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A trick to eliminate 2π (sometimes)

marci.gunyho.com

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Re: A trick to eliminate 2π (sometimes)

#31

Post by JdeBP »

gorkish wrote: post #15
NotYourLawyer wrote: post #5

Interesting, but I wonder if you’d get all the same benefits by just measuring angles in units of turns instead of radians. That seems cleaner than the weird dbar differential stuff.

This is more or less "the point" for those of us who argue for tau instead of pi. I should note that using this "trick" of prescaling rotations by 2pi so that they are in the 0..1 range is de rigueur in computer graphics programming.

For goodness' sake, don't tell Matt Parker that one can write computer programs where 2π is rescaled to 1!

We'll never hear the end of it. There'll be some Python program that takes four months to calculate 1.00000000000000000000000000000001 . (-:

Re: A trick to eliminate 2π (sometimes)

#32

Post by kccqzy »

NotYourLawyer wrote: post #5

Interesting, but I wonder if you’d get all the same benefits by just measuring angles in units of turns instead of radians. That seems cleaner than the weird dbar differential stuff.

That's actually exactly the question I asked my math teacher when I first learned about radians. I mean, I learnt degrees when I was very little, at an age when one tended not to question why, but I learned radians at an age old enough to question why. The answer I received was about making trigonometric identities cleaner: the derivative of sine becomes "just" cosine rather than a hypothetical turn-based sine (called usin by the article) having a derivative of a turn-based cosine multiplied by 2pi.

But this article seems to do a good job explaining that a lot of those 2pi factors appear when you deal with differentiation. So it seems useful to have both turn-based trigonometric functions and this new differentiation operator.

Re: A trick to eliminate 2π (sometimes)

#33

Post by sfpotter »

"[...], and it’s dimensionless, so you can’t easily check if you forgot to divide or multiply by it."

For what it's worth, it's often the case that a factor of 2pi is the difference between something being in terms of cycles/sec or rad/sec. In an experimental context, it usually isn't too difficult to judge which of these "units" a quantity you're looking at is in...

Re: A trick to eliminate 2π (sometimes)

#34

Post by anvuong »

nilstycho wrote: post #25
jovial_cavalier wrote: post #16

Earlier quoted context omitted.

I'm curious how blind people normally engage with math. For me, engaging with math almost always means conjuring up a visual representation in my mind. Failing that, an equation. Since visualization is so fundamental to doing math, and since mathematical symbols and equations are a written language for which there is no spoken analog, I really can't imagine engaging with math without my eyes. Even reading equations a…

I had a math graduate student teaching my linear algebra class. He taught dot and cross products entirely algebraically, never drawing vectors as arrows, but as arrays of numbers. When I suggested after class that teaching the visual representation might help some students, he pushed back. Visual understanding, he explained, was a crutch best avoided, because visual intuition could break down in higher dimensions. I…

This is quite true, especially true when talking about direction (gradient) in high dimensional space. I don't think this can be avoided, since after all we are creatures living in 3D space where left right up down are quite well-defined, just need to make a mental note every time you have to deal with more than 3 dimensions.

Re: A trick to eliminate 2π (sometimes)

#35

Post by H8crilA »

tgv wrote: post #20
H8crilA wrote: post #13

Also, π is the wrong constant. The very definition is awkward: the ratio of two radiuses to the circumference. How about one radius? It is much more natural to work with 2π. Some people use the letter τ (Tau) to denote 2π, and it simplifies almost all naturally occurring expressions. For example, what is more elegant? e^(π*i) = -1 e^(τ*i) = 1

τ = 0, got it.

Well, I can say π := 3π, and -1 still works. The point is exp() is that 2π*i periodic.

Re: A trick to eliminate 2π (sometimes)

#36

Post by Misdicorl »

movpasd wrote: post #12
NotYourLawyer wrote: post #5

Interesting, but I wonder if you’d get all the same benefits by just measuring angles in units of turns instead of radians. That seems cleaner than the weird dbar differential stuff.

Thing is, angles don't really have units (the technical term is they are dimensionless). They are a length (the subtended arc of a circle) divided by a length (the radius of the circle). When you want to do something like get a sine wave of period T, you inevitably have to include a 2π somewhere. Speaking as someone who had to write down many 2π's in university (especially as I find angular quantities like angular fr…

This is not true. Angles very much have units and it's why you can express the same concept with different numbers. Pi equals 180 degrees equals 0.5 turns.

1 radian has different units than 1 steradian and if they didn't there wouldn't be a need for two different words to denote them.

The quantity is a ratio of two lengths, and the length measure does "drop out". But it's not just any ratio, it's a very particular ratio, and the unit defines the particularness of that ratio.

Re: A trick to eliminate 2π (sometimes)

#37

Post by scythe »

I was with him until this point:

>This notation could be abused even further by denoting đx = 1/(2π) dx, which can then simplify some integral formulae,

But now you're screwing up all of your previous integral formulae!

Re: A trick to eliminate 2π (sometimes)

#38

Post by 6gvONxR4sf7o »

movpasd wrote: post #12
NotYourLawyer wrote: post #5

Interesting, but I wonder if you’d get all the same benefits by just measuring angles in units of turns instead of radians. That seems cleaner than the weird dbar differential stuff.

Thing is, angles don't really have units (the technical term is they are dimensionless). They are a length (the subtended arc of a circle) divided by a length (the radius of the circle). When you want to do something like get a sine wave of period T, you inevitably have to include a 2π somewhere. Speaking as someone who had to write down many 2π's in university (especially as I find angular quantities like angular fr…

Angles aren't dimensionless any more than lengths are dimensionless (feet per second makes just as much sense as rpm). It's just that angles have symmetries that lengths don't, which is where 2 pi comes in. Do you want units where your symmetries are expressed in multiples of 1, 2, or 2 pi (for turns, half-turns, and radians, respectively)?

Re: A trick to eliminate 2π (sometimes)

#39

Post by RogerL »

ctoth wrote: post #11

Unrelated to the content but complaining about the website is a popular thing to do here so I'd like to share my experience, as a blind user. Here is what my screen reader sees for this page: Recently, I came up with a trick that can get rid of in many cases. It’s pretty simple, but it has some interesting implications. This is the trick: I just define a new derivative operator, like so: That’s all. You just take the…

But what is the answer? What should I do differently? I get don't use images, but then what? I can't imagine all screen readers have the same capabilities, or that there is a base common ability, so what should we do?

Googling says MathML is the answer (e.g. https://www.washington.edu/doit/how-do-i-create-online-math-... this site uses MathML and your reader isn't handling it. So now what? (alt-tags? something else?)

Re: A trick to eliminate 2π (sometimes)

#40

Post by scythe »

NotYourLawyer wrote: post #5

Interesting, but I wonder if you’d get all the same benefits by just measuring angles in units of turns instead of radians. That seems cleaner than the weird dbar differential stuff.

You can always define angles in turns. But the problem is that it conflicts with the definition cos(x) = Re{e^(ix)}. Trig is not so easily separated from the rest of mathematics.
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