# Number of Fingers for a Math Oriented Race

Form follows function, or that's how the saying goes. Which isn't true for some things. That is to say, a large reason why humans use a base 10 numbering system is because we have 10 fingers, not that we have 10 fingers because we use a base 10 numbering system. Except as worldbuilders, we get to do the reverse and create an alien race with the precise number of fingers that would be the most useful.

In my sci-fi setting, I'd like the most advanced race to have an extra advantage - a superior numbering system, using base 16 instead of base 10. 16 is a perfect square, almost a perfect number, and uses squares of 2 as it's landmarks of higher numbers (it's 'ten's place' is 256, it's 'hundred's place' is 4096, etc.). It handles larger numbers far better than a Base 4 system would, and easily converts to binary.

So now comes the real question - how do I design the aliens such that a base 16 system is organically produced? After all, number systems aren't designed in the modern. Eight fingers on one hand is too much for my tastes, and a four-armed, four-fingered alien on two feet doesn't seem like it would naturally arise. And, considering this the 'smart' race, I'd rather not have a solution that comes up as a result of combat with each other, something which the four-armed variation definitely suggests.

Two points - one: Yes, I know there are ancient 16-bit numbering systems that have survived to the modern era. However, they don't see widespread use today. I'm looking for a way so that I don't have to handwave and say they just developed a system like that and took over. Second - an answer which proposes a different base system will be accepted, so long as you can definitively prove it superior to my base 16 for the purposes of a sci-fi setting.

EDIT: Reminder, I'm not asking which Base numbering system is superior. As far as I'm concerned, they've all got strengths and weakness, and 16 is what I find easiest for my setting. The question is a xenobiology question about alien design and development.

EDIT II: The most common numeral system used currently, and historically, is the Hindu-Arabic numeral system, as it is so called. It is Base 10, and spread throughout the world to become the backbone of math, which has only recently begun to commonly use other bases because of programming.

• Comments are not for extended discussion; this conversation has been moved to chat. – James Aug 28 '19 at 17:27
• What if the number of fingers wasn't standard across species (perhaps it varies by area in a similar way to hair colour with humans), so that individuals had to become adept at converting between different representations in order to communicate? This achieves the 'math-oriented race' part of your proposal without tying you to a specific base (perhaps in written mathematics it's normal to use whatever base is most convenient). Commenting as I can't post an answer. – dbmag9 Aug 28 '19 at 21:54

The same way humans can do base 12 on our hands, use the bones of the fingers, excluding thumbs (or using thumbs to keep track of sets completed).

So, take human physiology and either add a joint to each finger (16 bones on each hand) or remove a joint (8 bones on each hand, for 16 total).

• Way easier: count with the thump: index finger: fingertip (1), first knuklebone (part of finger) (2), 2nd part of finger (3), 3rd part of finger (4), continue to small Finger: fingertip (13), 1st part of finger (14), 2nd part of finger (15), 3rd part of finger (16). It is totalliy possible to count 1 - 16 with one hand (nomal 5 fingers) – Julian Egner Aug 29 '19 at 13:05

That is to say, a large reason why humans use a base 10 numbering system is because we have 10 fingers, not that we have 10 fingers because we use a base 10 numbering system.

Ask any anthropologist, physiology does not constrain counting base. We like to think that base 10 is natural because we have ten fingers, and it is common, but that's not the only way humans count.

Numberphile/Tom Scott has a great episode on how human societies around the world "naturally" count, and it's not all base 10. He demonstrates languages and body counting schemes of many bases, using varying parts of the body. Here's some ideas beyond just the fingers.

• Tips and joints on fingers
• Fist
• Knuckles
• Spaces between the fingers
• The palm
• Wrist
• Forearm
• Elbow
• Armpit
• Shoulder
• Collarbone

Fingers + fists is 12. You can get 16 or 32 using the tips and joints of the fingers, maybe using the thumb as something else. The spaces between fingers and thumb are 8, turn your hand over and its 16.

And humans have plenty of non-base 10 linguistic constructs. Here's some examples in just Western languages.

