Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

A trick to eliminate 2π (sometimes)

marci.gunyho.com

Post reply on HN
Print view View original

101–110 of 179 posts

Re: A trick to eliminate 2π (sometimes)

#101

Post by codazoda »

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…

Yes, I'm pretty sure it's that aria label. The first line of this code.

    
        
            
            2
            π
My first inclination was to simply run `say` on my Mac comamdn line and paste the line in. That read it correctly. The other commenter called out the `aria-hidden` attribute and I'd guess that's it. It's explicitly hidden.

Apparently this is all intentional as outlined in this bug report.

https://github.com/KaTeX/KaTeX/issues/38

Re: A trick to eliminate 2π (sometimes)

#102

Post by eigenket »

adrian_b wrote: post #100
eigenket wrote: post #70

Earlier quoted context omitted.

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…

> 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 seem that these two claims contradict each other.

I do agree that without some aditional work the power series argument I wrote above does seem to be wrong.

Re: A trick to eliminate 2π (sometimes)

#103

Post by bmacho »

garbagecoder wrote: post #10
orangecat wrote: post #2

Interesting. It's probably not worth defining a new constant for exp(2π), but this is a further demonstration of the Tau Manifesto's argument that 2π is much more of a fundamental value than π.

Fundamental sounds like a value judgment. Pi is transcendental. 2 isn't. That's really the distinction. Unless there were other finite factors in pi, that is the number that's always going to have to be approximated in computation.

Fundamental sounds meaningless to me.

Talking about statements and words with meaning I am sure, that Tau is the more useful circle constant out of these two: less symbols, conceptually clearer equations.

Also turn is generally a more useful measurement unit for angle than radian or degree, but radian has its own merits, and not disposable, unlike pi.

Re: A trick to eliminate 2π (sometimes)

#104

Post by SAI_Peregrinus »

mgunyho wrote: post #88
marcosdumay wrote: post #79

Earlier quoted context omitted.

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.

Even Newton's Principia has symbols, they're just different. His notation used dots and lines & boxes in various positions.[1]

[1] https://en.wikipedia.org/wiki/Notation_for_differentiation#N...

Re: A trick to eliminate 2π (sometimes)

#105

Post by eigenket »

6gvONxR4sf7o wrote: post #90
eigenket wrote: post #70

Earlier quoted context omitted.

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.

I would argue that dimensionful quantities are never used as arguments to power series, however you are correct that this does not imply that angles are dimensionless, since (like we do with other quantities, we can divide by whatever unit we like to get something dimensionless). I withdraw that argument.

A better argument that angles are dimensionless is that dimensionless quantities are formed by the ratio of two quantities with the same dimension, and the angle subtended by an arc in a circle is given by the ratio of the arc length and the radius.

Re: A trick to eliminate 2π (sometimes)

#106

Post by mgunyho »

codazoda wrote: post #101
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…

Yes, I'm pretty sure it's that aria label. The first line of this code. 2 π My first inclination was to simply run `say` on my Mac comamdn line and paste the line in. That read it correctly. The other commenter called out the `aria-hidden` attribute and I'd guess that's it. It's explicitly hidden. Apparently this is all intentional as outlined in this bug report. https://github.com/KaTeX/KaTeX/issues/38

But note how right above this span there is the , which is not aria-hidden, and which contains the MathML representation of the equations. I would have assumed that a screen reader would look at that, since it contains the semantical information.

Re: A trick to eliminate 2π (sometimes)

#107

Post by psychoslave »

marcosdumay wrote: post #74
psychoslave wrote: post #50

Earlier quoted context omitted.

It depends on what public you care about. If readability did general public is a concern, just get rid of all these symbols and go with plain prose text, possibly using images as preferred illustrations over any ideographic way to encode ideas.

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 appropriately with symbolic notations — be it astrological esoteric formula or mathematical abstruse formula.

An other one is how you chose to express ideas. Using symbols only is always possible, whatever the topic. Using prose only is always possible. Using multiple representations is also always possible, including audio record, alphabetical text, ideograms, pictures, video.

Note that I didn’t blame the author for any fault here. I just pointed out that, if one take as a goal to be the most accessible as possible to general public, using academic symbols is not the way to go. It doesn’t mean that people with some matching academic curricula might not prefer to have a document full of esoteric symbols, be it for real actual communication advantages or mere bigotry and vulgar elitism.

We all know here, I guess, that anything written with this kind of symbols can be just as well and without any ambiguity transcribed into a programming language which use exclusively mundane words or turned into a series of two signs that no one can grasp instantly.

That conversation make me think about comments in https://news.ycombinator.com/item?id=36433212

Re: A trick to eliminate 2π (sometimes)

#108

Post by NotYourLawyer »

scythe wrote: post #40
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.

Yeah, you’re right. And radians are what make the trig identities involving derivatives work out nicely.

Re: A trick to eliminate 2π (sometimes)

#109

Post by jacobolus »

gbear605 wrote: post #78
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.

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.

> mathematicians did it for thousands of years (just read Newton’s Principia)

What mathematicians did for thousands of years arose in a culture of oral proofs supplemented by prepared diagrams: Euclid's proofs were meant to be recited aloud in front of an audience while pointing at an image labeled only with single letters/numerals – not read in a book. The society was substantially illiterate, there was no access to paper or good pens, algebra had not yet been invented, and all arithmetic was done mentally or using fingers or physical tokens.

Compared to mathematical notation, natural language expressions are often incredibly large and cumbersome, can make following the argument extremely difficult, and make many kinds of symbolic manipulations all but impossible.

Providing a visual way to interpret and manipulate mathematical expressions was a revolution in mathematics without which most modern mathematics would never have appeared. Eliminating that is comparable to writing computer programs via punched cards because "that's how they used to do it".

Re: A trick to eliminate 2π (sometimes)

#110

Post by orangecat »

6gvONxR4sf7o wrote: post #68
eigenket wrote: post #53

Earlier quoted context omitted.

Units are not the same as dimensions, something can have a dimension of 1 (which is what we usually mean by "dimensionless") and still have different units, just as something can have a dimension of length but still be measured in meters or feet. As far as you three examples go, which is "correct" depends on what you are trying to calculate - if you want this to approximate the power series for sin close to 0 you sho…

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`…

you can compute e(-t) via power series too, adding seconds to seconds squared and seconds cubed

The argument of exp does have to be dimensionless, exactly because adding seconds to seconds squared doesn't work. If t has units of time, there has to be another factor with units of inverse time, for example continuous compound interest is exp(rate*time).

Post reply on HN