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Turns are better than radians

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Re: Turns are better than radians

#401

Post by ComputerGuru »

lordnacho wrote: post #8

That was quite convincing actually. I guess we all have this realization at some point in early math education. Why is it 360 degrees? Mainly because that's a nicely divisible number, no other good reason. Sometimes you find a 400 degree system on calculators but it doesn't seem to be taught anywhere (is it a French thing?) Then at some point you get shown radians, which relates the arc length to the radius. That som…

Regarding the 400 "degree" system: they're called gradians and it's part of the centesimal system of angular measures according to Wikipedia. And your French guess was right, they have their origins in the French Revolution. More here: https://en.wikipedia.org/wiki/Gradian

Re: Turns are better than radians

#403

Post by kragen »

kazinator wrote: post #384
kragen wrote: post #370

Earlier quoted context omitted.

If you're not using derivatives, integrals, or complex numbers, maybe you'd be better off using Wildberger's "rational trigonometry" with quadrances and spreads instead of angles? I haven't actually tried it myself. Wildberger's motivation is a sort of ultra-strict Platonism* mixed with the desire to extend analytic geometry to fields other than the real numbers, though, so it wouldn't be surprising if it wasn't actu…

So if I wanted to, say, calculate the height of a pole from the length of its shadow, I should use Wildberger's rational trig, because I don't need derivatives, integrals, or complex numbers? :)

You can do that without derivatives, integrals, complex numbers, sines, cosines, tangents, exponentials, square roots, or even addition and subtraction. If the pole is plumb, erect a pole smaller than you with a plumb bob, measure its length a and its shadow's length b, and then calculate the height of the original pole from its shadow's length c as ac/b.

If you have a slide rule, you can do this in a single motion: align c on the C scale over b on D and read off the answer on C above a on the D scale.

Re: Turns are better than radians

#404

Post by sicariusnoctis »

jacobolus wrote: post #351
pdpi wrote: post #112

Indeed, maths never "decreed that sine and cosine have to take radian arguments". But thinking that makes any sort of point is a fundamental misunderstanding of maths. There are infinitely many sinusoidal functions out there. You can just adjust amplitude, frequency and phase to your heart's content. Trigonometry basically requires that sine and cosine have specific amplitudes and phases, but gives not one shit about…

> Degrees are completely arbitrary Degrees are very natural in the context of ancient astronomy/astrology, where you have (1) ~365 days in a year, so that if you look at the path of something that takes a year you get about one degree change per day but with a number that is more easily divisible. (2) approximately 4y, 10y, 8y, 15y, 12y, 30y cycles for the moon and various planets. (3) A calendar with 12 months, 12 z…

All those approximations have error and the behaviour you describe depends on where you are on Earth. Moreover, degrees are not natural from a mathematical perspective.

Re: Turns are better than radians

#405

Post by fluoridation »

bonzini wrote: post #358
fluoridation wrote: post #176

Earlier quoted context omitted.

1° is actually 0.0174... radians . 1 radian is 57.295...°. The choice of unit to specify an angle is arbitrary.

But the angle is an adimensional unit (it's the ratio of two distances, one along the circumference and one along the radius) so 1 rad = 1. Therefore 1 degree is 0.0174... radians but it is also just 0.0174.

No, you're describing one particular way to measure angles. Radians express such a ratio, but degrees don't. 1° is not a ratio between distance along a circumference and radius, it's a ratio between amount rotated and complete revolution. 1° actually stands for 1/360 (of a revolution).

Which is why it's important to add the unit after the measurement. If someone tells you an angle measures 1, can you tell whether it's 1/360 of a revolution or the angle that would be formed by traveling along a circumference a distance equal to the radius of the circle?

Re: Turns are better than radians

#406

Post by jrockway »

jVinc wrote: post #357
kazinator wrote: post #326

> But math never decreed that sine and cosine have to take radian arguments! That is not entirely true. It comes from the relationship between those functions and the complex numbers via the Euler formula. ix e = cos x + i sin x There may be arithmetic/numerical inconveniences, but that's not all there is to "math". Let's define ncos and nsin ("nice cos, nice sin") as follows: nsin x = sin 2πx ncos x = cos 2πx So the…

I see where you're coming from, if the formulas end up having weird numbers like 535.4916 or numbers like 2.718 or 6.28318 then obviously there's something suspicious about the equation. But small correction though. You got the number wrong, it's actually much more weird than any of those mentioned. The actual equation you come to for ncos an nsin is: (-1)^(2x) = ncos(x) + i nsin(x) And yes, -1 is a very weird number…

> if the formulas end up having weird numbers like 535.4916 or numbers like 2.718 or 6.28318 then obviously there's something suspicious about the equation.

Well, 2.718 is different than those numbers, because the derivative of 2.718^x is 2.178^x, which is a very interesting property of 2.718. The same cannot be said about 535.4. (6.283 is the ratio of a circle's, diameter to radius, which is just something intrinsic to the universe. I think it even transcends the universe, but that's hard for me to reason about. But basically, both 2*pi and e are fundamentally interesting.)

Re: Turns are better than radians

#408

Post by IshKebab »

gjm11 wrote: post #75
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.

`cost` is a normal variable name in a lot of code though, whereas `cosh` definitely isn't! In fact I bet you can't find a single use of `cosh` in code that isn't talking about hyperbolic cosine.

Re: Turns are better than radians

#409

Post by eesmith »

ComputerGuru wrote: post #399
eesmith wrote: post #171

Earlier quoted context omitted.

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?

Re: Turns are better than radians

#410

Post by kazinator »

scythe wrote: post #396
jVinc wrote: post #357

Earlier quoted context omitted.

I see where you're coming from, if the formulas end up having weird numbers like 535.4916 or numbers like 2.718 or 6.28318 then obviously there's something suspicious about the equation. But small correction though. You got the number wrong, it's actually much more weird than any of those mentioned. The actual equation you come to for ncos an nsin is: (-1)^(2x) = ncos(x) + i nsin(x) And yes, -1 is a very weird number…

>The actual equation you come to for ncos an nsin is: >(-1)^(2x) = ncos(x) + i nsin(x) Try to formally define this procedure, though. You end up going in circles. Here's another version: lim[N->infinity] (1 + ix/N)^N = cos(x) + i sin(x) Now there are no "weird numbers", and both sides of the equation can be calculated directly , even by hand if you wanted. If all you're teaching students is a bunch of formulas to be…

> Try to formally define this procedure, though. You end up going in circles.

The cos(x) + isin(x) formula gives us a way to find the point on the complex plane's unit circle corresponding to an angle x, given in radians. (Plus it does more, because the argument is complex valued.)

The new formula with ncos and nsin does the same thing for an angle given in turns. E.g 0.25 (90 degrees): -1^(0.5) = i. It's understandable in terms of roots of -1.

When you want to know the principal N-th root of number on the complex plane, you can simply divide its argument (i.e. angle) by N. The other roots are then equidistant points around the circle. So for instance, the square root of -1, which is sitting at 180 degrees, is found at 90 degrees, and is therefore i.

We can use -1 as the reference for measuring angles. The turns unit (one circle) is twice as far around the circle as as -1, so that's where we get the 2. Because 90 degrees in turns isn't 0.5, but 0.25.

We could use 1 directly, but then we need the first complex root of unity. For instance, here is the Wikimedia diagram of the fifth roots:

https://en.wikipedia.org/wiki/Root_of_unity#/media/File:One5...

That root which is close to i, has an angle which is exactly 1/5 turns. There is a relationship between turns and roots of unity, because N roots occupy N equidistanct points on the circle spaced by 1/N turns.

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