal hand has 4 fingers (excluding the thumb) with 3 segments, along with 5 digits on the other hand to be used as segment pointers. This gives 3 x 4 x 5 = 60 unique configurations.

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The thumb is not a finger? :) –  Ｊ. Ｍ. Nov 3 '10 at 15:58
I suppose the sexagesimal counting didn't include the thumb because it doesn't have 3 natural and visible segments. It is sufficient to use 4 fingers with 3 segments on one hand and all five digits on the other. –  user02138 Nov 3 '10 at 16:04
Thanks, J.M. ;) –  user02138 Nov 3 '10 at 16:07
@NicolasManzini: Then you are counting incorrectly. Use your right hand (all five digits) as a pointer and touch only one segment (of three segments) on any of the four fingers on your left hand. The possible combinations is 60. If you doubt the validity of this counting, look it up -- it's pretty common knowledge. –  user02138 Jul 5 '12 at 23:54
@Ｊ.Ｍ. Well, if you define that finger has 3 joints, a thumb is not a finger. In fact, it is more rational to assume that thumb is something else as saying, that finger sometimes has 2 joints, sometimes 3. –  РСТȢѸФХѾЦЧШЩЪЫЬѢѤЮѦѪѨѬѠѺѮѰѲѴ Jun 25 '13 at 19:10

I am tempted to answer "for the same reason as this forum is in English" - ie human convention for effective communication and calculation. However there is another anthropomorphic aspect to this, in that there are advantages for a high base (compact encoding of numbers) and for a low base (smaller number of addition/multiplication facts to learn, fewer number 'symbols' to recall and write distinctly without confusion).

Binary and binary related computations are used in computing because it was technologically easier to encode '0' and '1' than to work with a higher base than 2, and computing conventions were created when computing resources and speed had to be optimised. The available length of string then restricts the size of number which can be stored or manipulated. Many of these reource constraints no longer exist in the same way (my computer has more capacity than I generally need).

So I think there is some form of rough optimisation with base 10, given the recall and ability of human beings, this was a good compromise. And we do not always use it when there is an advantage to be had in using another. And note that the Octal and Hexadecimal representations within computing are the ones closest to base 10 ....

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Because these ancient folks didn't fully foresee the glory of modern computer technology. Else they would have choosen a base that would be more compatible with computers binary number system: 8!

Generations of computer science students would it have so much easier and everything would be much better:

A Byte would have 10 bits (with 2^10 possible values)

Computer technology would have evolved from

And we would not need funny things like mebibyte:

• 1kilobyte = 1000 Byte (not 2000 as it is now :)
• 1MB=1,000,000 Byte
• 1GB=1,000,000,000 Byte
• 1TB=1,000,000,000,000 Byte

OK, for one terabyte you would only get 7% of the capacity compared to our current system, but who cares.

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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 !
1. 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 ,..

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