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A trick to eliminate 2π (sometimes)

marci.gunyho.com

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Re: A trick to eliminate 2π (sometimes)

#121

Post by adrian_b »

eigenket wrote: post #102
adrian_b wrote: post #100

Earlier quoted context omitted.

> They are dimensionless, but they still have units. The concepts are orthogonal. The concepts are not orthogonal, they are incompatible. A dimensionless quantity (which angles are not) is by definition the ratio of two quantities that are measured with the same unit. When you compute the ratio by division, the two identical units disappear from the result, therefore the result is indeed dimensionless. There is no wa…

Ok, you've replied to quite a lot of my comments, with some fairly confusingly worded responses. Some of your claims appear to be incompatible as far as I read them. For example you claim that angles are not dimensionless, but that dimensionless quantities are formed by the ratio of two quantities with the same unit. Since the angle subtended by an arc in a circle is the ratio of the arc length and radius, it would s…

No, even if you have reproduced a definition of the plane angle that is encountered in many textbooks "the angle subtended by an arc in a circle is the ratio of the arc length and radius", this definition is very incorrect.

This very wrong definition forced upon many students is the root of all misconceptions about plane angles.

The reason why this definition is wrong is because some words are missing from it and after they are added it becomes obvious that its meaning is different from what many teachers claim.

First there is no relationship whatsoever between the magnitude of a radius and the magnitude of an angle. What that definition intended to say was:

"the angle subtended by an arc in a circle is the ratio of the arc length and of the length of an arc whose length is equal to the radius".

By definition, a radian is defined as the angle subtended by an arc whose length is equal to the radius.

To explain how plane angles are really defined would take more space, but the only thing that matters is that the characteristic property of plane angles is that the ratio between two plane angles subtended by two arcs of a circle is equal to the ratio of the lengths of the two arcs.

Introducing this characteristic property of the angles in the so-called definition from above reduces it into the sentence "the angle is measured in radians". This is either a trivially true sentence when the angle is indeed measured in radians, or it is a trivially false sentence when the angle is measured e.g. in degrees. It certainly is not a definition.

If that had been the definition of plane angle, that would have meant that the plane angle was discovered only in the second half of the 19th century, together with the radian, while in reality plane angles have been used and measured with various units for millennia.

To measure plane angles, it is necessary to first choose an arbitrary angle as the unit angle, for instance an angle of one degree.

Then you can measure any other angle by measuring both the length of the corresponding arc and the length of the arc corresponding to the chosen unit angle. Then the two lengths are divided, giving the numeric value of the measure of the angle.

Re: A trick to eliminate 2π (sometimes)

#122

Post by Misdicorl »

jacobolus wrote: post #118
Misdicorl wrote: post #36

Earlier quoted context omitted.

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 ra…

The reason it is confusing is because an angle measure is a kind of logarithm of a rotation, and logarithms (sort of) have a unit: the base. The appropriate canonical representation of a rotation is a unit-magnitude complex number z = exp iθ = cos θ + i sin θ , which has a planar orientation (whatever plane i is taken to represent; if you want to represent a 3D rotation you can replace i with an arbitrary unit bivect…

I think I fundamentally disagree with you. Angles do not have a "base" any more than meters do (being embedded inside some metric space could be considered a base I suppose).

But you seem to be drawing a distinction between meters and angles in your analogy where I assert none exists. The base of a number system only affects representations.

This is not true for divisions of lengths. 1 meter divided by 2 meters is 0.5 as a number. But it is only 0.5 radians under (1ish) specific arrangements of those lengths in a particular metric space

Re: A trick to eliminate 2π (sometimes)

#123

Post by kandel »

adrian_b wrote: post #111
scythe wrote: post #40

Earlier quoted context omitted.

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.

