# Numeral system for a species with 8 fingers

In a nutshell, I've an 8-fingered bat-like humanoid (let's call them AX) and a 10-fingered 'standard humanoid' species (let's call them BX) sharing a planet.

AX was the first to develop things such as agriculture and animal keeping. They were the first to trade as well, and thanks to being able to cover vast distances by wing they worked as traders for millennia. With all this in mind I'd expect them to have been the first ones to decide on numbers, and have decent amount of influence on BX when it comes to counting, etc.

However, they only have four fingers per hand and I don't know how that'd affect things. It makes me wonder whether they'd be more likely to gravitate towards a base 8/9 (fingers/phalanges) counting system, for example? Or even a base 36? Would it be likely they'd end up coming up with a base 12 regardless?

I don't want to handwave anything or make uneducated decisions, so I'm hoping someone more educated than myself could help me out.

• edit: I've clearly made a mistake in not describing AX in more detail. [They're bat-like humanoids], are 2.5m tall and capable of flying with 60kg cargo... hence why I'd thought they could make ok traders. But these are a fairly new creation so I don't mind changing things that don't make sense.
• If A was the first to trade, and only traded in base 2 or base 4, then chances are B would already do the same. You could make it similar if they used a finger to count the joints in their fingers. E.g. A has 4 joints in each finger. They use a single finger to count their joints to track of how large the number is. B is a standard human with 3 joints. They use the thumb, to count 4 across to get the same base as A. (I'm doing a bad job explaining, but A uses thumb to count individual joints in finger, B uses thumb to count joints across the fingers.) Jan 8, 2018 at 6:18
• Having wings wouldn't make them good traders. Ability to move big cargo is what makes good trade. Why to bother trading gemstones if you can't buy food? Would be a hobby, not profession, if they literally can't earn a living that way. Jan 8, 2018 at 6:43
• As a side-note: We do still have remains of a Base12-Thinking. Think 24-hour-clocks (2*12), or the use of a "dozen". Even the words for numbers still show this in English (and German) at least: The "-teen" ending only starts on Thirteen, after all: No Oneteen, Twoteen before the thirteen! Jan 8, 2018 at 8:06
• @Renan with carts and boats wings are encumbrance, not help. That's my point. Jan 8, 2018 at 17:10
• I agree with @Molot that wings are a hindrance. Just look at the x-ray of a bat. davidmaybury.ie/wp-content/uploads/2013/07/… Opposable thumbs are impossible, and the bones are so lightweight that you can't pick up anything heavy. Jan 9, 2018 at 0:14

They could use base 8 without any handwaving, and it would be quite interesting. There is more to it than you may think, though.

We humans in the real world tend to think that everyone uses base 10 because that's just natural, or because it comes with having ten fingers. And Tom Lehrer (famous math teacher and musician) once said that base 8 is just like base 10, if you are missing two fingers.

In practice, though, base 10 stuck more for being practical than for being natural. It makes things easy for the bulk of the calculations that a regular person will have to do during their lives.

But there are so many things that are easier with radixes other than 10! That's why computers work with a binary base. IT professionals and computer scientists will often use base 16 in their jobs, and once you get used to it you can do mental and napkin calculations in hexadecimal.

Also notice that the usage of alternative radixes is not something that came up with recent technology. See Wikipedia's entry on vigesimal for a plethora of old and ancient mesoamerican, african and european cultures that used base 20. And the traditional weight measures of the chinese is built on base 16 and used to this day and age.

Back to your species A and B. They may each use their own numeric base, and the way they convert things from one system to another would feel like a simplification of conversions from metric to imperial (and vice versa). This would seem realistic.

But if you think that this adds little to your world and stories, you can write it off and weave your stories so that one numeric base was either imposed by force or selected for practicity. This would open the possibility for interesting stories on how the current system came to be.

All possibilities are equally belieavable and require no hand waving nor suspension of disbelief.

If you want more food for thought, see my answer for Aliens doing algebra. It is about how other species may abstract numbers in different ways, and how it would feel for each species to learn each other's algebra.

