# Would there be any advantages in having a numerical system other than base ten ? **URL:** \ **Category:** Factual Questions\ **Created:** [January 29, 2013, 11:15pm UTC](https://boards.straightdope.com/t/would-there-be-any-advantages-in-having-a-numerical-system-other-than-base-ten/648752 "2013-01-29T23:15:01Z")\ **Posts on this page:** 16\ **Page:** 2
**Author:** ![MichaelEmouse](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/michaelemouse/32/101_2.png) [@MichaelEmouse](https://boards.straightdope.com/u/MichaelEmouse)\ **Post date:** [January 30, 2013, 3:57am UTC](https://boards.straightdope.com/t/would-there-be-any-advantages-in-having-a-numerical-system-other-than-base-ten/648752/21 "2013-01-30T03:57:12Z")
8 would seem pretty good: It follows the binary system. 3 doublings is easy enough to follow mentally for most people. It allows easy calculation in 3 dimensions which is useful for volume. We can still count on our fingers and use our thumbs to count up to 24. ---
**Author:** ![Sunspace](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/sunspace/32/1250_2.png) [@Sunspace](https://boards.straightdope.com/u/Sunspace)\ **Post date:** [January 30, 2013, 4:20am UTC](https://boards.straightdope.com/t/would-there-be-any-advantages-in-having-a-numerical-system-other-than-base-ten/648752/22 "2013-01-30T04:20:25Z")
> [@Jragon](#): > > Here’s a [Numberphile](http://www.youtube.com/watch?v=U6xJfP7-HCc) video on why Base 12 is easier… > > 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)… And you can follow “eighteen” and “nineteen” with “dekteen” and “elteen”, and also follow “eighty” and “ninety” with “dekty” and “elty”. “100” is still “one hundred”, but we’d have “dek hundred” and “el hundred” before reaching a thousand. Thus, 15χ3 = “one thousand five hundred dekty-three”. In base ten, (1\*(12^3)) + (5\*(12^2) + 10\*(12^1)) + (3\*(12^0)) = 1728 + 720 + 120 + 10 = 2578. ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [January 30, 2013, 4:54am UTC](https://boards.straightdope.com/t/would-there-be-any-advantages-in-having-a-numerical-system-other-than-base-ten/648752/23 "2013-01-30T04:54:47Z")
> [@Sunspace](#): > > And you can follow “eighteen” and “nineteen” with “dekteen” and “elteen”, and also follow “eighty” and “ninety” with “dekty” and “elty”. “100” is still “one hundred”, but we’d have “dek hundred” and “el hundred” before reaching a thousand. Thus, 15χ3 = “one thousand five hundred dekty-three”. In base ten, (1\*(12^3)) + (5\*(12^2) + 10\*(12^1)) + (3\*(12^0)) = 1728 + 720 + 120 + 10 = 2578. I believe the reason they say things like “do-one” and “do-two” instead of “ten” and “eleven” is because the names are misleading. “Nineteen” refers to a specific value in decimal that is not equal to the value that looks like 19 in dozenal, so they opt for “do-nine” because it’s clear then that they’re not the same number. Same with using “two-do” instead of “twenty” for 20. Same reason we don’t say “ten” for 10 in binary. Specifically it’s: one, two, three…, dek, el, do (pronounced “doh”), do-one, do-two, do-three… do-dek, do-el, two-do … three-do… dek-do, el-do, gro (12^2 or 144 in decimal), two-gro… mo (12^3). Not sure what the names are for powers of 12 above mo. ---
**Author:** ![Iggy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/iggy/32/3364_2.png) [@Iggy](https://boards.straightdope.com/u/Iggy)\ **Post date:** [January 30, 2013, 9:34am UTC](https://boards.straightdope.com/t/would-there-be-any-advantages-in-having-a-numerical-system-other-than-base-ten/648752/24 "2013-01-30T09:34:50Z")
> [@Bones\_Daley](#): > > … > 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 )… Well, the obvious advantage of base 13 is that we would know the question to the answer to the ultimate question of life, the universe and everything. What do you get if you multiply six by nine? 6[sub][SIZE=“1”]13[/sub] x 9[sub]13[/sub] = 42[/SIZE][sub]13[/sub] Or [maybe not](http://en.wikipedia.org/wiki/Base_13). ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [January 30, 2013, 12:42pm UTC](https://boards.straightdope.com/t/would-there-be-any-advantages-in-having-a-numerical-system-other-than-base-ten/648752/25 "2013-01-30T12:42:05Z")
Here is a very odd (and totally coincidental) advantage of a base 8 (or 16) system. When you look closely to the length of the year, you find that leap years ought to be omitted not every 133 years as we currently do, but every 128 years, which is 200 in base 8. The current calendar theoretically omits a leap year every 4000 years (it ought to be every 3200). But in general I think the advantages of base 12 have been exaggerated. BTW, the Babylonians actually used a mix of base 10 and 6 to get their base 60. ---
