# Developing Counting Systems [duplicate]

This question already has an answer here:

It’s common knowledge that (most) modern societies use arabic numerals, and moreover use base 10 for counting. That is, you count from 1 to 9 (unless you’re a programmer and you start at 0) and then you use 10, which uses that 10's place.

I’ve learned about base 12 counting systems recently, and even about how Klingon isn't that alien - because of its base 10 system! Making a counting system which is alien to humans infers not using one that has been used by humans. I can't guarantee an alien counting system unless I know what has been used.

In the interest of making alien math more believable and actually alien, what counting systems have been used by humans, and why?

## marked as duplicate by James♦, Frostfyre, Tyrabel, Rob Watts, CyrusNov 13 '15 at 5:49

• Just remember, every base is base 10: rustywalrus.com/images/lg/et.png – KeithS Nov 12 '15 at 15:43
• You may want to look at en.wikipedia.org/wiki/Radix_economy. – Alice Nov 12 '15 at 16:07
• – JDługosz Nov 12 '15 at 23:27
• The Mayans used base 20 en.wikipedia.org/wiki/Maya_numerals – sdrawkcabdear Nov 13 '15 at 1:14
• The other question is purely about base 12. This question is only about base 12 tangentially. They are related, but this question is not a duplicate of the other question. – Brythan Nov 14 '15 at 13:41

Base 1 (aka Unary) tally marks. The first form of mathematical notation. Widely used over time and geography.
Base 3 (aka Ternary) for some experimental computers and research programs (rarely used).
Base 8 (aka Octal) for computer programming (very common usage in computer programming).
Base 16 (aka Hex, hexadecimal) for computer programming and the SF RPG game Traveller (very common usage in computer programming).
Base 13 (aka Hitchhiker's Guide to the Galaxy) in which $6 \times 9 = 42$

• Doesn't really answer the question ("what are reasons to choose one counting system or base over the other in an invented culture?") at all. Merely lists different systems, and provides a link that lists more systems. – DSKekaha Nov 12 '15 at 18:19
• @DSKekaha That’s an answer to a previous version of this question that got rightfully closed. – Crissov Nov 12 '15 at 19:21
• "trinary," not "ternary?" Also, I believe pigs' ears are notched in ternary in large agricultural establishments. – nitsua60 Nov 13 '15 at 2:39
• @nitsua60, If you notice on the wiki link, "Analogous to a bit, a ternary digit is a trit (trinary digit)." "Trinary" and "ternary" are one and the same, although ngrams shows a huge preference for "ternary". – MichaelS Nov 13 '15 at 5:43

Oksapmin, base-27 body part counting

The Oksapmin people of New Guinea have a base-27 counting system. The words for numbers are the words for the 27 body parts they use for counting, starting at the thumb of one hand, going up to the nose, then down the other side of the body to the pinky of the other hand, as shown in the drawing. 'One' is tip^na (thumb), 6 is dopa (wrist), 12 is nata (ear), 16 is tan-nata (ear on the other side), all the way to 27, or tan-h^th^ta (pinky on the other side).

Based on Jim2B’s link it looks like most Earth numbering systems are based upon fingers or fingers and toes. Base 5, Base 10, Base 20. The next largest themed grouping is Duodecimal (Base 12) and was most often used/devised in time keeping. Hours, months. There is also the Kaugel culture that uses base 24 (again perhaps time?) Out of the recorded cultures there are 6 recorded here that use some other form. So for creating a race with a different base system most would be based upon their physical appendages, some on their planetary orbits/revolutions. A small minority would have originated in some alternative. And those who are scientifically minded might have changed their base system post industrial/scientific revolutions or even post renaissance to something that was more mathematically elegant.

Some examples. A four limbed species with 3 digits per limb could use base 3,6 or 12 with base 6 being most common if they follow human trends.

A 6 limbed creature with 3 digits per limb might use 3,9,18. Though I feel they would tend toward 3. A 12 limbed creature with no digits would most likely use base 12. However if each appendage could be coiled 13 times (counting coils instead of limbs) they would use base 13.

• But as in our society, the numbering system in common usage may be set aside and a different one used for specific applications. Like Base 10 for common usage & Base 2 for programming in our culture. It is likely that certain numeric systems would span species/cultures. I think the bases 1,2,6,8,12,16,&24 are the ones most likely to do this. – Jim2B Nov 12 '15 at 22:33

Sadly the list of reasons for choosing one numerical system over another is too long to make a good answer, so I'm going to boil it down to the essence of numbers, historical intervention and the ability to do maths in the abstract.

A number, when written or used in mathematics, is in essence a way to denote a quantity of something. It doesn't matter whether that thing is real, imaginary, concrete or abstract, a number is a representation of a quantity of 'thing'. That gives a good starting point. What things are important to your creatures? Do they have a x moons that they might want to count? Are there any animals that hunt in packs of x number? Have they got x digits on which to count (this is a bit less important than you might think, but I'll get to that). Once you've worked out what things are important, you can move on to the next bit:

History is important to numerical systems. There are a whole slew of quantities in the world that are counted in certain groups because one set of people or another though that that number of things was important. Our entire world is a hodgepodge of things that have been slammed into base 10 when really they should be base 12 (hours, months) or 60 (minutes, seconds), or 14 (pounds. Don't even get me started on ounces). If the history of your world features a war between two races that use two different methods of counting, then only the victor's method is going to come out on top.

