A trick to eliminate 2π (sometimes)(marci.gunyho.com) |
A trick to eliminate 2π (sometimes)(marci.gunyho.com) |
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.
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.
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
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.
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 guess I find that programming is somewhat more word-oriented than pure math. For instance, how do you think about complex exponentiation, or a rotation matrix? Do you bring to mind the sensation of spinning around? For myself, I bring to mind the image of the entire complex plane rotating and stretching along a spiral, but I'm led to believe that those who are blind from birth aren't really capable of doing that.
Harder examples might include fourier series, convolution, gradient descent, etc.
I think you could almost consider the visualization of these things like a crutch. I wonder if not being able to visualize them might remove preconceived notions about how they behave, and give you different insights.
Why not use Darians? http://proper-pi-manifesto.com
The synathroesmic writing style and inapt analogies obfuscates the claims along with distracting and detering others from refuting them, but does not make them true.
Angles are defined in the abstract realm of mathematics not physics, which use them to describe physical phenomenon. In Cartesian space, the basis dimensions are defined using fixed perpendicular oriented lines, called coordinate lines on coordinate axes. Angles are not a basis dimension and as defined in Cartesian space are dimensionless. Angles are a dependent measure. Arc length is measured based on the units of the system. x radians or x degrees cannot be physically measured in and of themselves. In coordinate systems using an angle as a basis like spherical coordinates, angles exist as a dimension and the unit for them is free to chosen. Once the units are chosen, then the angles can be physically measured.
Pi has a definition for a good reason. Sometimes a discipline has to put up with perceived oddities until the real, deep problem is surfaced, grappled with and kicked in the nuts until it gives up and allows a paper or two to emerge without ridicule. With luck it might really show some ankle and a Nobel heaves into view 8)
This isn't it for (n)Pi n Phy Sci.
These two issues get solved by aria-description where the description exists entirely in an attribute on the element rather than as a reference to another node in the DOM tree. But it's still in the draft for ARIA 1.3 so it's not fully adopted yet.
about the attribute https://developer.mozilla.org/en-US/docs/Web/Accessibility/A...
CanIUse info (tl;dr only in chromesque browsers) in the property info https://developer.mozilla.org/en-US/docs/Web/API/Element/ari...
I think this really is just a redefinition of sin and cosine (and consequentially the exponential). My feeling is that we now have to deal with different derivative operators for periodic and non-periodic functions and a lot of other weird disconnects, e.g. we don't have a relation between a cycle and the radius (diameter, circumference ...) anymore. I'm not convinced that the (minor IMO) conveniences that gives us is worth it.
We can compute the principle value of ln(e^2πi)=ln(1) + i(2π + 2πk) (where k is an integer) so therefore
(e^2πi)^x = e^(xln(e^2πi)) = e^(x2π(k+1)i)
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 accessibility linters are nonfree and thus will never be used by site devs who aren't worth suing.
It's worth remembering that accessibility helps everyone, not just the disabled. In fact one of the more popular arguments against accessibility is that it allows non-disabled persons to do more than intended.
So while there's a good argument to be made for being charitable to those with a frankly really lousy condition, if you just want to be self-centered you still benefit from properly tagged data.
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.
(Indeed, we should entirely scrap the cross product in undergraduate level technical instruction and replace it with the wedge product; one happy effect will be replacing students' misleading spatial intuitions with better ones.)
Now, except perhaps for school exercises, anything complicated is done with a computer and this advantage is much less important.
The increased accuracy and simpler formulas in other places when measuring angles in cycles a.k.a. turns vastly outweigh the advantage of radians for differentiation.
In real applications you almost always compute the derivative of sin(a*x), not of sin(x), so you have to carry a multiplicative constant through derivations anyway and the single advantage of the radian vanishes.
The main reason is that the reduction of the argument to the range where a polynomial approximation is valid becomes much simpler.
Also, the primary inputs or the final outputs of any really complete computation are never in radians, because in physical devices it is not possible to realize radians with high accuracy, but only the angles that are in a rational relationship with the cycle. This is true both for geometric angles and for the phase angles of oscillations and waves.