**Author:** ![Mangetout](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mangetout/32/19_2.png) [@Mangetout](https://boards.straightdope.com/u/Mangetout)\ **Post date:** [January 30, 2013, 1:14pm UTC](https://boards.straightdope.com/t/would-there-be-any-advantages-in-having-a-numerical-system-other-than-base-ten/648752/26 "2013-01-30T13:14:39Z")
> [@colonial](#): > > The Maya used a base-20 (vigesimal) system for several centuries during the heyday of their civilization. Yeah, but that made the world end just before Christmas. Hardly an advantage, I think. ---
**Author:** ![Doug\_K](https://avatars.discourse-cdn.com/v4/letter/d/b5ac83/32.png) [@Doug\_K](https://boards.straightdope.com/u/Doug_K)\ **Post date:** [January 30, 2013, 3:32pm UTC](https://boards.straightdope.com/t/would-there-be-any-advantages-in-having-a-numerical-system-other-than-base-ten/648752/27 "2013-01-30T15:32:39Z")
> [@friedo](#): > > (Four bits always equals exactly one hexadecimal digit or two octal digits; not true for base 10.) Almost, but two octal digits equals exactly six bits, not four. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [January 30, 2013, 3:45pm UTC](https://boards.straightdope.com/t/would-there-be-any-advantages-in-having-a-numerical-system-other-than-base-ten/648752/28 "2013-01-30T15:45:50Z")
> [@](#): > > Quoth **Sunspace** : > > 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. This seems to be implying that “the old systems” preferentially used 12 as a base. They did sometimes (inches in a foot, or ounces in a Troy pound), but certainly not always. Rather, they seemed more to just pick random numbers as bases for each application, and just coincidentally happened to stumble upon convenient numbers sometimes. You can justify 12 inches in a foot as having many factors, or 16 ounces in a pound as being a power of 2, but how do you justify 14 pounds in a stone? And don’t even get me started on 231 cubic inches in a gallon. ---
**Author:** ![Lucas\_Jackson](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lucas_jackson/32/303_2.png) [@Lucas\_Jackson](https://boards.straightdope.com/u/Lucas_Jackson)\ **Post date:** [January 30, 2013, 4:10pm UTC](https://boards.straightdope.com/t/would-there-be-any-advantages-in-having-a-numerical-system-other-than-base-ten/648752/29 "2013-01-30T16:10:31Z")
> [@friedo](#): > > I look forward to algebra students having to use χ as a digit. 3χY? Or 3XY? And then Satan said, “Let’s put X in math…” ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [January 30, 2013, 4:23pm UTC](https://boards.straightdope.com/t/would-there-be-any-advantages-in-having-a-numerical-system-other-than-base-ten/648752/30 "2013-01-30T16:23:24Z")
> [@White SIFL](#): > > And then Satan said, “Let’s put X in math…” I’m so tired of having to find everyone’s “x”. Math’s “x”. The professor’s “x”. The equation’s “x”. Look, it’s over, you assholes. Leave your poor x alone, you look fucking desperate and pathetic. MOVE ON. Next time I have to find an “x”, I’m politely suggesting a restraining order to it. ---
**Author:** ![gazpacho](https://avatars.discourse-cdn.com/v4/letter/g/6f9a4e/32.png) [@gazpacho](https://boards.straightdope.com/u/gazpacho)\ **Post date:** [January 30, 2013, 4:37pm UTC](https://boards.straightdope.com/t/would-there-be-any-advantages-in-having-a-numerical-system-other-than-base-ten/648752/31 "2013-01-30T16:37:00Z")
I don’t think that the Babylonians used base sixty in the way we think of bases. If you look at the wiki page on Babylonian Numbers they are sort of using a base ten system but not exactly because they are using the position of the figure to indicate a base to the n times the number as a value. There is a separate character for 10 and the double and triple etc to indicate 20,30,40 etc. This sort of thing is common in most old counting systems. I also don’t think that you can have useful multiplication tables without the idea of 0 and the position of the digit giving it a value. > **[Babylonian cuneiform numerals](https://en.wikipedia.org/wiki/Babylonian_numerals)** > > Assyro-Chaldean Babylonian cuneiform numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record. > The Babylonians, who were famous for their astronomical observations, as well as their calculations (aided by their invention of the abacus), used a sexagesimal (base-60) positional numeral system inherited from either the Sumerian or the Akkadian civilizations. Neither of the pred... ---
