VIII:Alphabetic numerals of the Greeks

VIII : Alphabetic numerals of the Greeks

     

         [Contd. A Journey to the Wonderland of Math. by Ajay Kumar Chaudhuri.]

Now, let us explore another famous ancient civilisation, famous for its valuable contribution of mathematical principles and for their unique numerals. This is the Greek civilisation, which flourished in the 8^th century BC and it is normally said the glorious period ended when Greece fell to the Romans in 146 BC.

Greece is considered to be the cradle of Western civilisation, having been the birth place of democracy, Western philosophy, the Olympic Games, Western literature, historiography, political science, major scientific and mathematical principles and western drama, including both comedy and tragedy. The Greeks were first unified under Philip of Macedon in the fourth century BC. His son Alexander the great rapidly conquered much of the ancient world, spreading Greek culture and science from the eastern Mediterranean to the Indus River. Annexed by Rome in the second century BC, Greece became an integral part of the Roman Empire and its successor, the Byzantine Empire.

Greece is strategically located at the crossroads of Europe, Asia and Africa. Situated on the southern tip of the Balkan Peninsula, it shares land borders with Albania to the northwest, the Republic of Macedonia and Bulgaria to the north and Turkey to the north-east.

To ancient Greeks, their alphabets had two roles to play: One for writing and the other to express numerals. The Greek numbering system was uniquely based on their alphabets. But there were no single Greek national standards in the first millennium BC, since the various island states adhered to their individual independence. Naturally they each had their own currency, weights, measures etc. and there by it lead to small differences in the number system among different States. However, we may ignore those small differences but rather we will look at its general structure. It is also remarkable that ancient Greeks used different systems for expressing cardinal and ordinal numbers. (A cardinal number is a number that says how many of something are there, such as one, two, three, four etc. Whereas an ordinal number is a number that tells the position of something in a list such as 1^st, 2^nd, 3^rd, 4^th, etc,), So we must look carefully at what we mean by Greek number system.

How Greeks used their alphabets to symbolise numbers are shown in the Table no.5 below with their sounds, using lower case ones, as it is easier to recognize, though strictly speaking is inaccurate.

Table No.5

It is evident from the table that 27 numbers to be alphabetically represented. But there are only 24 classical Greek alphabets. So, they had to find 3 extra symbols for the missing numbers of 6, 90 and 900. To supplement this, they used 3 archaic or obsolete letters which used to be in the alphabet but had been dropped as they were no longer required.

Since both the upper and lower case versions of 24 classical Greek letters were used to symbolise numbers, it is important to be familiar with both the versions of those classical letters. They are as follows:

    Table No-6

                                       Greek alphabets upper and lower cases.

This alphabetic numerals of the Greeks was hard for arithmetical calculations, for someone had to remember that alpha + alpha = beta, iota + iota = Kappa, rho + rho = sigma etc. What a lot of symbols to keep track of! Multiplication in the Greek system was even worse. They would have to use abacuses for arithmetical calculations.

As the same alphabet may represents a letter as well as a number, so there was   a chance of ambiguity for using the same letters for making words as also for numbers. The Greeks sometimes enjoyed this as a fun with letters.

  Suppose the name of a young girl is Airlea. So in Greek letters it would be αւρλεα          whose numeric value is 1+10+100+30+5+1=147. But it will be very difficult to say, what are the names whose numeric value would be 147; for 147 may be split in many many number of ways whose sums will be 147 and hence may represent a host of names.

In the book of Revelations, a book of the New Testament that occupies a central place in Christian eschatology, we find, “Let him that hath understanding count the number of the beast for it is the number is Six hundred three score and Six”. There had been many suggestions as to what the name might be that would produce this value of 666. The Roman Emperor Nero was the most likely candidate for his boundless brutality. The full name of this emperor was Nero Claudias Augustus Germanicus and the sum of the numeric values of the word “Claudias” is 20+30+1+400+4+10+1+200 which is exactly 666.

Although the Greek alphabet began with only majuscule or upper case form (using capital letters), surviving papyrus manuscripts from Egypt show that unical or majuscule and cursive minuscule (lower case form or small letter form) forms began early. These letter forms sometimes replaced the former ones, especially in the case of the obscure numerals. For example, the numeral for 6 changed several times.

In Modern Greek, a number of other changes have been made. Instead of using overbar over an entire number, the Keraia is marked to its upper right, a development of the short marks formerly used for single numbers and fractions.

Now let us have a look at the first Greek numerals system dated back to first millennium BC. This was acrophonic system which means that the symbols for numerals come from the first letter of the name of the numbers. So the symbol has come from an abbreviation of the word which is used for the numbers.

In Greek language the numbers five, ten, hundred, thousand and ten thousand are Pente, Deka, Hekaton, Khilioi and Murioi respectively. So the acrophonic forms of those numbers are as in Table No.7(a).
    Table No.7a.

                                              Acrophonic forms of some Greek numbers.

In this acrophonic system there were no symbols for the numbers from 1 to 4, obeying the above rule, however, these numbers were expressed by simple strokes like I, II, III, and IIII which was an obvious notation not coming from the initial letter of the name of a number. So, the numbers from 1 to 10 in acrophonic system based on additive system will be as in Table No.7(b).

             Table No.7b.

                                              

There awaits one puzzle for us for the symbol of 5 which should be P, if it was derived from the first letter Pente. However this is simply a consequence of changes to Greek alphabet after the numerals coming from these letters had been fixed. By that time the symbols for numbers were probably not thought of as coming from the letters so there was no attempt to change them with changes to the symbols for the letters. One interesting thing to note here that the original form of π (pi) was G and so ,Pente was Gente.

But there were some short comings of this system. If base 10 is used with an additive system without intermediate symbols then many characters are required to express certain numbers. For example the number 9999 would require 36 symbols in this system which is very cumbersome.

In Greek acrophonic numbers there was a special symbol for 5, which undoubtedly cut down characters for representing numbers and perhaps arose from counting on fingers, as we have 10 fingers and 5 on each hand. What is a bit surprising is that the system had intermediate symbols for 50, 500, 5000 and 50000 but they were not new characters, rather they were composite symbols made, from 5 and the symbols for 10,100, 1000 and 10000 respectively, which is shown in the Table No 7(c).

Table No 7(c)

                                Higher numbers and combining acrophpnic numerals.

One thing is to be noticed here that since there was no positional aspect of the system, so there was no need for zero as an empty place holder.

Modern Greeks still use these types of numbers for formal use, whereas we use Roman numbers, although they use Arabic numbers for calculations. However, forms of the archaic letters in particular have changed since classical Greek times.

In our modern mathematics, we use letters of various Greek alphabets to represent numbers as well, but in a completely different way. In Algebra, we use letters to represent unknowns, or variables in a general formula.

In ancient Greek number system, specific numbers were denoted by particular letters. In fact, modern mathematicians also use certain Greek letters to mean specific constants, but these have different values than the ones above. For instance, π (pi) is used widely now-a-days to represent the ratio of a circle’s circumference to its diameter, not the number 80, as in the old Greek system, Similarly φ represents the famous golden ratio (not 500), which has a magical role in beauties of nature. We will see its magnificence subsequently.

            References: Internet:
            Credit to:
                        Greek alphabets upper and lower cases :Free- Online-Bible Study.Org
          ( https://www.free-online-bible-study.org/greek-alphabet.html)
                      Acrophonic forms of some Greek numbers: Greek number systems (Table No.7a,7b,7c)
            Prof. E.F.Robertson
(http://www-history.mcs.st-and.ac.uk/HistTopics/Greek_numbers.html)

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