Greek Numerical System and Mathematics

[Northern Neil]

This topic was inspired by one started by M. Porcius Cato, on reviewing books about Pythagoras, but mindful of straying off topic, I decided to start this separate, but related thread on the system of mathematics and numbers used by the Greeks. I ask a question: What numerical system did the Greeks use? Did they have a written notation for it? If so, was it similar to the Roman numerical system? Did they have a concept of 'zero'?

[Gaius Octavius]

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?

[Andrew Dalby]

Gaius Octavius said:

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?

 

Yes, the system GO mentions uses the 24 letters of the alphabet plus 3 others that were not in the normal Greek alphabet. That makes 27 in all, covering 1-9, 10-90, 100-900. The highest number you can notate using this range is 999. For numbers above that you can add an accent above or before the letter; the accent means "multiply by 1000", so that takes you up to 999,999.

 

Earlier, in 5th century BC Greek inscriptions for example, there was a quite different system, very much like Roman numerals, with symbols for 1, 5, 10, 50, 100, 500, etc.

 

The Greek scientist Aristarchus wrote a book whose title translates into English as "The Sand-Reckoner". Its aim is to describe a system of notation for very large numbers.

[Gaius Octavius]

Drs A.D. and MPC, zero and calculus?

[Andrew Dalby]

Drs A.D. and MPC, zero and calculus?

 

Just as you said above, the concept of "nothing" existed but I am aware of no mathematical notation for it. They apparently did not see the need.

 

As for calculus, I can't do calculus so I'm not qualified to pronounce on whether the Greeks could or not ...

[Rameses the Great]

The concept of 'zero' came from India by the way of the Arabs who gave it to the West.

[Gaius Octavius]

Earlier, in 5th century BC Greek inscriptions for example, there was a quite different system, very much like Roman numerals, with symbols for 1, 5, 10, 50, 100, 500, etc.

 

Is it known why they changed?

[Andrew Dalby]

Earlier, in 5th century BC Greek inscriptions for example, there was a quite different system, very much like Roman numerals, with symbols for 1, 5, 10, 50, 100, 500, etc.

 

Is it known why they changed?

 

Not by me. I wondered about this question, as I was writing my earlier posting, but I have no answer ready. Maybe someone else knows.

 

I have a suggestion, though. The later Greek system is admittedly less handy than "ours" (because you have to learn 27 characters instead of 10; 9 differs from 90 and again from 900) but it still makes it fairly easy to add up columns of figures and to do other similar operations, subtraction etc. With a system like the Roman or the earlier Greek, such calculations on paper are more difficult to do.

 

I say "ours" in quotes because Rameses reminds me that "our" system is borrowed from the Arabs, who borrowed it from the Indians.

[M. Porcius Cato]

Re: calculus. A recently discovered writing from Archimedes suggests that he had the basic concepts of calculus (e.g., infinity). See this article. For a slightly more detailed explanation, also see here.

 

FWIW, Mrs. Cato is a much better mathematician than I am, and when we watched a special on NOVA about the Archimedes Palimpsest, she was utterly convinced that Archimedes was using calculus in his proofs. I'm a bit more conservative, but I do grant that Archimedes could have developed his logic to surpass Newton's discoveries.

[Gaius Octavius]

Nice work MPC & AD. Now, I wish that I had paid some attention to calculus.

 

Completely off topic, and from MPC's link above:

 

"In other words, it is impossible to pigeonhole ancient Greek thought or, for that matter, the intellectual culture of any civilization. People like to simplify forces that shape history, and this can lead to conceptually crude, underdeveloped ideas -- such as that the Victorians were sexually repressed."

[caldrail]

The concept of 'zero' came from India by the way of the Arabs who gave it to the West.

Via the crusaders too - When they weren't haplessly slaughtering arab mathematicians that is.

[Gaius Octavius]

The concept of 'zero' came from India by the way of the Arabs who gave it to the West.

Via the crusaders too - When they weren't haplessly slaughtering arab mathematicians that is.

 

The 'concept' or the 'notation' and usage? It would seem certain that Greeks and Romans and Egyptians knew that two less two equalled 'zero' or nothing.

[M. Porcius Cato]

The 'concept' or the 'notation' and usage? It would seem certain that Greeks and Romans and Egyptians knew that two less two equalled 'zero' or nothing.

 

Good point. Even a 2-year-old understands the concept "all gone". The question is when people started thinking of "all gone" as a number, like 1 or 1/2, and when people started to symbolize the concept.

[Gaius Octavius]

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

Edited February 21, 2007 by Gaius Octavius

[M. Porcius Cato]

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?