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My aliens have 2 thumbs for a stronger grip and the 2 thumbs are on opposite sides of the hand. However since there are 6 fingers total per hand and 2 hands my aliens naturally use base 12.

+1 for base 12

The humans on the generation ship use base 10 naturally because they have a total of 10 fingers except in rare cases.

+1 for base 10

Base 12 has more terminating decimals than base 10

+1 for base 12

Base 12 is hard for people to understand

+1 for base 10

Base 12 saves digits when it comes to really big numbers

+1 for base 12

Base 10 is confusing for the aliens

+1 for base 12

4 for base 12 and 2 for base 10.

Basically I can see a lot of arguments towards using base 12 and a lot of arguments towards using base 10. There is a lot of conflict in my brain going on as to whether or not I should use base 12 or base 10. I mean addition and subtraction are easy in any base but multiplication and division you would have to reteach to someone using base 10. If I could have a real earth-like planet all to myself and my family, I wouldn't mind teaching them base 12 arithmetic.

But I can't so I have to make do with what I have.

Anyway, should I use base 12 or base 10 in my generation ship story? I have seen a lot of people say "base 10" but those people probably didn't even take into consideration the advantages of base 12.

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – Monica Cellio
    Dec 16, 2016 at 3:54

9 Answers 9

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Step back from the context of worldbuilding for a moment, and look at this from the context of storytelling. What purpose does the number system serve in your story? In-universe a different base might have advantages, but does the reader care about that?

Are you trying to extol the virtues of a dozenal system to your readers? Is the difference in number system a source of conflict between humans and aliens? In-universe you can use whatever system you like. Whether you use base-10, base-12 or base-π is entirely up to you and 100% opinion based. But in the story, giving the reader an unwanted math lesson and forcing him to do unfamiliar math in his head will just be frustrating, possibly frustrating enough to make him drop your story. Conventions are useful. Unless it's actually relevant, stick with the decimal system we all know and love.

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Use Base Twelve!

There are actual people out there that think we, in real life, should count in base 12. It's called the dozenal system. People have historically counted in other units, like the Aztecs counting in 20.

A Numberphile video with Prof. James Grime claims:

  • the number of factors that go into 12 make it easier for multiplication and division
    • Multiples of Three in Dozenal: 3, 6, 9, 10, 13, 16, 19, 20, 23, 26, 29, 30 ...
    • Multiple of Four in Dozenal: 4, 8, 10, 14, 18, 20, 24, 28, 30 ...
    • See the patterns? Most of your times tables you learn as a kid are now easy!
  • You "don't" have .99999.... = 1 abuses and run into as many infinitely long fractions happening in dozenal (because 4/10 = 0.3)
  • "Everyday" math becomes easier because of these built in factors
  • Base-12 won't change the "serious maths" all that much. (The numbers just look different.)
  • The dozenal system "would be easy" to teach to kids, it's just us who grew up with the base-10 who would be confused.

More Arguments for Base 12:

The historic use of base-12 systems explains some of the oddities of the imperial unit system. (In dozenal) 10 inches is 1 foot? That's obvious! Why would you use some odd system like (in dozenal) 84 cm in 1 m? That makes no sense!

Also, English has words to count up to twelve. Beyond that we need special combinations for numbers. It was (likely) built on a base-12 system.

Count your knuckles on your fingers (on one hand), and you get 12. If we were supposed to use base-10, why don't we have 10 knuckles on our fingers? We'd loose two knuckles, and that doesn't seem natural at all.

Besides, Zommuter points out we already use base twelve for months in a year and hours in a day. Why bother with only occasionally using base 12, and just use it all the time?

I just included these arguments to impress upon you that many objections to base-12 systems can be turned around and used on base 10. The base of a numerical system is pretty arbitrary, although there are consequences when you do choose a system. The least of which is, you need new symbols for "ten" and "eleven." (See the numberphile video for some people's take on it.)

Why Base-10?

We only bother with base-10 because of the French and arabic (actual hindu) numerals. If history was a little different, then we may be counting in dozenal, or maybe some other numeral system.

And the final reason to use base 12... people won't accuse you of being uncreative when it comes to the math and language of these aliens. You're building an work of science fiction, go for it.

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    $\begingroup$ "Count your knuckles on you fingers, and you get 12" Citation needed? I assume you are looking at only one hand? $\endgroup$
    – cobaltduck
    Dec 14, 2016 at 21:27
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    $\begingroup$ You have the same 0.99999... = 1 abuse, just use your highest digit instead of the 9. $\endgroup$
    – Paŭlo Ebermann
    Dec 14, 2016 at 22:50
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    $\begingroup$ @cobaltduck There are 2 ways I could interpret that. On one hand you look at your 4 fingers and each has 3 joints/segments leading to 12, but that's ignoring the 2 joints on your thumb. Or possibly look at the spaces around each knuckle and you get 6 for each hand (4 between fingers and 2 for left and right of pinky and thumb). But that seems less intuitive. I do agree that it's confusing and I think trying to count your knuckles to 12 is far far harder than just counting your fingers to 10. $\endgroup$
    – Virusbomb
    Dec 14, 2016 at 22:56
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    $\begingroup$ base 12! = base 479001600. Take care with exclamation marks and maths! $\endgroup$
    – Zommuter
    Dec 15, 2016 at 10:05
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    $\begingroup$ @Virusbomb It's indeed referring to the segments of your fingers. The thumb isn't counted because, when counting on one hand in dozenal, you use the thumb to point to each finger segment in turn. It may be easier to count small numbers in base 10, but you can only go up to 5 on one hand, and to 10 using two hands. In dozenal, you can count to 12 on one hand, and to 144 using two. $\endgroup$
    – cbh
    Dec 15, 2016 at 15:52
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Can't the aliens use B12, the humans use B10 and computers change their display depending on who is using them?

