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Re: The Sand Reckoner in Your ‘Langs

From: Eldin Raigmore Sunday, April 5, 2009, 21:43
```On Sun, 5 Apr 2009 01:55:14 -0400, Alex Fink <000024@...> wrote:
>On Sat, 4 Apr 2009 16:18:49 -0400, Eldin Raigmore
><eldin_raigmore@...> wrote:
>[snip]
>Perhaps surprisingly for a math person like me (or perhaps not?) I don't
>even know how to name vaguely-large numbers in most of my serious
>conlangs.
> Pjaukra is base 12 and I don't know the word for 12^3 (_undu_ 12, _sarda_
>12^2).  In Sabasasaj I haven't decided whether to use base 120 (subbase 10,
>though using lots of fractions so not the canonical subbase system) or no
>exponential base at all but something more irregular, and in particular
>_phian-tu_ 120 is my largest number word.
Adpihi is also base-twelve, and I also haven't necessarily got the answers to

I've mentioned on this list bases with at least as many factors as any smaller
number, which would include both twelve and 120 (as well as several others).

>[snip]
>In A:jat he-Heloun,
>[snip]<http://000024.org/conlang/AhH/05-11.html#su.numbers>
>[snip]
>This is of course inspired by the contrast of the short and long count
>systems for the English "illion" numbers.
Thanks.

>[snip]
>I'd disagree, if you use "natural" with the sense it has in "natural
>language" here.
OK, I think I should have said "as natural as".
The existing systems were formed a bit at a time; first people didn't need
really big numbers, then they needed some moderately biggish ones, then
some bigger yet, then some yet bigger; and it was probably the fifth or sixth
or seventh step before they started to get systematic.
Viewed from that perspective the successive-squaring system and the
geometric-sequence system could be equally "natural" and equally likely to
evolve; though I don't know how much of it could really be considered "natural
language" already, since mostly people (though not "most people") know who
made the innovation, and mostly that was in historical times, and mostly
common people don't have to, and in fact don't, have a clear notion of what
the higher numbers in the system mean.

>I analyse this pattern of having a geometric sequence as
>just another exponential base.
I have been defining things this way; maybe some other way is standard,
maybe my way is just as good for a "standard" as anything anyone else has

A number-name B is a "multiplicative base" if at least half of the number-names
from B to 2B are quasi-transparently derived from "B+n" for 1 <= n < B, and at
least half of the number-names from B to B^2 (or from B to the next higher
multiplicative base) are quasi-transparently derived from "mB+n", for 1 < m < B
and 0 <= n < B.

In English "ten" is a multiplicative base.

(I say "at least half" because of two phenomena; (1) in some languages some
numbers have their own names that don't fit into that system (e.g. eleven,
twelve) and (2) in some languages numbers close to the next multiple of the
base are named "anticipatorily", for instance "forty save one" instead of "thirty-
nine".)

A number-name C is a "sub-base" for B if C < B and at least half of the number-
names from C to B are quasi-transparently derived from "mC+n" for 1 <= m < C
and 0 <= n < C; _and_, at least half of the number-names from B to B^2 (or
whatever the next higher-than-B multiplicative-base is) are quasi-
transparently derived from "(jC+k)B+(mC+n)".

In English, for numbers from somewhere around 1010 up to 9999, "ten" is a
subbase for "hundred". (That is, we're likelier to say "nineteen forty-six"
than "one thousand nine hundred forty-six".)

In English, both "ten" and "hundred" are sub-bases for "thousand".

In the long-count system, "ten" and "hundred" and "thousand" are all sub-
bases for "million".

Mostly a subbase C is not less than the square root of the base B; mostly it
isn't more than twice such a square root.  That is, B <= C^2 <= 2B.

A number-name C is a "super-base" for B if at least half of the number-names
from C up to BC (or the next-higher superbase) are quasi-transparently
derived from (kC + mB + n).

In English, hundred is a superbase of ten, thousand is a superbase of both
hundred and ten, and million is a superbase of thousand and so on.

An "exponential base" system is a geometric sequence of superbases; there's a
starting point (call it S), and each next-larger superbase is a common ratio
(call it R) times the next-smaller one.  The number-name R, in such cases, is
called an "exponential base".  One prefers that the name for (R^n times S) be
quasi-transparently derived from n (or n-1 or n+1 or some other number close
to n).

English has the -illion system, which can have either S=R=10^6, or S=10^6
and R=10^3; and it also has the Indian system, with S=10^3 and R=10^2.

>You'll know of natlangs with base-subbase
>systems, say base 20 and subbase 5 (so that the numbers less than 20 are
>formed as 5a+b); so short-count English just has base 1000 and subbase 10.
By what I said above, short-count English has:
smallest base is ten;
hundred = ten-squared is a superbase for ten, and for most numbers less than
hundred-squared ten is a subbase for hundred;
thousand is a superbase for hundred and for ten, and both hundred and ten
are sub-bases for thousand;
there is a geometric sequence of multiplicative bases, each of which is a
superbase for all the smaller ones, but none of which is a subbase for any of
the larger ones, beginning with thousand and using a common-ratio of
thousand.  Each of ten and hundred is a sub-base for each of those
multiplicative bases.

>Do you know Donald Knuth's -yllion notation?
>  http://en.wikipedia.org/wiki/Knuth_-yllion
>This is the only instantiation of your suggestion I can think of that I've
>seen before.
No! Thanks a lot! I'd never seen that before.

>But really, big number names are just cumbrous, aren't they?  For powers of
>the base, "ten to the fifteenth" seems an easier way to go than
>"quadrillion".  So when I'm wearing my non-naturalist hat I can't think I'd
>bother with big number names either; better just a sufficiently lithe way to
>express exponentiation.  (And I probably wouldn't use them in number names
>as throughgoingly as English does either.  For example in my language with
>Robert Barrington Leigh, which was base 16, no positive powers of the base
>had names; numbers were purely and simply digit strings.  Alright, this is
>maybe bad in that you might lose count of how long a number is and so not
>know its order of magnitude, if you care about that.)
(I don't much like bases that are just powers of primes.)

To me, the successive-squares system is both as likely to evolve and as easy
to use.

>Alex
Thanks!
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