Octal number system

[Richard W]

What are you having trouble with? It's just like decimal, except you can only count to 7 before needing another digit.

I can imagine interesting interference from base 10 or even base 12. The numbers for 30₈ (=24₁₀) and 50₈ (=40₁₀) might be vulnerable to suppletion from base 10 languages, and i can imagine 120₈ (=80₁₀) being used as a sort of 'long hundred'.

[Ryan of Tinellb]

What are you having trouble with? It's just like decimal, except you can only count to 7 before needing another digit.

I can imagine interesting interference from base 10 or even base 12. The numbers for 30₈ (=24₁₀) and 50₈ (=40₁₀) might be vulnerable to suppletion from base 10 languages, and i can imagine 120₈ (=80₁₀) being used as a sort of 'long hundred'.

The other trouble I have is that all numbers are 10 in their own base, which means that the subscript x₁₀ seems ambiguous.

[StrangerCoug]

Having written a paper for a grad seminar on numeric systems once, I've practically come to believe you can find a natural language somewhere that uses any number system, if it even bothers to use numbers at all.

Even base nine?

Is that different from base three?

Technically, even though it's pretty trivial to convert between base nine and base three if you recognize the pattern since one is a power of the other:

Base 10	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
Base 9	1	2	3	4	5	6	7	8	10	11	12	13	14	15	16	17	18	20	21	22
Base 3	1	2	10	11	12	20	21	22	100	101	102	110	111	112	120	121	122	200	201	202

[Qwynegold]

Getting large numbers right can be tedious, so if you want to skip that, or double-check your results, you can use Wolfram Alpha.

Even easier if you are using Windows at least. Because you can just attach the Calculator to your taskbar for easy access, and then use that all the time. You have to choose "programmer" from the menu, and then you can convert between base 8, base 16, decimal and binary.

[KathTheDragon]

What are you having trouble with? It's just like decimal, except you can only count to 7 before needing another digit.

I can imagine interesting interference from base 10 or even base 12. The numbers for 30₈ (=24₁₀) and 50₈ (=40₁₀) might be vulnerable to suppletion from base 10 languages, and i can imagine 120₈ (=80₁₀) being used as a sort of 'long hundred'.

The other trouble I have is that all numbers are 10 in their own base, which means that the subscript x₁₀ seems ambiguous.

The ambiguity is imagined, as the subscript is always to be read as base-10, due to being a meta-notation.

[Salmoneus]

On the parallel thread on the other board, THC claims that the subscript is always to be written in letters, never in numerals. While I find this unconvincing as a statement of fact about common practice, it does make sense to me as a way of reducing the potential for confusion.

[Vijay]

Apparently, Tundra Nenets used to be base 9 (see second slide on p. 20 here).

[alice]

According to Wikipedia, early computers used words containing 6, 12, 24 or 36 bits, making octal (with 3 bits per digit) an ideal choice. Of course, today hexadecimal is more popular, with computers having 32- or 64-bit words.

I used to work next to someone who had programmed these things, and it was proper machine-code programming, with none of today's namby-pamby full-stack high-level language nonsense.

Of course, *real* conlangers use number systems with different prime-number bases for each gender, with special rules for mixed-gender plurals.

[mèþru]

Apparently, Tundra Nenets used to be base 9 (see second slide on p. 20 here).

Well that's a relief; I have base nine as the basis of the dominant numeral system in a large area of my conworld.

[Vijay]

Apparently, Tundra Nenets used to be base 9 (see second slide on p. 20 here).

Well that's a relief; I have base nine as the basis of the dominant numeral system in a large area of my conworld.

I don't think you're the only one. This isn't from your conworld, is it?

Anyway, (some?) Yuman languages are supposed to have base 3, so base 9 doesn't exactly seem far-fetched.

[akam chinjir]

I've been browsing A Grammar of Komnzo by Christian Döhler, and came upon this (pp.16-17):

For many of the customs described above, it is important to record the exact quantity of tubers. For the counting ritual a special base-six numeral system is used...

The counting procedure involves two men who move the yam tubers from a prepared pile. They take up three yams each, move a few meters and deposit them together in a new pile. One of the two is the designated counter and he shouts out näbi näbi näbi ‘one one one’. This means that they have moved the first unit of six. Without pause they take up again three yams each and move them over, while the counter shouts out yda yda yda ‘two two two’. Now two lots of six or 12 tubers have been counted. Again they pick up three yams each shouting ytho ytho ytho ‘three three three’. The two men continue with this process until they reach nibo ‘six’. Now 36 yams have been counted and the bystanders and observers cheer in agreement. This amount corresponds to one fta or 6². Each fta is marked by putting a single yam on the side of the new pile. The two men continue until all yams have been counted, and the little pile on the side which indicates the amount of fta slowly grows. Next, this pile is counted in the same fashion, only that each counting yam, that is put to the side, now markes one taruba, which corresponds to 216 or 6³. One may continue in the same fashion. Six taruba make up one damno corresponds to the amount 1,296 or 6⁴. For example, one damno is amount of yams that a man should store in order to bring his family through the year. Six damno make up one wärämäkä corresponding to 7,776 or 65. Finally, six wärämäkä make up one wi corresponding to 44,656 or 6⁶. I should add that nobody in Rouku remembered the last time this number was actually reached. The recursive counting procedure gives rise to the senary system.

