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Comment by eesmith | Hacker News Reader

Also, John Cook recently blogged about it at https://www.johndcook.com/blog/2023/01/06/rational-trigonome... and wrote:

> I became interested in rational trigonometry when I was working on a project to formally verify drone collision avoidance software. Because all calculations are purely algebraic, involving no transcendental functions, theorems are easier to prove. ...

> Another advantage is accuracy. Because a computer can carry out rational arithmetic exactly, rational trigonometry applied to rational points yields exact results. No need to be concerned about the complications of floating point numbers.

> A less obvious advantage is that the theory of rational trigonometry is independent of the underlying field

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> Because all calculations are purely algebraic, involving no transcendental functions, theorems are easier to prove. ...

Personally I would have said manipulation of trig identities and such is also purely algebraic.

They meant algebraic in the sense of https://en.wikipedia.org/wiki/Algebraic_operation.

Sure, but to be honest I'm a little bit boggled at the idea of a phd mathematician finding trig manipulation hard enough to be worth commenting on.

This review of Wildberger's book at https://philarchive.org/archive/FRADPR says the point isn't that the PhD mathematician finds trig hard, but rather to beginners:

> ... very becoming-numerate generation invests enormous effort in the painful calculation of the lengths and angles of complicated figures. Surveying, navigation and computer graphics are intensive users of the results. Much of that effort is wasted, Wildberger argues. The concentration on angles, especially, is a result of the historical accident that serious study of the subject began with spherical trigonometry for astronomy and long-range navigation, which meant there was altogether too much attention given to circles. ...

> Having things done better is one major payoff, but equally important would be a removal of a substantial blockage to the education of young mathematicians, the waterless badlands of traditional trigonometry that youth eager to reach the delights of higher mathematics must spend painful years crossing

Whether that's correct is different. https://handwiki.org/wiki/Rational_trigonometry seems to give a good description of opposing views.

I don't have a particularly high opinion of John D. Cook but I still wouldn't call him a 'beginner'.

I apologize. I misread the context of the thread. I mistakenly though you were commenting on Wildberger's reasoning for promoting rational trigonometry, rather than on Cook's decision to use that approach.

If "formal verification" involves using a software-based proof assistant or automatic theorem prover then perhaps that easier to encode for those tools via rational trigonometry?

Why not ask him directly?

Cook isn't proving results about trig functions/identities; they're proving results about software. Software can never correctly implement trig functions; it must always approximate, and that is the reason it's hard to verify.

In contrast, arithmetic on rational numbers can be implemented exactly; at least, up to the memory limits of the machine (e.g. using pair of bignums for numerator/denominator)

>A less obvious advantage is that the theory of rational trigonometry is independent of the underlying field

What sort of interesting results come out of using a non-real field here?

You'll need to ask an actual mathematician. Cook recently wrote that blog post, so it's much better to ask him in the comment section of that post.

Just because I can do a Google Scholar query using "rational trigonometry" and find papers saying things like:

"Explicit tough Ramsey graphs" at https://arxiv.org/pdf/0807.2692.pdf . "We present here two new explicit constructions ... The first construction is based on a remarkable new approach of Wildberger to trigonometry and Euclidean geometry."

does not mean I am qualified to judge what is "interesting", or figure out which use non-real fields.

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