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    $\begingroup$ This is certainly what I would do if I were an engineer on that ship. As a human, I would quickly and easily program the human's computers to accept, parse, format, and display numbers in either base-10 or base-12. My alien counterpart would do the same for his computers. It's called "localization" and is standard software engineering practice everywhere. $\endgroup$
    – A. I. Breveleri
    Dec 16, 2016 at 3:14
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The readers of your story are most likely going to be a regular user of base 10 (i.e. a human from Earth), so if you want your story to be readable you should use base 10 in any normal story telling, (i.e. the 23 aliens entered the room vs the 1B aliens entered the room), if you just start throwing out 1B ($23_{10}$) or A ($11_{10}$) as numbers you are just going to confuse people.

However if you were telling the tale from an alien's perspective a separate base could be useful for creating a definite sense of alien-ness in the narrative, but the numbering difference should be explained; have the human and alien characters have a brief discussion of the differences in numbering systems.

If you do choose to go base 12, for better readability I would recommend not using the normal method of showing alternate number systems (using letters to show numbers after 9 as I did above), but come up with names and or symbols to represent these numbers. One, two... eight, nine, snarg, blart; or 1,2...8,9,Ԇ,מ... you get the idea.

A simple answer for which number system would dominate in the ship is whoever built the ship would have used their number system throughout, which could cause confusion for the characters using the alternate system and make a nice place for a short discussion on the uses of alternate numbering system in the story. Remember keep any explanation short, unless your story is really about the virtues of alternate number systems and not an actual story.

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  • $\begingroup$ Hint: subscripts are commonly used to indicate number base. For example, $1\text{B}_{16} = 27_{10} = 33_8 = 11011_2$ says 1B in base 16 (hexadecimal) is equal to 27 in base 10 (decimal) is equal to 33 in base 8 (octal) is equal to 11011 in base 2 (binary). Of course you still need to agree on the symbols used to represent the values of each digit in whatever base you are using. $\endgroup$
    – user
    Dec 14, 2016 at 22:33
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What language are you writing in? What language is used on the ship, in universe?

The base of the number system is part of the language and if you are using English then you should use base 10. However if on board the lingua franca is that of the aliens then it would be understood that when a character says "15 days" it is actually a translation of $13_{12}$ '27 hour activity periods'. Since on board, days are longer and base 12 is being used.

You can hint at the use of base twelve on board by making references to dozens and gross. "I'll need five dozen soldiers" "3 gross! Thats a lot of mycocakes"

For the inhabitants, learning to use a different base would just be part of learning a new language. However unless you are writing in an alien con-lang, just use base 10.

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express base 12 within normal base 10 notation

You are writing in English (or some other human language) so everything is already translated. Look at words like “million” in languages that don’t group things the same way: Chinese will naturally refer to, for example, 2 (units of 10000) but you can’t translate that the way he said it because English doesn’t have a common word for 10000. So you would just write it as the value 20,000 which is the correct integer but doesn’t give the particular flavor of how he said it: two wàn. Don’t forget, he didn’t say “two” either, but 二 which sounds like “èr” so you are presenting a translation.

I recall a Niven short story where a crew had a complement of “eight and five” which shows the way the captain (using base 8) was thinking about it. But 13 is the same integer, the way we normally write it.

I suggest that you follow the literary convention of writing natural numbers in the language you are writing in. So if 37 appears in print we know it is decimal notation. But “round numbers” give a feeling of roundness that we lose if you just write 288. When that is important, and often enough to remind people, espress base 12 using the “round” units that English does have for this. So “2 gross”, not 12#200.

So if you want to convey that 37 is one more than a nice round number (like if I’m buying donuts) espress it as “three dozen plus one.” But that is not necessary for every use of every number.

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Use Base 10!

If you're worried about aliens using base 12 because they have 12 fingers rather than for plot relevant reasons, it's easy enough to justify the aliens independently developing base 10 arithmetic if you really need to bring it up. Here are two possibilities:

  • They count on the spaces between fingers, rather than on the fingers themselves. Some humans do this too, and their language uses octal!
  • It's awkward to manipulate all six fingers independently (e.g. physically uncomfortable to lower one thumb but not the other), so they can only count 0-5 rather than 0-6 using comfortable hand positions.

Or even more drastic approaches; if having 12 fingers isn't actually relevant to the story, make your aliens have 10 fingers, each hand having 3 normal fingers and two thumbs.

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You should probably go with whatever fits your story best. The reason we use base 10 is because it was the system used by mathematicians in the middle East during a time when the middle East was the greatest center of learning and culture in the world. One of the races must have been at least somewhat more advanced than the other at some point in the history of your world. You should use the base used by the race that had the most mathematical or scientific knowledge, as using that base would be a likely consequence of learning things from writings that use that base.

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Use base 10, you will makes less errors

With writing and writing again it will be difficult to be consistent on numbers and dates, so keep it simple.

If you give few citations on base 12, with translation in base 10, I think it would be enough for the readers, without confusing them (and without confusing because some introduced errors)

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