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From "Rhind Papyrus" from 1600 BC we know that the Egyptians had an estimate for pi, namely 3.16, meaning they knew only 2 digits of pi. According to this article they knew more digits, at least 4 digits of pi. Around 200 BC Archimedes estimated pi to 22/7 which is 3 digits of pi. This indicates that the Egyptians knew more digits 2000 years before Archimedes, however, it's not clear to me how many digits they actually knew.

http://www.bcamt.ca/wp-content/uploads/2015/03/Yochim.pdf

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    This looks to be phrased a bit unlucky: "knowing digits" before the concept of decimal system was known? My guess here is that you might mean something like precision / of an approximation / "equivalent to digits"? – LаngLаngС May 15 '18 at 11:54
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    I think you might find some of the answers to this question on the History of Science and Mathematics SE helpful. – andejons May 15 '18 at 13:04
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    Having an approximation that resolves to more decimal places does not necessarily mean that have a more accurate approximation. – KillingTime May 15 '18 at 13:23
  • The question is how much of pi they knew. Decimals is just one way to put it. – Ole Petersen May 15 '18 at 18:15
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    I'd advise rewording this question to omit the concept of "digits" - As far as I can tell you're asking the precision of the Egyptian knowledge of Pi. Digits is confusing - It is like asking how many aircraft carriers the Egyptians had as a way of understanding their sea power. – Mark C. Wallace May 19 '18 at 23:10
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The ancient Egyptians at the time of the Rhind Papyrus didn't really have the concept of Pi. The method they described for finding the area of a circle was to inscribe it within a square, and apply the ratio of 64/81 to the area within the square. However, we know today this is mathematically equivalent to using a Pi of 256/81. That's a hair smaller than 3.1605, which on Wikipedia's timeline page amounts to having it right to one decimal place.

The Ancient Babylonians and Indians at roughly the same time had their own heuristics which worked out to a Pi of 3 + 1/8 and 25/8 respectively, or 3.125 (exactly). That was a wee bit closer, but also accurate to only one decimal place. Nobody else is known to have widely established a significantly better estimate until Archimedes' 2 decimal places nearly 2000 years later.

The paper you linked is making several speculations and extrapolating from them. I don't wanna give the guy short shrift: they are some fascinating speculations. I find the idea of the pyramid builders rolling around a trundle wheel to plot out the four corners particularly compelling. But at its base that paper is just a lot of personal speculation and math fun, built around a core of historical and mathematical fact. It is of course quite possible to be using Pi without knowing it; that's exactly what our trundle wheel users would have been doing.

There was an Egyptologist who argued as early as 1940 that the Egyptians were also using 22/7, but that argument does not appear to be widely accepted today. I'm not sure how closely his arguments match to the paper you linked.

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    In terms of accuracy, "digits of pi" isn't meaningful until you develop a decimal system of notation. The Egyptians had a "pi" that was 0.6% too high, while the Babylonians were 0.5% too low. – Mark May 16 '18 at 0:54
  • I appreciate your answer, but it does not explain how they came to 3,141. We can conclude that they knew at least four digits. Possibly more, as there is a measurement uncertainty. – Ole Petersen May 16 '18 at 12:48
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    @OlePetersen - It doesn't explain that, because the current narrative is that most likely they didn't. By the time of Archemedes the Egyptians were more or less an integrated part of a Mediterranean knowledge community, and thus any discovery by a dude in Sicily became theirs to use as well. Likewise any discovery an Egyptian had already made would have been available to Archimedes, and he could have spent his time working on something else. – T.E.D. May 16 '18 at 13:50
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    @Mark - The use of decimal digits is obviously a modern affectation, but one that is very useful to us in quantifying an order-of-magnitude accuracy for their estimates of Pi. Honestly, the ancients' fractional method of doing math is arguably more accurate, as it can express some values that decimal notation has trouble with. Of course that advantage is utterly negated when we are dealing with a transcendental number like pi. – T.E.D. May 16 '18 at 13:57
  • Let us continue this discussion in chat. – T.E.D. May 16 '18 at 18:36

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