Greek Numerical System and Mathematics

[Julius Ratus]

I believe that the Greeks used the letters of their alphabet for numerals. As for zero, I have always believed that the ancients had a concept of nothing, but had no use for it as a number and therefor no notation.

 

Back to MPC. 'Lunes' = calculus?

 

 

The Medieval Russians used letters for numbers as well. Seeing as that they got their alphabet from the (Byzantine) Greeks it would make sense that they got their numbers from them as well.

[Gaius Octavius]

Now, are you just being cruel?

 

Yup! Would you expect less of your tormentor?

 

[Gaius Octavius]

The Medieval Russians used letters for numbers as well. Seeing as that they got their alphabet from the (Byzantine) Greeks it would make sense that they got their numbers from them as well.

 

That is interesting. It leads me to this question: Weren't the 'Byzantines' using Roman notation?

Edited February 21, 2007 by Gaius Octavius

[Andrew Dalby]

The Medieval Russians used letters for numbers as well. Seeing as that they got their alphabet from the (Byzantine) Greeks it would make sense that they got their numbers from them as well.

 

That is interesting. It leads me to this question: Weren't the 'Byzantines' using Roman notation?

 

No, it seems to go with language (or rather, with alphabet). As Greek became the Byzantines' main language (and had already been the main language for science and mathematics) their normal numeral system was the Greek one (the later, alphabetical, version). This was to be expected because, as I've already said, it was more practical than the Roman system anyway. No doubt, as Julius Ratus says, the idea was borrowed from the Byzantines by the Slavic peoples in the same way as they borrowed (and adapted) the Greek alphabet.

[Gaius Octavius]

Wasn't it St. Cyril who adapted and introduced the 'Cyrilic' alphabet to the Slavs?

For a 'cruel' imponderable, why didn't the Slavs and 'Greeks' adopt Arabic notation? Seems easier to use.

*Bump* Fractions?

[Andrew Dalby]

Wasn't it St. Cyril who adapted and introduced the 'Cyrilic' alphabet to the Slavs?

For a 'cruel' imponderable, why didn't the Slavs and 'Greeks' adopt Arabic notation? Seems easier to use.

*Bump* Fractions?

 

Oh, yes, Cyril was the prime mover. Wouldn't want to deprive him of his credit. One of the few people who can claim to have invented two alphabets (Cyrillic and Glagolitic).

 

As to your second question, I don't know at what period the Arabs adopted their numeral system from India. Until that date, it wasn't available for the Byzantines to borrow! Does anyone know the answer to that?

[Ruthe]

Speaking of "1/2" or "Pi", how did the Romans, etc., notate such fractions?

 

According to Ifrah's The Universal History of Numbers, the Babylonians used a positional notation system to denote fractions, and the Egyptians used an eR-symbol to denote the k in k/n. Unfortunately, he doesn't say anything about the Roman and Greek notation for fractions, although fractions were certainly implicit in the abacus they used.

 

Now, are you just being cruel?

 

Look at the thread titled "Roman Mathematics" in this forum, last post on Aug 30th 2006. This thread answers the question of Roman fractions. The were very adept at calculations with fractions, though as you will see, their fractions used a base of 12, not 10 simply because it was a more practical basis as halves, quarters and thirds were more common in daily use than just halves and fifths in a base 10 system. Had the Romans or even the French adopted a base 12 number system we would have been better served by it than our base 10 system. Don't take my word for it, ask a mathematician.

[M. Porcius Cato]

And how did the Roman notate their fractions?

[Ruthe]

And how did the Roman notate their fractions?

 

The best description of Roman Fractions is given in Numbers by Graham Flegg. This lists each fraction with its name and symbol. The text that goes with this table is copied here.

 

An excellent example of how completely these were integrated into Roman mathematics is shown by the references to two documents in this article in Speculum, the Journal of the Medieval Academy of America.

 

A more direct example is given by The Aqueducts of Rome by Sextus Julius Frontinus. In particular, paragraphs 24-63 illustrate the degree to which these fractions were used in this one particular area of technology.

 

I hope this helps your understanding.

[Gaius Octavius]

I am afraid to ask how the Greeks notated fractions?

[M. Porcius Cato]

Ruthe, the link you provided tells us that there was a special notation for uncial fractions, but then it doesn't show us the symbols. What we're looking for aren't the names of the fractions (e.g., one-fourth) but the symbols used (e.g., 1/4).

[Gaius Octavius]

MPC, aren't those symbols in the 'Numbers' link above?

[Ruthe]

MPC, aren't those symbols in the 'Numbers' link above?

 

Exactly Gaius!!!

 

The second column shows symbols for each fraction and shows different variations at different periods for the same fraction.

As for the Greeks, I have no information on their practices. It could be they didn't use fractions. If I find anything I'll pass it on.

 

P.S. Just so you don't have to go back to the previous thread I mentioned, I've included a link to a picture of a copy of a Roman pocket calculator ( a small portable abacus with two colums for fractions) that shows an example of the symbols for fractions. The second column from the right is for 'unciae' while the first column has three separate slots for fractions of an uncia. The accepted values are from top slot to bottom, 1/2, 1/4 and 1/3 of an uncia. But I claim the bottom slot with its two beads is for twelfths of an uncia. My argument for this is that if the bottom slot is used for thirds of an uncia, you could only represent the following fractions of an uncia:

 

0/12, 3/12 (1/4), 4/12 (1/3), 6/12 (1/2), 7/12 (1/4 + 1/3), 8/12 (2/3), 9/12 (3/4 = 1/2 + 1/4), 10/12 ( 1/2 + 1/3), 11/12 (1/4 + 1/3 + 1/3), 13/12 (1/2 + 1/4 + 1/3) and 17/12 ( 1/2 + 1/4 + 1/3 + 1/3)

 

This arrangemant does not allow for representing 1/12, 2/12 (1/6) or 5/12 (1/4 + 1/12 + 1/12).

 

If the bottom slot is used for twelfths of an uncia it would allow all other twelfths up to 11/12 to be represented. This is a far more logical progrssion and would allow these two columns to represent smaller duodecimal fractions by the simple expedient of a mental shift of the magnitude which is in effect the same as moving the decimal point in our 10 based system. Note that the decimal system is not unique since every base system allows for this same shift when multiplying or dividing by the base itself.

 

The abacus is is a copy using diagrams and pictures and is on the site of the Online Abacus Museum. Note that the first two columns are decribed as ounces ans fractions of ounces, but in fact these are more generaly twelfths and fractions of twelfths. The Romans used he same names for weights that were parts of an 'As' and for fractions of a 'Pes' or foot. The Latin word 'uncia' is the origin of both 'inch' and 'ounce' ( the pound was previously 12 ounces before the introduction of the 16 ounce Avoirdupois pound).

Edited February 24, 2007 by Ruthe

[M. Porcius Cato]

MPC, aren't those symbols in the 'Numbers' link above?

 

Ooops! Thanks GO (and Ruthe). This is great. I wonder where these fractional notations were found.

[Gaius Octavius]

MPC, my well known clemency spares you.