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 derivative and then divide by
. I call this derivative operator with the bar on the upper
the reduced derivative.Now, why is this interesting? To start off, we’ll note that the unique function
...
The unfortunate truth is a ton of math content on the web reads like this. It has crippled me as a blind user who would like to appreciate math for over a decade. In university I was forced to pursue a degree other than CS because the math program used software which produced output like this and refused to change.
There has been technical progress, and many sites are starting to work better--Wikimedia most fantastically, but this old bugbear made me want to speak up and beg people to try and review their math content with a screen reader before publishing (I think MathJax has some built-in accessibility now?).
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 content entirely, even though it's the most basic <body><div id="content"><p> ... structure possible. I have basically zero web dev experience so I don't know how to fix this, maybe I need to tweak the KaTeX settings.
I think it's quite sad that math is so difficult on the web. While setting up the blog, I looked around and it seemed like FF is the only browser with proper MathML support, but I think that was also being phased out because it's apparently buggy and hard to maintain. IMO, the screen reader version should just basically be the LaTeX source, which is probably kind of awful when read out loud, but at least it would be unambiguous.
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?
What screen reader do you use? As a sighted person with no screen reader experience I tried Apple's VoiceOver on some math heavy Wikipedia pages and found that it completely mangled the formulas there, e.g. not distinguishing between numerator and denominator of fractions, not giving any indication of the difference between a coefficient versus an exponent, pronouncing invisible formatting commands, and so on.
Are there any websites with extensive, complicated mathematical formulas which are accessible to screen reader users? What's the current state of the art?
In general, would you prefer a typeset formula navigable in the usual way by a screen reader, or an explicit English language fallback explicitly pronouncing the formula the way a lecturer would read it to a class?
I ask because from what I can tell there are few if any screen reader users among authors of Wikipedia technical articles. I at least have quite a poor understanding of how to make those articles accessible.
Do you mind if I email you to ask questions about screen readers and mathematical formulas?
> review their math content with a screen reader before publishing
I second this for anything you put on the web. I opened the webapp I work on with a screen reader, and it was an incredibly valuable experience. You get to see your site from a different perspective, and various issues stand out like a sore thumb, and I honestly found fixing the accessibility issues extremely satisfying.
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 aloud verbatim is not reliable. "X plus B squared" can mean (x + b)^2 or x + b^2
Wow, I'd be interested in knowing how to fix this. I don't currently write math on the web but if I had to, I would be tempted to do it exactly like this: bare HTML with MathML, no JavaScript. And would do it thinking about blind people relying on accessible content to read it.
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?
Do you have any additional examples of webpages that fail to be readable for you (math-related or otherwise)? I work on an accessibility tool for web developers, and would like to see if we are detecting those bad cases correctly.
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, commutation with derivatives), and it turns out to be possible by using the Lebesgue measure divided by sqrt(2π) in all the integrals. Thus it may seem that defining đx=dx/sqrt(2π) can be a better choice.
> 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 Fourier transforms if you keep the 2pi in the exponential (or hide it inside Θ). Does this not apply also to the elementary formulas you mention?
Anyway, I love the choice of theta because a while ago I came up with a nice notation for sin and cos and this fits it really well. When I first learned trig, it was by way of skipping into physics early. I only understood cos as the magic button for getting x components from angles, and y as the button for y components. So my notation is based on this very literal brute understanding. All the symbols are circles with lines on the appropriate sides.
sin = -O- (should be overbar)
cos = O|
-sin = _O_
-cos = |O
Why did I make symbols for the negative versions of the same functions? Is minus sign too good for me? No. I did it because you can differentiate by just rotating the symbols clockwise and integrate by rotating counter clockwise. d/dx O| = _O_.
The way you defined theta, and the graphical depiction of theta, fits nicely.
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.
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 frequencies ugly and unintuitive to work with), I think this notational trick would've been very useful!
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.
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.
I often wonder if someday when we meet alien intelligences, they'll have a completely different set of constants, derivable from our own but different. Θ=535.491... may be such an example.
I've occasionally played with the notations that maybe ə = e^i or even 1 = e^{2πi} to simplify these sorts of expressions before. So for example a forward moving wave can be written,
1^{x/λ – νt}
with no particular ambiguity or even parentheses. The choice of constants should give you some pause though—we don't have a great way to talk about "true wavenumber" k so we have to talk about wavelength, and we use "f" for a lot of other things while Greek nu looks like an English V so that can sometimes be confusing... it's not _bad_ but it's weird enough that it's not obviously better.
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?
> I’m not entirely sure about the intuitive meaning of “taking the derivative and dividing by 2π”. Is there some sort of fundamental connection to periodic functions?