We've put these into base 10 because we find it easy to manipulate base 10 numbers in abstract because we have 10 fingers. The ability to turn abstract mathematics into concrete mathematics is an important Thing for you to consider, but the number of digits on your hand isn't that important, as long as you can get the numbers in your head into a concrete system you can use for maths. A simple abacus takes care of that. Make an abacus with the same number of beads on each row as the number of moons in the sky and suddenly you're doing abstract maths in base x.

One final thought for you: The Romans wrote all their numbers down using a combination of bases 2 and 5. The Inca used a complex series of knots in strings that still hasn't been fully explained. Your race is alien. Go nuts.

## Five fingers

The system of tally marks that I learned had a special symbol for every fifth number. The first four were vertical lines, something like ||||, but the fifth was a diagonal line across the previous four.

Roman numerals had special behavior in groups of five and ten.

The Mayans used a base 20 system -- the total number of fingers and toes.

And of course the modern system that we use is base 10.

## Computers

Binary. This is the simplest number system to implement for a computer. This will be true regardless of the number of appendages. However, we are bilateral. Two eyes; two ears; two nostrils; two arms; two legs; as well as two sexes, etc. That might have mattered.

Octal and Hexadecimal. These are more complicated. They are natural alternatives to binary. Each digit can be represented by three or four binary digits (bits) and covers all possibilities. But why those two? Why not base 4 or base 32? The most likely explanation is that we found these two most natural because we already used base 10 and these are the two closest that have direct translations to binary. On the other hand, we did choose hexadecimal over octal, even though octal is closer to having the same number of digits as decimal. From a binary perspective, sixteen is a nicer number than eight. $10^{100}$ rather than $10^{11}$.

Unary. A simple system, but wasteful to record. One unary digit can only hold two numbers (zero or one). Two unary digits can hold three values, etc. Sixty-four unary digits can only only sixty-five values. Contrast that with binary where sixty-four bits is enough to hold eighteen quintillion values. An advanced alien race might understand unary, but they seem unlikely to use it for anything.

Ternary. There are three types of charge in nature: positive from protons; negative from electrons; neutral from neutrons. Aliens that had more of a connection to the number three might use positive/neutral/negative rather than our on/off (binary). This would be really alien to us.

## Others

Duodecimal. The first four counting numbers all divide twelve. This is true regardless of base. This makes twelve a natural number for a base that transcends number of appendages. That said, I think that the advantage is small enough that an alien race is unlikely to use it unless they have some stronger connection (e.g. twelve fingers). It's interesting that the main way we use this is to tell time, where divisions are most natural and useful.

Sexagesimal. Divides all of the first six numbers evenly. However, it doesn't divide the next three and will divide fewer and fewer numbers. Again, this seems like a number system that mathematicians would naturally consider but not one that would become popular on its own merits. The Babylonians sort of had a sexagesimal system, but it's organized as six groups of ten.

## Conclusions

The most likely number system to be understandable by aliens is binary. Its on/off nature makes it easy to represent plus it is one of the two most natural number systems, regardless of appendages. The only number system that might block binary is ternary. So if you're going for a sharable number system, binary would be our best chance.

If instead your goal is to describe an alien mathematics and you want it to be as alien as possible, try ternary. Dividing things in two is natural to us. What if an alien race found dividing into threes more natural? Three sexes. Trilateral. Or even just not bilateral. Maybe our prejudice towards binary is caused by our nature. Maybe a unisexual species without identical parts at all might think of things differently. Hard for us to tell, as we're the only intelligent species whose development we know.

Both binary and ternary are systems that computers could use. This makes them natural for advanced civilizations. They would almost certainly develop at least one of them. Of course, we don't use binary as our normal number system. We use decimal, based on our number of manipulative digits (fingers). Another species might use the same basis for their most common number system.

What might a species that does not have a fixed number of digits pick as a number system? I think that binary and ternary are still the natural number systems when they get past unary. After that, they may well pick a base like sixteen or twenty-seven as a convenient abbreviation. If they are exceptionally mathematically oriented (number theory and/or geometry), they might pick something like sexagesimal before developing computers. Maybe.

Note that this is all speculation. We have no real basis for saying how someone not like us would develop number systems. Our only example is us. And we don't actually know why we chose to do things the way that we did. If we were different, would we have made different choices? Or the same choices--possibly for different reasons?

• Actually, binary isn't "fully expressive", even for fairly trivial cases. There's no way to directly and exactly represent 1/3 in binary, for example. Binary is however fully expressive in the case of integers (though negative integers can be represented in different ways while remaining a "binary" notation; compare one's complement and two's complement, for example) and as such lends itself very well to anything that can be expressed as appropriately-sized integers (including fixed precision decimal numbers). – a CVn Nov 13 '15 at 13:24
• @MichaelKjörling You can represent one third in binary the same way you can in decimal. Either as a fraction (1/11) or as something that repeats: .010101... But I delete fully expressive as it wasn't making the point that I wanted, that binary is much terser than unary and not that much more verbose than ternary. – Brythan Nov 13 '15 at 15:16