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A trick to eliminate 2π (sometimes)

marci.gunyho.com

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Re: A trick to eliminate 2π (sometimes)

#81

Post by enriquto »

mgunyho wrote: post #73
enriquto wrote: post #17

You can also eliminate the constant in some of the integral formulas by using đx instead of dx. I'm surprised the author does not propose this. However, some constants will still remain. Most conspicuously, the 2π constant in the very definiton of the Fourier transform. I once took a personal crusade to eliminate all such constants in the elementary Fourier formulas (plancherel-parseval, convolution theorems, commuta…

> You can also eliminate the constant in some of the integral formulas by using đx instead of dx. I'm surprised the author does not propose this. In the post I propose doing that for Gauss' theorem and Cauchy's formula, because there it's convenient, heh. But to me it feels better to use Θ^ix than a scale factor in front, since the 2pi is always present in the exponential, while the prefactor can be avoided in Fourie…

> Does this not apply also to the elementary formulas you mention?

Oh, you are right! I disliked the 2pi factor in the exponential because it messes with derivatives. But if you define your scaled derivative then the factor disappears again. So cool!

Re: A trick to eliminate 2π (sometimes)

#82

Post by mgunyho »

zerocrates wrote: post #72
mgunyho wrote: post #55

Earlier quoted context omitted.

Author here, I'm sorry to hear that it doesn't work well with a screen reader. I tested it with the reader mode of Firefox, which renders MathML perfectly, although I don't know how that would translate to a screen reader. Safari reader mode renders the math inline, like this: I just define a new derivative operator, like so: dxđ f(x)≡2π1 ⋅dxd f(x). That’s all. while Chrome's reader mode just fails to recognize the c…

I bet the "reading" modes of the browsers are just using the HTML that's visible on the page, but that's marked explicitly as hidden from screenreaders, using the aria-hidden attribute.

From what I can tell it's the opposite (on Firefox): the math is in the source HTML twice, once in , which is hidden using CSS in the non-reader mode but is rendered in the reader mode, and , which is what FF displays normally, and it's missing from the source when I inspect it in reader mode. Having two spans like this comes directly from KaTeX, whose authors I'm sure have thought about accessibility. I imagined that MathML would be somewhat standard and screen readers would understand it.

Re: A trick to eliminate 2π (sometimes)

#83

Post by linuxdude314 »

broses wrote: post #56

People have proposed introducing a symbol for 2π before, most often τ. I like to go a step further and introduce a symbol for 2πi. I use pi with a dot above it, pronounced "pi dot". Pi dot can be defined as the period of the exponential function (which can be defined in terms of its Taylor series). Then 2π is pi dot / i, and π is pi dot / 2i. Of π, 2π, and 2πi, 2πi is probably the most natural, even though it's imagi…

This is the way!

Introduce all the constants you need.

Don’t redefine functions and operators for syntactic sugar.

Re: A trick to eliminate 2π (sometimes)

#84

Post by adrian_b »

eigenket wrote: post #47
6gvONxR4sf7o wrote: post #38

Earlier quoted context omitted.

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)?

Angles are absolutely more dimensionless than lengths are. For an easy check you can't add quantities where the dimension differs, which means it doesn't make sense to add a length to its cube. On the other hand it does make sense to add an angle to its cube - this is a necessary component of computing sin(angle) by the power series sin(angle) = angle - (angle^3)/6 + ...

Your argument is wrong.

Any physical quantity, for instance length, can appear as an argument of a nonlinear function that can be developed in a Taylor series. So your example would be identical for any other quantity not only for angle. I can make an analog computing element where a voltage is equal to the sinus of another voltage, so after your theory, voltage is dimensionless.

The reason why this is possible is that the arguments of such nonlinear functions are either explicitly or implicitly not the physical quantities, but their numeric values, i.e. the ratios between those quantities and their units, which are dimensionless.

In the case of the nonlinear sinus function, what is usually written as sin(x) is just one member of a family of functions where the arguments are angles implicitly divided by units of plane angle:

sin(x) is the sinus function with the angle implicitly divided by 1 radian

sin(x * Pi/2) is the sinus function with the angle implicitly divided by 1 right angle

sin(x * Pi*2) is the sinus function with the angle implicitly divided by 1 cycle a.k.a. turn

sin(x * Pi/180) is the sinus function with the angle implicitly divided by 1 sexagesimal degree

It is very sad that the logical thinking about angles of most people has been perverted by what they have been taught in school, which is just a bunch of nonsense copied again and again from one textbook to another.

Re: A trick to eliminate 2π (sometimes)

#85

Post by paulddraper »

kccqzy wrote: post #32
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 (calle…

That's a really good answer.

