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.
A trick to eliminate 2π (sometimes)
91–100 of 179 posts
Re: A trick to eliminate 2π (sometimes)
#92Earlier quoted context omitted.
I bet the "reading" modes of the browsers are just using the HTML that's visible on the page, but that's marked explicitly as hidden from screenreaders, using the aria-hidden attribute.
From what I can tell it's the opposite (on Firefox): the math is in the source HTML twice, once in , which is hidden using CSS in the non-reader mode but is rendered in the reader mode, and , which is what FF displays normally, and it's missing from the source when I inspect it in reader mode. Having two spans like this comes directly from KaTeX, whose authors I'm sure have thought about accessibility. I imagined tha…
Huh, you're right about what the Firefox reader mode does. I'm somewhat surprised it uses the MathML. I'd just assumed what was happening from the 2pi in the header becoming plain text but that header is probably a special case and just from the page title rather than the h1.
Re: A trick to eliminate 2π (sometimes)
#93Earlier quoted context omitted.
Angles are absolutely more dimensionless than lengths are. For an easy check you can't add quantities where the dimension differs, which means it doesn't make sense to add a length to its cube. On the other hand it does make sense to add an angle to its cube - this is a necessary component of computing sin(angle) by the power series sin(angle) = angle - (angle^3)/6 + ...
Your argument is wrong. Any physical quantity, for instance length, can appear as an argument of a nonlinear function that can be developed in a Taylor series. So your example would be identical for any other quantity not only for angle. I can make an analog computing element where a voltage is equal to the sinus of another voltage, so after your theory, voltage is dimensionless. The reason why this is possible is th…
> sin(x) is the sinus function with the angle implicitly divided by 1 radian
This to me sounds like the most natural explanation. For example, in a sibling comment someone mentioned that "you can calculate e^(-t)", but I disagree: in physics it's always e^(-t / T), where T is some time constant, so that the argument of the exponential is dimensionless. Same applies to sin(x): usually we write something like sin(2pi f t), where the units of f and t cancel out, and the 2pi is there to cancel out the invisible implicit 1 radian. sin(ft) would be wrong, at t = 1 / f you wouldn't have advanced by a full cycle.
Re: A trick to eliminate 2π (sometimes)
#94Also, 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
Coincidentally, happy Tau Day! (6.28)
Not a coincidence :)
Re: A trick to eliminate 2π (sometimes)
#95Earlier quoted context omitted.
From what I can tell it's the opposite (on Firefox): the math is in the source HTML twice, once in , which is hidden using CSS in the non-reader mode but is rendered in the reader mode, and , which is what FF displays normally, and it's missing from the source when I inspect it in reader mode. Having two spans like this comes directly from KaTeX, whose authors I'm sure have thought about accessibility. I imagined tha…
Huh, you're right about what the Firefox reader mode does. I'm somewhat surprised it uses the MathML. I'd just assumed what was happening from the 2pi in the header becoming plain text but that header is probably a special case and just from the page title rather than the h1.
Yes, I manually put 2π in the html , while the has the KaTeX rendered math in it :)
Re: A trick to eliminate 2π (sometimes)
#96Earlier quoted context omitted.
How can we compute angle - (angle^3)/6? 360 - (360^3)/6 = -7M degrees or is it this? 2*pi - (2 * pi)^3 / 6 = -35 radians = -2k degrees Or maybe this? 1 - (1^3)/6 = 0.8 turns = 300 degrees They're wildly inconsistent because I'm not taking the units into account and we have to take the units into account.
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…
Actually the dimensions are by definition the same as the fundamental units, i.e. the units that are chosen freely, independently of all other units.
A dimensional formula of a quantity just writes its unit as a function of the fundamental units.
In any equality of physical quantities, in the two sides not only the dimensionless numeric values must be equal, but also the units must be equal, which is usually expressed by saying that the dimensions must be the same, and it is verified by writing in both sides the dimensional formulae, i.e. the units of both sides as functions of the fundamental units.
A dimensionless quantity is a ratio of two quantities that are measured by the same unit, so that the units simplify during the division.
There may be different but related dimensionless quantities, which are differentiated by different definitions of those quantities, but a dimensionless quantity cannot have different units.
This is just meaningless mumbo-jumbo that has been sadly introduced in the documents of the International System of Units, in 1995, after a shameful vote of the delegates, who have voted automatically, without thinking or discussing, a vote equivalent with establishing by vote that 2 + 2 = 5.
Re: A trick to eliminate 2π (sometimes)
#97Earlier quoted context omitted.
I think most people are fine with the idea that some of the costs (monetary and otherwise) arising from disabilities should be transferred from the people suffering from them to society at large. But I don't think it's reasonable for producers and users of broken screen readers to expect everyone (including authors of private blogs posts) bending over backwards to accommodate them, nor is it a remotely efficient use…
If there is a bug in the screen reader then that's on the vendor. However the overwhelming number of cases where a website is unusable with a screen reader are due to lack of proper semantic tagging, like alt text and so on. The last thing a screenreader user wants to hear on a site is "button," "button," "button..." There's certainly room for tooling to help though. I definitely think it sucks that many accessibilit…
Did you even bother looking at the website source before writing this?
Re: A trick to eliminate 2π (sometimes)
#98I 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.
Re: A trick to eliminate 2π (sometimes)
#99I 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.
Re: A trick to eliminate 2π (sometimes)
#100Earlier quoted context omitted.
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`…
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 way to choose a unit for a dimensionless quantity in the usual sense.
At most you could define a new different dimensionless quantity, as the ratio of two dimensionless quantities, i.e. as a ratio of ratios, but because it needs a different definition this should better be viewed as a different quantity, not as the same quantity with a different unit.