• English: Four score and seven years ago
• French: soixante dix-neuf (60 + 10 + 9)
• Welsh: pedwar ar bymtheg a thrigain (4 + (5 + 10) + (3 × 20))

You can use this for an interesting little scene where, perhaps, a human observes your aliens have X digits and so "obviously" they must use base X. Only to have the alien incredulously demonstrate their own obvious-to-them method of counting.

• This is the answer I was looking for. OP's question rests on something that isn't actually true. – Michael W. Aug 28 '19 at 15:52
• This. Also, with 10 fingers we can represent 11 symbols, so our natural base is base 11. – Blueriver Aug 28 '19 at 17:56
• @Blueriver So much more than 11. I'll update. – Schwern Aug 28 '19 at 18:14
• Well, I don't think there is any other reason for base 10 to be most common than us having 10 fingers, because it is not a particularly practical choice in any other aspect. Base 12 is way more practical. – Jan Hudec Aug 28 '19 at 19:49
• @JanHudec We don't always do what is practical nor optimal. – Schwern Aug 29 '19 at 6:29

Just go with those four-fingered-hand aliens. Let me remind you, despite having 10 fingers, for quite some time, base 12 was very popular parallel system and there were even attempts to discard base 10 in favour of base 12. It still survived alongside base 60 in time measurement, and word "dozen" is used till this day. It doesn't need to develop very organically, and 8 fingers is already good enough start. Just assume that originally, they used base 8, but they also used base 16 for certain things like dividing their day into hours, etc.

However, base 16 and base 8 aren't as good as you'd expect. They're indeed good now in days of computing, but base 12 and base 10 had two big advantages over offshoots of binary - division. Base 12 allows simpler division for 2, 3, 4, 6, 8, 9 and base 10, while worse, still allows 2, 4, 5, 8, 10.

It's more likely that such a species would go for base 12, to eventually replace it in some areas by base 16 for math and science, but base 12 would likely stick around as something normal people wouldn't easily leave behind.

A reminder of how people can get stuck in their ways is how, we could easily divide day into 20 'hours' of 100 'minutes' each, redefining meaning of hour and minute, but we don't do it.

• You can argue that 12 is not a good candidate for a base, as it needlessly includes the factor 2 twice. I'd expect either a use of base 6 (simpler, more efficient) or 30 (include 2, 3, and 5 as factors). – cmaster - reinstate monica Aug 28 '19 at 1:31
• I'm sorry, but I have to disagree with you. To understand why 12 is a good candidate, you need to understand where was base 12 used. Base twelve used dozens and grosses(144), and was used by people who mostly operated with these numbers. For example, they brought six dozen and five eggs. Or a merchant might have four grosses three dozens and two pieces of currency. Base 12, unlike base 6, is large enough to cover most numbers people used with units, dozens and grosses. Meanwhile to achieve similar coverage, you'd require 1, 6, 36, and 216. 30 is impractical - too many numbers. – Failus Maximus Aug 28 '19 at 2:01
• @cmaster, if you consider how often you have to divide something 2,3,4 or 5 ways that second factor of 2 makes good sense rather than something to be removed. Hence 12, 20 and 60 being recurrent themes. – Separatrix Aug 28 '19 at 9:28
• @Separatrix 60 is the smallest base in which dividing it by 2, 3, 4, 5, 6 all result in an integral number of elements. The "duplicated" 2 there ends up improving things. Dropping to 30 means that dividing by 4 gives you a half-unit, which is a pain. – Yakk Aug 28 '19 at 14:16
• @cmaster: 12 is a very natural candidate for a base if your alien race cares about music; the sequence of notes-per-octave that are unreasonably good goes something like 7, 12, 19, 31, 53, 171, ... depending in part on how you weight the fundamental ratios (this list gives 3/2 weight 8, 5/4 weight 6, 5/3 weight 4, 7/4 weight 2, 7/6 and 7/5 weight 1, no other ratios considered). Of those 12 is a nice small even highly-divisible number whose biggest problem is a pretty sharp major third (+14 cents when most people can tell tones at ~5-10 cents) – CR Drost Aug 28 '19 at 20:15

Your aliens are smart. So they use the four fingers they have on one hand to represent all their 16 digits by using a different combination for each:

• 0 = closed fist
• 1 = only thumb stretched out
• 2 = only index finger stretched out
• 3 = index + thumb
• 4 = only ring finger (there is no middle finger!)
• 5 = ring + thumb
• 6 = ring + index
• 7 = ring + index + thumb
• 8 = only pinky
• 9 = pinky + ...

