Numbers and Numeral Systems

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Aszev
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Numbers and Numeral Systems

Post by Aszev » Tue 11 Oct 2011, 19:51

The following post was written and posted by Torco on the zbb. It's reposted here by me with his permission:

Sometimes I get the feelings some people seem to think numeral systems have exactly one defining trait: their base. This is based on a number of gross misconceptions.

Mind you numeral systems are linguistic as well as graphic, and when someone think of the way cultures write their numbers they're probably not thinking about "the number system of a certain language" but to the way a large culture -almost everyone in the planet by now- *writes* numbers. but languages aren't about writing stuff, they're about saying stuff. So what's the numeral system of english?

hint: no its not "base 10", dummy, that's *notation*. it's complicated: it has a lot of names for numbers 1 to 12 [one, two, three], then another set of words for numbers 13-19 which are more or less non-derived [three-thirteen], or at least there's a special affix that marks [+plus ten], /tI:n/, thirteen, nineteen. Then again you have a special word for numbers 1 to 9 times ten (twenty, thirty, seventy) of which the first four are kind of irregular and the rest are regular, having a [+times ten] affix -ty. Finally you have words for the quantity ten times ten, hundred, and you work that one differently than you work the quantity one times ten: one times ten times a small number gets its own affix: twenty, thirty, forty, fifty. ten times ten times gets a different word: hundred, and you say one hundred. that's funny because strings of numerals aren't obvious in English: if I say two nine I'm not saying an obvious number: two times nine? two plus nine? twenty nine? If I say five eleven is it 5+11? 5 times 11? 511 ? but if I say two other unrelated word numbers it's really clear: four hundred. BUT successive number words don't indicate multiplication: thirty is 30 and seven is 7 and 37 is thirty seven (I think it's written thirty-seven), so there number+number is a sum. how do you know when to sum number words and when to multiply them? maybe when the small one comes first you multiply ? let's see:

"two ten"

... no, that doesn't sound like twenty

"hundred million"

yes, that sounds like 1E8. (a billion, you call that? that's cause you're weird.)

"three -- a hundred and two"

nope, that's not it, that's not 306, more like 302... even more like three hundred and two.

so it only works with small number + word for a power of ten. That's weird. Of course I'm not gona describe english numerals in its entirety, but the above complexity *CANNOT* be summed up as "English is base 10" because it's just... NOT base 10, it's base 10 and 12 with weird sign-values at 1-12, and at powers of 10, so it's roughly base 10, but not quite. it also is additive, starts with the biggest quantity and ends with the smallest, and has these complicated ways to know whether to multiply or add two sign-values together, it doesn't have ways to divide or substract, just add and multiply (in number construction, at least. the romans did; IV is five minus one, there's no such construction in English Spoken Numeral System). Notation systems tends towards being more orderly, even positional [which is a supremely elegant solution], and even then post-positional like the scientific notation (2,1752E9, 2 + 1752 divided by 10*10*10*10 and then all that times 10(*10)-nine times, yeah, intuitive isn't it ?)

For all their complexity, our writing system is pretty straightforward: the last number plus the second-to-last number times 10 plus the third-to-last number times 100 plus blablabla. Actual language's numeral systems, on the other hand, are MUCH more FUN. and they're almost never positional: outside of things like "Beverly Hills nine oh two one oh"... still, if I say "nine oh one two four" you have no inmediately intuitive understanding of what quantity that is. what's nine oh one two four times six ? but what's nine thousand twenty four times eleven? easy, that's six thousand and... geddit?

Numbers are a part of our life, and the arabic positional base 10 system, along with our complicated, arbitrary additive system [three (by) hundred (plus) nine], are so integrally engrained in our lives that it's hard to think of the concept of number in itself. A number is, basically, a representation of quantity, and quantity is not as obvious as one might think: after all most real things are not really discrete but, at least to some degree, continuous. If we make the exercise of forgetting the ways to count we know (which is made especially hard by the fact that, for most of us, mindlessly repeating words labeled as numbers. [wan], [t_hu], [Tri] is one of the first feats of mindless repetition of meaningless strings we're asked to perform as children ) the problem of quantity becomes especially complicated. the first logical way to solve it is tally marks.

|, ||, |||, ||||, |||||, ||||||, |||||||, ||||||||, ||||||||||

this is called unary system because every glyph represents one. The question of one what isn't trivial, and numeral systems might use different drawings or graphemes for different kinds of ones: for instance, our numeral system uses slightly different representations for the same quantities in the numerator and the denominator: three thirds, for instance: the fraction has the same quantity in the numerator and the denominator, and yet we call one of those quantities three and the other quantity third. Similarly, a system might use different representations to the same quantity depending on whether that quantity is a factor, or a rough estimate (four-ish, five-ish). The idea that all fours should always be represented the same way is quite narrow.

