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Comment by tromp | Hacker News Reader

> Some time ago, much effort was expended to convince people to replace approximations of “pi” (3.14159…) with approximations of “tau” (6. 28318…).

IMO the effort was simply to replace the use of pi with the use of tau. What does approximation have to do with it?

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The author is a software engineer and when he says "people", he implicitly means other software engineers. When he says "replace" he means in code, not in equations.

It's a general challenge of writing on the web that you don't know what context the author assumes and the author doesn't know what context the reader assumes.

In this case, the blog title "Computer, Enhance!" and the article subtitle "Switching away from radians makes code simpler, faster, and more precise." sends a pretty clear signal that this is about programming and not pure mathematics.

For any given article on the web, you can always generate valid criticisms based on the author assuming some context that may not be true for all possible readers. You can't say, "Ice cream is cold" without some commenter pointing out that you're doing a disservice to astronauts for whom ice cream is freeze dried and room temperature.

I find the best way to extract value from writing on the web is to simply try to understand the author's assumed context and go from there.

Right, it is a funny way to say it.

Computationally, we all only ever work with approximations, but when doing mathematics, pi is exact all the way out to the infinity-th digit. To multiply by pi (or any irrational, but particularly transcendental) in a pure mathematical context is to audaciously specify an infinitely long computational process. It is dizzying to contemplate, almost mystical. Sort of like modular arithmetic with an infinitely-precise irrational modulus.

Is any number mod infinity equal to itself?

In the surreal numbers [1], the definition of division and hence modulo applies to all numbers, finite and infinite alike. And for any finite number -omega < x < omega you would indeed have x mod omega == x.

[1] https://en.wikipedia.org/wiki/Surreal_number

You might already know about it, but Ehrlich claims[1] that the Surreals form an ordered field which is a maximal hyperreal number system in NBG, and therefore all the results of non-standard analysis "work" (up to isomorphism) in the Surreals.

[1] Philip Ehrlich. "The absolute arithmetic continuum and the unification of all numbers great and small." Bull. Symbolic Logic 18 (1) 1 - 45, March 2012. https://doi.org/10.2178/bsl/1327328438 Theorem 20

What do you want mod infinity to mean?

The point is PI and it’s brother TAU, being transcendental, is always approximated. Which is important to think about if you care about accuracy.

And I think that's the most important part of the argument here. By using radians in this case, you do extra calculation steps only to reduce the accuracy. That seems like a poor investment of the processor's time.

It's a correct description. (Perhaps further refined as "technically correct".) The following are not exact values of π or τ, only approximate ones:

  >>> import math
  >>> math.pi
  3.141592653589793
  >>> math.tau
  6.283185307179586

The effort wanted to replace all uses of Pi (including the ones in math formulas) and not just their use in (floating-point) computer programs where we implement approximations of those formulas.

Sure.

But just because there was {a lot of work to replace all uses of pi with tau} doesn't mean there wasn't {a lot of work to replace all approximate uses of pi with approximate uses of tau in computer programs}.

Right, which makes it an effectively incorrect description. It is like if I said I wanted to change all green fruits to orange ones and you reported that as "my undertaking to change green apples to orange apples." It's the truth, but it's not the whole truth.

As I noted parenthetically: Perhaps further refined as "technically correct".

Nothing in the rest of the article deals with pi or tau as transcendental numbers, only about a bit patterns and (implicitly) floats or doubles.

If your focus is IEEE 754 apples, then perhaps other sorts of fruit aren't so important?

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