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