• Programmers used octal when computers used 8 bit bytes, since it was a short version of Base2 for larger numbers. Same with hexadecimal when we went to 16-bit computers. Even after computers went to 32- and 64- bit, we stuck with hex, which I think was more to do with not "inventing" more symbols for the larger number systems than anything. These are all still essentially Base2, just condensed. Jan 8, 2018 at 14:33
• What arithmetic is easier in base 10 than in other bases? I really can't think of any significant difference between bases. It could even be argued that base 10 is less convenient, because it has fewer factors than other bases like 12 or 30. Jan 8, 2018 at 15:04
• You state that "base 10 stuck more for being practical than for being natural". Why? How is it more practical? If everybody was using base 8 or 12 I don't quite see what would be more practical about base 10. On the contrary, base 12 has the advantage of easier division by 2, 3 and 4, which is definitely more practical for many situations. Jan 8, 2018 at 17:14
• @computercarguy octal was actually used on computers like the PDP-8 that had a word length of 12 bits. Octal encodes 3 bits, so it's useful on systems where the native word length was a multiple of 3. Hex (which encodes 4 bits) has always been preferred on 8-bit systems and multiples thereof. Jan 9, 2018 at 7:05
• "In practice, though, base 10 stuck more for being practical than for being natural. It makes things easy for the bulk of the calculations that a regular person will have to do during their lives." Circular reasoning alert! The reason base 10 makes those calculations easy is because we use base 10! If we had eight fingers and used base 8, the calculations would be just as easy in that base because it would be the natural number base we would be accustomed to. Jan 9, 2018 at 9:13

### Use whatever you want, no hand-waving needed

Historically there is no strong relationship between the number of fingers and the base of the number system used. The Mayans base was 20, the Babylonians base was 60 - factors other than fingers tend to be dominant.

number of fingers != base of number system

It has taken the world thousands of years to converge on decimal and metric. There are still many people that say the base 12-ish imperial system is better than metric. The English had a non-decimal monetary system till 1971. Roman numerals are still used.

I’m sure if trained from birth humans could easily handle a wide range of bases. I think the limit would end up being human working memory rather than fingers.

If the bat race had a good base 8 system humans would use it for thousands of years without any complaint.

• Your interesting answer doesn't discuss the fact that base 20, base 60 and the Roman numerals contain the prime factor 5, most probably caused by the number of fingers of a hand (or toes of a foot). So that anatomic feature had quite some impact on our way of expressing numbers. Jan 8, 2018 at 18:16
• Base 60 might actually have been chosen because it was a Highly Composite Number. But...that observation wouldn't be noticed unless there was an existing number system (and then everyone switched over). Jan 8, 2018 at 20:24
• @RalfKleberhoff they also both contain 4 and 2 as prime factor. As Draco18s mentioned using a highly composite number as a base is quite helpful. It’s one of the reasons people use to support the imperial system. 12 has more factors than 10 which can make doing maths in your head easier. In this case of OP, the proposed 8 has the same number of factors as 10. Also 8 being a power of 2 would likely make its interactions with binary systems easier too.
– Nath
Jan 8, 2018 at 22:55
• @Nath The big advantage for using 12 as a base for measurements instead of 10 is that the human brain is really good at estimating halves and can do thirds with a little practice. As such bases that have prime factorizations of only two and three lend themselves more easily to reasonably precise measurements by eye or by feel. For a modern system base 8 or 16 would make a lot more sense since they require only halves and get along nicely with binary computers, but we've got what we've got. Jan 8, 2018 at 23:27
• This is the best answer IMO. There's a base-27 counting system in one (human) culture where they count up from one hand, using parts of the arm and face and down the other arm and hand. Past 27 it's effectively "one-man-plus-[x]". mentalfloss.com/article/31879/… Jan 9, 2018 at 16:52

The base one chooses for numbering is purely a convention. A similar convention is used for alphabets.

