If the rules of arithmetic are universal, then we might expect aliens to use similar counting systems to those used on Earth, albeit adjusted to local anatomy and brain capacity. For example, we could imagine that alien centipedes would count in base 100 (assuming they each have exactly 100 legs to count on). The centipede would have the advantage of requiring fewer brain cells per leg to rack up large numbers.
However, we have to ask if base 100 would be a useful number system in practice, relative to the probability distribution of numbers actually encountered.
That is, of all numbers ever used, the smaller numbers will be encountered most frequently, at least, in the rational galaxies. The number one will be the most numerous number, followed by two, then three, and so on decaying in a curve, so that very large numbers (such as 1,233,367,564,455,489) are rarely if ever used for anything. This tendency is likely to be a pretty universal trait, though it will be overlain, or punctuated, by other more local considerations: on Earth, in base ten, multiples or powers of 10 form local peaks in probability -- the number 20 will be encountered more often than 19 or 21, and there is another local peak at 100. The peaks will not all be regular, but could cluster around locally culturally specific features, such as dates (peaking around the 1900s and 2000s).
It should be no coincidence that the local base ten peaks have unique labels attached to them (one hundred, one million -- even an ichi man here or there). It works both ways: they are named because they are useful milestones (in base ten), and they are used more all the more frequently because they have handy - named - numeric handles. Handy, at least, if you have ten digits to count with.
But what of our intelligent alien centipede? With base 100, it would be lumbered with having to devise 100 numerals, one for each leg. Most of those numbers wouldn't be used very much. Yet, to accommodate all the necessary numerals, a complicated system of symbols would have to be devised. (At least, for the smart centipede, the most commonly used numbers could be allocated the simplest symbols). But also, think of the size of keyboard an alien centipede's typewriter would have to be (presumably a linear device, set in banks of two, one key per leg) -- and think how little used some of the keys would be (unless they were also used as letters....). This would not be an efficient use of material or mental resources.
Thinking of the clumsiness of having 100 symbols, our intelligent alien centipede might go to the other extreme, opting not for base 100, but binary. Here, the number of numerals is minimised. Here, 0 and 1 could be represented by the right legs and the left legs, and so the whole centipede can store numbers up to 2 to the power of 50 (that is, 1,125,899,906,842,624)! It might need to be a pretty smart centipede to juggle in its head the numbers it held in its body.
Of course, binary has its own disadvantages: it needs relatively long strings of digits to represent modest numbers. Unless you happen to be a centipede, this may be a problem. In binary, the number 70 has to be represented by 7 digits (1000110, or 7 pairs of legs) compared with decimal's two, or base 100's one (leg no. 70). In fact, there must be a trade-off between the number of 'digits' required and the number of numbers that actually need representing -- given the observed decay in frequency of numbers as they get larger. You could theoretically calculate a number base that optimised the number of numeric symbols required and the number of digits required to represent typical numbers -- all normalised to local anatomy.
For many life forms, base 10 may turn out to be a handy compromise; handy if you have 10 fingers, also workable if you have 100 legs. So, if you should come across a race of intelligent alien centipedes, you should not be surprised if they knew the ten times table like the back of your hand.
The author welcomes any refutation of the above arguments. s.marshall[at]asahi.email.ne.jp
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