The Sand Reckoner in Your ‘Langs

The Sand Reckoner in Your ‘Langs

See:
<http://en.wikipedia.org/wiki/The_Sand_Reckoner#Naming_large_numbers>


Questions:
How do you name very-large numbers in your conlangs and natlangs?
For as many of the following three lists of ten numbers as you like, how, in
your ‘langs, do you say them?

Successive squares:
* base
* base^2
* base^4
* base^8
* base^16
* base^32
* base^64
* base^128
* base^256
* base^512

Successive cubes:
* base
* base^3
* base^9
* base^27
* base^81
* base^243
* base^729
* base^2187
* base^6561
* base^19683

Succesive cubes interpolated with squares (there may be a better way to say
that):
* base
* base^2
* base^3
* base^6
* base^9
* base^18
* base^27
* base^54
* base^81
* base^162



Discussion:

<http://pages.prodigy.net/jhonig/bignum/qauniver.html>
says (or at least I take it to imply) that one is unlikely to have to count more
than about 10^80 individual things (that being around the number of atoms in
the universe); and one is unlikely to need more than about 40 significant digits
of precision in anything, since the angle subtended by an atom in the
Andromeda Galaxy, as seen from here, is more than 10^(-40) degrees. And
before the Middle Ages common people never (or at least rarely) needed
numbers bigger than a thousand or a myriad, and before modern times never
(or at least rarely) needed number larger than a million.

Nevertheless there may be some easy-ish ways to name numbers up to very
large limits.

Successive Squaring

In a language with an “exponential base” (see
<http://wals.info/feature/description/131>
, “Chapter 131 Numeral Bases” by Bernard Comrie, the last paragraph of “1.
Introduction”), a natural-seeming way to name larger numbers, it seems to
me, is to name “units” (in the sense Archimedes used) by successive squaring
of the base.  The first ten such units would be:
* the base
* the square of the base, base^2
* the square of the number above, base^4
* the square of the number above, base^8
* the square of the number above, base^16
* the square of the number above, base^32
* the square of the number above, base^64
* the square of the number above, base^128
* the square of the number above, base^256
* the square of the number above, base^512

For instance, if the base is ten, these would be:
* ten (10 in decimal)
* one hundred, ten^2 (100 in decimal)
* one myriad, the square of one hundred, ten^4 (10,000 in decimal)
* Archimedes’ “unit of the second numbers”, ten crores or one hundred millions
or one thousand lakhs, the square of one myriad, ten^8 (100,000,000 in
decimal)
* the square of the number above, ten^16 (10,000, 000,000, 000,000 in
decimal)
* the square of the number above, ten^32
* the square of the number above, ten^64
* the square of the number above, ten^128
* the square of the number above, ten^256
* the square of the number above, ten^512

The first ten powers of ten could be named using only the first four of those
words. For instance, assuming the name for 10^8 is “megamyriad”, we would
have;
* 10 ten
* 100 hundred
* 1,000 ten hundred
* 10,000 myriad
* 100,000 ten myriad
* 1,000,000 hundred myriad
* 10, 000,000 ten hundred myriad
* 100, 000,000 megamyriad
* 1,000, 000,000 ten megamyriad
* 10,000 000,000 hundred megamyriad

Successive Cubing

Another natural-seeming way to organize the names of large numbers might be
in successive cubes instead of successive squares.
* the base
* the cube of the base, base^3
* the cube of the number above, base^9
* the cube of the number above, base^27
* the cube of the number above, base^81
* the cube of the number above, base^243
* the cube of the number above, base^729
* the cube of the number above, base^2187
* the cube of the number above, base^6561
* the cube of the number above, base^19683

For instance, if the base is ten, these would be:
* ten (10 in decimal)
* one thousand, ten^3 (1,000 in decimal)
* one milliard, the cube of one thousand, ten^9 (1,000, 000,000 in decimal)
* the cube of one milliard, ten^27 (1,000, 000,000, 000,000, 000,000,
000,000 in decimal)
* the cube of the number above, ten^81
* the cube of the number above, ten^243
* the cube of the number above, ten^729
* the cube of the number above, ten^2187
* the cube of the number above, ten^6561
* the cube of the number above, ten^19683

The first ten powers of ten could be named using only the first three of those
words;
* 10 ten
* 100 ten ten
* 1,000 thousand
* 10,000 ten thousand
* 100,000 ten ten thousand
* 1,000,000 thousand thousand
* 10, 000,000 ten thousand thousand
* 100, 000,000 ten ten thousand thousand
* 1,000, 000,000 milliard
* 10,000 000,000 ten milliard