>The only disadvantage of this approach is that symbolic differentiation and integration are more complicated, by multiplication with a constant.
There is more to it than that. The exponential function is not really e^x, it is lim_{N->∞} (1 + x/N)^N. The advantage of this definition is that, since it is valid everywhere on the complex plane, it allows the rigorous definition of real powers without recourse to inverse functions. We still teach students to prove things.
But also, isn't this a little contradictory? The goal of the original blog post was to simplify various differential equations. If you aren't simplifying symbolic differentiation, then what are you simplifying?
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
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.
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".
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
Please bear in mind that mathematical notation is a language, and that mathematical formulas are perfectly valid plain prose in that language.
I imagine that some screen readers will fail gracelessly when faced with Chinese script or Hindu as well. Especially if they're not unicode compliant.
But once you hit that threshold you are simply in a battle of dueling accessibility concerns. Not everyone is sighted, but neither does everyone rely on English as a primary language.
Nor should they as the English language was not designed to convey all concepts accurately. It excels mostly in conveying concepts germane to anglophone cultures. And three guesses what concepts are not popularly relevant to your standard anglophone? That's right.. calculus and theoretical physics.
That's not to bash on English as a language. It really is a flexible beast with a far reaching vocabulary. But there literally exists no language that is ideal for encapsulating every single idea under the sun. One must have a way to support many of the most diverse ones at the same time. And modern mathematical notation belongs on that short list along with English.
Equations and mathematical notation aren't simple illustrations. They're in many ways more important than the stuff around them.
The appropriate canonical representation of a rotation is a unit-magnitude complex number z = exp iθ = cos θ + i sin θ, which has a planar orientation (whatever plane i is taken to represent; if you want to represent a 3D rotation you can replace i with an arbitrary unit bivector) but is unitless.
Such a rotation z can be thought of as the ratio of two vectors of the same magnitude: z = u / v satisfies zv = u, i.e. is the object by which you can multiply v on the left to obtain u. Whatever original units your vectors u and v had gets divided away.
This is similar to the way the "ten" in "scale by ten" is unitless, but if you take the logarithm you get "scale by 10 decibels" or "go up by 3 octaves and 3.9 semitones", which have the base of the logarithm as a kind of unit.
But you seem to be drawing a distinction between meters and angles in your analogy where I assert none exists. The base of a number system only affects representations.
This is not true for divisions of lengths. 1 meter divided by 2 meters is 0.5 as a number. But it is only 0.5 radians under (1ish) specific arrangements of those lengths in a particular metric space
This is analogous to the way a scalar logarithm can have a base of "octaves" (doublings), "decibels", or "powers of the golden ratio" (as found in the Zometool construction toy). Or pick your favorite other logarithmic system.
Both are "units" in a certain sense, but neither one is quite the same kind of "unit" as light years or foot–pounds or amperes.
> 1 meter divided by 2 meters is 0.5 as a number. But it is only 0.5 radians under (1ish) specific arrangements of those lengths in a particular metric space
Just as 1 meter straight ahead divided by 2 meters straight ahead is the unitless scalar number 0.5, we can likewise treat angles (i.e. rotations) as ratios: 1 meter straight ahead divided by 1 meter to the right has the unitless bivector-valued ratio i, oriented like the ground you are standing on. You can multiply this bivector by some other coplanar vector to rotate it a quarter turn. For example, you can multiply it by the vector «3 inches due North» to get the new vector «3 inches due West»; notice how the units do not change because our bivector i is unitless.
Essentially, I think that whatever angles are, they are not like other dimensionful physical quantities. I have two arguments.
The first: Someone mentioned symmetries in a reply. I wanted to mention them too but didn't have time to structure my thoughts into a coherent argument. But the gist of it is that dimensionality is just a kind of scale invariance, and the scale invariance of angles is fundamentally different from that of linear quantities due to their periodicity — to apply a unit transformation, you have to scale the quantity _and the period_.
