A trick to eliminate 2π (sometimes)

https://marci.gunyho.com/theta.html

by mgunyho • 3 years ago

195 172 3 years ago

marci.gunyho.com

orangecat2• 3 years ago

Interesting. 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.

garbagecoder2• 3 years ago

Fundamental 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• 3 years 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?

bmacho• 3 years ago

Fundamental 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• 3 years 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• 3 years ago

I 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_b11• 3 years ago

While 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• 3 years ago

I'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• 3 years ago

Clever!

NotYourLawyer3• 3 years ago

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.

movpasd2• 3 years ago

Thing 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!

Misdicorl6• 3 years ago

This 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.

6gvONxR4sf7o4• 3 years ago

Angles aren't dimensionless any more than lengths are dimensionless (feet per second makes just as much sense as rpm). It's just that angles have symmetries that lengths don't, which is where 2 pi comes in. Do you want units where your symmetries are expressed in multiples of 1, 2, or 2 pi (for turns, half-turns, and radians, respectively)?

adrian_b11• 3 years ago

The claim that plane angle, solid angle and logarithms are dimensionless quantities is a horrendous mistake and many generations of physicists have been brainwashed by being taught this aberration without ever stopping to think whether this claim can be proved.

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).

gorkish• 3 years ago

This is more or less "the point" for those of us who argue for tau instead of pi.

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.

JdeBP2• 3 years ago

For goodness' sake, don't tell Matt Parker that one can write computer programs where 2π is rescaled to 1!

We'll never hear the end of it. There'll be some Python program that takes four months to calculate 1.00000000000000000000000000000001 . (-:

kccqzy• 3 years ago

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 (called usin by the article) having a derivative of a turn-based cosine multiplied by 2pi.

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.

paulddraper2• 3 years ago

That's a really good answer.

And justification in general for "why radians" vs degrees, gradians, turns, whenever

linuxdude3147• 3 years ago

Due to the fact differential operators are linear and the nature of the accumulation of constants of integration it’s pretty easy to prove that the differential operator proposed is in fact equivalent to the standard derivative operator.

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.

scythe4• 3 years ago

You can always define angles in turns. But the problem is that it conflicts with the definition cos(x) = Re{e^(ix)}. Trig is not so easily separated from the rest of mathematics.

NotYourLawyer3• 3 years ago

Yeah, you’re right. And radians are what make the trig identities involving derivatives work out nicely.

adrian_b11• 3 years ago

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 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.

alecst• 3 years ago

Responding 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• 3 years ago

FWIW 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.

phonebucket• 3 years ago

Easy trick to eliminate difficulties arising from pi: fix them via legislation [0].

[0] https://en.wikipedia.org/wiki/Indiana_Pi_Bill

ctoth2• 3 years ago

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 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_cavalier2• 3 years ago

I'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• 3 years ago

The 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.

nilstycho2• 3 years ago

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 thought that was a surprising perspective from a math graduate student, of all people.

ctoth2• 3 years ago

In terms of reading the notation out loud, if there is verbal ambiguity, remove it?

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.

curtis3389• 3 years ago

review their math content with a screen reader before publishing

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.

noman-land• 3 years ago

Thank you for sharing your experience. Web accessibility has a long way to go and reminders like these help.

sebzim4500• 3 years ago

The equations are in mathml, I feel like the issue is with the screen reader.

RogerL• 3 years ago

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?)

bjornasm• 3 years ago

One option would be to do testing with screen-readers and then find out if the content works, and if not find out why.

psychoslave4• 3 years ago

It depends on what public you care about. If readability did general public is a concern, just get rid of all these symbols and go with plain prose text, possibly using images as preferred illustrations over any ideographic way to encode ideas.

1-more2• 3 years ago

When I view source (not the DOM) I can see that all of your math has aria-hidden="true". That seems to carry over into the parsed/generated math markup: run `document.body.innerHTML += '<style>[aria-hidden="true"]{ outline: 3px solid red; }</style>'` to see all the hidden-from-screenreader things highlighted. That may be enough.

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>

Then you make sure that the element that wraps up your first equation has aria-describedby="definition-of-reduced-derivative" so that the SR reads out that content. I think you may need to not have "aria-hidden" on that math wrapper, but I'm not sure.

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.

mgunyho9• 3 years ago

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 content entirely, even though it's the most basic <body><div id="content"><p> ... structure possible. I have basically zero web dev experience so I don't know how to fix this, maybe I need to tweak the KaTeX settings.

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.

zerocrates2• 3 years ago

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.

codazoda• 3 years ago

Yes, I'm pretty sure it's that aria label. The first line of this code.

    <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>

My first inclination was to simply run `say` on my Mac comamdn line and paste the line in. That read it correctly. The other commenter called out the `aria-hidden` attribute and I'd guess that's it. It's explicitly hidden.

