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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)?

m0nhawk
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Samrat Patil
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    Because we have ten fingers. – J. M. ain't a mathematician Nov 03 '10 at 06:34
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    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. ain't a mathematician Nov 03 '10 at 06:42
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    +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 03 '10 at 06:55
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    @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 03 '10 at 07:06
  • We use binary on computers merely because it's easy to represent 0 "off" and 1 "on", and those two other systems because it takes so much binary digits to represent medium-sized numbers. – J. M. ain't a mathematician Nov 03 '10 at 07:08
  • +1 @J.M this is the only reason. We count in number system 10 because we have 10 fingers. – Djaian Nov 03 '10 at 08:08
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    ObTomLehrer: Base-8 is just like Base-10 ... if you're missing two fingers. (from "New Math") – Blue Nov 03 '10 at 11:28
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    @J.M.: Are you sure it's not because of our ten toes? – user642796 Aug 13 '12 at 20:41
  • @Arthur, the cultures I know count with their fingers. I suppose there is a tribe somewhere that prefers the use of the toes for counting, but I would think an inconvenient amount of contortions would be required... – J. M. ain't a mathematician Aug 14 '12 at 05:16
  • 36 in base 8 is 44 ..now try to divide it by 5 ..you will yourself understand why decimal .! – arnab Nov 22 '13 at 21:15
  • @J.M. Why do you think `duodecimal` is better? Very curious. – AGamePlayer Jan 18 '14 at 01:25
  • Also, [every base is base 10](http://math.stackexchange.com/questions/166869/is-10-a-magical-number-or-i-am-missing-something)... ;) :) – anishsane May 06 '14 at 09:06
  • @Aw, I suggest reading [this part](http://books.google.com/books?id=HqeoWPsIH6EC&pg=PA20) of Underwood Dudley's book; some of the reasons given there are essentially the same as mine. – J. M. ain't a mathematician May 01 '15 at 14:20
  • @J.M. I think if that was the reason, we would probably use base 11 (http://smbc-comics.com/index.php?id=4010) ... I like this answer: http://math.stackexchange.com/a/465128/33914 – bodo Feb 07 '16 at 22:21
  • @J.M. Some cultures (historically) use base twelve systems with finger counting - counting the non-thumb joints on each finger of one hand: https://en.wikipedia.org/wiki/Duodecimal#Origin – T.C. Proctor Dec 18 '16 at 18:01
  • It wouldn't be that difficult to imagine counting up to 10 and then imagining 2 more, or perhaps using our fist (thumb, index, middle, ring, pinky, close hand) X 2 – DomenicDatti Dec 28 '17 at 16:08
  • "we don't have 12 "digits" on our hands." Yes we do, we have 12 phalanges on our fingers – Joeri May 11 '21 at 11:42

8 Answers8

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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.

J. M. ain't a mathematician
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Andrey Rekalo
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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. ain't a mathematician Nov 03 '10 at 11:10
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    **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](http://www.ams.org/notices/200201/rev-dauben.pdf), [part2](http://www.ams.org/notices/200202/rev-dauben.pdf) by the eminent mathematical historian Joseph Dauben. – Bill Dubuque Nov 03 '10 at 13:36
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    @Bill Dubuque: Thanks for the comment. I have not been aware of the controversy. – Andrey Rekalo Nov 03 '10 at 13:58
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    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
  • In Indo-European the situation *seems* confused: there is (weak) internal evidence suggesting that they counted in eights earlier. The fact that the word for eight seems to be dual in form (Latin and the earliest Germanic languages) may not be significant, but the indisputable fact that “nine” and “new” are very similar in Latin, Greek, and the early Gmc languages (even in modern German!) suggests that nine was the new number, displacing eight and letting ten appear as the basis. – Lubin Aug 13 '12 at 22:01
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    "BEWARE Ifrah's book has received highly critical reviews by experts" So what then would those experts have to say on the matter? – The_Sympathizer May 16 '13 at 05:38
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    @J.M. The word digit is used to refer to the numerals below ten because you can count them on your fingers, yes, but using that as evidence for *why* we selected ten as a base in the first place is bogus. Using the multiple meaning of "digitus" as evidence that we originally selected base ten because of our fingers is like answering "Why do we use strings of numerals to assign unique identifiers to individual phone lines?" by pointing out that "digits" is common slang for "phone number". Certainly the slang came after the choice, not the choice to justify the slang. – Marcel Besixdouze Sep 27 '14 at 20:29
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    @Marcel, I was merely pointing out a linguistic note, and was in no way implying that it was, as you say, "evidence". – J. M. ain't a mathematician May 01 '15 at 13:39
  • But there isn't really an advantage for using a base 10 system over, say base 16, right? Because, yes, until dec10 you can count with your fingers, but that's not a property that is bound to the numeric system... – Nearoo Jan 27 '17 at 18:30
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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" nonsense, 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.

