Octal number system

[xxx]

What makes base 5 more ‘universal’ than base 10? Surely if you want a universal base it would be better to go with 2, 3, 6 or 12?

it seems that humans (and animals!) are able to distinguish, precisely, without computing, only numbers from 0 to 4...

[Ryan of Tinellb]

What makes base 5 more ‘universal’ than base 10? Surely if you want a universal base it would be better to go with 2, 3, 6 or 12?

it seems that humans (and animals!) are able to distinguish, precisely, without computing, only numbers from 0 to 4...

I don't know what they're called in the literature, but I coined "degunative number" to refer to those numbers able to be intuited at a glance.

[akam chinjir]

I don't know what they're called in the literature, but I coined "degunative number" to refer to those numbers able to be intuited at a glance.

That sort of intuiting is called subitising. I don't know if there's an established term for the numbers involved.

[Richard W]

We could even imagine some insane language in which:
- for k < 13, there is no base (each number is non-decomposable)
- for 12 < k < 80, there is multiplicative base 10
- for 79 < k 100, there is a multiplicative mixed base where x=10 and y=20, with priority to the highest multiple of y
- for 99 < k < 180, there is multiplicative base 10
- for 179 < k < 200, there is a mixed base again, and so on.

But surely such a system would never catch on...

Is it long-established in the British Islands?

[Richard W]

Or is it just a written language?

[zompist]

So if we take the numbers 4, 44 and 214, and assume that w=x=y=10, and a,b,c etc are all <10:
- in a pure additive system these are [4], [10+10+10+10+4], and [10+10+10+10+10+10+10+10+10+10+10+10+10+10+10+10+10+10+10+10+10+4]
- in a pure multiplicative system these are [4], [10x4 + 4] and [10x10 + 10x10 + (1x)10 + 4]
- in a pure exponential system these are [4], [10x4 +4] and [2x10^2 + (1x)10^1 + 4]

This is reasonable, but I think too hung up on Western arithmetical notation.

Very narrowly, if you look at the system I outlined, you can't say whether it's additive, multiplicative, or exponential— because you don't know what the (implied) operators are, and because it just doesn't count high enough.

Natlangs cobble together their number systems in an ad hoc, inconsistent way. Sometimes you have explicit operators, mostly you don't. E.g., Chinese 十三 is 10+3 while 三十 is 3x10. That's overloading concatenation, you might think, but then English has sixteen vs. sixty— here concatenation in the same order is given two meanings.

So when you see [2][2] in an Indo-Pacific or Amazonian language, you don't really know if it's 2+2 or 2x2. In some languages you could find out by looking at higher numbers. E.g. [2][3] in Toba means 6 (multiplicative), while in Western Desert Language it's 5 (additive). But again, many languages overload concatenation. (Occasionally it expresses yet another operation, subtraction— e.g. [2][10] = 8.)

So far as I know, no natlang expresses powers of the radix as numerical notation does— that is, as some combination of [1] and several [0]'s. They all just create names for the powers, in a completely ad hoc fashion— "one, ten, hundred, thousand", sometimes resorting to reduplication or augmentatives ("million" is etymologically "big thousand"). So if you want to be pedantic, no natlang can be exponential in your system. But if you relax a bit and allow words representing the powers of the radix, then many natlangs are obviously exponential for high numbers. And in many natlangs, we just can't say, because people don't count that high, or don't do so in front of linguists.

The other thing is, how we name numbers (counting) isn't the same as how we use them in arithmetic. E.g. if you look at a French number like 13,491 it's a mess:

treize mille quatre cent quatre vingt dix un
13 1000 4 100 4 20 10 1

But when French people do arithmetic, they don't use any of the concepts etymologically inherent in their numbers; they use a "pure exponential system".

[bradrn]

What makes base 5 more ‘universal’ than base 10? Surely if you want a universal base it would be better to go with 2, 3, 6 or 12?

it seems that humans (and animals!) are able to distinguish, precisely, without computing, only numbers from 0 to 4...

Good point xxx — I forgot about subitising!

[TomHChappell]

Sumerian/Akkadian had base 60, and WALS adds Ekari. The Mesopotamian languages also make use of tens. But they did have a positional notation, in which (say) a sequence of four numbers meant four different powers of 60, and this extended to fractions. (That is, you could write a number as <units> <multiples of 1/60> <multiples of 1/3600> <multiples of 1/216000>.)

Whoever I’m thinking of would call that an “exponential base”.

Hmm? It's exactly like our system!

Well, yes. English and French and German and Spanish and Russian and all languages of techno-industrious-scientific cultures in modern real life, have what these writers would call exponential-base systems, where the language’s numeral system has a sequence of superbases in which all the sufficiently high bases are the consecutive powers of one of the early bases.
English’s base system is ten, hundred, thousand, million, billion, trillion, ..., decillion, ..., centillion.
Every base bigger than a thousand is a power of a thousand.

Exponential-base systems “are more sophisticated” than (most?) other multiplicative-base systems because speakers can intelligibly express larger numbers that they’ve never expressed before to addressees who understand them even though they have never heard those numbers before, using a lexicon of comparatively few number-words and a numeral-phrase composition technique with comparatively few and simple rules.

———

[xxx]

other thing about bases, names of numbers can follow a base that is not used in mathematics transcriptions...

