How did ancient Greeks do mathematics?

The ancient Greeks did some amazing things with mathematics: calculating pi, making Pythagoras famous, etc.

My question is this: <font face=“arial,helvetica” color="#4040FF">What kind of numbering system did they use?</font>

I can’t imagine that they could have calculated pi using Roman Numerals.

<font size="-1">(Hmm… III . I IV I V IX II XI V III V …)</font>

At the same time, I’m pretty sure they didn’t use Arabic numerals.

So how did they do it?

Afraid to fly? Hey, I’ve been there!


This may also be relevant:

Back off, man. I’m a scientist.

Fingers & little stones, drawing in the sand.

From “A History of Mathematics” by Carl B. Boyer:

The Attic and Ionian notations have both been found in inscriptions, but the Ionian system prevailed arond the third century BC.

(Attic notation seen from 454 to 5 BC)
(Ionian notation probaby used from 5th century BC, possibly as early as 8th century BC).

The Ionian notation used an older version of the Greek alphabet with three extra letters: vau (aka digamma or stigma), koppa and sampi.

One letter was used (in alphabetical order) for 1-9, 10-90, 100-900.

1 = alpha, 2 = beta, 3 = gamma, …, 10 = iota, …, 100 = pi

The three older letters occurred in the following position:
6 = vau, 90 = koppa, 900 = sampi

For 1000 to 9000, the first 9 letters were used again, with the letter preceded by a stroke or accent. (In this book, the stroke or accent looks somewhat like a comma.)

So for example the number 8888 would be written as
,eta omega pi eta

For numbers larger than 10000, you would write the letter M (greek myriad) with an integer symbol above or after it, separated by a dot from the the rest of the number, to indicate that the number next to the M should be multiplied by 10000.

e.g. 88888888 would be written as
M,eta omega pi eta . eta omega pi eta

Yo could apply the same principals for the double myriad, 100000000

Fractions are another topic.

J’ai assez vécu pour voir que différence engendre haine.

Timothy Campbell: I can’t imagine that they could have calculated pi using Roman Numerals.

3.14159265358979 =

<font face=“arial,helvetica” color="#000000">Arnold</font>: Thanks! That was most informative. To be honest, though, their system doesn’t sound a whole heck of a lot easier than Roman Numerals.

<font face=“arial,helvetica” color="#000000">AWB</font>: I’m sorry, but I don’t understand the point of your post. Were you trying to express pi in Roman Numerals? I did that in the original post.

Actually, since the decimal “point” seems to rely on a decimal numeric system, the next question is “How did they represent fractions?”


Timothy Campbell says:

Actually it’s closer to our modern notation than the Roman Numeral system. Notice that with the Myriad prefix you can easily write large numbers without many more characters than in the modern arabic notation. The only thing they didn’t think of is having the position of the digit represent a different multiple of a certain base.

tomndebb says:

Firstly, the decimal point may seem a misnomer when used with other bases, but you can use that notation with any base.

base 3: 1/3 = 0.1
base 10: 1/3 = 0.3333333…
base 16: 1/3 = 0.5555555…

Secondly, you don’t need a decimal point to write fractions. e.g. to write 0.125 I can write 1/8.

Originally, the Greeks wrote fractions using a mark similar to a quote mark following the number. so for example
1 / 32 would be gamma betta ’

However, that could also be confused with 3.5 or 3 1/2.

Later on they came up with a different system that I will have to look up later.

J’ai assez vécu pour voir que différence engendre haine.

P.S. In my original post I made two mistakes.

100 = rho, not 100 = pi


written as
M,eta omega pi eta . ,eta omega pi eta

J’ai assez vécu pour voir que différence engendre haine.


Another mistake in my post on fractions!

1 / 32 would be lambda betta ’

However, that could also be confused with 30.5 or 30 1/2.

J’ai assez vécu pour voir que différence engendre haine.

Hmm; we may have the number of the Beast here! What greek letters correspond to 666?

So pi would be upsilon eta ’ ?

I remember my old supervisor telling me that he’d learnt at school how to carry out multiplication and division of roman numerals. I have no idea how this is/was done. Are there a set of rules or something?

OK, back to EB. They discuss how the Egyptians multiplied. They had a similar system as the Romans, so I’m guessing it applies here too.

Pay attention class!

Hope that came out right.

Back off, man. I’m a scientist.

For divination purposes, it is not necessary to use the “correct” letters. Indeed, it wouldn’t be possible to get 666, because the letter for “6” is digamma (a.k.a. vau), the ancestor of “F”, which was no longer used in the Greek language (and you won’t find it in modern listings of the Greek alphabet). It is only necessary that the letters add up to the right total.

But it is 99% certain that these Greek numerals were where 666 comes from. “Nero Caesar” (spelled in Greek) makes 666. Some ancient manuscripts say 616 instead of 666, and that is produced by “Caesar-God”.

In Hebrew, by the way, which uses a system like the Greek, certain numbers are written in odd ways (I don’t have a reference to hand to get the actual numbers, but it would be like calling “31”, “twenty-eleven”) to avoid accidently spelling YHWH, the unspeakable Name of God.

John W. Kennedy
“Compact is becoming contract; man only earns and pays.”
– Charles Williams

Slight correction on that 666 thing can be found here:

The Satanists have a good faq too. :slight_smile:

I think the “twenty-eleven” thing was actually fifteen, written as 9-6 instead of 10-5.

I can understand that Hebrew thing, but why do the French say “quatre-vingt” (“four-[times]-twenty”) for 80 and “quatre-vingt-dix” (“four-[times]-twenty-[plus]-ten”) for 90?!

This should explain it:

“It is lucky for rulers that men do not think.” — Adolf Hitler