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

Jens
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    "why do we all use base 10?" ... but "we" *didn't* all use base 10. Very perceptive. – futurebird Nov 03 '10 at 12:23
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    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 04 '10 at 06:22
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    @Sumrat: http://en.wikipedia.org/wiki/Numeral_system and related pages. – Jens Nov 04 '10 at 07:13
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    @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) – KutuluMike Jul 05 '12 at 15:10
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    "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. – Danubian Sailor Jun 25 '13 at 19:07
  • @MichaelEdenfield The positional system was new, yes, and made arithmetic much easier. But the choice of ten as a base goes back to Mesopotamian civilizations, and this particular choice doesn't make arithmetic easier at all. – Marcel Besixdouze Sep 27 '14 at 20:21
  • 60 is a superior highly composite number,which is why it's so useful. – Phonics The Hedgehog Aug 08 '16 at 15:21
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Because it makes the metric system so much simpler :).

David E Speyer
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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.)

PPJ
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    There was the russian Setun computer system among others. – ogerard Apr 13 '11 at 14:59
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    There's a Wikipedia article on [radix economy](http://en.wikipedia.org/wiki/Radix_economy) that gives the argument for base $e$ being the most efficient. – Simon Nov 21 '11 at 04:25
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    @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 06 '13 at 09:16
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    @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 08 '13 at 10:30
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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.

user02138
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    The thumb is not a finger? :) – J. M. ain't a mathematician Nov 03 '10 at 15:58
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    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 03 '10 at 16:04
  • I know, I was just obliquely pointing out something in your phrasing, which you now have corrected. :) – J. M. ain't a mathematician Nov 03 '10 at 16:06
  • 2 years later i will say, sorry but it's a bit out of reality. if you count your 4 fingers and the space between why yould you use your five finger on the other side. i just did it and it sounded completely wrong. – Nicolas Manzini Jul 05 '12 at 21:02
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    @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 05 '12 at 23:54
  • @user02138 i said the math is right, it's just not very practical – Nicolas Manzini Jul 06 '12 at 08:17
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    @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. – Danubian Sailor Jun 25 '13 at 19:10
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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 ....

Mark Bennet
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The reason is history and tradition. The decimal system is a convention that was adopted long ago and is so widespread and used that it would be enormously difficult to change it for any other system, no matter how advantageous it may be. This is not the only example, we have the gregorian calendar (rather crude), and the british imperial system of units, which one could argue to be "unnatural". Attempts have been made to adopt better systems, but as far as I know they have failed on account of the effort it would take to make such change.

Rogelio Molina
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  • There were many, many systems of units like the British imperial units. E.g . there were many feet, all slightly different just in Britain. Most of that nonsense was swept away by the French Revolution in France, adopting a "modern, rational" system of units (metrical system), which Napoleon's conquests spread over most of Europe. Today it is slowly working it's way across even recalcitrant countries like Britain and the US. It has already become universal in science. – vonbrand Feb 27 '16 at 01:25
  • @vonbrand : I was not aware of the rôle played by the French Revolution on spreading the metric system, thanks for pointing that out. On a side note, I am all in favour of the metric system and I look forward to see it adopted globally. Also, I now remember hearing somewhere that the foot for instance, changed in Britain with the king, but was more less constant in ''space'', whereas the unit lengths in Germany were constant in ´´time´´ but changed regionally, hence not in ''space''. Just something curious. – Rogelio Molina Feb 27 '16 at 14:51
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    In Germany I know there were several different measures in use, often for example the length of the lower arm of the tailor doing the measuring of cloth. In Britain I assume everybody used *their* foot as a rough measure, that was standardized sometime when precision became a necessity. Same for inch (in Spanish "pulgada", i.e., "length of the thumb"), and it's relation to the foot. – vonbrand Feb 27 '16 at 15:49
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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.

Alois Heimer
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  • "Not 2000 as it is now"? – JDługosz Nov 12 '15 at 23:32
  • 1kB would be 1000 Bytes in octal system (= 512 Bytes in decimal) not 2000 (in octal) as it is now (=1024 decimal). 'was just joking... – Alois Heimer Nov 13 '15 at 12:51
  • I see: all values after the first paragraph are in base 8. I don't think they would use 3-digit groupings and commas, though. When writing in other bases I always use a thin-space for grouping and group 4 digits typically. Historicly, octal computers did not use 8 bit words, but 9. Values were 18 and 36 bit sizes iirc. – JDługosz Nov 13 '15 at 13:20
  • The point I wanted to make is: Many programmers would have had an much easier job if they wouldn't have to recalculate decimal values into hexadecimal. – Alois Heimer Nov 17 '15 at 18:14