In case of the alphabet, Phoenicia was the first to develop a versatile system and spread it into the Mediterranean Sea with their commerce. Most of the civilizations born afterwards in that area used then the same alphabet (even I do while I write this answer).

Using an octal numerical system is as good as any other base, and considering they have 8 fingers is the most likely to be chosen.

Regarding other numerical bases, their mathematicians can of course discover them, as we did. They will not be knowingly used by the masses, though. (there are 10 types of persons: those who understand the binary, and those who don't)

• The base is a little more than convention. Your implementation of operations differs between bases, and some hardware can implement some bases better than others. That's why computers universally use base 2, rather than base 10 (which would be more convenient for us humans using those computers). The hardware implementation of math using current silicon technology is easier to do in base 2 than any other base. Jan 8, 2018 at 15:52
• The definition of operations does not change between bases (since bases are just an encoding of a number, and mathematical operations are defined on numbers, not their encodings). The implementation is not important unless it gives the wrong result (and modern computers, unless carefully designed to avoid it, do give the wrong result for values like 0.1, but we still get along fine). Jan 9, 2018 at 7:14

The idea that water takes the path of least resistance is fundamentally true for everything. No one would count digits/not thumbs, flanges, knuckles, etc. That's too much work. Your bats would develop base-8 and the humanoids would simply deal with it.

But, if you think about it, human counting hasn't always been base-10. We've done odd things for odd reasons. Like hours in a day, or pre-decimalization British money and pieces of eight.

So, while your batoids would most likely count base-8 and that will thow an amazingly interesting curve ball into the mix. But you need to focus as well on how numbers are used because that will affect things, too. These influences include the whims of kings, religion, practicality, and a host of other non-count-on-my-fingers reasons. Your humanoids can easily accomodate base-8.

But, if you think this is fun... wait until your society needs to leave the gold standard because the good, honest folks have discovered clipping coins is more valuable than the original coins, themselves!

• I like this, have them use base 7 becasue the great King Batafrax III 1600 years ago lost a digit in battle and people switched from base 8 to base 7 to honor him. All dates BB (Before Batafrax) must be calculated differently, as do census numbers etc etc. Jan 8, 2018 at 8:16

The Mayans used base twenty, so it is possible that your aliens might use a base like sixteen. That counts fingers and toes (assuming they have the same number of toes as fingers).

They could also go the other way. If you want both races to use the same numbering system, it sounds like both have four limbs. So make it base four. That would probably be an easier system to share than either base eight or base ten. That would help reinforce the idea that the two are interdependent with a long history of working together.

Having them use base eight or base ten would indicate that historically one has dominated the other. So one would adjust to the other.

Having each use their own base would indicate that they are mostly independent of each other.

Another argument in favor of base four would be if they would use their feet to manipulate things rather than hands. Consider that the "hand" of a bat is the wing. Note so good at manipulation. But bats use their feet to grab things. And a typical bat would need to use one foot to maintain its perch. So situations when it could use both feet at once would be rare. Therefore, they might count with just one foot.

• The more comments I get the more I'm more worried for AX's design makes no sense. :'D They're meant to have 5 toes a foot and 6 a hand (two contribute to flight, the rest form a fairly human-ish hand) and they stand upward. They're engineered is my excuse. -am writing more, just didn't know 'enter' means 'submit'-
– Haan
Jan 8, 2018 at 8:40
• - continuation form last comment: You gave me some good stuff to think about. The two species are meant to be more interdependent than not, so from that point of view the base 4 would really make sense. Thanks for your input!
– Haan
Jan 8, 2018 at 8:54

Larry Niven's Kzin had 3 fingers and a thumb, and counted in base-8. So there is fictional precedence for this.

Here on Earth, the Mayans were not the only people counting in base-20 either. Traditional English counting systems also used it, evolving independently of other European systems of numbers.

As other answers have said, the AX will develop something appropriate to them. When the BX become numerate, they'll learn to do it the AX way.