[TomHChappell]

On the parallel thread on the other board, THC claims that the subscript is always to be written in letters, never in numerals. While I find this unconvincing as a statement of fact about common practice, it does make sense to me as a way of reducing the potential for confusion.

Your last (compound) sentence seems right to me.
I’ve seen it done often, and it’s the way I was taught in elementary school and junior high. Maybe it’s not common practice, but I think 10₁₀ means any base. You just can’t tell which base is meant by 10₁₀ unless you already know what 10 means. You can tell it’s a base-and-place numeral-writing system, and reasonably guess it’s a whole number greater than 1, but that’s about it.

Zompist has, or at least used to have, a list of numeral bases attested by RL natlangs, with usually one or two example languages per base. I remember there weren’t any binary languages, but almost every natural-number base from three to seventy-six (or was it seventy-eight?) was represented IRL.
Of course for most bases much over twenty, no natural language uses it as a multiplicative base, as defined by WALS.

[TomHChappell]

According to Wikipedia, early computers used words containing 6, 12, 24 or 36 bits, making octal (with 3 bits per digit) an ideal choice. Of course, today hexadecimal is more popular, with computers having 32- or 64-bit words.

The Space Cadet keyboard depended on nine-bit bytes.
And there was a machine with eighteen-bit words; the two extra bits were for data integrity and security.

[zompist]

Zompist has, or at least used to have, a list of numeral bases attested by RL natlangs, with usually one or two example languages per base. I remember there weren’t any binary languages, but almost every natural-number base from three to seventy-six (or was it seventy-eight?) was represented IRL.

You're probably thinking of someone else's page. What I have is here— search for 'base'. Bases other than 2, 10 and 20 are rare.

Of course for most bases much over twenty, no natural language uses it as a multiplicative base, as defined by WALS.

this chapter doesn't have that definition... 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>.)

[Vijay]

Some uncommon numeral systems are relatively common in Melanesia. Ekari/Ekagi/Kapauku/Mee has base 60. EDIT: Never mind, you just mentioned Ekari

[TomHChappell]

You're probably thinking of someone else's page.

Must be, unless you’ve just forgotten, which seems unlikely!

What I have is here— search for 'base'.

Thanks!
You include examples of bases four and five and six and eight; and maybe others.
I never heard of a natlang with base two, except that Yagua has a two-five-ten-twenty-forty numeral system. It’s the only natlang I’ve ever read about that violates the rule that if a language has two bases, one must be a multiple of another; five is not a multiple of two (I’ll bet you already knew that!).

Of course for most bases much over twenty, no natural language uses it as a multiplicative base, as defined by WALS.

this chapter doesn't have that definition...

By the “base” of a numeral system we mean the value n such that numeral expressions are constructed according to the pattern ... xn + y, i.e. some numeral x multiplied by the base plus some other numeral.

That’s the definition I was thinking of. You’re right it doesn’t emphasize (nor even mention!) the difference between “base” and “multiplicative base”.
I think the less-sophisticated-than-that kind of base was called an “additive base” or some such phrase, on the spinnwebbe ZBB. Or somewhere. The one with the “buri buri buri buri” thread. (I think.)
In WALS feature 131 those are represented by e.g, the “Extended Body-Part” Systems. IIANM.

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”.
Interestingly, as Ray Brown once mentioned on CONLANG-L, the Romans’ Latin used an exponential base of ten for whole numbers, but used an exponential base of one-twelfth for fractions.

BTW I’m pretty sure there is or was or are or were (a) natlang(s) with (a)
three — six — twentyfour, and/or
four — twelve — one-hundred-twenty, and/or
five — twenty — three-hundred-sixty,
base-system(s); or at least one of them.

Needed citation missing, I know; sorry !

—————

Thanks! Interesting stuff!

[zompist]

You include examples of bases four and five and six and eight; and maybe others.

There's a couple of base 12 in there-- Maktalu, Nimbia.

I never heard of a natlang with base two,

Well, there's quite a few languages which we can call, neutrally, 2-centric. The pattern tends to be

[1]
[2]
[2] [1]
[2] [2]
[2][2][1]
[2][2][2]
...

Now, this isn't how we expect a binary system to work based on computer notation. But it is if we accept that [2][2] is the word for four!
After all, we don't insist that English isn't decimal because the PIE word 'hundred' is etymologically 'ten-(something)'. (Where the 'something' is unknown.)

So far as I know, all such systems are used only for small numbers by our standards. On the other hand, such systems are what make me dubious about claims that certain people "can't count above two". The commonness of the above system makes me think that many "one, two" systems are really extensible the same way, only the original researchers didn't realize it because they were only expecting to find a word "three".

