ANCIENΤ GREEΚ ALPHABETICAL NUMBERS

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THE GREEK ALPHABET IS ABSOLUTELY FUNDAMENTAL FOR THE HISTORY OF WRITING AND FOR WESTERN CIVILIZATION AS A WHOLE (IFRAH, 2000). AS C. HIGOUNET (1969) EXPLAINS THE GREEK ALPHABET, QUITE APART FROM ITS HAVING SERVED TO TRANSMIT ONE OF THE RICHEST LANGUAGES AND CULTURES OF THE ANCIENT WORLD, FORMS THE “BRIDGE” BETWEEN SEMITIC AND LATIN SCRIPTS.

HISTORICALLY, GEOGRAPHICALLY, AND ALSO GRAPHICALLY, IT WAS AN INTERMEDIARY BETWEEN EAST AND WEST; EVEN MORE IMPORTANTLY, IT WAS A STRUCTURAL INTERMEDIARY TOO, IN THE SENSE THAT IT FIRST INTRODUCED REGULAR AND COMPLETE REPRESENTATIONS OF THE VOWEL SOUNDS.

CADMOS THE LEGENDARY FOUNDER OF THEBES, IS SAID TO HAVE BROUGHT THE FIRST SIXTEEN LETTERS FROM PHOENICIA; PALAMEDES WAS SUPPOSED TO HAVE ADDED FOUR MORE DURING THE TROJAN WAR; AND FOUR MORE WERE INTRODUCED LATER ON BY A POET, SIMONIDES OF CEOS.

THE ANCIENT GREEK CONSISTS OF THE TWENTY-FOUR LETTERS OF THE CLASSICAL GREEK ALPHABET, PLUS THE THREE OBSOLETE LETTERS, DIGAMMA, KOPPA AND SAN.

THE 27 LETTERS ARE (UNIVERSITY OF ST ANDREWS, SCHOOL OF MATHEMATICS AND STATISTICS, 2001):

THESE TWENTY-SEVEN SIGNS ARE DIVIDED INTO THREE CLASSES.

THE FIRST, GIVING THE NUMBERS 1 TO 9, USE THE EIGHT LETTERS OF THE CLASSICAL ALPHABET, PLUS DIGAMMA, INSERTED IN THE SEQUENCE TO REPRESENT THE NUMBER 6.

THE SECOND CLASS CONTAINS THE EIGHT FOLLOWING LETTERS, PLUS THE OBSOLETE KOPPA, TO GIVE THE SEQUENCE OF THE TENS, FROM 10 TO 90.

FINALLY THE THIRD CLASS GIVES THE HUNDREDS FROM 100 TO 900, USING THE LAST EIGHT LETTERS OF THE CLASSICAL ALPHABET PLUS THE SAN (FOR THE VALUE OF 900).

INTERMEDIATE NUMBERS ARE PRODUCED BY ADDITIVE COMBINATIONS.

FOR 11 TO 19, FOR INSTANCE, YOU USE IOTA, REPRESENTING 10, WITH THE APPROPRIATE LETTER TO ITS RIGHT REPRESENTING THE UNIT TO BE ADDED.

LARGER NUMBERS WERE CONSTRUCTED IN THE SAME SORT OF WAY.

FOR EXAMPLE HERE IS 269:

TO DISTINGUISH THE LETTERS USED AS NUMERALS FROM “ORDINARY” LETTERS, A SMALL STROKE IN PLACED OVER THEM.

NOW THIS NUMBER SYSTEM IS COMPACT BUT WITHOUT MODIFICATION IS HAS THE MAJOR DRAWBACK OF NOT ALLOWING NUMBERS LARGER THAN 999 TO BE EXPRESSED.

COMPOSITE SYMBOLS WERE CREATED TO OVERCOME THIS PROBLEM.

THE NUMBERS BETWEEN 1,000 AND 9,000 WERE FORMED BY ADDING A SUBSCRIPT IOTA TO THE SYMBOLS FOR 1 TO 9.

HOW DID THE GREEKS REPRESENT NUMBERS GREATER THAN 9,999?

WELL THEY BASED THEIR NUMBERS LARGER THAN THIS ON THE MYRIAD WHICH WAS 10,000.

THE SYMBOL "M" WITH SMALL NUMERALS FOR A NUMBER UP TO 9,999 WRITTEN ABOVE IT MEANT THAT THE NUMBER IN SMALL NUMERALS WAS MULTIPLIED BY 10,000.

HENCE WRITING "β" ABOVE THE "M" REPRESENTED 20,000:

SIMILARLY "ρκγ" WRITTEN ABOVE THE "M" REPRESENTED 1,230,000:

OF COURSE WRITING A LARGE NUMBER ABOVE THE "M" WAS RATHER DIFFICULT SO OFTEN IN SUCH CASES THE SMALL NUMERAL NUMBER WAS WRITTEN IN FRONT OF THE "M" RATHER THAN ABOVE IT.

