Greek Numbers and Arithmetic
The earliest numerical notation used by the Greeks was the Attic system. It employed the vertical stroke for a one, and symbols for ``5", ``10", ``100", ``1000", and ``10,000". Though there was some steamlining of its use, these symbols were used in a similar way to the Egyptian system, being that symbols were used repeatedly as needed and the system was non positional. By the Alexandrian Age, the Greek Attic system of enumeration was being replaced by the Ionian or alphabetic numerals. This is the system we discuss.
The (Ionian) Greek system of enumeration was a little more sophisticated than the Egyptian though it was non-positional. Like the Attic and Egyptian systems it was also decimal. Its distinguishing feature is that it was alphabetical and required the use of more than 27 different symbols for numbers plus a couple of other symbols for meaning. This made the system somewhat cumbersome to use. However, calculation lends itself to a great deal of skill within almost any system, the Greek system being no exception.
Greek Enumeration
and
Basic Number Formation
First, we note that the number symbols were the same as the letters of the Greek alphabet.
where three additional characters, the 
(digamma), the
(koppa), and the
(sampi) are used. Hence,
Larger Numbers
Larger numbers were also available. The thousands, 1000 to 9000, were represented by placing and apostrophe ' before a unit. Thus
The letter M was used to represent numbers from 10,000 on up. Thus
Alternatively, depending on the history one reads
As should be evident this system does not allow very large numbers to be expressed. Archimedes extended the system in his book The Sand Reckoner where he computed the number of grains of sand to fill the universe (of Aristarchus).
Fractions
The Greeks used fractions, as did earlier civilizations. Their notation, however, was ambiguous and context was crucial for the correct reading a fraction. A diacritical mark was placed after the denominator of the (unit) fraction. So,
but this latter example could also mean
.
More complex fractions could be written as well, with context again being important. The numerator was written with an overbar, the denominator with the diacritical mark. Thus,
Numerous, similar, representations also have been used, with increasing sophistication with time. Indeed, Diophantus (who came along very late in Greek mathematics) uses a fractional form identical to ours but with the numerator and denominator in reversed positions.
Calculation
The arithmetic operations are complex in that so many symbols are used. Multiplication was carried out using the distributive law. For example:
Remarkably, division was performed in essentially the same way as we do it today.