Is there anything about the decimal number system that conveys any advantages over any other number system?
So is it any more likely that some other alien race would use base-10 numbers in everyday life or are they just as likely to use say... base-8 or 12 (8 fingers and no thumbs, or 12 joints in the fingers)? base-3? 16? 29? 100? Or is there some useful property in base-10 numbers beyond us being able to count with our fingers?
It's likely that the aliens would develop a system with a radix (base) of whatever number of fingers/toes/tentacles that they had easily available to count with. And as TheBlackCat pointed out, there are many options beyond that.
There's even different base systems among humans. For instance, the Babylonians used a base-60 system, the french used a base-20 (they say 4*20 instead of 80 even today), and the Wikipedia article for base-6 says that several cultures have adopted that system.
If you wanted to, you could even make it a plot point that different cultures use different bases. Perhaps two major cultures view their particular radix as having religious significance, and the fact that they use different bases is a point of contention and occasionally war between them.
And there's no real advantages to the number 10 either. In fact, there are people who think we should switch to a different radix so that we'd have an easier time dividing. As discussed on reddit, the main tradeoff between having a low or high radix is the number of digits you have to memorize vs how long the resulting numbers are. Low radix means few digits but long numbers, and high radix means lots of memorization but short numbers.
But 10 is when double digits start!
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Even on Earth, we routinely use other bases. Computer scientists use binary (base 2), hexadecimal (base 16), and octal (base 8), as well as decimal, very routinely. Various world cultures (past and present) have used number systems with all kinds of bases. There is a base-12 number system called Dozenal (or Duodecimal) that has some real advantages over base-10 for us (such as having more factors, so it can be more easily divided).
We use base-10 mainly because we have ten fingers. Whoopee. Base-12 is a better system in many respects, but probably the main reason we aren't converting is that base-10 simply is too well established. And you could make the same case for other counting systems with other advantages.
For the above stated reasons, aliens would not be any more likely to use base-10 than any other reasonable (to them) radix.
You didn't specifically mention a need to communicate with these aliens, but as a sort of bonus answer, here are some points to consider if communication is required:
Often people advocate just using binary or unary (base-1. Think hash marks) as sort of a lowest common denominator. But really, any base will do as long as you define it before you start communicating. Alien language is a whole different topic, but if you can convey something like the following, you're set:
apple
banana |
kiwi ||
plum |||
You now have established a link between your base-4 number system (apple = 0, banana = 1, etc.) and its base-1 equivalent. If your extraterrestrial buddies can't do that conversion, they probably stole someone else's spaceship.
Others have immediately pointed out that base 10 is a human thing. Note that bases 12 and 60 are more “for a reason” and may show up in the aliens’ cultures too! (More on that below.)
But let me point out that a “base” (positional system) is not the only way to go. Even we know about Roman numerals. I’ve seen authors be “more alien” by defying the very concept of a base.
For example, Robert L. Forward had one alien culture use a balanced ternary code, which is a base but not as we know it.
It is positional, but each position can have positive and negative values. This (not necessarily ternary) could show up in the way abacuses are made and merchants compute, and give an early apprciation for zero and negative numbers relative to our own development!
They might have an irregular hierarchy of bases rather than the same base in every power, with weird culturally significant rules. The writing system may obscure the fact that they are in fact using a “modern” positional system, because they retrofit it into their old writing system which originally used an irregular system.
Murphy gave an example of this based on anatomy. But I’d like to point out that we do that with “traditional” units in many cultures, and the idea can be formallized and refined using number theory:
Look at the idea of an anti-prime (look for the Numberphile video. A subset of anti-primes such that each is a integer multiple of the previous gives (1, 2, 4, 12, 60, 180, 360, 720, 5040, …) Now we continue using grouping hieararchies like that, but write each coefficient in decimal (like 36 minutes, 22 seconds). If they had a writing system more like syllabic glyphs, you can imagine the number being written as an ornate picture.
This might evolve into a refined “modern” system that’s base 60 in its mathematical properties, but written with digit groupings having hiarchial clumpings reflecting the subsequent antiprimes.
They may not associate small natural numbers with unique glyphs and use multplication (whether by positions or other notation) to form bigger numbers.
