A trick to eliminate 2π (sometimes) | Hacker News Reader

A trick to eliminate 2π (sometimes)

marci.gunyho.com

195 pointsmgunyho172 comments

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?).

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

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.

Using postfix operations in the Way of Forth would be unambiguous and of course otherwise superior as well as is well known [citation needed]. “x b plus squared” vs “x b squared plus”. Well, at least as long as it’s agreed on whether “x b” means two variables or one with a two-letter name. But the latter don’t really exist in math. You just expand to new alphabets when you run out of letters.

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.

This is quite true, especially true when talking about direction (gradient) in high dimensional space. I don't think this can be avoided, since after all we are creatures living in 3D space where left right up down are quite well-defined, just need to make a mental note every time you have to deal with more than 3 dimensions.

Seems like an odd choice when talking about the cross product, since the cross product is only a thing in 3D. You can define analogous things in other dimensions but it becomes clearer and clearer that it’s not meaningfully a ‘product’.

So it doesn’t matter if your visual intuition for a cross product breaks down in higher dimensions - a cross product is only a thing in three.

This is very off topic, but the wedge product absolutely is "meaningfully a product", generalizes fine to arbitrary dimension, and has a perfectly reasonable visual/spatial/geometric interpretation.

(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.)

Or bivectors, or k-vectors, or blades…

Good point; you're right. I might be misremembering the cross product. I do remember that he didn't even teach the geometric interpretation of vector addition.

I can see where he's coming from. Geometric intuition is an useful tool but in this semester's linear 2 class I developed more because I stopped using it. It's too strong of a tool and blots out "dryer" intuition and methods, and also as you progress you find more and more places where it's not useful. What's the geometrical intuition for whether two circles intersect in Q^2? who knows?

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.

>I'm a bit confused to your original point about visualization

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.

> 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.

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

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

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?)

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

I think most people are fine with the idea that some of the costs (monetary and otherwise) arising from disabilities should be transferred from the people suffering from them to society at large. But I don't think it's reasonable for producers and users of broken screen readers to expect everyone (including authors of private blogs posts) bending over backwards to accommodate them, nor is it a remotely efficient use of societal resources. It also creates completely perverse incentives.

If there is a bug in the screen reader then that's on the vendor.

However the overwhelming number of cases where a website is unusable with a screen reader are due to lack of proper semantic tagging, like alt text and so on. The last thing a screenreader user wants to hear on a site is "button," "button," "button..." There's certainly room for tooling to help though. I definitely think it sucks that many 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.

Did you even bother looking at the website source before writing this?

What part of if there’s a bug in the screen reader it’s on the vendor did you fail to understand?

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.

You can't discuss math while getting rid of the math symbols. That's not a reasonable proposal.

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.

It’s not entirely impossible; mathematicians did it for thousands of years (just read Newton’s Principia). It does use very specific language to do so, and some of the language might not exist for some advanced mathematical concepts, but I think that this whole article could be written that way.

Yeah, let's talk about the difference in accessibility between the text on this site and Newton's Principia...

I think parent is referring here to the fact that math used to be written without symbols (other than numbers) up to the 1300s according to Wikipedia: https://en.wikipedia.org/wiki/History_of_mathematical_notati... (very interesting article!) However, I would say that there is a reason why notation tends towards terse symbols: it's much more efficient and unambiguous.

Even Newton's Principia has symbols, they're just different. His notation used dots and lines & boxes in various positions.[1]

[1] https://en.wikipedia.org/wiki/Notation_for_differentiation#N...

Maybe ctoth or some other user of screen reader might have a look at https://en.wikisource.org/wiki/The_Mathematical_Principles_o... and tell us how it goes.

> mathematicians did it for thousands of years (just read Newton’s Principia)

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".

> You can't discuss math while getting rid of the math symbols. That's not a reasonable proposal.

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

Why is "plain prose" so logically the litmus test for what can be discussed? It certainly wasn't designed around the idea it should cover every possible concept, or do so remotely efficiently either, so the repeated assertion it logically is the only way to know if a concept can be discussed is a bit hard to follow. It does cover most concepts though, and conveniently.

