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Turns are better than radians

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Re: Turns are better than radians

#71

Post by fbanon »

doliveira wrote: post #52
fbanon wrote: post #32

>But math never decreed that sine and cosine have to take radian arguments! Ummm, actually it did. The Taylor-series of sine and cosine is the simplest when they work with radians. Euler's formula (e^ix = cosx + isinx) is the simplest when working with radians. Of course you can work in other units, but you'll need to insert the appropriate scaling factors all over the place. "Turns" don't generalize to higher dimens…

sin(x) ~~ x only in radians, so honestly that's reason enough. Once in a while we get programmers wanting to disrupt mathematical notation for whatever reason... Worst I've seen so far was one arguing that equations should be written with long variable names (like in programming) instead of single letters and Greek letters. Using turns because it's a little easier in specific programming cases is just as short-sighte…

Or just defining the result of division by zero as zero "for safety": https://www.hillelwayne.com/post/divide-by-zero/

It boggles the mind, truly!

Re: Turns are better than radians

#72

Post by ttoinou »

WastingMyTime89 wrote: post #67
fbanon wrote: post #32

>But math never decreed that sine and cosine have to take radian arguments! Ummm, actually it did. The Taylor-series of sine and cosine is the simplest when they work with radians. Euler's formula (e^ix = cosx + isinx) is the simplest when working with radians. Of course you can work in other units, but you'll need to insert the appropriate scaling factors all over the place. "Turns" don't generalize to higher dimens…

The writer don’t seem to realise that radian is not an arbitrary unit but a dimensionless one which is defined so that 1rad is actually just 1. Reading the submission and the comments here, I’m under the impression that trigonometry is not extensively taught in middle schools and high schools in the USA. While I’m slightly envious you might not have to suffer developing powers of cosine and sine but that would explai…

Correct, radians are a "fake" unit made up to understand better formulas (the same way we use types in programming languages)

Re: Turns are better than radians

#73

Post by version_five »

fbanon wrote: post #32

>But math never decreed that sine and cosine have to take radian arguments! Ummm, actually it did. The Taylor-series of sine and cosine is the simplest when they work with radians. Euler's formula (e^ix = cosx + isinx) is the simplest when working with radians. Of course you can work in other units, but you'll need to insert the appropriate scaling factors all over the place. "Turns" don't generalize to higher dimens…

Right, radians are the "natural" units of angle, others generally just make a circle into some integral number of units for convenience, but you always have to go back to radians to actually do calculation.

In the next installment, maybe he'll propose that turns can be limiting because diving up a circle requires the use of fractions, and suggest instead of 1 turn per circle, we make a number that's easily divisible into many integer factors. Maybe 216, or I don't know, 360?

Re: Turns are better than radians

#75

Post by gjm11 »

IshKebab wrote: post #10

This is quite convincing but it would have been more convincing if he'd acknowledged the downsides and explained why it is in radians in the first place. (On balance I think he's probably still right.) Perhaps we can make new named functions that operate in turns, along the same lines as ln/log. sint, cost, etc. Ok maybe not cost.

We manage OK with cosh for the hyperbolic cosine even though "cosh" is a word. For that matter, "sin" and "cos" are both words, though of course "sin" and "sin" are pronounced differently.

Re: Turns are better than radians

#76

Post by hutzlibu »

doliveira wrote: post #52
fbanon wrote: post #32

>But math never decreed that sine and cosine have to take radian arguments! Ummm, actually it did. The Taylor-series of sine and cosine is the simplest when they work with radians. Euler's formula (e^ix = cosx + isinx) is the simplest when working with radians. Of course you can work in other units, but you'll need to insert the appropriate scaling factors all over the place. "Turns" don't generalize to higher dimens…

sin(x) ~~ x only in radians, so honestly that's reason enough. Once in a while we get programmers wanting to disrupt mathematical notation for whatever reason... Worst I've seen so far was one arguing that equations should be written with long variable names (like in programming) instead of single letters and Greek letters. Using turns because it's a little easier in specific programming cases is just as short-sighte…

Whats wrong with long variable names?

Re: Turns are better than radians

#77

Post by exo-cortex »

Is there a Rust library that also takes "turns" (or cycles or something similar) that I can use? I have been wondering (very low-key-ish) about this problem for some years and now that I know this I want to use turns in my code.

Re: Turns are better than radians

#78

Post by q-big »

adrian_b wrote: post #63
fbanon wrote: post #32

>But math never decreed that sine and cosine have to take radian arguments! Ummm, actually it did. The Taylor-series of sine and cosine is the simplest when they work with radians. Euler's formula (e^ix = cosx + isinx) is the simplest when working with radians. Of course you can work in other units, but you'll need to insert the appropriate scaling factors all over the place. "Turns" don't generalize to higher dimens…

The simplicity of the Taylor series of sine and cosine is irrelevant, there are no important applications for those series. There is only one consequence of those series that matters in practice, which is that when the angles are expressed in radians, for very small angles the angle, its sinus and its tangent are approximately equal. While this relationship between small angles, sinuses and tangents looks like an arg…

> The simplicity of the Taylor series of sine and cosine is irrelevant, there are no important applications for those series.

The addition theorems for trigonometric functions can easily be shown by the multiplication theorem for Taylor series (and adding two Taylor series). This proof would be more convoluted if the Taylor series were not so easy.

Also, because of the simplicity of their Taylor series, one immediately sees that sin and cos are solutions of the ODE y'' = -y.

Another application of the Taylor series is that by their mere existence, sin and cos (as real functions) have a holomorphic extension.

Re: Turns are better than radians

#79

Post by anonymoushn »

WastingMyTime89 wrote: post #58
anonymoushn wrote: post #51

Earlier quoted context omitted.

The post says that programmers convert to radians to call APIs that take radians and then immediately divide by pi before computing the sine. So doing everything in radians would still involve an extra floating point divide compared to the alternative.

No, you are confused. That’s a conversion back to degree after computing the sine. You don’t randomly divide by pi for no reason.

so you're telling me 4/pi is the conversion ratio between degrees and radians, and i am confused?

If you would be so kind, point out the part of the sine implementation that works in degrees: https://github.com/reyoung/avx_mathfun/blob/be617bbcf66993c4...

Re: Turns are better than radians

#80

Post by Aardwolf »

doliveira wrote: post #52
fbanon wrote: post #32

>But math never decreed that sine and cosine have to take radian arguments! Ummm, actually it did. The Taylor-series of sine and cosine is the simplest when they work with radians. Euler's formula (e^ix = cosx + isinx) is the simplest when working with radians. Of course you can work in other units, but you'll need to insert the appropriate scaling factors all over the place. "Turns" don't generalize to higher dimens…

sin(x) ~~ x only in radians, so honestly that's reason enough. Once in a while we get programmers wanting to disrupt mathematical notation for whatever reason... Worst I've seen so far was one arguing that equations should be written with long variable names (like in programming) instead of single letters and Greek letters. Using turns because it's a little easier in specific programming cases is just as short-sighte…

Those perfect radians use 2*pi, aka tau, though, a different math notation issue, where mathematicians have chosen the wrong option (imho) and a case for disrupting that part of math notation, to make radians easier to teach: 1/4th of a circle could be tau/4 radians, 1/8th could be tau/8, etc..., instead of confusing halved factors with radians expressed as amount of pi.

Regarding long variable names: I'd rather have long variable names, than a mathematician using some greek symbol in formulas without telling what the meaning of it is (and it could be different depending on their background). But I have no issues with the single letter variables if they're specified properly.

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