With their two hands, they can either remember two different digits at the same time, or put them together to form a 2-digit number. This allows them to count up to 255 with their fingers.

Your aliens actually envy the humans for our fifth finger, realizing that humans are able to count up to 1023 on their fingers alone... if only they would recognize the full power of their ten fingers!

(Full disclosure: Even humans can learn to use their fingers efficiently, counting mindlessly from zero to 31 on a single hand, and I'm proof of that. I only need to make sure that I don't inadvertendly show someone a 4... 1 is ok, 6 is ok, 18 is reserved for concerts...)

• I've used this method to do binary arithmetic for years. 0-15 on 4 fingers, 0-31 on one hand, up to 1023 on both. It's... well, handy :) – Corey Aug 28 '19 at 4:12
• Unfortunately our ring finger doesn't work properly so counting above seven is difficult. In addition, you binary representation is difficult to learn, a Gray code is far more likely - closed fist, thumb, index+thumb, index, ring+index, ring+index+thumb, ring+thumb, ring, pinky+ring, pinky+ring+thumb, pinky+ring+index+thumb, pinky+ring+index,, pinky+index, pinky+index+thumb, pinky+thumb, pinky – JCRM Aug 28 '19 at 10:47
• @JCRM The thing about the ring finger is training. Also, the counting rule for binary is simpler than the counting rule for Gray code: With binary you always set the first finger that's not set and clear all the ones before it. This rule easily enters muscle memory. With Gray codes you would need to remember/compute somehow which finger is the next to toggle, the rule is much harder to get into muscle memory. As I said, I regularly count with binary. Usually, I count without thinking about it, then I look at my hand and say "oh, that were 23 frobs that I counted". – cmaster - reinstate monica Aug 28 '19 at 11:41
• I've tried counting binary on one hand with each finger representing a single bit - it's not ideal. The one's place finger has to go up and down every single time. I often use a decimal alternative where each hand represents one decimal digit - thumbs are worth 5 and the other fingers are 1 each. Very intuitive for decimal species. With 1 less finger per hand, you can convert this method to octal very easily (thumbs worth 4 instead). From there, it's just a matter of doubling to get hexidecimal. – Darrel Hoffman Aug 28 '19 at 18:01
• @DarrelHoffman "The one's place finger has to go up and down every single time." That's why you use the thumb as the least significant bit, for both hands. I won't deny that it needs some getting used to, but it definitely works well once you've trained it a bit. It's one of these silent skills like touch typing that you just don't learn in a day or two, but which come in very handy once you've got the hang of them. The hybrid approaches that you explain may work reasonably as well. You loose a significant amount of number space (1024 -> 100) for the sake of increased decimal compatibility. – cmaster - reinstate monica Aug 28 '19 at 18:54

## Give them prehensile feet

4 fingers on each of two hands.
4 toes on each of two feet.

The feet don't even have to be fully prehensile. Just usable enough that they'd think to count them. After their civilization creates their math system, their culture can also move to one that wears shoes and so forth. The feet don't have to be highly visible today.

## Make it a cooperative culture

If the math system comes from two people working together, then two pairs of hands—each with 4 fingers—gives you your perfect 16.

## Count other body parts

When my daughter was learning math, I encouraged her to count on her body to get the process started. For numbers higher than 10, we sometimes used my fingers, sometimes her toes, and other times her face.

8 fingers + 2 ears + 2 eyes + 2 lips + 2 nostrils = 16

Or use any other body parts you desire, including ones that might be particular to this species.

The base in which numbers are represented doesn't change, ease, or inform mathematics. All it changes is arithmetic.