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, except of course that we have an explicit decimal point. Numbers are technology, and we benefit from new inventions! Same reason we have a word for 'spaceship' and Akkadian doesn't. But when you get into representing fractions, you're really talking about arithmetic, not numbers.)

[xxx]

why not a non-decimal base more universal: 5...
under 10 bases are quite simple for small numbers, but quickly the spreadsheets and their functions of conversions between bases are welcome...

[bradrn]

for a priori reasons, I use a non-decimal base more universal: 5...

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?

[Salmoneus]

I never heard of a natlang with base two,

Well, there's quite a few languages which we can call, neutrally, 2-centric. The pattern tends to be

[1]
[2]
[2] [1]
[2] [2]
[2][2][1]
[2][2][2]
...

Now, this isn't how we expect a binary system to work based on computer notation. But it is if we accept that [2][2] is the word for four!
After all, we don't insist that English isn't decimal because the PIE word 'hundred' is etymologically 'ten-(something)'. (Where the 'something' is unknown.)

I think that's cheating a little - there's a difference between the etymology of a word and its synchronic meaning. So intuitively there's a difference between a phrase synchronically meaning 'two plus two' and a word that merely etymologically meant 'two plus two' once upon a time. Also, 'hundred', even diachronically, is still multiplicative, unless the 'something' meant 'ninety'...

I think there is a valid distinction to draw between what Tom calls additive, multiplicative and exponential bases, although I've no idea if, or with what terminology, it's drawn in the literature.


Here's a rough attempt at a formula for counting: numbers within the range k are expressed with the formula aw_f + bx^g + cy^h ...

The 'standard' decimal is produced when, for all k, w=10, x=10, y=10, etc, and f,g,h etc equal 1, 2, 3 etc, and a,b,c etc are all <10.

If you make a, b, c etc all equal either 1 or 0, and f,g,h etc all equal 1 or 0, then you have an 'additive' system.
If you allow a,b,c etc to not always equal either 1 or 0, but f,g,h etc all equal 1 or 0, then you have a 'multiplicative' system.
If you allow not only a,b,c but also f,g,h to not equal 1 or 0, then you have an 'exponential' system.

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]

All of these systems can in some sense be called 'decimal', 'base ten' or 'ten-centred' or the like, but are very different. Only the exponential system is what we might more properly call 'radix 10', or decimal in the narrow sense.

Normally, w=x=y (etc); otherwise, you have a mixed base system. For instance, in a 10:60 multiplicative system, 214 would be [3x60 + 3x10 + 4].

Mixed base systems also enable systems in which a,b,c etc are all =< 1, but f,g,h aren't. For instance, 214 could be expressed as [5^3 + 2^3 + 7^2 + 5^2 + 7^1] (where x,y,z are primes under 10). I doubt any human language uses such a system, though.

It's also notably possible to have what we might call a 'mixed regime' system, in which the parameters of the formula change depending on k. For instance, a system might be additive base 10 up to 20, but then multiplicative base 5 above 20. So 18 would be [10 + 8], but 28 would be [5x5 + 3].

Interesting thigns can then also happen with a, b, c etc. In standard exponential decimal, each of these must be < 10, which is convenently =x. But you could also have systems where a,b,c can be greater than x. Tolkien's hobbits have such a system, in which 110 is [11x10] (eleventy-ten) - that is, their a,b,c are <12, even though their x,y,z are 10.

In such a system, there are multiple ways to express a number, and there may or may not be rules to say which is preferred - larger 'big end' (higher powers) or large 'small end' (lower powers) numbers. So 111 could be in one system [11x10 + 1] (maximising the big end) or in another [10x10 + 11] (maximising the small end). Or in some systems there may be a choice.

Similarly, when a,b,c < x,y,z, you likewise get a choice. So 28 could be [5x5 + 3], or [4x5 + 8] (big end vs small end).

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...

Finally, there can also be cases where a,b,c etc may themselves be products. This is only really an etymological quirk with exponential system, but it provides an interesting variation to multiplicative systems. So, while a normal multiplicative system may have 210 as [10x10 + 10x10 + 10] or (if a < 21 even though x=10) [20x10 + 10], a product system could have [3x7x10].

So far as I know, all such systems are used only for small numbers by our standards. On the other hand, such systems are what make me dubious about claims that certain people "can't count above two". The commonness of the above system makes me think that many "one, two" systems are really extensible the same way, only the original researchers didn't realize it because they were only expecting to find a word "three".

Pratchett's trolls are derided for being unable to count to three - they just count 'one, two, many'.

This is because humans don't bother listening beyond that point. How the trolls actually count is "one, two, many, many-one, many-two, many-many, many-many-one, many-many-two, many-many-many, lots!" [that is, they have an additive system where x is 1, 2, 3 or 10. This is notably different from a mixed radix multiplicative or exponential system with bases of 3 and 10, because the former would give 'two-many' (for 4) and 'many-many' (for 9), while the latter would give 'two-many' and 'many-cubed'...]