# Greek Numbers and Arithmetic

**1 ****Introduction**

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.

**2 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 adiacritical mark ' before a unit. Thus

In other sources we see the diacritical mark placed as a
subscript before

the unit. Thus

The uses of a M was used to represent numbers from 10,000
on

up. Thus

Alternatively, depending on the history one reads

Archimedes, in his book The Sand Reckoner, calculated the
number

of grains of sand to fill the universe. This required him to develop

an extention the power of Greek enumeration to include very large

numbers.

**Fractions**

In the area of fractions, context was crucial for
correctly 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. Thus,

Numerous, similar, representations also have been used,
with increasing

sophistication over time. Indeed, Diophantus uses a fractional

form identical to ours but with the numerator and denominator in reversed

positions.

**3 Calculation**

The arithmetic operations are complex in that so many
symbols are

used. However, as you can imagine, addition amounts to grouping and

then carrying. For example not terribly
unlike

what we do. Multiplication was carried out using the distributive law.

For example:

Remarkably, division was performed in essentially the same
way as we

do it today.