If you have a function f(x) where x is measured in radians, and there are 2pi radians per turn, then you can change variables.
Let t represent turns. One turn is 2*pi rad, and you want t = 1 when you've gone all the way around in x, so t = x/2pi.
By the chain rule,
df(x)/dx = df(t)/dt dt/dx = 1/2pi * df(t)/dt
So I think this might be the meaning you're looking for when you do the rescaling of the derivative.
You're using turns as units instead of radians. cos(x=2pi)=cos(t=1)=1, and so on.
> I like this, because it kind of eliminates the need for radians: the x in usin(x) has the unit of “turns”. I think this is conceptually much simpler.
I wonder if this would help for elliptic integrals. They are notoriously hard to solve, and I keep hitting them in my hobbyist calculations around magnetic fields. This is maybe the best video I've found to make them approachable:
I've thought of this as well, and sorta agree. But I think the real source of the 2pi-s is a bit more subtle? Which is basically that anywhere 2pi shows up is a quantity that is supposed to have different 'units' than the rest of the equation it's in, but for whatever reason we have erased all the units so we keep finding quantities multiplied together with a conversion factor of 2pi. Now one way to handle that is to write Tau or something in its place... but another is to, somehow, erase all the 2pis entirely but keep track of the 'units' on everything.
In fact it is very hard to find places in math where 2pi shows up 'on its own', added to other quantities that are not also in angular units of some sort. That is, most of the pis show up in calculations that involved pi, or circles, in some way (often quite sneakily). Of course it shows up on its own in the circumference/area of a circle, but you can express the area in terms of the circumference, so really it's just about the circumference that has a special value. And I wonder about whether it's possible to just... pick a different value for the circumference, an arbitrary symbol with no value, and then expressing every other use of pi in terms of that one without ever being forced to pick its value.
(Of course when you tie a string around a circle and measure it and it comes out to 2 pi r, yeah, you're forced to pick the correct value. Oh well.)
"[...], 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...
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 imaginary. I suppose that depends on the type of math you're doing though.
On a similar note, when doing quantum physics, I like to introduce h dot, which is i × h bar. There are tons of formulas where you either get i × h bar or -i / h bar, but these are just h dot and 1 / h dot, so this removes a little sign confusion and saves a little handwriting.
People will argue that real constants are more natural, but maybe they're not. Maybe radians are naturally imaginary, so if h bar is meant to have dimensions of energy time per radian, then it's better to use the imaginary h dot.
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.
I was once lucky enough to take a physics class taught by the head of the department, and I remember one of his policies on tests or homework was that if you got an answer that was off by 1/2 or 2*pi or anything like that, he'd nonetheless issue full credit because "you got all the physics right."
To address the problem they discuss at the end with defining Θ = e^2πi, they could instead define Θ(x) = e^2πix, the circular analog to the exponential function exp (which is really more fundamental than exp(1) = e anyways).
Note there's an existing notation which I've mostly seen in lower-class settings like high-school textbooks: r <angle-sign> theta, for the complex number r e^(i theta). Optionally leave out the r.
So you have that "most beautiful formula in all of mathematics":
I once read a book that proposed a "new" trigonometry that iirc worked with the hypotenuse squared and maybe the sin^2 of an angle as it's base quantities, and the author showed how easy it was to do stuff. This feels about the same. Not wrong, but not really useful once you've learned the usual way to do it, not easier to learn, and you'll be forever confused if it's all you learn.
Trigonometry is really about circles and rotations. The triangles are just an artifact of static diagramming. Zeroing in on triangle properties suggests a lack of fundamental understanding.
I’m surprised by the number of positive replies. Obviously the current notation is made up like all notations, but this is just a waste of time, 2п naturally arises in so many places. The h-bar for the Plank constant is just a product of not agreeing what constant to denote. sin(x)=x+o(x) is just too nice to give up. Switching units needs a way greater benefit than this.
This is amateurish math at best. Defining a new derivative operator that is mathematically equivalent to the normal derivative operator and then reformulating physics equations as some way to prevent human error is beyond absurd.
dbar is pretty common in lattice field theory notes (though I've never seen it in a book), where it is used because fourier integrals naturally come with a 1/2π.
The argument about why not to include the i in 2 pi i is incorrect. The problem is he says (e^x)^i2pi = e^i2pix does not work because ln(e^i2pi)=0. But he needs to use the complex logarithm. And for the complex logarithm ln(e^z) =z for z in C.
If that wasn't the case calculation rules of logarithms and exponentials would depend on if arguments are complex or real, a lot of physics would become much more complicated suddenly.
