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The paper isn't cited, but I was able to find it in less than 5 minutes. Cite: http://www.sciencedirect.com/science/article/pii/S0315086017300691 http://www.sc

by trebor 9y ago

The paper isn't cited, but I was able to find it in less than 5 minutes. Cite: http://www.sciencedirect.com/science/article/pii/S0315086017300691 http://www.sciencedirect.com/science/article/pii/S0315086017... I think the reason they're calling it superior, is because it uses fractions/ratios instead of decimal numbers. Ours are rounded (usually to to 4-7 digits), but if they didn't have to round it and could resolve division afterward it may have been more accurate. > Babylonian exact sexagesimal trigonometry uses exact ratios and square ratios instead of approximation and angles. But I'm not a mathematician and cannot verify these claims.

gus_massa 9y ago
Mathematician here. I skimmed the article, I hope I get most of the details right. The press article is totally overhyped. Just ignore it. In the original tablet they (some Babylonians) cherrypick a few interesting angles (~50 values?) that are like 360°/(2^x 3^y 5^z). You can´t use 40°, you must use 40°15'. This angles have the nice property that tan^2(angle) is a nice rational number. [I'm not sure, please check the exact property.] So they have a table that uses fractions instead of decimal numbers. So for some angles you get exact representations, but operating with a long fractions is painful. The current method of using floating points calculation is better 99% of the time. (Using fractions for the trigonometric values is nice for some theoretical results in algebra, and is a well known trick in that area.) Unless I'm missing something, this is a nice historical result but it's not an improvement for the current calculations.
- trebor 9y ago
  Yeah, the press loves to inflate things. Thanks for a mathematician's opinion on it. I think it's pretty clever of them (the Babylonians), since I don't remember them having a decimal system. I guess by cherry-picking specific angles that resolve to rational numbers, it let them speed up some of their calculations.
- jacobolus 9y ago
  Your summary isn’t quite right. There are no degrees (or angle measures) involved here. Instead, we have the lengths of the short side of various rectangles and the lengths of the corresponding diagonals, along with (what the author’s claim is) the squared ratio between the diagonal and the long side of each rectangle – the squared secant – each of which happens to be a sexagesimally regular number with the rows organized in order based on this squared ratio (and which can conveniently also be interpreted as the squared ratio between the short side and the long side – the squared cotangent – by subtracting 1). The authors claim that such a table can be used (along with linear interpolation) as a very effective tool for solving trigonometry problems. I thought the paper was quite interesting and worth studying for anyone interested in number theory, metrical geometry, high school level math education, or mathematical history. I’ll agree that the press is a bit ridiculous though. Nobody is seriously suggesting that we should give up on computers and do all of our arithmetic in base sixty.

by trebor 9y ago

gus_massa 9y ago

Mathematician here. I skimmed the article, I hope I get most of the details right. The press article is totally overhyped. Just ignore it. In the original tablet they (some Babylonians) cherrypick a few interesting angles (~50 values?) that are like 360°/(2^x 3^y 5^z). You can´t use 40°, you must use 40°15'. This angles have the nice property that tan^2(angle) is a nice rational number. [I'm not sure, please check the exact property.] So they have a table that uses fractions instead of decimal numbers. So for some angles you get exact representations, but operating with a long fractions is painful. The current method of using floating points calculation is better 99% of the time. (Using fractions for the trigonometric values is nice for some theoretical results in algebra, and is a well known trick in that area.) Unless I'm missing something, this is a nice historical result but it's not an improvement for the current calculations.

trebor 9y ago
Yeah, the press loves to inflate things. Thanks for a mathematician's opinion on it. I think it's pretty clever of them (the Babylonians), since I don't remember them having a decimal system. I guess by cherry-picking specific angles that resolve to rational numbers, it let them speed up some of their calculations.
jacobolus 9y ago
Your summary isn’t quite right. There are no degrees (or angle measures) involved here. Instead, we have the lengths of the short side of various rectangles and the lengths of the corresponding diagonals, along with (what the author’s claim is) the squared ratio between the diagonal and the long side of each rectangle – the squared secant – each of which happens to be a sexagesimally regular number with the rows organized in order based on this squared ratio (and which can conveniently also be interpreted as the squared ratio between the short side and the long side – the squared cotangent – by subtracting 1). The authors claim that such a table can be used (along with linear interpolation) as a very effective tool for solving trigonometry problems. I thought the paper was quite interesting and worth studying for anyone interested in number theory, metrical geometry, high school level math education, or mathematical history. I’ll agree that the press is a bit ridiculous though. Nobody is seriously suggesting that we should give up on computers and do all of our arithmetic in base sixty.