Look at the theoretical formation of naturql numbers — it doesn’t look like a “base” at all, does it?
Knuth wrote a very cool and interesting book on surreal numbers from the point of view of two people decyphering markings in rocks.
IV were added in more modern times - the Romans simply wrote IIII; so you just slapped the two numbers together, and if you had IIIII, you replaced it with a V and so on; IV actually meant the same thing as VI - 1 + 5. We still use the system in pubs - that's what "tally marks" are, really.
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I like some of the other answers but let's go weirder.
Technically you can even have a base system which varies along a number or has its digits in a complex order.
Imagine a number system where, in order of evaluation:
The first digit is base 8
The second digit is base 2
The third digit is base 10
The fourth digit is base 12
And when being written down they order them
[third][fourth][first][second]
So the number 5 would be
0050
9 would be
0011
16 would be
1000
160 would be
0100
(hopefully I've not messed any of those up)
Each position can even use entirely separate symbol sets or they could use the same symbols but where 1 symbol can mean different things changing by position.
And then loop for arbitrarily large numbers. perhaps it's a species with 8 tentacles, 10 fingers, 8 toes and 2 trunks who see it as an obvious system and order their numbers by the positions of the appendages on their bodies and their relative ease for use in counting.
For context I once had to write an encoder which could accept a number an translate it using arbitrary symbols, order etc and found it perfectly logically consistent if somewhat confusing.
I would call a system like this slightly less likely than a simpler system using a single base but it's an option.
Most of the answers make an assumption that isn't quite correct. The assumption is that a base is required; in a universe with no rules about how life can develop, this is not the case.
Why do humans have bases?
One of our major limitations as a species is related to counting. Most people can distinguish between 2 and 3 items without counting. Many, but not most people, can distinguish between 5 and 6 items without counting. (this is called Subitizing) But nobody has the neurological apparatus to distinguish 21 and 22 items without counting. So we are hard wired to tell which tree has more fruit if there are just a few apples, but not if the tree is filled with fruit.
Contrast this with out ability to talk and listen. When speaking, we can issue 1400 muscle instructions/second. When listening, we can identify instantly which of the 75,000 words or sounds we know have been used. Even more computational ability is required to catch a baseball on a windy day in the outfield, or block a soccer/football penalty shot.
My point is our minds have to ability to work in numbers larger than 10$^6$, we just don't have the ability do it with counting. We evolved complex systems for communication, vision, image processing, and movement, but there was no evolutionary reason to develop a very complex counting and mathematical brain. We can view this as a major weakness of our species.
What about non-terrestrial creatures?
We have English words for numbers we are all familiar with: one, two, three, four, five, six, seven, eight, nine. Imagine an alien that had a mind that was powerful enough to distinguish between 124,523 and 124,524 ping pong balls instantly. It is likely that they would have a different word for each number. (err, if they used words)
Source
A book called "What counts" by Brian Butterworth. A very good read. He says because we don't have a math processing center like we have for vision, emotion, or memory, the brain uses a work around. Our number skills are based in the portion of the motor cortex related to finger control (p244), which is why children use their fingers to learn counting, and why we must use a base. (Note, because we have only 10 fingers, this is why human bases are generally <100). He also mentions that we learn addition before subtraction, and multiplication before division, and our susceptibility to the Stroop effect. Even more interesting the numerical stroop effect.
Edit:
Butterworth also reviewed anthropological literature about the origin of a counting; and found that many tribes did not use a base; one tribe (I think in Australia? Need to look it up later) they only had words to count to 3. After that, they would use the fingers, toes, eyes, ears, nose,etc. of themselves, and if they needed more, other people as well. They didn't need words to understand relatively large numbers. If I recall correctly, the anthropologist didn't report what happened when the number was too large.
There are a few important things to note, in addition to all the things mentioned.
As said, 10 isn't special, other than humans having 10 fingers.
In addition, 60 isn't special, either.
360 is special, but only to humans, because that's really close to the number of days in a year. Early calendars were based on 360, and (some) early number systems co-evolved with their calendars. (If you've ever played with the GRAD setting on a scientific calculator, having 400 GRADs in a circle would make much more sense than the 360 we ended up with.) The Sumerians liked 60 because it divides so nicely into 360, and 12 is nice because we have 5 fingers on our hand, and 5 12s gets us back to 60.
