There were no single Greek national standards in the first millennium BC. since the various island states prided themselves on their independence. This meant that they each had their own currency, weights and measures etc. However we will not go into sufficient detail in this article to examine the small differences between the system in separate states but rather we will look at its general structure.
The first Greek number system we examine is their acrophonic system which was use in the first millennium BC. 'Acrophonic' means that the symbols for the numerals come from the first letter of the number name, so the symbol has come from an abreviation of the word which is used for the number. Here are the symbols for the numbers 5, 10, 100, 1,000 and 10,000.
We have omitted the symbol for 'one', a simple '|', which was an obvious notation not coming from the initial letter of a number. For 5, 10, 100, 1,000 and 10,000 there will be only one puzzle for the reader and that is the symbol for 5 which should by P if it was the first letter of Pente. However this is simply a consequence of changes to the Greek alphabet. The original form of was G and so Pente was originally Gente.
Now the system was based on the additive principle in a similar way to Roman numerals. This means that 7 is simply G||, the symbol for five followed by three symbols for one. Complete this table with the numbers 1…10 in Greek acrophonic numbers.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
 |  |  |  |  |  |  |  |  |  |
The system had intermediate symbols for 50, 500, 5,000 and 50,000 but they were not new characters, rather they were composite symbols made from 5 and the symbols for 10, 100, 1,000 and 1,0000 respectively. Here is how the composites were formed.
Notice that since there was no positional aspect of the system, there was no need for zero as an empty place holder. The symbol H represented 100 as no problem is created in the representation by the number having no tens or units.
Now this is not the only way in which such composite symbols were created. We have already mentioned that different states used variants of the number system.
We now look at a second ancient Greek number system, the alphabetical numerals. As the name 'alphabetical' suggests the numerals are based on giving values to the letters of the alphabet. They placed an accent at the top right to indicate that it was no longer a letter, but a numeral, and that was it. The alphabetical order was kept, but, as twenty-seven numerals were needed, three more signs were added :
| Alpha | | Iota | | Rho |  |
| Beta | | Kappa |  | Sigma |  |
| Gamma | | Lambda |  | Tau |  |
| Delta | | Mu |  | Upsilon |  |
| Epsilon | | Nu |  | Phi |  |
| Digamma | | Ksi |  | Chi |  |
| Zeta | | Omicron |  | Psi |  |
| Eta | | Pi |  | Omega |  |
| theta | | Koppa |  | San |  |
Now numbers were formed by the additive principle. For example 21, 42, 77, 15, 98, 36, 269 were written :
| 21 | 42 | 77 | 15 | 98 | 36 | 269 |
  |   |   |   |   |   |    |
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 or superscript iota to the symbols for 1 to 9.
First form of 1,000 ... 9,000.
Second form of 1,000 ... 9,000.
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 above the M represented 20,000 :
similarly 
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. For example Aristarchus wrote the number 71,755,875 :
M
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 in fact historically Archimedes made his proposal nearly 50 years before Apollonius).
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 above it represented 10,000, M with 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 = 104 as the basic number which was raised to various powers he used 100,000,000 = 108 raised to powers. The first octet for Archimedes consisted of numbers up to 108 while the second octet was the numbers from 108 up to 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 1064.
2005