That's not his argument; he says defining Θ = e^2πi is not useful because e^2πi = 1, so Θ and Θ^x would also be 1. That's why he defined Θ = e^2π instead, so that Θ^x (or possibly Θ^ix) is a useful operation.
This notation maybe makes some things in trigonometry or Fourier analysis easier to do. Then a wild polynomial appears and all of the sudden we have to write
Arguing that a literal mathematical equivalence (e^ix = e^2piix) that you then use to “redefine“ trig functions so you can reformulate physics to mitigate making errors dividing my a constant is completely absurd.
In situations when there are strings involving 2pi where this makes any kind of sense typically a new constant is introduced to incorporate it.
ctoth|2 years ago
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 derivative and then divide by
. I call this derivative operator with the bar on the upper
the reduced derivative.Now, why is this interesting? To start off, we’ll note that the unique function
... The unfortunate truth is a ton of math content on the web reads like this. It has crippled me as a blind user who would like to appreciate math for over a decade. In university I was forced to pursue a degree other than CS because the math program used software which produced output like this and refused to change.
There has been technical progress, and many sites are starting to work better--Wikimedia most fantastically, but this old bugbear made me want to speak up and beg people to try and review their math content with a screen reader before publishing (I think MathJax has some built-in accessibility now?).
mgunyho|2 years ago
I think it's quite sad that math is so difficult on the web. While setting up the blog, I looked around and it seemed like FF is the only browser with proper MathML support, but I think that was also being phased out because it's apparently buggy and hard to maintain. IMO, the screen reader version should just basically be the LaTeX source, which is probably kind of awful when read out loud, but at least it would be unambiguous.
RogerL|2 years ago
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?)
jacobolus|2 years ago
Are there any websites with extensive, complicated mathematical formulas which are accessible to screen reader users? What's the current state of the art?
In general, would you prefer a typeset formula navigable in the usual way by a screen reader, or an explicit English language fallback explicitly pronouncing the formula the way a lecturer would read it to a class?
I ask because from what I can tell there are few if any screen reader users among authors of Wikipedia technical articles. I at least have quite a poor understanding of how to make those articles accessible.
Do you mind if I email you to ask questions about screen readers and mathematical formulas?
sebzim4500|2 years ago
curtis3389|2 years ago
I second this for anything you put on the web. I opened the webapp I work on with a screen reader, and it was an incredibly valuable experience. You get to see your site from a different perspective, and various issues stand out like a sore thumb, and I honestly found fixing the accessibility issues extremely satisfying.
jovial_cavalier|2 years ago
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 aloud verbatim is not reliable. "X plus B squared" can mean (x + b)^2 or x + b^2
noman-land|2 years ago
jraph|2 years ago
unknown|2 years ago
[deleted]
xingped|2 years ago
hoten|2 years ago
enriquto|2 years ago
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, commutation with derivatives), and it turns out to be possible by using the Lebesgue measure divided by sqrt(2π) in all the integrals. Thus it may seem that defining đx=dx/sqrt(2π) can be a better choice.
mgunyho|2 years ago
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 Fourier transforms if you keep the 2pi in the exponential (or hide it inside Θ). Does this not apply also to the elementary formulas you mention?
scythe|2 years ago
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
BlueTemplar|2 years ago
IIAOPSW|2 years ago
Anyway, I love the choice of theta because a while ago I came up with a nice notation for sin and cos and this fits it really well. When I first learned trig, it was by way of skipping into physics early. I only understood cos as the magic button for getting x components from angles, and y as the button for y components. So my notation is based on this very literal brute understanding. All the symbols are circles with lines on the appropriate sides.
sin = -O- (should be overbar)
cos = O|
-sin = _O_
-cos = |O
Why did I make symbols for the negative versions of the same functions? Is minus sign too good for me? No. I did it because you can differentiate by just rotating the symbols clockwise and integrate by rotating counter clockwise. d/dx O| = _O_.
The way you defined theta, and the graphical depiction of theta, fits nicely.
smaddox|2 years ago
Or put the modifier on the denominator so that the product and chain rules are obvious (the modifier only persists on the dx, not on the dy)
JBorrow|2 years ago
NotYourLawyer|2 years ago
kccqzy|2 years ago
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.
movpasd|2 years ago
Speaking as someone who had to write down many 2π's in university (especially as I find angular quantities like angular frequencies ugly and unintuitive to work with), I think this notational trick would've been very useful!
gorkish|2 years ago
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.
scythe|2 years ago
mabbo|2 years ago
I often wonder if someday when we meet alien intelligences, they'll have a completely different set of constants, derivable from our own but different. Θ=535.491... may be such an example.