Why are there 7 days in a week? Because 7 divides so cleanly into the 28 days in a "month" - i.e. the number of days in our lunar cycle.
I did a research project on the evolution of number systems as part of my undergrad work. From 1 to 0 was the seminal work on the topic. Number words and number symbols was also a big chunk of my research.
An alien species (assuming, like us, it couldn't divest itself of what it's been used to for millenia of growth) will base their numbers on whatever is relevant to them. Number of appendages is important, but so is working with constants given to you by your environment. You may get a really interesting system by there being multiple moons or stars in their system.
One last interesting note - humans and other animals have an ability to easily recognize groups of up to 4. After that, it becomes increasingly difficult. Even if we had 8 fingers per hand, it's possible tally marks would be broken up in 4s or 5s, just because it's easiest to read them that way.
EDIT
I feel a need to add a bit more; the comments to my answer are arguing over minutiae, and it's obscuring the important point.
The number system of your society will form extremely early on - it will coincide with the birth of your society and the formation of their earliest languages. To know what your alien's number system will be, you have to think of how their society was first created, and what phenomena they would observe while doing so.
Number words start with the simple, familial concept: "There is me (1), us (a few), and others (many)."
The next immediate growth is grouping: "I thought there were more of my family. Is someone missing? How many are we?" At this point, you start grouping using the number of some appendage. For humans, the easiest thing is the five fingers on your hand. They're always around, and you can pick up and put down each finger as needed: "There's me. There's Ugg. There's Ock. There's Uga. There's Gug. That's one hand. And there's Gaa. That's one-hand, one-finger. We were one-hand, two-finger yesterday. Where's Kaa? WHERE'S KAA????"
Number concepts don't evolve beyond this much (humans went from one hand to two, then stayed there) until you go from familial to tribal, and have enough safety for there to be someone who focuses on observing. This would be a priest, or a medical man. They'll notice something that recurs on a predictable scale. It has to be a predictable scale, really, because there is not sufficient level of intellectual sophistication to find a non-regular repeating pattern. At this point of time, the scientific discovery is that the pattern exists. That "ability to predict the future" is part of what will make this observer powerful. Whether (s)he will "make the sun/moon come back" or will "make water erupt from the ground" (a la Old Faithful), their knowledge will make them powerful, so they'll develop it. Whatever number this thing recurs at will likely become the society's radix. It's likely that this counting will be of something astronomical - either day cycles or moon cycles - because there aren't much more precise measurements. Hours won't exist for tens of thousands of years yet.
This is what you're contending with when you look at a number system - concepts that go back as far as society itself, at the formation of a species' first words.
It's important to realize that numbers weren't chosen by the society. They found the numbers based on themselves and their environment that best helped them to survive, and their culture, math, and science all grew up based on these values. The cultural inertia of such numbers is really hard to overcome.
Number words - and the numerical systems that derive from them - must have their origins in observable phenomena. At the time that the number words are formed, there just isn't enough strength in the culture for them to form any other way. There will be little decision-making in this process - it would have to be an easy enough decision to make that society would actively make it over tens of thousands of years of the most primitive culture possible.
A spacefaring society will have developed their mathematics enough to realize there are other, likely more efficient bases. Their specialists will use them in their specialist tasks (as we do for computers, and for speaking in abstract number theory about better bases.) Particularly, their rocket scientists could use number constructs that most others do not. But the people will still use whatever number system their society grew around, based on their physiology and their local astronomy.
I would say any type of science is based on the consciousness of the entities developing/using it. For example, most people conclude many humans use base 10 numbering because that's how many fingers we perceive oursevles as having through our five senses. Our perceptions through our senses populate our consciousness with content, and our thinking coagulates around content present in our awareness. So, any sciences developed by our consciousness have a good chance of being developed based on the most common systematic/reliable perceptions found in consciousness.
In this way, one can say science is based on consciousness, on our perceptions.
For example, it could be argued that it is not so much that we have 10 fingers, but that 10 fingers are very prominent in our awareness--because they permeated our awareness so fully during our evolution, many humans chose to or inadvertantly began to think in terms of tens.