For instance English is docenalism-ready, who use different names from 0 to 12... which is a natural way on earth (month, year...) and an old use (dozen, gross,...) and easy for divisions...

In fact whatever your "linguistic base" you can use any "mathematics base" to compute...

[bradrn]

For instance English is docenalism-ready, who use different names from 0 to 12... which is a natural way on earth (month, year...) and an old use (dozen, gross,...) and easy for divisions...

I don’t quite see this… I know that 12 is a ‘baker’s dozen’, but is there a word for 11?

In fact whatever your "linguistic base" you can use any "mathematics base" to compute...

What exactly is the difference between those two terms?

[Richard W]

I don’t quite see this… I know that 12 is a ‘baker’s dozen’, but is there a word for 11?

What's wrong with 'eleven'?

A ‘baker’s dozen’ is 13.

[Richard W]

Natlangs cobble together their number systems in an ad hoc, inconsistent way. Sometimes you have explicit operators, mostly you don't. E.g., Chinese 十三 is 10+3 while 三十 is 3x10. That's overloading concatenation, you might think, but then English has sixteen vs. sixty— here concatenation in the same order is given two meanings.

You're overlooking the postfix operator. In so far as phonetic attrition is removing what's left of it, as a complete number 'sixty' is being replaced by 'six oh'.

As an example of ad hoc systems, Thai 'phan haa' is a good example. 'Phan' = 1,000, 'haa' = 5. While formally 'phan haa' is 1005, in practice 'phan haa' means 1500.

[xxx]

For instance English is docenalism-ready, who use different names from 0 to 12... which is a natural way on earth (month, year...) and an old use (dozen, gross,...) and easy for divisions...

I don’t quite see this… I know that 12 is a ‘baker’s dozen’, but is there a word for 11?

11 in base 10 is eleven, 11 in base 12 is a dozen and one...
100 in base 10 is eight dozen and 4,
144 is a gross...

In fact whatever your "linguistic base" you can use any "mathematics base" to compute...

What exactly is the difference between those two terms?

the positional numeral system base is often confused with the linguistic base which allows the composition of the number names... many natural languages have composite linguistic bases but use the positional numeral system in base 10 (cf English above, French between 70 and 99,...)

[zompist]

For instance English is docenalism-ready, who use different names from 0 to 12...

Not really: 11 and 12 are 'one left over', 'two left over' in Germanic, worn down enough so that the derivation is forgotten.

Which points out another fun number-building method— not just implied operators, but implied operands. (i.e. 11 = "One left over [ten]".)

[xxx]

For instance English is docenalism-ready, who use different names from 0 to 12...

Not really: 11 and 12 are 'one left over', 'two left over' in Germanic, worn down enough so that the derivation is forgotten.

okay, English is not the best example, but the forgotten etymologies argument is weak enough to allow it... especially with the dozen system...

Which points out another fun number-building method— not just implied operators, but implied operands. (i.e. 11 = "One left over [ten]".)

the" linguistic bases" do not have the regularity of mathematics, they allow all imaginable combinations to the natural languages...

what about your conlangs...

[xxx]

I don’t quite see this… I know that 12 is a ‘baker’s dozen’, but is there a word for 11?

What's wrong with 'eleven'?
A ‘baker’s dozen’ is 13.

a baker's dozen is a "normal" dozen and one (except for the weight of bread...)
so a baker's dozen, in base 12, is 11 ...

[zompist]

okay, English is not the best example, but the forgotten etymologies argument is weak enough to allow it... especially with the dozen system...

dozen < douzaine 'set of 12 things' < douze '12' < duodecim 'two-ten'

There's got to be some natlangs with an underived root for '12', but derivations are going to be way more common. Even base-20 Maya has lahca < lahun-ca 'ten-two'.

But, base 12 is awesome. It's a little late for us to change, but it's a far better choice than base 10. (But if you absolutely need to divide things easily by 5, there's base 60.)

I wonder if anyone's played with base 210. There'd be a lot of symbols to learn, but you don't actually need 210 graphemes. Then you could easily divide by 2, 3, 5, or 7.

what about your conlangs...

I'm extremely fond of using different bases— Almeans so far have used 6, 8, 10, 12, and 18. My explanation is that they have 18 digits, which is just awkward enough that it gets people thinking about different bases more than Earth humans do.

[xxx]

what about your conlangs...

as for me, who discovers an a priori language (in the strong sense), I am a prisoner of choices made forty years ago...

and which go beyond: with its use without double articulation, it amalgamates positional numeral system and pronouncement of numbers, plus syntactic rules and mathematical notation rules...

for example, I noticed that it was changing the order of operations...

but as always in conlanging, you should have to undertake everything to really discover how your conlang runs... and I'm not mathematician...

[Neon Fox]

You're overlooking the postfix operator. In so far as phonetic attrition is removing what's left of it, as a complete number 'sixty' is being replaced by 'six oh'.

In the words of Wikipedia, "citation needed". I have never heard anyone say six-oh in place of sixty outside of highly-marked contexts like "talking to the control tower".

[alice]

An interesting question is how, if your conworld uses a different number base, you represent this in fiction. If you have base seven, for example, do you refer to "sixty-nine" or something like "one forty-nine, two sevens, and six"?