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Roman numerals don't have 0. I was taught that the arabs introduced 0 in their arabic numerals and it is depicted as a decimal point. The arabs, in turn, got their number system from india in sanskrit where zero is also used.

If the ancient Greeks did not have zero, they would have had great difficulty with some of their calculations such as the circumference of the Earth.

This article says Europe did not get zero until 1200AD.

So how did the Greeks perform these calculations without it, or did they have it back then?

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  • $\begingroup$ Why do you think that not having a zero would have caused difficulties to Eratosthenes calculating the circumference of the earth? $\endgroup$ – ShreevatsaR May 26 '17 at 0:26
  • $\begingroup$ @ShreevatsaR well, try it. You would need pi to very precise decimal places and how can you have non-integers without zeros or decimal points? $\endgroup$ – 0tyranny 0poverty May 26 '17 at 0:35
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    $\begingroup$ @ShreevatsaR my bad, didn't need pi, he used geometry. Eratosthenes used a bit more geometry to reason that the shadow's angle would be the same as the angle between Alexandria and Syene as measured from the Earth's center. Conveniently, 7.2 degrees is 1/50th of a full circle ( 50 x 7.2° = 360° ). Eratosthenes understood that if he could determine the distance between Alexandria and Syene, he would merely have to multiply that distance by 50 to find the circumference of Earth! $\endgroup$ – 0tyranny 0poverty May 26 '17 at 0:51
  • $\begingroup$ Modern notation can be very tempting. You don't need decimal points even to know π precisely. First, you realize that the circumference is proportional to (scales linearly with) the diameter. Then, you just have to determine the constant of proportionality, for which any notational system will do. For example, Āryabhaṭa (despite being in India) represented the value of π by saying that the circumference of a circle with diameter 20000 is close to 62832. (In Roman numerals, to lower precision: “a circle of diameter MM has circumference close to MMMMMMCCLXXXIII” — less convenient, but workable.) $\endgroup$ – ShreevatsaR May 26 '17 at 0:52
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During the classical and early Hellenistic period (until 200 BC) Greeks did not use any positional system, they had their own which was decimal but not positional. The units from 1 to 9 are assigned to the first nine letters of the archaic Greek alphabet from $\alpha$ to $\theta$. Each multiple of ten from 10 to 90 was assigned its own separate letter from the next nine letters from $\iota$ to ϙ. And so on, see Greek numerals. This worked fine for practical and even some scientific calculations. In the 3rd century BC Eratosthenes and Aristarchus, Archimedes's contemporaries, did calculations for estimating the circumference of the Earth and distances to the Moon and the Sun without the benefit of not only zero but even of a positional system.

However, there weren't enough letters to express very large numbers (which is where placeholder zeros may come handy), no more than the myriad ($10^4$) could be expressed. When the need arose Archimedes in Sand Reckoner invented his own number system capable of expressing them. He introduced "orders" based on the myriad, which was the "unit of the first order". The myriad myriads ($10^8$) was unit of the second order, and so on. Archimedes did not have zero either, but one can manage without it even for something like a positional system. Archimedes's was akin to the system with base $10^8$, but it did not take.

After 200 BC Hellenistic astronomers, such as the father of astronomy Hipparchus, started using the sexagesimal system (with base $60$), imported from Babylon along with long term precise astronomical data Greeks lacked. This gave a great boost to the development of mathematical astronomy, and the sexagesimal system was sometimes used for doing astronomical calculations, in particular by Ptolemy in Almagest in the second century AD (presumably he was a Greek ethnically, but he lived and worked in Alexandria, Egypt). However, the sexagesimal system made no advances beyond a very narrow circle of practitioners, not even all astronomers used it. Zero was used as a punctuation mark for purely notational purposes, not as a number, and the symbol for it, when not simply blank space, was $\omicron$ for "obol" (a coin of smallest value). For details and further developments see Was the concept of zero ever developed without relation to positional number systems?

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It depends on the definition of the "Ancient Greeks". Ptolemy, who lived in 2 century AD, lived on the territory of the Roman empire, but wrote in Greek, and probably worked in Alexandria (modern Egypt) most of the time. He is usually considered Greek. He used zero with sexagesimal Babylonian system (base 60, and digits were Greek letters). Zero was used as a place holder (which is actually its main function) and was denoted by Greek letter $\omicron$ (omicron).

The work of Ptolemy, or some related/derived work spread to the East eventually reaching India, as seen from its influence on the later Arab and Indian astronomy. But I do not know whether invention of zero in India was dependent o this or not.

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  • $\begingroup$ very interesting. they used base 60 instead of base 10, why? I'm having great difficulty understanding having zero as just a decimal point (assuming that's what place holder means) and not actually including it as one of the digits. $\endgroup$ – 0tyranny 0poverty May 25 '17 at 19:48
  • $\begingroup$ Place holder is not a decimal point. Place holder is what we need to distinguish between 2, 20 and 200. Base 60 came from Babylon where astronomy originated. 60 is convenient because it has many factors. We still use base 60 in measuring angles. $\endgroup$ – Alexandre Eremenko May 26 '17 at 6:24
  • $\begingroup$ Another good base is 12 which also comes from Babylon (Zodiac signs) and also used nowadays for time measurement. The only advantage of 10 is that we have 10 fingers, but 12 and 60 are much more convenient for mental calculation. $\endgroup$ – Alexandre Eremenko May 26 '17 at 6:33

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