So, if an alien race more often has something other than tens in their awareness, their numbering systems will most likely be based on that content. Given the enormous range of types of concepts possible to perceive, and tens only being one of those possibilities, it might stand to reason that 10 base numbers are only one out of an infinite number of possible numbering systems.
In addition, if we take the term "alien" as meaning "different from us", then the more alien a race, the more alien their numbers will be to us.
There is a range of concepts humans do not consider alien/foreign because they are considered within our range of plausibility. However, any alien race dealing with concepts outside that range of familiarity will likely begin to develop sciences and numbers outside our range of recognition. So, an alien race that is numerous standard deviations away from human (i.e. more alien than we can comprehend) will likely develop sciences based on concepts we cannot even relate to easily. For example, it is extremely difficult to imagine an alien race developing sciences without numbering systems, yet technically speaking, it is possible they use concepts too different for use to comprehend instead. Our human consciousness defaults to think in terms of numbers as being an unalienable foundation of science, something science cannot exist without. However, if one extrapolates based on the idea that standard deviations in thinking can be infinite, and yet human can only comprehend thoughts within a finite range of deviation, then it becomes almost a certainty that somewhere in the infinity of possible universes in an infinite omniverse, there would be sciences with numbering systems too alien for us to comprehend or even begin to imagine.
There is nothing special with the decimal system, except for being directly based on the number of fingers for humans.
Alas, most other numbering systems (duodecimal, sexagesimal, etc) also have some underlying ties with our fingers. See: you can count using only one hand up to 12 by using your thumb to point to each of the other fingers bones. If you use your other hand, you can count how many times you've done this (which is actually like counting in decimal in the second hand) so that you reach 60.
To be more thorough, in modern times we still use many numbering systems (positional or not). We still use roman numerals for some things, while we also use both duodecimal and sexagesimal systems to count time. If we are talking about subsecond measurement, we also use decimal, and if talking about multiple days we usually also use decimal. For angles, we also use a mix of sexagesimal and decimal.
Talk about legacy development! That says something about the Sumerian influence on our culture, from thousands of years ago! By the way, even Mayan or even Prehistorical numbering systems also rely on some form of base-5 to facilitate counting with your fingers and that show when writing the numbers.
Remember that ease of learning is a big part of the success of a numbering system. If some people find math hard today, imagine how it was before it permeated our lives.
By the way, Robert Heinlein briefly touches the subject of contact with an alien society with a 3-based numbering system in Stranger in a strange land.
Don't forget divisibility rules. Writing in base ten makes it easy to distinguish between odds and evens. And identify multiples of five. And multiples of three, too, by adding the digits.
So base ten is good for quick identification of multiples 2, 3 and 5.
Is it a coincidence that these are the lowest three primes? Probably.
Other than that, besides easy divisibility-checking for the first three primes, there's nothing special (to my limited human mind, anyway) about the base 10 system.
In fact, a culture could pick a few other bases, were they interested in divisibility.
base 6 and base 16 both allow the same methods as above. base 36 or base 66 are arguably even more useful for this, but perhaps too many symbols to memorize (see emo bob's post).
I think it may depend largely on the alien's physical makeup and/or cultural history.
As others have pointed out, humans use base-10 (because of number of fingers), base-60 (historically for easier calculations) etc.
So it is completely conceivable that an alien race may prefer base-3 (ternary). Perhaps this alien race has an Artificial Intelligence origin (intelligent machines based on computers) and some analyses here on earth have shown that a base-3 system would be the most economical way to build a digital computer (but only slightly more so than base-2 (binary) or base-4). Or maybe they have some special evolution that makes their brains very much like a biological computer, and selection selected the most economical brain structure (ternary) due to resource limitations.
Thomas Fowler built a balanced ternary (-1, 0, +1) computer out of wood in 1840, and had this to say:
I often reflect that had the Ternary instead of the denary [decimal] Notation been adopted in the Infancy of Society, machines something like the present would long ere this have been common, as the transition from mental to mechanical calculation would have been so very obvious and simple.