And justification in general for "why radians" vs degrees, gradians, turns, whenever

Re: A trick to eliminate 2π (sometimes)

#86

Post by User23 »

patrec wrote: post #75
bjornasm wrote: post #48

Earlier quoted context omitted.

One option would be to do testing with screen-readers and then find out if the content works, and if not find out why.

I think most people are fine with the idea that some of the costs (monetary and otherwise) arising from disabilities should be transferred from the people suffering from them to society at large. But I don't think it's reasonable for producers and users of broken screen readers to expect everyone (including authors of private blogs posts) bending over backwards to accommodate them, nor is it a remotely efficient use…

If there is a bug in the screen reader then that's on the vendor.

However the overwhelming number of cases where a website is unusable with a screen reader are due to lack of proper semantic tagging, like alt text and so on. The last thing a screenreader user wants to hear on a site is "button," "button," "button..." There's certainly room for tooling to help though. I definitely think it sucks that many accessibility linters are nonfree and thus will never be used by site devs who aren't worth suing.

It's worth remembering that accessibility helps everyone, not just the disabled. In fact one of the more popular arguments against accessibility is that it allows non-disabled persons to do more than intended.

So while there's a good argument to be made for being charitable to those with a frankly really lousy condition, if you just want to be self-centered you still benefit from properly tagged data.

Re: A trick to eliminate 2π (sometimes)

#87

Post by enriquto »

scythe wrote: post #46
enriquto wrote: post #17

You can also eliminate the constant in some of the integral formulas by using đx instead of dx. I'm surprised the author does not propose this. However, some constants will still remain. Most conspicuously, the 2π constant in the very definiton of the Fourier transform. I once took a personal crusade to eliminate all such constants in the elementary Fourier formulas (plancherel-parseval, convolution theorems, commuta…

If you're really slick about it, you even "fix" the Gaussian integral this way. Let é = e^sqrt(2pi), déx = dx/sqrt(2pi), and we have int_{R}(é^(int_0^x(t dét)) déx) = int_{R}(e^(sqrt(2pi) x^2/(2 sqrt(2pi))) dx/sqrt(2pi)) = 1/sqrt(2pi) int_{R}(e^(x^2/2) dx) = sqrt(2pi) / sqrt(2pi) = 1

> exp(sqrt(2pi))

Now, this is a number that I don't recall having seen before. The letter é seems strangely fitting for it

é = 12.2635111...

Re: A trick to eliminate 2π (sometimes)

#88

Post by mgunyho »

marcosdumay wrote: post #79
gbear605 wrote: post #78

Earlier quoted context omitted.

It’s not entirely impossible; mathematicians did it for thousands of years (just read Newton’s Principia). It does use very specific language to do so, and some of the language might not exist for some advanced mathematical concepts, but I think that this whole article could be written that way.

Yeah, let's talk about the difference in accessibility between the text on this site and Newton's Principia...

I think parent is referring here to the fact that math used to be written without symbols (other than numbers) up to the 1300s according to Wikipedia: https://en.wikipedia.org/wiki/History_of_mathematical_notati... (very interesting article!) However, I would say that there is a reason why notation tends towards terse symbols: it's much more efficient and unambiguous.

Re: A trick to eliminate 2π (sometimes)

#89

Post by jameshart »

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…

Seems like an odd choice when talking about the cross product, since the cross product is only a thing in 3D. You can define analogous things in other dimensions but it becomes clearer and clearer that it’s not meaningfully a ‘product’.

So it doesn’t matter if your visual intuition for a cross product breaks down in higher dimensions - a cross product is only a thing in three.

Re: A trick to eliminate 2π (sometimes)

#90

Post by 6gvONxR4sf7o »

eigenket wrote: post #70
6gvONxR4sf7o wrote: post #68

Earlier quoted context omitted.

Sure, but that's orthogonal to the "angles don't really have units" assertion and the "it does make sense to add an angle to its cube" assertion, which are the ones I'm responding to. As another example for the second assertion, you can compute e(-t) via power series too, adding seconds to seconds squared and seconds cubed, etc, which comes up all the time. But that doesn't mean `dimensionless + seconds + seconds^2`…

If you say "angles have units" I agree with you - obviously you can measure them in degrees or radians or whatever you want. I was responding to the claim > Angles aren't dimensionless any more than lengths are dimensionless They are dimensionless, but they still have units. The concepts are orthogonal. As for the question about adding an angle to its cube, I would say the enormous usefulness of computing trig functi…

> I would say the enormous usefulness of computing trig functions by power series suggests strongly that this is meaningful.

That argument holds just as true for dimensional quantities frequently computed by power series though, which means it can't be valid.

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