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.
This post was sourced from https://worldbuilding.stackexchange.com/q/153914. It is licensed under CC BY-SA 4.0.
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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.
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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.
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