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CONLANG Home

CONLANG Home

CONLANG  October 2004, Week 3

CONLANG October 2004, Week 3

Subject:

OT: Re: Calénnawn and a manic digression on numbers

From:

"H. S. Teoh" <[log in to unmask]>

Reply-To:

Constructed Languages List <[log in to unmask]>

Date:

Thu, 21 Oct 2004 09:46:30 -0700

Content-Type:

text/plain

Parts/Attachments:

Parts/Attachments

text/plain (84 lines)

On Wed, Oct 20, 2004 at 11:51:58AM -0400, Sally Caves wrote:
[...]
> I'd like to get hold of David Peterson's zero-less base eight.  I wonder if
> making an even number the base makes a subtle difference.  What I discovered
> about my novenus (a number not chosen arbitrarily...nine is particularly
> potent) is its groupings of threes, of course; multiples of threes produces
> perfect numeric symmetry, visually and conceptually; would such a society
> make a distinction between "even" and "odd" numbers"?  Or would triads be
> the unit?  What different mathematical concepts would they come up with?  Or
> could they?  Is mathematics pure logic in every basal configuration?  Even
> without zero?
[...]

That's an interesting observation. I have a feeling that odd/even
would not be so important as multiples of 3. In fact, such a system of
categorization does have interesting properties, as I have discovered
myself while considering base 3 systems for the EbisÚdi. I will
arbitrarily call the 3 categories even, over, and under. "Even" here
refers to multiples of 3. Under this system, 3, 6, 9, 12, ... would be
considered "even" (written as 3_N, 6_N, 9_N, 13_N, 16_N, etc.). An
"over" number is an even number plus 1, i.e., 1_N, 4_N, 7_N, 11_N,
14_N, ... . An "under" number is an even number plus 2 (or an even
number minus 1, hence the name): 2_N, 5_N, 8_N, 12_N, ... .

Why the funny names? Because over + under = even; even + over = over;
and even + under = under. (You can see this for yourself by applying
the same definitions to decimal numbers.) Even + even = even, as in
the 2-multiple case; but two overs make an under, and two unders make
an over. So you have the following equivalences:

	E + E = E
	E + O = O
	E + U = U
	O + O = U
	U + U = O

(Where E, O, U, stand for even, over, under, respectively.)  A lot
more interesting than the traditional odd/even, where the only
equivalences are:

	E + E = E
	E + O = O
	O + O = E

(Where E, O, are the traditional even and odd, respectively.)

If you wish to pursue this subject further, you might want to read up
on Gauss' modular numbers (also known as integer rings). Essentially,
odd/even is just a system based on integers modulo-2, which means that
you divide everything by 2 and look at the remainder. If it's 0, it's
even, if it's 1, it's odd. You can add/subtract these remainders at
will, as long as you divide by 2 at the end and take the remainder.
For example, the rule "odd + odd = even" can be explained as 1 + 1 = 2
(because 1 defines oddness in this system), but the remainder of 2
divided by 2 is 0, which is even, hence odd + odd = even.

Similarly, the even/over/under system I described above is simply
modulo-3 instead of modulo-2. That just means you divide everything by
3 and look at the remainder. For example, the remainder of 17 divided
by 3 is 2.  Hence, it's an "under" number. Adding two unders is
equivalent to 2+2=4, which gives a remainder of 1 upon division by 3.
Hence, we derive the rule "under + under = over". In the same way,
"over" numbers have a remainder of 1 when divided by 3, and so adding
two overs is equivalent to 1 + 1, which gives 2, i.e., an "under"
number. Hence, over + over = under. That over + under is even is
simple: since over is equivalent to a remainder of 1, and under is a
remainder of 2, adding them gives 3, which has a remainder of 0 upon
division by 3.  Because "even" is equivalent to a remainder of 0,
adding "even" numbers (in the sense of being a multiple of 3) always
gives a remainder of 0. Hence, even + even = even.

One could, of course, go much farther and examine systems modulo-4,
modulo-5, and so forth, but the farther one goes, the more complexity
is introduced, which reduces the utility of having a simple
categorization scheme in the first place. There, it ceases to be a
simple way to categorize numbers, and becomes more of a mathematical
tool for performing much more complex tasks.


T

--
It's amazing how careful choice of punctuation can leave you hanging:

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