131–140 of 179 posts
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? e^(π*i) = -1 e^(τ*i) = 1
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…
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? 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
.sr-only {
position:absolute;
left:-10000px;
top:auto;
width:1px;
height:1px;
overflow:hidden;
}
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.
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.
Earlier quoted context omitted.
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
e^(x + i*y) = 2^(x/ln2) * 1^(y/2Pi)
Most formulae from textbooks are written in such a way to be simpler with e^x and its inverse, but it is almost always possible to move the constants ln2 and 2Pi between various equations so that in the end they will disappear from most relations, with the exception of the derivation or integration formulae.
In most applications, more equations are simplified than those which become more complicated.
A very important advantage of 2^x and 1^x versus e^x is that for the former the reductions of the argument to the principal range where the function is approximated by a polynomial can be done with perfect accuracy and very quickly, unlike for the latter. Moreover, for the former it is easy to verify the accuracy of any approximation, because for any argument that is represented as a binary number the functions 2^x and 1^x can be computed with a finite number of sqrt invocations (based on the formulae for half angle) and sqrt can be computed with any number of desired digits. Computing e^x with an arbitrary precision is trickier, because it requires criteria for truncation of an infinite series.
It should be noted that 1^1.0 = 1, 1^0.5 = -1, 1^0.25 = i, 1^0.75 = -i
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…
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.
Earlier quoted context omitted.
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.
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…
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.
Earlier quoted context omitted.
Mind explaining how to express some simple functions? Im interested
All mathematical formulas can be inter-converted based on the identity: e^(x + i*y) = 2^(x/ln2) * 1^(y/2Pi) Most formulae from textbooks are written in such a way to be simpler with e^x and its inverse, but it is almost always possible to move the constants ln2 and 2Pi between various equations so that in the end they will disappear from most relations, with the exception of the derivation or integration formulae. In…
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…
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…
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.
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).
So you have that "most beautiful formula in all of mathematics":
tau = 1
