Xjyuk

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.

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).

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.

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.

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.

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.

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...)