> 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.

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.

>Please bear in mind that mathematical notation is a language, and that mathematical formulas are perfectly valid plain prose in that language.

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 don't think it's really viable to just stop posting equations altogether. Yeah it's more accessible, but it would also completely cripple any sort of discussion between mathematicians, physicists, etc.

Equations and mathematical notation aren't simple illustrations. They're in many ways more important than the stuff around them.

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.

How does this interact with Find? Select text?

solid point: it kind of wrecks them. I just checked in a codepen. Copying the text around the hidden paragraph will also copy the hidden paragraph. Maybe that's a feature? Searching for the content will also show that it's on the page but that you cannot meaningfully get to it, which kind of stinks.

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...

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.

I bet the "reading" modes of the browsers are just using the HTML that's visible on the page, but that's marked explicitly as hidden from screenreaders, using the aria-hidden attribute.

From what I can tell it's the opposite (on Firefox): the math is in the source HTML twice, once in <span class="katex-mathml">, which is hidden using CSS in the non-reader mode but is rendered in the reader mode, and <span class="katex-html" aria-hidden="true">, which is what FF displays normally, and it's missing from the source when I inspect it in reader mode. Having two spans like this comes directly from KaTeX, whose authors I'm sure have thought about accessibility. I imagined that MathML would be somewhat standard and screen readers would understand it.

Huh, you're right about what the Firefox reader mode does. I'm somewhat surprised it uses the MathML. I'd just assumed what was happening from the 2pi in the header becoming plain text but that header is probably a special case and just from the page title rather than the h1.

Yes, I manually put 2π in the html <title>, while the <h1> has the KaTeX rendered math in it :)

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

But note how right above this span there is the <span class="katex-mathml">, which is not aria-hidden, and which contains the MathML representation of the equations. I would have assumed that a screen reader would look at that, since it contains the semantical information.

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.

This does seem like a bug/missing feature in Chrome's a11y tree, which you can see in the DevTools with an experimental feature enabled: https://i.imgur.com/ckWnrlL.png

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.

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?

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.

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?

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.

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.

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) ?

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

> 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...

> 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?

> 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!

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.

> 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)

It says that in TFA :)

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.

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!

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.

Yeah, 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.

The 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.

I think I fundamentally disagree with you. Angles do not have a "base" any more than meters do (being embedded inside some metric space could be considered a base I suppose).

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

The logarithmic base for an angle is something like "degrees", "radians" or "turns".

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.

I understand your analogy, but I reject it's validity. Degrees/radians/turns map to meters/feet/angstrom. Decibels and octaves are true "number" multipliers. You could for instance talk about degrees in octaves or in dB if you like. It's just not particularly useful for the domain

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

You absolutely can measure an octave or decibel. It's just a relative quantity, in just the same way any quantification of orientation is relative.

(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.)

I think this will devolve at this point. But you should consider that you are perhaps far too confident in your position here. My last attempt is due you to consider what happens when you

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.

Multiplying degrees together is not a meaningful concept.

But you can compose rotations.

Multiplying decibels together is also not meaningful, but you can compose scaling.

> Multiplying degrees together is not a meaningful concept.

Curious what you think of the d_theta * d_phi term in a spherical coordinates integral...

You can compute solid angle (a.k.a. spherical excess, normalized spherical surface area) by taking a surface integral, but the units are not "square degrees" or "square radians", but instead an entirely new type, usually just measured in radians ("steradians"). Some people have defined https://en.wikipedia.org/wiki/Square_degree but that is a stupid unit.

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.

Jeez, I've really kicked off quite the heated discussion in the replies... I don't want to get bogged down in lengthy arguments, but I feel I should explain my reasoning in more detail.

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.

The solution to your quandary is to realize the division operator is massively overloaded in your expression. What you actually want is to specify s and r as true line segments and define a new operator which takes two segments and outputs the angle between them. This operator happens to reduce to division of magnitudes in certain circumstances.

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

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)?