**Author:** ![yabob](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/yabob/32/2821_2.png) [@yabob](https://boards.straightdope.com/u/yabob)\ **Post date:** [January 30, 2013, 4:48pm UTC](https://boards.straightdope.com/t/would-there-be-any-advantages-in-having-a-numerical-system-other-than-base-ten/648752/32 "2013-01-30T16:48:21Z")
> [@Hari\_Seldon](#): > > Here is a very odd (and totally coincidental) advantage of a base 8 (or 16) system. When you look closely to the length of the year, you find that leap years ought to be omitted not every 133 years as we currently do, but every 128 years, which is 200 in base 8. The current calendar theoretically omits a leap year every 4000 years (it ought to be every 3200). > > But in general I think the advantages of base 12 have been exaggerated. BTW, the Babylonians actually used a mix of base 10 and 6 to get their base 60. The Gregorian calendar omits 3 leap years every 400. If you are referring to the further refinement which has been suggested from time to time to add a 4000 year rule, it’s never been accepted, AFAIK. Concerning leap years: A while back, I got curious as to the best leap year rule which would work by having a fixed number of leap years in a fixed length cycle of not too many years. So I used [continued fractions](http://en.wikipedia.org/wiki/Continued_fraction) to find the best rational approximation of the year length with a reasonably small denominator, and came up with 12053/33. Implying a leap year rule which would have 8 leap years in a 33 year cycle. It does indeed have less than half the error of the Gregorian rule (and since it corrects on a 33 year cycle rather than 400, it lets solstices and equinoxes wander off far less before getting adjusted). I then discovered I’d just invented the non-observational form of the Iranian [Jalali calendar](http://en.wikipedia.org/wiki/Leap_year#Solar_Hejri_calendar). Of course, nobody is going to stand for having to divide the year by 33 and look at the remainder to see if it’s a leap year or not. Just think, if we used base 33, we could have a more accurate leap year rule which simply depended on the last digit of the year … ---
**Author:** ![Saint\_Cad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saint_cad/32/18907_2.png) [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)\ **Post date:** [January 30, 2013, 6:18pm UTC](https://boards.straightdope.com/t/would-there-be-any-advantages-in-having-a-numerical-system-other-than-base-ten/648752/33 "2013-01-30T18:18:32Z")
One point is that a higher base has more compact numbers. For example 1000000 is 7 characters in base 10 but only 5 characters (F4240) in hexadecimal and only 4 characters in base 60. ---
**Author:** ![scr4](https://avatars.discourse-cdn.com/v4/letter/s/59ef9b/32.png) [@scr4](https://boards.straightdope.com/u/scr4)\ **Post date:** [January 30, 2013, 6:28pm UTC](https://boards.straightdope.com/t/would-there-be-any-advantages-in-having-a-numerical-system-other-than-base-ten/648752/34 "2013-01-30T18:28:52Z")
> [@Saint\_Cad](#): > > One point is that a higher base has more compact numbers. For example 1000000 is 7 characters in base 10 but only 5 characters (F4240) in hexadecimal and only 4 characters in base 60. And it’s only 1 character in base 1000001. At some point it becomes impractical. A base-1,000,001 number system will need 1,000,001 different characters for numbers, and a multiplication table will have over one trillion entries - good luck memorizing that. Even a base-16 multiplication table is more than 2x larger than base-10. ---
**Author:** ![The\_Hamster\_King](https://avatars.discourse-cdn.com/v4/letter/t/8edcca/32.png) [@The\_Hamster\_King](https://boards.straightdope.com/u/The_Hamster_King)\ **Post date:** [January 30, 2013, 6:31pm UTC](https://boards.straightdope.com/t/would-there-be-any-advantages-in-having-a-numerical-system-other-than-base-ten/648752/35 "2013-01-30T18:31:27Z")
If we used base 12 instead of base 10 then percentages would work off a base of 144 instead of 100. That would be nice for gaming because you could roll 2d12 to generate percentages and any excuse to roll more d12’s is a Good Thing. ---
**Author:** ![Greg\_Charles](https://avatars.discourse-cdn.com/v4/letter/g/839c29/32.png) [@Greg\_Charles](https://boards.straightdope.com/u/Greg_Charles)\ **Post date:** [January 30, 2013, 6:39pm UTC](https://boards.straightdope.com/t/would-there-be-any-advantages-in-having-a-numerical-system-other-than-base-ten/648752/36 "2013-01-30T18:39:45Z")
> [@Bones\_Daley](#): > > 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 … You know how it’s easy to tell if a number is even or divisible by 5, even a number with a dozen digits or more? You just have to look at the final digit and you know. If we used base 12, we could do the same to determine if a number is even, divisible by 3, 4, or 6, though not 5. The flip side of that is that it’s easier to do thirds and fourths as fractions. A fourth of a foot is 3 inches and a third is 4 inches. A fourth of a meter is 2.5 decimeters and a third is 3.33333… It’s easier to find a fifth of a meter than a fifth of foot, but that’s less useful than being able to find thirds and fourths. The ideal case would be to have a base 12 number system, and a metric system based on 12s as Sunspace suggested. That will never happen though. The effort it would take to transition the basis of our numbering system would be truly staggering. [Previous page](https://boards.straightdope.com/t/would-there-be-any-advantages-in-having-a-numerical-system-other-than-base-ten/648752.md?page=1)