## Just counting fingers is too base-ic

Our base 10 number system is theoretically there because we have 10 fingers but for the same reason, that being physical traits, there are (or were) societies that have other counting bases. 20 is a common one to bring up because toes exist but iirc there was a small country or tribe that had a base 24 because that is the number of sections on your fingers (no thumb included) and yet another that was base 18 because they counted up their arm across their shoulder and then onto the other hand rather than just using their fingers (fingers, wrist, forearm, upper arm, shoulder, repeat down other side).

With this in mind you can see that just counting the number of fingers is a very base way to count and actually one of the most boring. You can find any number of interesting ways that fit within your story to give them a weird number system or even share the same number system with their more standard humanoid friends. Maybe the bat people came up with base 10 even with 8 fingers because they count like so: 1,2,3,4,wing(5),6,7,8,9,wing(10), which could give them a base 10 system with a naming scheme based off 5s (or even a base 5 number system).

AX-AB Relations

Assuming that the different species evolved all the way from small cells and upwards, then there was a vast amount of time where neither species communicated in a helpful manner, such as before each species developed complex spoken language. I buy into the developmental argument and think that humans tend to, when left alone, develop a base ten counting system (generally but not always).

Furthermore, as @Renan pointed out, Base 10 is quite useful. If the AX species formed complex society and trade, then it is possible they discovered base 10 on their own and use it for certain fields of work (much like Computer Scientists use base 2 and base 16 today). Furthermore, if they were exposed to human cultures using base 10, then it is quite reasonable to think that they may have adopted it for human-developed projects (say humans got to calculus first or something similar). In the same vein, traders that do business with humans would certainly know base 10 (given humans using it), so they could join mathematicians and educators as AX beings fluent in base 10.

This would work the other way around too. BX/Human merchants, educators, administrators, and translators would have significant use for knowing base 8 or whatever the AX species uses. This co-mingling of numerical systems could be an early path into the growth of mathematics in your universe. Systems used by traders to convert from base 8 to base 10 might be the earliest form of mathematical formula or computational systems, leading to further discoveries and societal progress.

If the AX species is dominant globally or in certain regions, you can bet their number system will also be dominant in these areas. But I think that both number systems could co-exist if you wish. On the other hand, you could use this difference as a wedge to be driven between the species, causing conflict. I love this little world your building!

First not all numerical bases are not equals. Powers of 2 are great in computer world because the basics of all computer is binary: there is some current or not. But for human beings, acceptable bases are ones that can be easily processed, that is where it is easy to multiply and divide by small numbers. 12 is one of these bases because 12 is divisible by 2, 3, and 4. But a non injured human being has five digits on each hand, so divisibility by 5 is important.

Thus, it is not by accident if many civilizations have used at a moment a base 60 system, because it is the smaller number that can be divised by 2, 3, 4 and 5 - BTW you gain divibility by 6. Other common bases were 12 (2, 3, 4, 6) (12 hours on a clock, 12 pence in a shilling in old monetary systems), 20 (2, 4, 5) 20 shillings in a pound, both being divisors of 60. For the other divisors, numbers 6 and below were too small, 15 is not divisible by 2, and 30 while mathematically interesting because is is the smallest number divisible by 3 primes, is rather great and not divisible by 4 when dividing in 4 is a common operation (think of a pie...)

So for a species with 8 fingers, the number 5 is likely to be less important, so the natural bases should be 8, 12 and 24

Beware of the number of digits. Even if base 2 is really convinient, the number of digits needed to write a number escalate really quickly, this is one reason why we don't use this kind of small base. Base 5 (like on one hand or 4 for your bat-like humanoïd) follow the same logic.

On the other hand, with a base 36, you will need 36 symbols to represent each digits. Our alphabet doesn't have that many symbole, it could be really tricky to calculate with this.

That being said, I don't really know how bats works or how the AX mind could work, but they realistically could develop a setup to uses those radixes or already have a setup to come up with it

• +1 for mentioning the issue with number of digits needed to represent a number vs. number of symbols needed to represent all numbers. Jan 10, 2018 at 13:17