17 ms·
A trick to eliminate 2π (sometimes)
- orangecat 3y agoInteresting. 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.
- garbagecoder 3y agoFundamental 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.
- ohwellhere 3y ago"Tau is transcendental. 0.5 isn't." It's definitely a value judgment. But the value judgment is: Is the radius or the diameter more fundamental to a circle?
- BlueTemplar 3y agoAs the Tau manifesto points out, tau/4 = pi/2 = lambda is pretty fundamental too ! (The "orthogonality constant" ?)
- garbagecoder 3y agoGeometry isn't the only application of pi. Nor is the geometric interpretation the only way to determine what is "fundamental." Why not use Darians? http://proper-pi-manifesto.com http://proper-pi-manifesto.com
- bmacho 3y agoFundamental sounds meaningless to me. 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.
- mabbo 3y ago`Θ^i = 1` does make this really slick, imho. 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.
- munchler 3y agoI don't think Θ makes for a good fundamental constant, though, because it's composed of pi and e, which must persist as distinct concepts in the end.
- adrian_b 3y agoWhile pi and e must persist as two distinct concepts, they may be substituted by a constant pair that is much more useful for practical purposes is 2*pi and ln 2 ("natural" logarithm of two). 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.
- crdrost 3y agoI'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.
- thumbuddy 3y agoClever!
- NotYourLawyer 3y agoInteresting, 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.
- movpasd 3y agoThing 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!
- Misdicorl 3y agoThis is not true. Angles very much have units and it's why you can express the same concept with different numbers. Pi equals 180 degrees equals 0.5 turns. 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.
- jrockway 3y agoYeah, units that algebraically reduce to 1 are always very interesting to me. Consider a chart showing how many CPU seconds you're consuming per second. The unit is seconds/second, which is equal to 1, but it is still a distinct concept from radians.
- jacobolus 3y agoThe reason it is confusing is because an angle measure is a kind of logarithm of a rotation, and logarithms (sort of) have a unit: the base. 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.
- alecst 3y agoResponding to this part in the article: > 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 3y agoFWIW they mention this in the article: > 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.
- deleted 3y ago[deleted]
- phonebucket 3y agoEasy trick to eliminate difficulties arising from pi: fix them via legislation [0]. [0] https://en.wikipedia.org/wiki/Indiana_Pi_Bill https://en.wikipedia.org/wiki/Indiana_Pi_Bill
- ctoth 3y agoUnrelated 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?).
- jovial_cavalier 3y agoI'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
- kybernetikos 3y agoThe way I've heard those distinguished in spoken math is x + b^2 is said "x plus b squared" and (x + b)^2 is said "x plus b all squared. There's a similar approach for divide "x plus b over 8" vs "x plus b all over 8". That was often enough but if it wasn't you'd be reduced to pronouncing brackets.
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- H8crilA 3y agoAlso, 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
- JdeBP 3y agoLots of focus on the Greek letters, at the expense of an important Latin one, there. (-:
- bobbylarrybobby 3y agoe^pi + 1 = 0, of course
- tgv 3y agoτ = 0, got it.
- ajkjk 3y agoI'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.)
- enriquto 3y agoYou 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.
- BlueTemplar 3y agoA bit similar to how pi is only half of a turn, so you need to apply the transformation twice (so square root) to get back where you were(ish) ?
- scythe 3y agoIf you're really slick about it, you even "fix" the Gaussian integral this way. 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
- enriquto 3y ago> exp(sqrt(2pi)) Now, this is a number that I don't recall having seen before. The letter é seems strangely fitting for it é = 12.2635111...
- mgunyho 3y ago> 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?
- IIAOPSW 3y agoYou should call it d/dx bar. 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.
- sfpotter 3y ago"[...], 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...
- scythe 3y agoI was with him until this point: >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 3y agoIt’s insane… I’m still shocked this isn’t trolling.
- hinkley 3y agodang, the article says 2𝞹, not 2n
- broses 3y agoPeople 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.
- dan_fornika 3y agoWould it be more appropriate to call 2πi "tau dot"?
- broses 3y agoThe problem with Tau is that it's already used for a lot of other things. I like to use π with a line through it for 2π ("pi cross"). But that's less likely to catch on since Tau for 2π is already well known.
- linuxdude314 3y agoThis is the way! Introduce all the constants you need. Don’t redefine functions and operators for syntactic sugar.
- DavidSJ 3y agoTo 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).
- abecedarius 3y agoNote 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": <angle-sign> tau = 1
- DavidSJ 3y ago> <angle-sign> tau = 1 Who knew that if you go around in a circle you get back where you started?
- dTal 3y agoAnd here it is in Unicode glory: ∠τ = 1
- abecedarius 3y agoThanks, I should've figured there'd be a character.
- cycomanic 3y agoThe 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.
- twiss 3y agoThat'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.
- cycomanic 3y agohe writes: > 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).
- cycomanic 3y agojust to expand on this the issue is not that e^2πi = 1 but that (e^z)^x =/= e^(zx) when z is complex, because in general z^w . instead it would be e^(x ln(e^z)), where the logarithm of a complex number is a multivalued function. 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)
- mgunyho 3y agoHmm, this indeed seems to be the case! I think I was confused by ln(1), since 1 is a real number, but the multi-valued complex logarithm of 1 is indeed k 2pi (and now I see the appropriate notation should be "log" instead of "ln"). Perhaps the whole thing could indeed be reduced to "1^x" with an asterisk that 1^x means complex exponentiation. I'll have to update this section in the post.
- zackmorris 3y agoI 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: https://www.youtube.com/watch?v=SJtbeg_PZ30 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) https://en.wikipedia.org/wiki/Nome_(mathematics)
- deleted 3y ago[deleted]
- linuxdude314 3y agoIs this really not an elaborate troll? 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.
- failuser 3y agoI’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.
- linuxdude314 3y agoSame. 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.
- version_five 3y agoI 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.
- kevin_thibedeau 3y agoTrigonometry 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.
- icapybara 3y agoMuch ado about nothing. Hiding 2pi behind an abstraction layer just makes things harder to debug later.
- gunnihinn 3y agoThis 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 (d bar) x = 1 / 2 pi.
- evanb 3y agodbar 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.
- raldi 3y agoI 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."
- sweezyjeezy 3y agoIt's a common idea in analysis / number theory in mathematics - see for example https://en.m.wikipedia.org/wiki/Exponential_sum https://en.m.wikipedia.org/wiki/Exponential_sum
- hgsgm 3y agoWhy do physicists care so much about dimensions but then pretend that their dimensionless quantities don't have units?
- linuxdude314 3y agoA lot of people in the comments here don’t seem to understand the difference between the two.