The second: Consider units from a "type theory" perspective instead. If you are considering exclusively linear trigonometry (no arcs), it's trivial to assign a dimensional type structure to expressions (e.g. cos takes angle type and maps it to dimensionless type). But as soon as you allow arc lengths, it becomes cumbersome to type common expressions.
I think these distinctions form the crux of the disagreement. Ultimately, it depends on your intuitive notion of what "dimensionality" actually means, and how it ought generalise to other kinds of quantities.
Here is an example to highlight my point. Let there be a circle C of centre O and radius r. Let A be a point on the circle. Let there be a point M outside the circle such that (AM) is tangent to C. Let B be the intersection of C and [OM]. Let s be the arc length along C from A to B. Then we want to write AM = r tan(s/r).
How does one get s/r to resolve to an angular dimension? Ought we instead ascribe s dimensions of length-angle? Imagine, then, that the circle is in fact a pulley, and we wish to measure a change x in length of rope as the pulley rotates through the angle of the arc from A to B. We would want to write x = s. But this is now dimensionally inconsistent.
It's certainly possible to make all these expressions correctly typed by introducting appropriate conversion constants. But this seems to me to be cumbersome. Since in physics, arc and linear lengths can convert freely into one another, it seems more economical to just let angles be dimensionless.
Edit: in other words, you've encoded tons of information in the problem statement about the relationship between r and s and you aren't properly encoding that in your type system allowing s/r to output an angle
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 that the arguments of such nonlinear functions are either explicitly or implicitly not the physical quantities, but their numeric values, i.e. the ratios between those quantities and their units, which are dimensionless.
In the case of the nonlinear sinus function, what is usually written as sin(x) is just one member of a family of functions where the arguments are angles implicitly divided by units of plane angle:
sin(x) is the sinus function with the angle implicitly divided by 1 radian
sin(x * Pi/2) is the sinus function with the angle implicitly divided by 1 right angle
sin(x * Pi*2) is the sinus function with the angle implicitly divided by 1 cycle a.k.a. turn
sin(x * Pi/180) is the sinus function with the angle implicitly divided by 1 sexagesimal degree
It is very sad that the logical thinking about angles of most people has been perverted by what they have been taught in school, which is just a bunch of nonsense copied again and again from one textbook to another.
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.
360 - (360^3)/6 = -7M degrees
2*pi - (2 * pi)^3 / 6 = -35 radians = -2k degrees
1 - (1^3)/6 = 0.8 turns = 300 degrees
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 should use radians. Otherwise you use something else.
sin(x)
= x/(1 radian)! - x³/(3 radians)! + …
= x/(1 radian) - x³/(1 radian × 2 radians × 3 radians) + …
There is a natural reason for pi occurring in physics that makes little sense to ignore.
Treating it as something to be dealt with misses the forest for the trees.
Radians are the naturally occurring Euclidean unit of angular measurement.
In that sense its more a DSL than a generalist language. And anything you can express in a DSL, you can also express it with a more generalist language. Most likely the DSL will provide a far more compact way to communicate what it allows to express, but that compactness doesn’t come for free: it takes time to compress and decompress and you have to also communicate the DSL specification in some preexisting medium.
On my side, I suppose "discuss" to mean we can talk on the topic face to face with spontaneous expressions means like oral expression and listening comprehension, or sign language. I don’t mean that there is no other mean to discuss, but when we can’t transpose the topic in such a medium, we are probably beyond the realm of discussion. For example when we make love, there is more happening than what can either hope to realize through mere talk.
Once again, I’m not again DSLs and so on. I just mean that there are not the proper tools for maximizing accessibility.
I have said that the so called "natural" exponential, normally written as e^x or exp(x) of either real or complex argument and its inverse, the hyperbolic a.k.a. natural logarithm, and any other functions derived from it are neither needed nor useful when computations are done by computers, as opposed to computations done with pen and paper.
