Re: Baseless (ahem) speculation about ternary counting

```On Mon, Jul 08, 2002 at 11:12:21AM +0000, Andrew Strader wrote:
> marqoS wrote:
> >wa'         =  1[3] = 1[10]
> >cha'        =  2[3] = 2[10]
> >wa'ej       = 10[3] = 3[10]
>
> "Klingon originally had a ternary number system; that is, one
> based on three. Counting proceeded as follows: 1, 2, 3; 3+1,
> 3+2, 3+3; 2*3+1, 2*3+2, 2*3+3; 3*3+1, 3*3+2,
> 3*3+3; and then it got complicated." (TKD 5.2)
Yeah, that would be the section which I noted was hard to infer useful data
from.  First, it says that Klingon originally had a ternary number system;
then it gives an example of counting that is not quite ternary:
the "+3"s are what makes it weird.

If we counted in base ten according to the above description of
counting in base three, we would need an extra digit (say "X")
which means "ten" and is only used for even multiples of ten.
It would effectively replace "0": although we would still need a
symbol to represent the actual number zero, we wouldn't need a
placeholder for other numbers:

1,2,3,4,5,6,7,8,9,X, 11,12,13,14,15,16,17,18,19,1X, 21,22,23,24,25, . . .
. . . ,96,97,98,99,9X, X1,X2,X3,X4,X5,X6,X7,X8,X9, XX

That last number XX (10*10+10) represents 110 in our usual notation,
and is the point after which things presumably get complicated.
The only real complication is the need to introduce a notation for
powers in the written-out form; the next number would be written
the same way we write it now - 111 - and could be written out
Okrandian-style as "10^2+10+1".  Similarly, the next number after
"3x3+3" would be "3^2+3+1".

However, in my previous message, I was operating under the
assumption that the explanation of counting in base three in TKD was
oversimplified for the benefit of a non-mathematical reader and that
the Klingons actually counted in base three the way we would expect:
1, 2, 3, 3+1, 3+2, 2*3, 2*3+1, 2*3+2, 3^2, etc.  Or, in ternary
notation, 1, 2, 10, 11, 12, 20, 21, 22, 100.

> As much as I like your current hypothesis, it's unlikely that "wa'ej" was
> related to the notation for 3, given that the Klingon system needed to have
> native roots for one, two, and three. If anything, "wa'ej" should have meant
> 4.
Hm. I don't follow that.  If the current system followed that logic, then
<wa'maH> would be "eleven" (and "ten" would be a "basic counting numeral",
possibly bare <maH>, possibly something else . . .)

> According to Okrand, Klingon had basic counting numerals thru 3. When they
> converted to decimal, they probably just borrowed 4-8 from their music, and
> nine from somewhere else,
Yup.  That was my starting point.  I just added the supposition that
the "basic" counting numeral for three was, originally, a compound
formed analogously to the way ten is formed now.  Over time the
pronunciation blurred from *<wa'ej> into just <wej>, possibly even
before the switch to base ten.   Maybe other numbers were similarly
blurred so that they pronouned 6 as *<chej> instead of *<cha'ej>.

> maybe from Hutlh because nine was lacking, or else
> from Hu' because they wanted their number system to get up. :)
Heh. :)

> All this is pure, unbridled speculation.
Most definitely!

> Or maybe Okrand has said more about it, and I've missed out.
Could be, but if so, so have I.  And Okrandian proclamations tend to be
broadcast on this list fairly quickly. :)