AN EXAMPLE FROM ARISTARCHUS:

FOR MOST PURPOSES THIS NUMBER SYSTEM COULD REPRESENT ALL THE NUMBERS WHICH MIGHT ARISE IN NORMAL DAY TO DAY LIFE. IN FACT NUMBERS AS LARGE AS 71,755,875 WOULD BE UNLIKELY TO ARISE VERY OFTEN.

ON THE OTHER HAND MATHEMATICIANS DID SEE THE NEED TO EXTEND THE NUMBER SYSTEM AND WE NOW LOOK AT TWO SUCH PROPOSALS, FIRST ONE BY APOLLONIUS AND THEN BRIEFLY ONE BY ARCHIMEDES.

ALTHOUGH WE DO NOT HAVE FIRST HAND KNOWLEDGE OF THE PROPOSAL BY APOLLONIUS WE DO KNOW OF IT THROUGH A REPORT BY PAPPUS.

THE SYSTEM WE HAVE DESCRIBED ABOVE WORKS WITH PRODUCTS BY A MYRIAD.

THE IDEA WHICH APOLLONIUS USED TO EXTEND THE SYSTEM TO LARGER NUMBERS WAS TO WORK WITH POWERS OF THE MYRIAD.

AN M WITH AN Α ABOVE IT REPRESENTED 10,000, "M" WITH "Β" ABOVE IT REPRESENTED M^2, NAMELY 100,000,000, ETC.

THE NUMBER TO BE MULTIPLIED BY 10,000, 10,000,000, ETC IS WRITTEN AFTER THE "M" SYMBOL AND IS WRITTEN BETWEEN THE PARTS OF THE NUMBER, A WORD WHICH IS BEST INTERPRETED AS "PLUS".

AS AN EXAMPLE HERE IS THE WAY THAT APOLLONIUS WOULD HAVE WRITTEN 587,571,750,269:

ARCHIMEDES DESIGNED A SIMILAR SYSTEM BUT RATHER THAN USE 10,000 = 10^{4}10,000=104 AS THE BASIC NUMBER WHICH WAS RAISED TO VARIOUS POWERS HE USED 100,000,000 = 10^{8}100,000,000=108 RAISED TO POWERS.

THE FIRST OCTET FOR ARCHIMEDES CONSISTED OF NUMBERS UP TO 10^{8}108 WHILE THE SECOND OCTET WAS THE NUMBERS FROM 10^{8}108 UP TO 10^{16}1016.

USING THIS SYSTEM ARCHIMEDES CALCULATED THAT THE NUMBER OF GRAINS OF SAND WHICH COULD BE FITTED INTO THE UNIVERSE WAS OF THE ORDER OF THE EIGHTH OCTET, THAT IS OF THE ORDER OF 10^{64}1064.

SOURCES

HIGOUNET, CH. (1969). L'ECRITURE. PRESSES UNIVERSITAIRES DE FRANCE - PUF

IFRAH, G. (2000). THE UNIVERSAL HISTORY OF NUMBERS FROM PREHISTORY TO THE INVENTION OF THE COMPUTER. NEW YORK: J. WILEY.

UNIVERSITY OF ST ANDREWS, SCHOOL OF MATHEMATICS AND STATISTICS. (2001). GREEK NUMBER SYSTEMS. (WRITTEN BY JJ O’ CONNOR AND W. B. ROBERTSON) RETRIEVED 23/9/2020, FROM MAC TUTOR: https://mathshistory.st-andrews.ac.uk/HistTopics/Greek_numbers/

REFERENCES

ARISTARCHUS WAS A GREEK MATHEMATICIAN AND ASTRONOMER WHO IS CELEBRATED AS THE EXPONENT OF A SUN-CENTRED UNIVERSE AND FOR HIS PIONEERING ATTEMPT TO DETERMINE THE SIZES AND DISTANCES OF THE SUN AND MOON.
APOLLONIUS WAS A GREEK MATHEMATICIAN KNOWN AS 'THE GREAT GEOMETER'. HIS WORKS HAD A VERY GREAT INFLUENCE ON THE DEVELOPMENT OF MATHEMATICS AND HIS FAMOUS BOOK CONICS INTRODUCED THE TERMS PARABOLA, ELLIPSE AND HYPERBOLA.
ARCHIMEDES WAS THE GREATEST MATHEMATICIAN OF HIS AGE. HIS CONTRIBUTIONS IN GEOMETRY REVOLUTIONISED THE SUBJECT AND HIS METHODS ANTICIPATED THE INTEGRAL CALCULUS. HE WAS A PRACTICAL MAN WHO INVENTED A WIDE VARIETY OF MACHINES INCLUDING PULLEYS AND THE ARCHIMIDEAN SCREW PUMPING DEVICE.
PAPPUS IS THE LAST OF THE GREAT GREEK GEOMETERS AND ONE OF HIS THEOREMS IS CITED AS THE BASIS OF MODERN PROJECTIVE GEOMETRY. HE WROTE COMMENTARIES ON EUCLID'S ELEMENTS AND PTOLEMY'S ALMAGEST.