The number of digits, arms, feet, eyes, knuckles, gill slits, or kinks in their prehensile tails won't change the brilliance, insight, and utility of the math your species develops.

• I get a little bit disgusted at how many entries in OEIS.org are all about properties of the decimal representations of numbers! – Anton Sherwood Aug 28 '19 at 20:01

Why not a primate like species with three fingers and an opposable thumb on each hand for a total of 16 digits. Primates are largely adapted to use both hands and feet for grasping with Hominids (those silly creatures we call humans) being an exception (they have less dexterous feet than most primates, as an adaptation for bipedial locomotion necessitated some design changes. Still, it's not unheard of for a human to be quite adapt at dexterous manipulation of tools with their feet and some practice. I once met a man with no arms, who played a mean guitar and had a specially adapted steering device he used to drive a car).

The counting system of base 16 would evolve from using all four limbs in counting, since their society would likely have equally used all limbs for movement.

I like the concept you are going for (an organic justification for a numbering system and how it influences the alien civilisation).

I would say (1): On what basis are you saying 8-fingers per hand is too many? Their hands might look more like spiders than human hands. (2) Why does four arms seem unlikely or imply combat specialisation? I don't "get" the reason why 4 seems intrinsically unlikely or violent - they could be for climbing.

(3) Biology dictating the base is cool, but it doesn't have to be fingers, or even any kind of body part any real animal actually has. As one example: perhaps your aliens breathe through their skin, and thus are unable to talk for communication. Instead perhaps they posses 16 nodules on their bodies that they can make change colour in chameleon style (or make glow or whatever) - they use these to "talk". In this setup it seems almost inevitable they would use base 16, their word for "two" might just be to make two of the nodules turn red.

If you wanted symmetric bodies like ours, you might consider having 3 phalanges and an opposable thumb-like thingy, but each digit has two independent tips. That gives you a natural 8 counters per hand, and with two hands you get to 16.

They'd look normal wearing mittens, normalish wearing gloves, and might be kind of a thing of nightmares barehanded. Like stubby octopi.

Make them truly double-jointed.

Some people count on fingers by starting with a closed fist, and opening them. Some people start with an open hand, and close the fingers. Your race instinctively do the latter, but their fingers bend both ways. So, you bend the fingers of one hand "in" (1-2-3-4), "out" (5-6-7-8), then shake them all about swap to the other hand (9-10-11-12, 13-14-15-16)

Thumbs are used for keeping track of "up", or for indicating positive/negative numbers

(You can simulate something similar yourself, by having 1-2-3-4 being "fingers as far down the palm as possible", and 5-6-7-8 being "finger curled over but fore knuckle protruding", for example)

# Start base 4 extend to base 16

Your alien race has 4 fingers on each hand and since the first invention of "counting" they got used to base 4 for everyday small stuff.

Once their knowledge grew and everyday use contained larger numbers they started grouping up their digits in pairs. For examle: 10.31.02 (= 1234 base 10)

After a while their symbols for the pairs of digits got mixed up and people started counting in base 16 using the combined simbols as if each of them are only a single digit. The words for the numbers might have already been combined, similar to how we have the words "eleven" or "twelve" instead of "ten and one" or "ten and two".

Notice how these changes could happen at any time and relatively early on in development. Despite having only 4 fingers, they organicaly developped a base 16 numbering system just like you asked for.

• The race starts thinking of "hands" and "fingers", so 1=1F 2=2F 3=3F 4=1H (0F) 5=1H+1F 6=1H+2F Eventually, they start writing numerals in a way analogous to Mesoamerican base-20: 1 = . 2 = .. 3 = ... 4 = _ 5 =_with a dot on top of it 7=_ with three dots 8= = 12= triple bar 15=triple bar with three dots above – Monty Harder Aug 28 '19 at 20:45

## 1. Bifurcated fingers.

First there is no reason you can't have eight fingers per hand the early terrestrial vertebrates had 9.

But you don't want that so how about bifurcated fingers that is fingers that split into smaller units as you get further away from the hand. There is even a condition in humans that causes it sometimes. I suggest going with something between II and V, personally Prefer IV just for the functional aspect.