crdrost|2 years ago
1^{x/λ – νt}
with no particular ambiguity or even parentheses. The choice of constants should give you some pause though—we don't have a great way to talk about "true wavenumber" k so we have to talk about wavelength, and we use "f" for a lot of other things while Greek nu looks like an English V so that can sometimes be confusing... it's not _bad_ but it's weird enough that it's not obviously better.
munchler|2 years ago
H8crilA|2 years ago
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
floatrock|2 years ago
basically the point of the Tau Manifesto: https://tauday.com/tau-manifesto
And would ya look at the date today... 6/28...
tgv|2 years ago
hughes|2 years ago
JdeBP|2 years ago
bobbylarrybobby|2 years ago
justincredible|2 years ago
[deleted]
alecst|2 years ago
> I’m not entirely sure about the intuitive meaning of “taking the derivative and dividing by 2π”. Is there some sort of fundamental connection to periodic functions?
If you have a function f(x) where x is measured in radians, and there are 2pi radians per turn, then you can change variables.
Let t represent turns. One turn is 2*pi rad, and you want t = 1 when you've gone all the way around in x, so t = x/2pi.
By the chain rule,
df(x)/dx = df(t)/dt dt/dx = 1/2pi * df(t)/dt
So I think this might be the meaning you're looking for when you do the rescaling of the derivative.
You're using turns as units instead of radians. cos(x=2pi)=cos(t=1)=1, and so on.
killthebuddha|2 years ago
> I like this, because it kind of eliminates the need for radians: the x in usin(x) has the unit of “turns”. I think this is conceptually much simpler.
zackmorris|2 years ago
https://www.youtube.com/watch?v=SJtbeg_PZ30
If anyone has any abstractions that might help, maybe around the nome the author mentioned, I'd love to hear them!
https://en.wikipedia.org/wiki/Nome_(mathematics)
ajkjk|2 years ago
In fact it is very hard to find places in math where 2pi shows up 'on its own', added to other quantities that are not also in angular units of some sort. That is, most of the pis show up in calculations that involved pi, or circles, in some way (often quite sneakily). Of course it shows up on its own in the circumference/area of a circle, but you can express the area in terms of the circumference, so really it's just about the circumference that has a special value. And I wonder about whether it's possible to just... pick a different value for the circumference, an arbitrary symbol with no value, and then expressing every other use of pi in terms of that one without ever being forced to pick its value.
(Of course when you tie a string around a circle and measure it and it comes out to 2 pi r, yeah, you're forced to pick the correct value. Oh well.)
sfpotter|2 years ago
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...
broses|2 years ago
On a similar note, when doing quantum physics, I like to introduce h dot, which is i × h bar. There are tons of formulas where you either get i × h bar or -i / h bar, but these are just h dot and 1 / h dot, so this removes a little sign confusion and saves a little handwriting.
People will argue that real constants are more natural, but maybe they're not. Maybe radians are naturally imaginary, so if h bar is meant to have dimensions of energy time per radian, then it's better to use the imaginary h dot.
dan_fornika|2 years ago
linuxdude314|2 years ago
Introduce all the constants you need.
Don’t redefine functions and operators for syntactic sugar.
orangecat|2 years ago
garbagecoder|2 years ago
raldi|2 years ago
DavidSJ|2 years ago
abecedarius|2 years ago
So you have that "most beautiful formula in all of mathematics":
<angle-sign> tau = 1
version_five|2 years ago
kevin_thibedeau|2 years ago
failuser|2 years ago
linuxdude314|2 years ago
This is amateurish math at best. Defining a new derivative operator that is mathematically equivalent to the normal derivative operator and then reformulating physics equations as some way to prevent human error is beyond absurd.
sweezyjeezy|2 years ago
scythe|2 years ago
>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!
linuxdude314|2 years ago
evanb|2 years ago
cycomanic|2 years ago
If that wasn't the case calculation rules of logarithms and exponentials would depend on if arguments are complex or real, a lot of physics would become much more complicated suddenly.
twiss|2 years ago
unknown|2 years ago
[deleted]
gunnihinn|2 years ago
(d bar) x = 1 / 2 pi.
hgsgm|2 years ago
linuxdude314|2 years ago
unknown|2 years ago
[deleted]
icapybara|2 years ago
linuxdude314|2 years ago
Arguing that a literal mathematical equivalence (e^ix = e^2piix) that you then use to “redefine“ trig functions so you can reformulate physics to mitigate making errors dividing my a constant is completely absurd.
In situations when there are strings involving 2pi where this makes any kind of sense typically a new constant is introduced to incorporate it.
phonebucket|2 years ago
[0] https://en.wikipedia.org/wiki/Indiana_Pi_Bill
thumbuddy|2 years ago
hinkley|2 years ago