More reading:
I remember reading an article about an electrical circuit that implemented base negative two math. Some examples with four "bits" valued (-8, 4, -2, 1):
0000 = 0
0001 = 1
0010 = -2
0011 = -1
0100 = 4
0101 = 5
0110 = 2
0111 = 3
1000 = -8
1001 = -7
1010 = -10
1011 = -9
1100 = -4
1101 = -3
1110 = -6
1111 = -5
It may be that fibonacci is an earthling sequence or it may be a universal law of biology, it's hard to say.
Decimal may be significant in that it is the product of two fibonacci numbers 2 * 5.
Perhaps aliens with a set of 6 fingers on 2 arms that evolve base 12 would be much less likely.
There's nothing special about base 10. It's just what we learned when we were kids, because we have 10 fingers. If you do a lot of binary or hexadecimal arithmetic, it feels just as natural.
However, let's assume that we count base 10 because it was passed to us by our caveman ancestors who learned to count with their fingers. Let's dig deeper. Why did we simply count fingers, and not use binary numbers? If we used our fingers to represent binary numbers, we would be able to count to 1023 using our 10 fingers. That's much more powerful than simply counting to 10.
Part of it is the anatomy of our fingers. It's hard to represent numbers like 1010101010 using our fingers. But an alien species might not have that dexterity problem.
Another limitation is the human brain. Counting to 10 doesn't use much brainpower. But our caveman ancestors probably didn't need to count to 1023. A more intelligent species probably wouldn't mind so much.
If an alien species only had 4 fingers, they'd only be able to count to 4 by counting fingers. But using binary, they'd at least be able to count to 15.
Now suppose there was an alien species who represented binary numbers when counting with fingers. When they invented pen and paper, their finger counting method would have naturally extended to writing binary on paper. They'd have the SAME method of counting with fingers as with paper and pen. They'd only have to teach their children one method!. Think about how it is for humans. We first teach our kids to count with fingers. Then, when they reached a certain age, we teach them the decimal characters, and teach them to convert what they've already learned to the new system.
Some people would argue that an alien with 8 fingers would use base 8. But using my argument above, I'd say that an intelligent species with dexterous fingers would have counted base 2 regardless of how many fingers they had.
Nothing special about any of your bases. They all are belong to us.
Different human civilisations used different bases. The Bablonians used base 60 the Maya base 20. https://en.wikipedia.org/wiki/List_of_numeral_systems shows how many possible bases are used by different languages.
Even within our own language we don't only use base ten. On side of a clock shows 12 hours. There are 2*12=24 hours per day and not 10 or 100 hours. There are 12*5=60 minutes per hour.
Having 60 minutes per hour means that it's 1/3 of an hour is a nice round number of 20 minutes. If the hour had 100 minutes 33.333 minutes wouldn't be a nice round number.
Rumours have it some tribes in America still uses a system of measuring lengths that uses base 12. In Europe the French tried to get everything normed to base 10 and the common base is useful but even the French didn't succeed in getting the 10 day week. They tried but people liked their 7 day week too much to switch.
As I write this, every answer declares there is nothing special about base ten. I think one point that's been missed is the value of a digit as an order of magnitude. For example, consider three interpretations of this statement:
10 + 1 ≈ 10
So base ten is nice because moving a digit one space to the right means you can neglect it in your calculations. However, if your aliens have spent millenia with technology that requires extreme precision, they may find that our one-part-in-ten estimation criterion isn't useful for any of the problems they're currently solving (like how the "three is approximately two" rule isn't really useful for us). So maybe they have a much larger radix.
Note that above, we make an argument for why base ten is more useful for humans than base two or base one-hundred, but we don't know that it's actually optimal- e.g. what about base nine or base eleven? One drawback of the decimal system (and a reason why imperial measurements are still used in cooking) is that humans are pretty good at splitting things into twos and okay with threes, but fairly poor at divisions that have larger prime factors. For example, suppose you have a volume-unit of some substance and you'd like to separate out some smaller unit:
Now, maybe your aliens have some exotic biology like five eyestalks or seven hands. They might then evolve a psychology that allows them to easily and accurately compare a higher number of quantities (e.g. "is the weight I'm holding in each hand about equal?"), which could in turn affect the radices in which they can efficiently perform mental math; they'd likely prefer to record their figures in a notation which reflects their bias.
