Thanks for the context - I was baffled at first how the Guardian would run with the tagline "a trignometric table more accurate than any".

But it's because the sine of 60 degrees is said by modern tables to be equal to sqrt(3) / 2, which Wildberger doesn't "believe in", he prefers to state that the square of the sine is actually 3 / 4 and that this is "more accurate".

The actual paper is at [1]:

[1] https://doi.org/10.1016/j.hm.2017.08.001

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kragen 4 days ago | parent | next [-]

Well, no, if you look at a trigonometric table, it doesn't say sin 60° = √3/2, because that isn't a useful value for calculation. It'll say something like 0.866025. But that has an error of a little more than 0.0000004. Instead Wildberger prefers saying that the spread (sin²) is ¾, which has no error. It is more accurate. There's no debate about this, except from margalabargala.

The news from this paper (thanks for the link!) is that evidently the Babylonians preferred that, too. Surely Pythagoras would have.

But how do you actually do anything useful with this ratio ¾? Like, calculating the height of a ziggurat of a given size whose sides are 60° above the horizontal? Well, that one in particular is pretty obvious: it's just the Pythagorean theorem, which lets you do the math precisely, without any error, and then at the end you can approximate a linear result by looking up the square root of the "quadrance" in a table of square roots, which the Babylonians are already known for tabulating.

For more elaborate problems, well, Wildberger wrote the book on that. Presumably the Babylonians had books on it too.

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dhosek 4 days ago | parent [-]

> √3/2 … that isn't a useful value for calculation

Some tables do indeed have that value and it is a very useful value for calculation, one that can be symbolically manipulated to get you an exact number (albeit one likely expressed in radicals) for your work. When I used to teach algebra, it was a struggle to get students to let go of the decimal approximations that came out of their calculators and embrace expressions that weren’t simple decimals but were exact representations of the numbers at hand. (Then there’s really fun things like the fact that, e.g., √2 + √3 can also be written as √(5+2√6) (assuming I didn’t make an arithmetic error there)).

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kragen 4 days ago | parent [-]

What do those tables say for 59°59'? I'm skeptical that what you're looking at is, strictly speaking, a trigonometric table.

If you want to know how many courses of bricks your ziggurat is going to need, given that the base is 400 cubits across and there are 10 courses of bricks per cubit, you're going to have to round 2000√3/2 to an integer. You can do that with a table of squares, or you can use a decimal (or sexagesimal) fraction approximation, and I guess you're right that it isn't clear that one is necessarily better than the other.

Incidentally, the fact that we write things like 59°59'30" comes about because the Babylonians at least weren't using Wildberger's "spreads" all the time.

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dhosek 3 days ago | parent [-]

Those tables don’t give a value for that. They only give values for angles whose trigonometric values can be expressed in terms of rational expressions with radicals (and angles as rational expressions in terms of π).

	▲	kragen 3 days ago \| parent [-]
		That sounds like a sort of "trigonometric table" you couldn't use in practice for calculation. You need to be able to look up whatever value you measure with your sextant or theodolite to within its measurement precision. You seem to be using the term "trigonometric table" in a way conflicting with the standard use explained in https://en.wikipedia.org/wiki/Trigonometric_tables perhaps as a sort of pun or joke, or perhaps as a sort of act of activism against hegemonic notions of trigonometry.

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amai 4 days ago | parent | prev | next [-]

What does Wildberger then think about i = sqrt(-1)? Is this also "not accurate" enough?

	▲	numpy-thagoras 4 days ago \| parent [-]
		Ultrafinitism does not rule out higher algebraic structures

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margalabargala 4 days ago | parent | prev [-]

> But it's because the sine of 60 degrees is said by modern tables to be equal to sqrt(3) / 2, which Wildberger doesn't "believe in", he prefers to state that the square of the sine is actually 3 / 4 and that this is "more accurate".

Personally I don't believe in either value. I prefer to state that the sine of 60 degrees is 2.7773. I believe that is more accurate.