Angles are absolutely more dimensionless than lengths are. For an easy check you can't add quantities where the dimension differs, which means it doesn't make sense to add a length to its cube. On the other hand it does make sense to add an angle to its cube - this is a necessary component of computing sin(angle) by the power series sin(angle) = angle - (angle^3)/6 + ...

How can we compute angle - (angle^3)/6?

    360 - (360^3)/6 = -7M degrees

or is it this?

    2*pi - (2 * pi)^3 / 6 = -35 radians = -2k degrees

Or maybe this?

    1 - (1^3)/6 = 0.8 turns = 300 degrees

They're wildly inconsistent because I'm not taking the units into account and we have to take the units into account.

Units are not the same as dimensions, something can have a dimension of 1 (which is what we usually mean by "dimensionless") and still have different units, just as something can have a dimension of length but still be measured in meters or feet.

As far as you three examples go, which is "correct" depends on what you are trying to calculate - if you want this to approximate the power series for sin close to 0 you should use radians. Otherwise you use something else.

Sure, but that's orthogonal to the "angles don't really have units" assertion and the "it does make sense to add an angle to its cube" assertion, which are the ones I'm responding to.

As another example for the second assertion, you can compute e(-t) via power series too, adding seconds to seconds squared and seconds cubed, etc, which comes up all the time. But that doesn't mean `dimensionless + seconds + seconds^2` implies seconds are dimensionless any more than sin's series with `angle + angle^3` implies that angles are dimensionless.

If you say "angles have units" I agree with you - obviously you can measure them in degrees or radians or whatever you want. I was responding to the claim

> Angles aren't dimensionless any more than lengths are dimensionless

They are dimensionless, but they still have units. The concepts are orthogonal.

As for the question about adding an angle to its cube, I would say the enormous usefulness of computing trig functions by power series suggests strongly that this is meaningful.

> I would say the enormous usefulness of computing trig functions by power series suggests strongly that this is meaningful.

That argument holds just as true for dimensional quantities frequently computed by power series though, which means it can't be valid.

I would argue that dimensionful quantities are never used as arguments to power series, however you are correct that this does not imply that angles are dimensionless, since (like we do with other quantities, we can divide by whatever unit we like to get something dimensionless). I withdraw that argument.

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.

> They are dimensionless, but they still have units. The concepts are orthogonal.

The concepts are not orthogonal, they are incompatible.

A dimensionless quantity (which angles are not) is by definition the ratio of two quantities that are measured with the same unit.

When you compute the ratio by division, the two identical units disappear from the result, therefore the result is indeed dimensionless.

There is no way to choose a unit for a dimensionless quantity in the usual sense.

At most you could define a new different dimensionless quantity, as the ratio of two dimensionless quantities, i.e. as a ratio of ratios, but because it needs a different definition this should better be viewed as a different quantity, not as the same quantity with a different unit.

Ok, you've replied to quite a lot of my comments, with some fairly confusingly worded responses. Some of your claims appear to be incompatible as far as I read them.

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.

No, even if you have reproduced a definition of the plane angle that is encountered in many textbooks "the angle subtended by an arc in a circle is the ratio of the arc length and radius", this definition is very incorrect.

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.

you can compute e(-t) via power series too, adding seconds to seconds squared and seconds cubed

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).

Actually the dimensions are by definition the same as the fundamental units, i.e. the units that are chosen freely, independently of all other units.

A dimensional formula of a quantity just writes its unit as a function of the fundamental units.

In any equality of physical quantities, in the two sides not only the dimensionless numeric values must be equal, but also the units must be equal, which is usually expressed by saying that the dimensions must be the same, and it is verified by writing in both sides the dimensional formulae, i.e. the units of both sides as functions of the fundamental units.

A dimensionless quantity is a ratio of two quantities that are measured by the same unit, so that the units simplify during the division.

There may be different but related dimensionless quantities, which are differentiated by different definitions of those quantities, but a dimensionless quantity cannot have different units.

This is just meaningless mumbo-jumbo that has been sadly introduced in the documents of the International System of Units, in 1995, after a shameful vote of the delegates, who have voted automatically, without thinking or discussing, a vote equivalent with establishing by vote that 2 + 2 = 5.

