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> I fail to see how it's wrong. If you call Math.sin(x) it will return a numerical approximation of sin(). Right. Now what's being argued here is that instead
by Lazare 9y ago
> I fail to see how it's wrong. If you call Math.sin(x) it will return a numerical approximation of sin().
Right. Now what's being argued here is that instead of measuring an angle X, calculating the sine of X, and then (probably) rounding the result to a convenient decimal approximation, we should measure an angle X, round it to an angle Y with a convenient non-repeating decimal representation, and calculate the sine of Y. (That's the whole point of Mansfield's and Wildberger's analysis of the tablet; that it was a trig table that only gave answers for angles with convenient answers, rather than, as modern trig tables do, giving it for all angles.)
So instead of measuring an angle as 29.88 degrees, calculating the sine of 29.88 as 0.49818510533, then rounding to 0.4982 or whatever, we should instead calculate the sine of 30 degrees, and get the wonderful totally accurate "exact fraction" 1/2.
1/2 is totally accurate representation of the sine of 30 in a way that 0.4982 is not a totally accurate representation of the sine of 29.88, yes... But 0.4982 is a much more accurate representation of the angle you've actually measured.
From an engineering point of view (which offhand seems like the only one where accuracy matters), normal trig is obviously, obviously better than rational trig, because normal trig is more accurate in the ways that matter. The more sigfigs I have in my measurement of my angle, the more decimal places I can leave in my numerical approximation of the answer. It may always be inexact in a way that 1/2 is not, but it'll also always be exact in a much more important way. :)
> Such as? 60 has more prime factorizations than 10
So it does, but that's not the claim being made. Rather, the claim was:
"We count in base 10, which only has two exact fractions: 1/2, which is 0.5, and 1/5."
To which one might note that:
1. All fractions are exact, by definition. 1/3 is an exact fraction; it's exactly 1/3, in exactly the way 1/2 is exactly 1/2. The fact it's decimal representation is repeating does not impact this.
2. Mansfeld apparently means "fractions with terminating decimal representations" when he says "exact", which is a weird and non-standard usage. But even so, you can construct tons of fractions in base 10 that have non-repeating decimal representations, like 2/5ths, or 5/8th.
3. Possibly he means "...which only allows you to construct two fractions using a numerator of 1 and a denominator consisting of a number smaller than the size of the base which has non-repeating decimal representations..." (which would be a pretty weird thing to focus on) it's still wrong; it's missing, eg, 1/4th.
4. Finally, he reckons that in base 60, 7/60 + 30/3600 aka 1/8 aka 0.125 is an "exact fraction". That sort of factorization is legitimate, but obviously if that's an exact fraction so is 2/10 + 5/100 aka 1/4th. Or come to that, 6/10 aka 0.6.
In short, the claim is gibberish. It's using a phrase with no clear definition in a way which is internally inconsistent. There is no way that 1/8th is an "exact fraction" in base 60, but only 1/2 and 1/5 are "exact fractions" in base 10.
(Also, you're using the word "irrational" wrong.)
- adrianratnapala 9y agoI agree it's absurd just say that the rational trig system is "more accurate" and leave it at that. The idea that we have, for thousands of years, been ignoring mathematical wisdom of the ancient Babylonians is pure click-bait. But the issue goes beyond the maths itself and into what humans want to do with it. Let's start with an apparent nitpick. You say "rather than, as modern trig tables do, giving it for all angles". But no table gives all angles, it's always sample. For modern tables, the samples tend to be regularly spaced, while the Babylonians apparently chose to sample the rational points. Which is better likely depends on the use case. It's not obvious that "normal trig" is better from an engineering point of view. At least not for all engineers. If you are designing a Babylonian temple then you have the freedom to make your rooms so-many-cubits by so-many. That is, it's fairly easy to make rational ratios between the opposite. It's then handy to know what mathematical properties the resulting angle has. Or else to be able to look at your table and pick ratios that have some useful property. You could not do this with a "normal" trig table, or even a modern a hand calculator.
- Lazare 9y ago> You say "rather than, as modern trig tables do, giving it for all angles". But no table gives all angles, it's always sample That's true, and my wording was poor. But during the heyday of trig tables, they were very thorough, giving you results for angles (typically) very, very close to whatever you were trying to calculate, and to usefully high number of decimal places. (I'm old enough - just - that I actually used them in school.) But more to the point, we don't use trig tables any more, we calculate the number on the fly to any arbitrary desired degree of accuracy. That's always going to be the best choice; there's no tradeoff here. > If you are designing a Babylonian temple then you have the freedom to make your rooms so-many-cubits by so-many. Also quite true, but of course, no temple wall is exactly N cubits high; the measured height is at best an approximation, so the resulting ratio of height to width (or whatever) is at best an approximation. All we need from our trig tables (or calculator) is enough accuracy. And in practice, that's not hard. The reality is that, as the linked article says, "it seems like a solution to a problem that doesn’t exist". When we actually used trig tables to build bridges, we didn't have issues with trig tables, and now we have a much better solution. Trig tables that, as you put it, sample the rational points just doesn't seem like it's better for any use case.