Apparently this is all intentional as outlined in this bug report.

https://github.com/KaTeX/KaTeX/issues/38

zamadatix2• 3 years ago

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.

hoten3• 3 years ago

You used the Katex library for rendering math symbols, which currently emits code that supports MathML and falls back to an HTML rendering (this is my understanding after reading the HTML for a bit). The MathML content is marked up correctly such that browser should be able to construct the a11y tree correctly. I didn't check, but apparently other browsers like FF and Safari construct the a11y tree from the MathML markup correctly.

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.

jacobolus8• 3 years ago

What screen reader do you use? As a sighted person with no screen reader experience I tried Apple's VoiceOver on some math heavy Wikipedia pages and found that it completely mangled the formulas there, e.g. not distinguishing between numerator and denominator of fractions, not giving any indication of the difference between a coefficient versus an exponent, pronouncing invisible formatting commands, and so on.

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?

jraph• 3 years ago

Wow, I'd be interested in knowing how to fix this. I don't currently write math on the web but if I had to, I would be tempted to do it exactly like this: bare HTML with MathML, no JavaScript. And would do it thinking about blind people relying on accessible content to read it.

xingped• 3 years ago

Hi! I'm trying to work on understanding how to make websites more accessible right now and probably the first thing people generally think of are blind users. However, I'm having a hell of a time figuring out how to use screen readers. Is there any screen reader or documentation/tutorial you might recommend for sighted users to learn how to use screen readers?

hoten3• 3 years ago

Do you have any additional examples of webpages that fail to be readable for you (math-related or otherwise)? I work on an accessibility tool for web developers, and would like to see if we are detecting those bad cases correctly.

H8crilA3• 3 years ago

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

tgv2• 3 years ago

τ = 0, got it.

H8crilA3• 3 years ago

Well, I can say π := 3π, and -1 still works. The point is exp() is that 2π*i periodic.

bobbylarrybobby• 3 years ago

e^pi + 1 = 0, of course

bauble• 3 years ago

e^i*tau = 1 is pretty elegant, too.

dTal2• 3 years ago

e^pi + 1 = 24.14069263...

JdeBP2• 3 years ago

Lots of focus on the Greek letters, at the expense of an important Latin one, there. (-:

floatrock• 3 years ago

t simplifies almost all naturally occurring expressions

basically the point of the Tau Manifesto: https://tauday.com/tau-manifesto

And would ya look at the date today... 6/28...

hughes• 3 years ago

Coincidentally, happy Tau Day! (6.28)

mgunyho9• 3 years ago

Not a coincidence :)

ajkjk• 3 years ago

I'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.)

enriquto3• 3 years 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.

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.

BlueTemplar2• 3 years ago

A 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) ?

scythe4• 3 years ago

If 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

enriquto3• 3 years 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...

mgunyho9• 3 years 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?

enriquto3• 3 years ago

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!

IIAOPSW• 3 years ago

You 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.

smaddox• 3 years ago

You should call it d/dx bar.

Or put the modifier on the denominator so that the product and chain rules are obvious (the modifier only persists on the dx, not on the dy)

JBorrow• 3 years ago

It says that in TFA :)

sfpotter• 3 years 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...

scythe4• 3 years ago

I 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!

linuxdude3147• 3 years ago

It’s insane… I’m still shocked this isn’t trolling.

hinkley• 3 years ago

dang, the article says 2𝞹, not 2n

broses2• 3 years ago

People 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• 3 years ago

Would it be more appropriate to call 2πi "tau dot"?

broses2• 3 years ago

The 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.

linuxdude3147• 3 years ago

This is the way!

Introduce all the constants you need.

Don’t redefine functions and operators for syntactic sugar.

DavidSJ2• 3 years ago

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).

abecedarius2• 3 years ago

Note 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

DavidSJ2• 3 years ago

<angle-sign> tau = 1

Who knew that if you go around in a circle you get back where you started?

dTal2• 3 years ago

And here it is in Unicode glory:

∠τ = 1

cycomanic4• 3 years ago

The 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• 3 years ago

That'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.

cycomanic4• 3 years ago

he 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).

zackmorris• 3 years ago

I 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

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)

linuxdude3147• 3 years ago

Is 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• 3 years ago

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.

linuxdude3147• 3 years ago

Same.

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• 3 years ago

I 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• 3 years ago

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.

icapybara• 3 years ago

Much ado about nothing. Hiding 2pi behind an abstraction layer just makes things harder to debug later.

gunnihinn• 3 years ago

This 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• 3 years ago

dbar 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• 3 years ago

I 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• 3 years ago

It's a common idea in analysis / number theory in mathematics - see for example https://en.m.wikipedia.org/wiki/Exponential_sum

hgsgm• 3 years ago

Why do physicists care so much about dimensions but then pretend that their dimensionless quantities don't have units?

linuxdude3147• 3 years ago

A lot of people in the comments here don’t seem to understand the difference between the two.

A trick to eliminate 2π (sometimes)

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