I’m not sure it will work everywhere, but for sin(x), one can write

  sin(x)
  = x/(1 radian)! - x³/(3 radians)! + …
  = x/(1 radian) - x³/(1 radian × 2 radians × 3 radians) + …

That makes the ‘radians’ units cancel out.

Your argument is wrong.

Any physical quantity, for instance length, can appear as an argument of a nonlinear function that can be developed in a Taylor series. So your example would be identical for any other quantity not only for angle. I can make an analog computing element where a voltage is equal to the sinus of another voltage, so after your theory, voltage is dimensionless.

The reason why this is possible is 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.

> sin(x) is the sinus function with the angle implicitly divided by 1 radian

This to me sounds like the most natural explanation. For example, in a sibling comment someone mentioned that "you can calculate e^(-t)", but I disagree: in physics it's always e^(-t / T), where T is some time constant, so that the argument of the exponential is dimensionless. Same applies to sin(x): usually we write something like sin(2pi f t), where the units of f and t cancel out, and the 2pi is there to cancel out the invisible implicit 1 radian. sin(ft) would be wrong, at t = 1 / f you wouldn't have advanced by a full cycle.

If you mean units instead of dimensions I agree.

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.

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).

> is a horrendous mistake and many generations of physicists have been brainwashed by being taught this aberration

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.

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.

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 . (-:

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.

That's a really good answer.

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

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.

I too fail to see a problem. There is a certain degree of trickiness in all walks of life. Sometimes there is a decent hack that simplifies things and sometimes there is a notion that is just as complex to deploy as the current state of the art and hence isn't worth pursuing.

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.

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.

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

The simpler derivation formula was important when such symbolic computation was done by hand.

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.

If you’re using a computer for symbolic algebra or whatever, none of this matters anyway. The whole post is about simpler notation for the sake of making things less error-prone when working by hand.

For a computer it matters because it is possible to compute with higher accuracy and speed the trigonometric functions with the argument in cycles instead of radians (similarly for the binary exponential and logarithm vs. the "natural" exponential and logarithm).

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.

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.

Mind explaining how to express some simple functions? Im interested

All mathematical formulas can be inter-converted based on the identity:

e^(x + i*y) = 2^(x/ln2) * 1^(y/2Pi)

Most formulae from textbooks are written in such a way to be simpler with e^x and its inverse, but it is almost always possible to move the constants ln2 and 2Pi between various equations so that in the end they will disappear from most relations, with the exception of the derivation or integration formulae.

In 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

so you express x (the simple polynomial) as ln(2^(x/ln2)? or as an infinite series expansion?

You have misunderstood my point. I have not said anything about polynomials.

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.

Ah interesting! I thought you were saying we can express all the other analytical functions with 1,2^x.

The function 1^x is constant everywhere. You can write (-1)^x to get the kind of effect you are looking for, but doing this with 1^x is madness.

>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?

`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.

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.

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.

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.

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

τ = 0, got it.

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

The original post missed the i...

No, even with i one of the solutions is \tau = 0

e^pi + 1 = 0, of course

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

e^pi + 1 = 24.14069263...

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

> 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...

Coincidentally, happy Tau Day! (6.28)

Not a coincidence :)

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.

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.

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)

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.)

"[...], 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...

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.

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

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.

This is the way!

Introduce all the constants you need.

Don’t redefine functions and operators for syntactic sugar.

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.

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.

"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?

As the Tau manifesto points out, tau/4 = pi/2 = lambda is pretty fundamental too ! (The "orthogonality constant" ?)

Geometry 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

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.

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."

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).

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

> <angle-sign> tau = 1

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

And here it is in Unicode glory:

∠τ = 1

Thanks, I should've figured there'd be a character.

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.

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.

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.

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.

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

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!

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

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.

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.

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.

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).

just 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)

Hmm, 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.

See my other reply down below. That said I agree with the final conclusion that including the i is not a good idea, although I'm a bit on the fence on if the whole thing (I'm convinced by the tau=2pi arguments on the other hand).

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.

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.

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

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

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

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.

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

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

Clever!

dang, the article says 2𝞹, not 2n

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