Altered Perspectives and Insight to Algorithm

           An Illustration of the Glyph Arithmetic Development

                          of Ethiopic Numbers




The Ethiopian numeral system was devised following the ancient Greek method of
modifying existing members from the character set for the spoken language.  The
characteristic over bars and under bars indicate a likely Roman influence as
well. Unlike ethier the Romans or Greeks the Ethiopic numberals show a greater
mutation from the spoken letters that they may have been based upon. Also, the
arrangment of the numerals in the formation of numbers does not have the 
same kind of cyclic behavior found in Greek, Roman, or the Arabic stems.  The
algorithm for the glyph arrangement is not immediately apparent and often
presents problems even for native users.

This paper will attempt to present graphically the arrangement cycles
intrinsic to the system.

The glyphs that make up the Ethiopian numeral system number 20 and are
presented in the following :

Ones
     0  1  2  3  4  5  6  7  8  9
                ߝ
Tens
      10   20  30  40  50  60  70  80  90
                       
Higher
      100     1,000       10,000
              /       

"" is often used by Amhara merchants for the value of one thousand, after its
word name in Amharic (ӝ).  The character will not be treated here as if
it were a numeral.

Table  shows the core of the 4 stepped development that makes up a single
cyclic period in numeric growth.
 
Table 
|------------|------------|--------------|------------|
|ΝΝΝΝ|ΝΝΝΝ|ΝΝΝΝ|ΝΝΝΝ|
|------------|------------|--------------|------------|
|  10        |  100       |  1000        |  1,0000    |
|------------|------------|--------------|------------|

Table  presents the growth over four periods, or powers of 10,000 ().
The Arabic equivalent for the end of the period, column 4, is provided
on the left hand side.

Table  
|------------|------------|--------------|------------|
|ΝΝΝΝ|ΝΝΝΝ|ΝΝΝΝ|ΝΝΝΝ|1,0000
|------------|------------|--------------|------------|
|ΝΝΝ|ΝΝΝ|ΝΝΝ|ΝΝΝ|  1,0000,0000
|------------|------------|--------------|------------|
|ΝΝ|ΝΝ|ΝΝ|ΝΝ|  1,0000,0000,0000
|------------|------------|--------------|------------|
|Ν|Ν|Ν|Ν|  1,0000,0000,0000,0000
|------------|------------|--------------|------------|

We can observe that the character in the 4th column of any row, will cary
over into the next row.  As if it were a coefficient on the right hand side
(RHS) of the new numeric sequences.  We can "factor" this RHS coefficient
glyph out of Table 1 to observe more readily the growth of the coefficient.
We do so in column 6 of Table  below and recognizing its growth from row to
row, we will treat it as a varaible across rows.

Table 
-----|------|------|--------|------|---------
() |Ν|Ν|Ν|Ν|
-----|------|------|--------|------|---------
() |Ν|Ν|Ν|Ν| 
-----|------|------|--------|------|---------
() |Ν|Ν|Ν|Ν| 
-----|------|------|--------|------|---------
() |Ν|Ν|Ν|Ν|  
-----|------|------|--------|------|---------

Columns 2 - 4 now reveal again the cyclic pattern within the numeral
development as seen first in Table .The orders of 10000 are factored to 
the right in column 6.  The right beingthe side that the glyphs () are 
appended to on the 4 cyclic variables.ΝThere was an implied coeffient of 1 
() living in the template and nowrevealed in the 1st column.  We have not
seen it before due to the natureof a decimal (base 10) series :

   1 x 10 = 10 
   xΝ= 

For Ethiopic numberals, this trait breaks down when the multiples of 10 up to 
90 are considered.  Each such multiple will have unique a character 
representations, which introduces a 2nd type of cyclic behavior to arrangment
of the numeral in numeric growth.

It is now wise to assign a new variable to left coefficient, we do
so with  and introduce another  that will alway be equal in value to
10 or more appropiately .  

Suppose the coefficient before the orders of 10 is no longer singular.
Let the coefficent be  (3) then  =  andΝ = ٝ = .  Table 
represents Table  with the LHS coefficient .  

Table 
|------------|--------------|--------------|--------------|
|ΝΝΝΝ|ΝٝΝΝΝ|ΝΝΝΝ|ΝٝΝΝΝ|
|------------|--------------|--------------|--------------|
|ΝΝΝ|ΝٝΝΝ|ΝΝΝ|ΝٝΝΝ|
|------------|--------------|--------------|--------------|
|ΝΝ|ΝٝΝ|ΝΝ|ΝٝΝ| 
|------------|--------------|--------------|--------------|
|Ν|Νٝ|Ν|Νٝ|
|------------|--------------|--------------|--------------|

Table  can be shown as the combination of two other tables, Table and
a second table holding the alternating LHS coefficients :

|------|------|--------|------|---------
|  |Ν|  |Ν|
|------|------|--------|------|---------
  |    |Ν|  |Ν| 
|------|------|--------|------|---------
  |  |Ν|  |Ν| 
  |------|------|--------|------|---------
  |  |Ν|  |Ν|  
  |------|------|--------|------|---------

                +

|------|------|--------|------|   Left coefficients
|Ν|Νٝ|ΝΝ|Νٝ|   added for each row.
|------|------|--------|------|


There remains a final reduction for our developing numerical composition
algorithm.   Table which shows the growth of the rows for increasing powers
of , may be reduced to a single row with an exponent given on the RHS
variable like so:

  |------|------|--------|------|-------
  |  |Ν|  |Ν|Ν^Ν(= 0 -> infinity )
  |------|------|--------|------|-------
                                            becomes the number of  that
ΝΝΝΝΝΝΝ+                          will be appended on the RHS.

|------|------|--------|------| 
|Νٝ|Ν|ΝٝΝ|Ν|  LHS Coefficients.
|------|------|--------|------|


Note here that the  coefficient appears in the columns where the number of
zeros ("0"s) is odd in Table , and that  appears in the columns showing
an even number of zeros.

A recombination follows but requires the special rule for omiting (1) as
a left hand side coefficient except for the second special case of ^0 which
occurs only in the in the first column of the first row. 

 Reduced Glyph Composition Algorithm :

     10     100      1,000   10,000  (10,000^)
  |------|--------|--------|--------|-------
  |Ν* |Ν|Νٝ|Ν|Ν^(= 0 -> infinity )
  |------|--------|--------|--------|-------

    = ם->ΝߝΝΝΝΝΝ(= 01 -> 09)    for even number of zeros.
ΝΝ= =>Ν->Ν    (= 10 -> 90)    for  odd number of zeros.

    * Special rule for combination with (1)


The Ethiopian numerals are not commonly used these days for more than giving
calendar years.  With 1987 as both a unitless number and as a year for an
example, we can demonstrate the arrangment of the numerals required to
construct each.

The Number 1987 :

  1987 =  1,000 + 9x(100) + 80 + 07
       =    + x   +  + 
       =  

The Year 1987 :

  1987 =  19x(100) + 80 + 07
       = (10+9)x(100) + 80 + 07
       =   x    +  +Ν
       =  ߝݝ