But note that sometimes a word has to be repeated. In order to make that
less of a problem, one might use the first five of the successive cubes, and
also the squares of those first five:
(1) the base
(2) the square of the base, base^2
(3) the cube of the base, base^3
(4) the square of the number in (3), base^6
(5) the cube of the number in (3), base^9
(6) the square of the number in (5), base^18
(7) the cube of the number in (5), base^27
(8) the square of the number in (7), base^54
(9) the cube of the number in (7), base^81
(10) the square of the number in (9), base^162

For instance, if the base is ten, these would be:
(1) ten (10 in decimal)
(2) one hundred, ten^2 (100 in decimal)
(3) one thousand, ten^3 (1,000 in decimal)
(4) one million, the square of one thousand, ten^6 (1,000,000 in decimal)
(5) one milliard, the cube of one thousand, ten^9 (1,000, 000,000 in decimal)
(6) the square of one milliard, ten^18 (1,000,000, 000,000, 000,000 in decimal)
(7) the cube of one milliard, ten^27 (1,000, 000,000, 000,000, 000,000,
000,000 in decimal)
(8) the square of the number in (7), ten^54
(9) the cube of the number in (7), ten^81
(10) the square of the number in (9), ten^162

The first ten powers of ten could be named using only the first five of those
words;
* 10 ten
* 100 hundred
* 1,000 thousand
* 10,000 ten thousand
* 100,000 hundred thousand
* 1,000,000 million
* 10, 000,000 ten million
* 100, 000,000 hundred million
* 1,000, 000,000 milliard
* 10,000 000,000 ten milliard



I don’t think that successive fourth powers or successive fifth powers, or
indeed successive nth powers for any n>3, are very likely to be very natural,
because of the need for either repeating a number-name three or more times
in a row, or inventing yet more “interpolating” number-names.

But IMO successive-squaring is probably more “natural” than the European
method of having a geometric sequence, each new “unit” being a set multiple
of the next smaller one.
For instance, the Indian system starts at one thousand and then multiplies
each unit by one hundred to get the next larger:
1,000 one thousand
1,00,000 one lakh
1,00,00,000 one crore
… etc. …

The “American” “short-count” system starts at one million and then multiplies
each unit by one thousand to get the next larger:
1,000,000 one million
1,000,000,000 one billion
1,000,000,000,000 one trillion
1,000,000,000,000,000 one quadrillion
… etc. …

The “traditional Continental (?)” “long-count” system starts at one million and
then multiplies each unit by one million to get the next larger:
1,000,000 one million
1,000,000, 000,000 one billion
1,000,000, 000,000, 000,000 one trillion
1,000,000, 000,000, 000,000, 000,000 one quadrillion
… etc. …


To get up to some very high limits, such as, for instance, 10^80, takes more
individual number-names for the geometric-sequence systems than for the
successive-squaring system.  For instance, using base ten, the “traditional
British” or “long-count” system requires thirteen steps; the “American”
or “short-count” system requires 25; and the Indian system requires 38.

Furthermore, the smaller the base is, the more steps are required. Suppose
your goal is N; the start of your sequence is base^S, and the common ratio is
base^R. You need approximately log(N/(base^S))/log(base^R) steps to get up
to N.  Assuming base^S and base^R are close to equal, the number of steps
needed rises rapidly as base^R decreases. For instance, to reach or exceed
10^80 in only 9 steps you’d need base^S=base^R to be at least sixty^5. If
base^S=base^R is one million or less you’d need at least 14 steps to get to
10^80; if base^S=base^R is a myriad or less you’d need at least 20 steps to
get to 10^80; if base^S=base^R is a thousand or less you’d need at least 27
steps; if base^S=base^R is a hundred or less you’d need at least 40 steps;
and so on.

But, using a successive-squaring or a successive-cubing system, the number
of such numerals needed to get up to a very high limit, such as 10^80, is semi-
independent of the base.

Take some high number N you want to reach or surpass.  Using successive
squaring, you could reach it in log(log(N)/log(base))/log(2) steps; using
successive cubing, you could reach it in log(log(N)/log(base))/log(3) steps.
So to equal or exceed 10^80, for instance, would take 6 steps of successive
squaring for any base from eighteen to three-hundred-sixteen; 7 for bases
from five to seventeen; 8 for bases three or four; and 9 for base two.
And to equal or exceed 10^80 would take 4 steps of successive cubing for
any base from ten to nine-hundred-eighteen; 5 for bases from three to nine;
and 6 for base two.

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