And again some early tetrapod hands look like this so there is no reason you can't evolve it. 4 bifurcated fingers per hand gives you 16 total.

## 2. use 4 fingers per hand and have exposed toes. if the alien uses its feet for grasping, say because it is a climber, boom base 16.

Humans don't only use recently other bases then base ten. Backwards people in the US have still inches and feet. Even in Europe we have a seven day week even when the French revolutionaries thought that a ten day week would make much more sense. We still have an hour of 60 minutes that each has 60 seconds because the Babylonians had a sexagesimal number system.

It rather that the cultures behind the languages that use base ten won and additionally rational efforts tried to standardize everything into a single base.

Use 6 fingers on each hand, for a base-12 system. It's superior to base ten (and 16) for simple maths as you can evenly divide 12 into halves, thirds and quarters, making it a convenient system to use for trading. Faint traces of this remain in our own number systems today, like a dozen, 12 inches in a foot, 12 hours on the clockface.

A higher base condenses data but a low base is easier to compute in the mind or on paper. Simplicity in manual computations allows for better oversight and error checking, with less memorization.

A higher base condenses the data. If you have sufficiently many distinct symbols you can create a high base system that requires only one written symbol to convey the value. With many distinct numeric symbols, you likely have many distinct symbols for words as well...akin to various Asian languages.

However, a high base system is not the going to be the easiest to teach mathematics with. Smaller bases would be easier to teach and easier to calculate in ones head or on paper as there are fewer states to memorize. Imagine trying to multiple 256 by 753 if your base system was 200,000. You'd have 200,000 distinct symbols to memorize and likely no way to keep track of your progress. Base 10 works because it is sufficiently large as a base to condense data and sufficiently small and conveniently mapped to our digits that we can easily keep track of computations.

I once created a base fifty numeric system with the story background being that the species using it possessed 50 tentacles clustered in groupings of 5. This allowed for a Roman numeral parallel (mix base 5 base 10) where the species had an earlier mixed system with base 5 (tentacles in a cluster), base 10 (tentacle clusters), and base 50 (total tentacles).

For computing Base 2 is used because of the simplicity of On/Off, 1s and 0s. However, it is conceivable to have multi-state computations - if one had voltage ranges and sufficiently precise equipment to distinguish voltage states, you could create a computer that operated on any base system your engineering could handle. You just have to create the right rules. Computers are great at memorization, so high base numeric system is more feasible on the memory side, but you'd have to support it with the hardware (voltage or other method of implementation).

Assuming you could get through the hardware restrictions, you could have a race that was sentient and machine...

Otherwise a race with memory that had nearly instantaneous perfect recall. Conceptually even biomachine oriented race - the "Warstrider" series comes to mind

Frame challenge: you'd get nearly all the benefits of using base 16 by using octal (base 8). Base 8 for people with 4 fingers per hand would feel natural to readers who count to 10 with 5 fingers per hand.

Numbers would of course be longer than decimal or hexadecimal numbers.

For info about the use of octal in computers, you can start at Wikipedia

Two ideas:

# They have 4 hands

Two at the end of each arm (like us) and one on each elbow

# They have 16 eyes

eg: 4 on the face, 1 on each shoulder, one at the end of each of their 8 fingers and two on other places/organs. They should have a powerful brain to handle that much information. They might have to choose which eye to open since opening all 16 eyes would drain their energy.

I find the idea of an alien race with four arms (with four fingers each) and capable of multi-tasking interesting (doing two different things at the same time with each pair of hand). Multi-tasking is a good sign of advanced intelligence. And can be a good reason to stay up.

Following up on Christian's comment about seven day weeks... your aliens could use any base you want, for the same reason weeks have seven days: because their deity/religion said to do that. (This doesn't require that they be religious in the time of your story. They may have even forgotten the religion that started their counting system; it's just been ingrained for that long.)

This has the advantage of rendering moot any logical objections to using whatever base you want them to use.

p.s. Clarke's Rama series had an octal-using species (Octospiders) and a hexadecimal-using species (Avians). The game based on Rama II requires the player to use these for some of the puzzles.