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?).
I just define a new derivative operator, like so: dxđ f(x)≡2π1 ⋅dxd f(x). That’s all.
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.
Chrome _just_ added support for MathML, so the lack of a11y support here is not surprising. I found this bug report which I believe covers this: https://bugs.chromium.org/p/chromium/issues/detail?id=103889...
In short - you didn't do anything incorrect, and AFAIK any temporary fix to improve a11y for Chrome users would involve some heavy lifting in the Katex library.
I suggest interested parties to star the above bug report.
___
Other commenters mentioned the aria-hidden property being set. This is intentional, as without it Safari/FF/compliant screen readers with MathML support would double-read the content. I'm honestly not sure what the ideal markup would be to support both types of browsers - should a solution exist, it may involve using JavaScript to change the markup based on the user agent detected.
<span class="katex-html" aria-hidden="true">
<span class="base">
<span class="strut" style="height:0.6444em;"></span>
<span class="mord">2</span>
<span class="mord mathnormal" style="margin-right:0.03588em;">π</span>
</span>
</span>
Apparently this is all intentional as outlined in this bug report.
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?)
For content that really cannot be written in a way that screen readers can handle, there is always the idea of Screen Reader Only content. It's a hassle, but let's jump in and give it a shot
For instance for the first math thing you can have
<style>
.sr-only {
position:absolute;
left:-10000px;
top:auto;
width:1px;
height:1px;
overflow:hidden;
}
</style>
<p class="sr-only" id="definition-of-reduced-derivative">
The reduced derivative with respect to x is denoted crossed-d over dx of the function f(x). It is equivalent to the the derivative with respect to x (denoted d over dx) of the function f(x) all over 2 pi.
</p>
This is not an authoritative answer; I'm just some asshole who writes front-end code a lot. More of a Cunningham's Law situation that anything really. You don't want to end up creating one experience for sighted users and completely different one for screen-reader and refreshable-braille-display users. But this can maybe get the wheels turning for how to address it? Also again maybe TOTALLY unnecessary once you un-hide the math markup.
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?
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.
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
For instance, for your example: "X plus B quantity squared" or "x + b squared"
You can also do things like change the pitch of speech as symbols are nested, play specific tones to represent symbols, pan things across the stereo field to represent groupings, and otherwise make the symbolic equation into a multimodal experience.
But of course, you can't do any of this if the semantic representation of the equation is lost and it is rendered as strictly graphics or whatever.
I'm a bit confused to your original point about visualization because how ever in the world could I program if I couldn't abstractly manipulate symbols? I suppose not visually, but there's something non-word-oriented happening in my head.
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.
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?
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!
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
Now, this is a number that I don't recall having seen before. The letter é seems strangely fitting for it
é = 12.2635111...
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.
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.
And justification in general for "why radians" vs degrees, gradians, turns, whenever
Source: used to tutor calculus and differential equations in college.
Generally speaking this is not a useful trick.
There are _plenty_ of amazing ways to leverage Euler’s identity but I fail to see how this is one of them.
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!
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.
I will discuss only the plane angle, because it is the most important, but the situation is the same for solid angle and logarithms.
The justification commonly given is that the plane angle is dimensionless because it is the ratio of two lengths, the length of the corresponding arc and the length of the radius. This justification is stupid, because that is not the definition of the plane angle, but it already includes the choice of a particular unit.
As formulated. this justification only states the trivial truth that the numeric value of any physical quantity is the ratio between that quantity and its unit. By the same wrong justification, length is dimensionless, because it is the ratio between the measured length and the length of a ruler that is one meter long.
Correct is to say that the plane angle is a physical quantity that has the property that the ratio between two plane angles is equal to the ratio between the lengths of the corresponding arcs.
This is a property of the same nature like the property of voltage that the ratio of two voltages across a linear resistor is equal to the ratio of the electric currents passing through the resistor. This kind of properties are frequently used in the measurement of physical quantities, because few of them are measured directly but in most cases ratios of the quantities of interest are converted in ratios of quantities that are easier to measure.
This property of the plane angle allows the measurement of plane angles, but only after an arbitrary unit is chosen for the plane angle. Because the choice of the unit is completely free, i.e. completely independent of the units chosen for the other physical quantities, the unit of plane angle is by definition a fundamental unit, not a derived unit.
The freedom of choice for the unit of plane angle is amply demonstrated by the large number of units that have been used or are still used for plane angle, e.g. right angle (the unit used by Euclid), sexagesimal degree, centesimal degree, cycle a.k.a. turn, radian.
The fundamental units of plane angle, solid angle and logarithms must never be omitted from the dimensional formulae of the quantities, otherwise serious mistakes are frequent & such mistakes have delayed the progress of physics with many years (e.g. due to confusions between angular momentum & action; the Planck constant is an angular momentum, not an action, as frequently but wrongly claimed). This is a problem especially for the unit of plane angle, which enters in the correct dimensional formulae of a great number of quantities, including some where this is not at all obvious (e.g. magnetic flux).
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.
We'll never hear the end of it. There'll be some Python program that takes four months to calculate 1.00000000000000000000000000000001 . (-:
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 symbolic differentiation and integration are more complicated, by multiplication with a constant.
In my opinion the simplifications that are introduced everywhere else are more important than this disadvantage.
The real and complex function e^x was preferable in the 19th century, when numeric computations were avoided as too difficult and simple problems were solved by symbolic computation done with pen and paper.
Now, when anything difficult is done with a computer, both the function e^x and associated units like the radian and the neper are obsolete.
When programming computations on a computer, using 2^x and 1^x results in simpler and more accurate computations.
Moreover, when doing real physical measurements it is possible to obtain highly accurate values when the units are cycle and octave, but not when they are radian and neper.
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.
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.
When using this alternative pair of constants (which are the ratios between two pairs of units, cycle vs. radian and octave vs. neper), there is no longer any need for pi or e.
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
basically the point of the Tau Manifesto: https://tauday.com/tau-manifesto
And would ya look at the date today... 6/28...
> 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.
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!
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.)
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...
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.
Introduce all the constants you need.
Don’t redefine functions and operators for syntactic sugar.
It's definitely a value judgment. But the value judgment is: Is the radius or the diameter more fundamental to a circle?
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.
So you have that "most beautiful formula in all of mathematics":
<angle-sign> tau = 1
Who knew that if you go around in a circle you get back where you started?
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.
>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!
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.
> where ln(e^2πi)=ln(1)=0,
that is incorrect because the logarithm of a complex number is multi-valued. He even cites the correct source on wikipedia, but his argument is incorrect (and because the exponent is 2πi he actually would get a meaningful results I believe).
(d bar) x = 1 / 2 pi.
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.