In all traditional formulae where the "natural" exponential function or functions derived from it occur, all occurrences can be replaced using a pair of functions of real argument, the function 2^x with real value and the function 1^x with complex value, either directly or with functions derived from this pair, e.g. the binary logarithm.
In computer programs this substitution results in both higher accuracy and higher speed and it has as a side effect that the units radian and neper are never needed.
It should be noted that even in the 19th century, when the "natural" exponential and logarithm and the trigonometric functions with argument in radians were useful for symbolic computations done by hand, they were never used for practical numeric computations.
All practical numeric computations were done using the function 10^x and the trigonometric functions with argument in degrees and their inverses, by using mathematical tables where the values of these functions were tabulated (or equivalently, by using slide rules).
The use of the "natural" exponential and logarithm and of the trigonometric functions with argument in radians for practical computations has become widespread only after the development of the electronic computers, after programming languages like Fortran have included them as standard functions.
I consider that this has been a mistake, similar to the use of decimal numbers in some computers. Both the use of decimal numbers and the use of the "natural" exponential and logarithm and of the trigonometric functions with argument in radians are sub-optimal in all their possible applications.
[1] https://en.wikipedia.org/wiki/Notation_for_differentiation#N...
Edit: another example difference. I can't measure an octave or dB. I can measure a degree
Edit2: we've reached reply limit but I concede you can measure a decibel. Point about dB degrees still stands though
(For example, you can measure an octave by marking out a particular fret on your guitar; it will make an octave change whichever particular note you start with.)
You can feel free to "reject" whatever you want. You'll just be wrong/confused. ;-) (But you'll be in good company. Most working engineers, scientists, and mathematicians don't have or need a particularly clear philosophical understanding of angles.)
A) multiply two degrees together
B) multiply two dBm values together
The output "units" in A change but do not in B. dB and angles are very different.
Edit: the units in B do change, but the dB part doesn't. Tired.
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.
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` implies seconds are dimensionless any more than sin's series with `angle + angle^3` implies that angles are dimensionless.
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).
> 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 functions by power series suggests strongly that this is meaningful.
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.
That argument holds just as true for dimensional quantities frequently computed by power series though, which means it can't be valid.
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.
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.
This very wrong definition forced upon many students is the root of all misconceptions about plane angles.
The reason why this definition is wrong is because some words are missing from it and after they are added it becomes obvious that its meaning is different from what many teachers claim.
First there is no relationship whatsoever between the magnitude of a radius and the magnitude of an angle. What that definition intended to say was:
"the angle subtended by an arc in a circle is the ratio of the arc length and of the length of an arc whose length is equal to the radius".
By definition, a radian is defined as the angle subtended by an arc whose length is equal to the radius.
To explain how plane angles are really defined would take more space, but the only thing that matters is that the characteristic property of plane angles is that the ratio between two plane angles subtended by two arcs of a circle is equal to the ratio of the lengths of the two arcs.
Introducing this characteristic property of the angles in the so-called definition from above reduces it into the sentence "the angle is measured in radians". This is either a trivially true sentence when the angle is indeed measured in radians, or it is a trivially false sentence when the angle is measured e.g. in degrees. It certainly is not a definition.
If that had been the definition of plane angle, that would have meant that the plane angle was discovered only in the second half of the 19th century, together with the radian, while in reality plane angles have been used and measured with various units for millennia.
To measure plane angles, it is necessary to first choose an arbitrary angle as the unit angle, for instance an angle of one degree.
Then you can measure any other angle by measuring both the length of the corresponding arc and the length of the arc corresponding to the chosen unit angle. Then the two lengths are divided, giving the numeric value of the measure of the angle.
But you can compose rotations.
Multiplying decibels together is also not meaningful, but you can compose scaling.
Curious what you think of the d_theta * d_phi term in a spherical coordinates integral...
While rotation is naturally oriented like a bivector (plane), solid angle is naturally oriented like a trivector (3-space).
The natural representation is as a kind of (unitless) ratio formed from 3 vectors, not the product of two vector–vector ratios.