There is no need for that definition. It is possible to completely remove the e^x function from mathematics without losing anything. It is possible to express everything using a pair of functions, the real function 2^x and the complex function 1^x. Then the cosinus and the sinus are the real and imaginary parts of 1^x (where x is measured in cycles a.k.a. turns). The only disadvantage of this approach is that symboli…

Mind explaining how to express some simple functions? Im interested

Re: A trick to eliminate 2π (sometimes)

#125

Post by zamadatix »

mgunyho wrote: post #55
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…

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…

Chrom* browsers just got decent MathML support finally, so I don't think it's on the way out quite yet. Some still like other solutions more though.

Re: A trick to eliminate 2π (sometimes)

#126

Post by zamadatix »

psychoslave wrote: post #107
marcosdumay wrote: post #74

Earlier quoted context omitted.

You can't discuss math while getting rid of the math symbols. That's not a reasonable proposal. Math on the web is broken, the the affected people should be up complaining about that. This site did the most accessible thing possible; the fact that every tool broke here, just like they do for every other method is not really the author's fault.

> You can't discuss math while getting rid of the math symbols. That's not a reasonable proposal. If we assert that some topic can’t be discussed with plain prose, then the only logical conclusion is that you can’t discuss the topic at all. > Math on the web is broken, the the affected people should be up complaining about that. There is really two different topic there. One is, how can screen readers deal appropriat…

Why is "plain prose" so logically the litmus test for what can be discussed? It certainly wasn't designed around the idea it should cover every possible concept, or do so remotely efficiently either, so the repeated assertion it logically is the only way to know if a concept can be discussed is a bit hard to follow. It does cover most concepts though, and conveniently.

Re: A trick to eliminate 2π (sometimes)

#128

Post by HappMacDonald »

psychoslave wrote: post #107
marcosdumay wrote: post #74

Earlier quoted context omitted.

You can't discuss math while getting rid of the math symbols. That's not a reasonable proposal. Math on the web is broken, the the affected people should be up complaining about that. This site did the most accessible thing possible; the fact that every tool broke here, just like they do for every other method is not really the author's fault.

> You can't discuss math while getting rid of the math symbols. That's not a reasonable proposal. If we assert that some topic can’t be discussed with plain prose, then the only logical conclusion is that you can’t discuss the topic at all. > Math on the web is broken, the the affected people should be up complaining about that. There is really two different topic there. One is, how can screen readers deal appropriat…

> If we assert that some topic can’t be discussed with plain prose, then the only logical conclusion is that you can’t discuss the topic at all.

Please bear in mind that mathematical notation is a language, and that mathematical formulas are perfectly valid plain prose in that language.

I imagine that some screen readers will fail gracelessly when faced with Chinese script or Hindu as well. Especially if they're not unicode compliant.

But once you hit that threshold you are simply in a battle of dueling accessibility concerns. Not everyone is sighted, but neither does everyone rely on English as a primary language.

Nor should they as the English language was not designed to convey all concepts accurately. It excels mostly in conveying concepts germane to anglophone cultures. And three guesses what concepts are not popularly relevant to your standard anglophone? That's right.. calculus and theoretical physics.

That's not to bash on English as a language. It really is a flexible beast with a far reaching vocabulary. But there literally exists no language that is ideal for encapsulating every single idea under the sun. One must have a way to support many of the most diverse ones at the same time. And modern mathematical notation belongs on that short list along with English.

Re: A trick to eliminate 2π (sometimes)

#129

Post by jameshart »

jacobolus wrote: post #112
jameshart wrote: post #89

Earlier quoted context omitted.

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.

This is very off topic, but the wedge product absolutely is "meaningfully a product", generalizes fine to arbitrary dimension, and has a perfectly reasonable visual/spatial/geometric interpretation. (Indeed, we should entirely scrap the cross product in undergraduate level technical instruction and replace it with the wedge product; one happy effect will be replacing students' misleading spatial intuitions with bette…

Or bivectors, or k-vectors, or blades…

Re: A trick to eliminate 2π (sometimes)

#130

Post by xingped »

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…

Hi! I'm trying to work on understanding how to make websites more accessible right now and probably the first thing people generally think of are blind users. However, I'm having a hell of a time figuring out how to use screen readers. Is there any screen reader or documentation/tutorial you might recommend for sighted users to learn how to use screen readers?
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