Unary systems are likely the oldest systems there are: when counting one can, say, make one mark on a stick for every sheep and voilá, you can count in unary: you don't need words for the numbers, you don't even need a notion of number: you just need to make one mark for every one sheep and you just represented the number of sheep you have. If you want to check later if you still have those sheep you can count them again in a similar manner and compare the marks on the stick: if there's one mark in the new stick for each mark in the old stick and no more and no less all is well and no one stole your sheep.

big numbers, however, aren't easy to manage in unary: consider a sheperd with 130 sheep. This is represented like so
ooooooooooooooooooooooooooooooooooooooooooooooo
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
so how many more sheep that is, compared to this?
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
The answer is three, if you were interested.

How do you solve that problem? well this is the prime problem of numeral systems, both written and spoken: abbreviation. Numeral systems are all abbreviations of the obvious way to represent quantity: one drawing, or one pebble, or one word, for each unit. Our ancestors, when discovering numbers, probably called them one and many, and counted like one, one, one, one, one, probably with their fingers or some other memory aid. And then something amazing happened; they gave other numbers names: two for ||, three for |||, and many for anything larger. Some basic mathematics and they could talk about many more quantities: two and three might have been the way to speak the quantity we call five. perhaps two times three times one times four, for example. This very rudimentary system with three numbers and larger quantities expressed as sums and multiplications is already a functional numeral system: it can represent in an unambiguous manner any integer imaginable... except, so did our unary tally marks. The three-numeral multiplication system might represent the number of circles here
ooooooooooooooooooooooooooooooooooooooooooooo
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
as

3*3*3*3
plus 3*3*3
plus 3*3*3
minus 2+3

that's 130 but, as you can see, it's unwieldy for large numbers. Maybe the people who used it didn't need to work large numbers: a small village, few transactions of quantifiable things or large engineering projects, their simple, elegant system worked well for small quantities. two and two times three. Eventually common operations become numbers, people forgetting that they're actually doing a multiplication and just calling the number nine as something like 3*3, say thuree or threetime. from this the notion that the affix -time means something squared, multiplied by itself, eventually takes hold, so onetime is 1, twotime is 4, threetime is 9. maybe then thureetime becomes nine squared: 27. And in such a manner more and more numerals arise. People attempt to find a common, consensual way to use those numerals: pronouncing the word for multiplication everytime in the string 3*3*3*3*3*3+3*2*3*3 would be tiresome, so maybe saying two number words in a row indicates that they're supposed to be multiplied. the order of resolution of our terms, what we call parentheses, is another complication. in the graphical representation of the word 130 above I use line breaks to mark separate terms... maybe our three-people use a special word in that role: so thureetime three then add thureetime then add thureetime substract two-and-three. not bad, considering our best shot at 130 in, say, roman numerals, was CXXX. hundred and three times ten and three times ten and three times ten. our system so far has the following words for numbers:

one
two
three
thuree - 9

and it can construct other number words with affixes.

twotime - 4
threetime - 9
thureetime - 27

maybe two-and-two, or twotime, becomes its own number word. like tutu. then

one
two
three
tutu - 4
twee - 5
thuree - 9
tututime - 16
thureetime - 27
(let's abbreviate thureetime to reetime, shall we? abbreviation is, after all, the stuff of math)

Eight numerals, eight signs for quantity: those can be used to construct all sorts of quantities. 130 then becomes reetime times twee minus twee, reetimes twee minus twee, if we don't mention the multiplication. Easy! maybe tutu times reetime plus tututime plus three and three again, if we feel like, as English does, not using substraction but only addition and multiplication. This is a pretty functional little system. And it isn't positional. The thing is that no one speaks positional, languages don't generally use positional systems because... I'm not sure, maybe because saying one, three, nine, eight, eleven, two, two, two doesn't convey the bigness of quantity that a hundred million something does.

The majority is numeral systems, both written and, more importantly, spoken, are positional. In fact, to my knowledge, there's like two positional systems that are widespread, arabic and babylonian sexagesimal. By contrast, most spoken systems are definitely sign-value, additive* and, to some degree, multiplicative and substractve. So random words that represent random quantities are thrown until you reach the desired quantity... it's like saying "this is like a thousand and then a hundred, another hundred, another hundred, and... yeah, add like two more"... we again see the theme of abbreviation upon abbreviation: a hundred, after all, is an abbreviation for
one, another, another, another, another, another, another, another, another, another, one, another, another, another, another, another, another, another, another, another, one, another, another, another, another, another, another, another, another, another, one, another, another, another, another, another, another, another, another, another, one, another, another, another, another, another, another, another, another, another, one, another, another, another, another, another, another, another, another, another, one, another, another, another, another, another, another, another, another, another, one, another, another, another, another, another, another, another, another, another, one, another, another, another, another, another, another, another, another, another, one, another, another, another, another, another, another, another, another, another.
*(hell, they might be substractive, for all one can imagine, right? isn't 1999 more easy to represent as E4-1... not much, but 189999999999993 is 1E15-1E14,-7).

Oh, also, dedicated drawings for numerals isn't the only way: A LOT of systems use alphabet letters: so

O - one
T - two
H - three
U - tutu - 4
W - twee - 5
R - ree - 9
U* - tututime - 16
R* - reetime - 27

sooo 130 is RW-W. O, T, H, U, W, HT, etctera.

Balls I'm probably gonna get Jankoed.
Sound change works in mysterious ways.

Image CE
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