Khan Academy now supports tau | Hacker News Reader

Khan Academy now supports tau

khanacademy.org

104 pointsspicyj66 comments

Tau is clearer in some domains and pi is better in others. The mental overhead from switching between the two is too much, so just pick one already. And let the rest of us know, so we can go back to having a clear standard and not requiring the mental overhead of switching while studying different authors.

Oh yeah, we went through that 2000 years ago and the winner was pi. I haven't seen any compelling reason to switch yet. (Says the guy who uses "j" for the imaginary constant. :P)

In which domains is pi better? And was there a discussion 2000 years ago that led to pi "winning"? (If so, it'd be really fun to read its reasoning.) [Both of these are honest questions; I'm not a mathematician.]

http://www.thepimanifesto.com/

The Pi Manifesto has a few examples of pi beating tau (statistics, polygons, and complex numbers mainly), while also pointing out how silly and biased some of the tau examples are. However, the argument isn't very convincing from either side.

And as to your second question, I have no idea. I just know the idea has been settled for a long time now and I think this whole debate is needlessly distracting. Not that it isn't fun to watch or think about, but it confuses people who are just trying to learn and use math.

Their argument about the normal distribution is completely off though. There is completely no reason to group the two with the standard deviation. Their biggest argument seems to be that the area of unit circle is exactly pi. Sadly, there is absolutely no need to work with fractions of areas of the unit circle, while if you're working with angles the unit circle is a most natural standardisation. The normal distribution has been a lot clearer to me since I've understood the two should be grouped with the pi.

Their argument about trigonometric functions is completely wrong and obviously so. Trigonometric functions work with angles and it's already shown (and pimanifesto readily admits) that tau shines there.

Their argument about Euler's identity is as inane as the tauists' is.

They don't understand quadratic forms.

I'm guessing this site is sarcastic.

Try http://www.thepimanifesto.com/ for a πle of cases where pi works out more cleanly, and other good arguments for pi.

> a πle of cases

τché!

We can start with any domain where torque is a thing. Torque is a particularly bad thing to conflict with.

I've also seen arguments that it's better when dealing with triangles (all angles in a triangle add up to pi radians). I don't think I buy that one, since everyone uses degrees for that anyway.

The mental overhead from switching between the two is too much...

Really? It's just two constants that are both easy to remember and trivial to convert between.

I don't have a horse in this race. I don't mind having two distinct constants to deal with.

Next time you write some code, try this:

#define ONE 1 #define TWO 2 #define THREE 3 ... etc.

Then use the words in place of integers wherever you need them. Hell, mix and match. And tell me if you are just as productive as you would be sticking with the regular integers you know and love.

I think that wouldn't be a meaningful bottleneck on my productivity. But I could be wrong.

x = ONETHREEZEROEIGHT;

That seems like a bad comparison, writing out multiple digits in place of a single digit should ALWAYS be less efficient.

Agree....wouldn't having more language for these things make it MORE efficient? I'm down for a symbol for pi/2 (tau/4)!

Sure. And in return, you can't use constants, but must instead type out 3.141592653... to whatever decimal place you deem important. Nor typedefs. Nor named flags (`FileRead | FileWrite` is just long-hand for `3`). At the end of the day we'll see who's pulling their hair more.

Then we'll go back in a couple weeks, and look at the same code again. Mine's wordy. Yours is cryptic, and requires looking up the use of everything.

Tau is clearer in some domains and pi is better in others.

This argument is addressed explicitly in the section "What is really going on here?":

http://tauday.com/tau-manifesto#sec:what_is_really_going_on

> Tau is clearer in some domains and pi is better in others.

I always thought Tau was better because it was easier to understand when teaching it. For using it, it doesn't much matter.

But when you are a kid learning about it, having a clear picture of why specifically that number is very important, and Tau is much more obvious.

After reading both the tau and pi manifestos I get the impression that it's quite an arbitrary choice: either the circumference is fundamental and angles are natural (tau), or the radius is fundamental and areas are natural (pi). It boils down to a trade-off and you can cherry-pick examples where either is "more natural", but in a mathematical sense it doesn't matter a single bit and that's why we should not waste time on this, it doesn't bring anything new to the table.

I don't ever remember ever reading anything about how 2000 years ago we had a debate on if pi would be better than tau (if 3.14 is better than 6.28).

goes online to see if he can find answer to question before asking... Nope, sorry.

Re: 2000 years. pi as standard notation goes back to Euler, though of course Archimedes and others calculated it much earlier.

I haven't heard of tau being considered pre-2001, though it must have happened here and there (I had the same idea independently, for example).

At uni, decades ago, I was told that whenever a "mathematician" appeared up on the telly, it was to present their new idea of definining a new constant equal to 2pi...

Not the same now, of course, with shows like Numb3rs and Big Bang Theory, and a greater public awareness of the importance of numeracy.

The proponents of tau that cite the ubiquity of the 2pi factor in maths seem to forget that it's an exceedingly huge swath of knowledge and thus any notion of ubiquity might as well be moot, or totally dependent on the region of maths this person frequents. I wouldn't be surprised if there are some domains of maths where even natural numbers are used rarely or almost never (formal logic springs to mind). That's why it's possible to cherry-pick favourable examples which support either side of pi-tau debate.

But you might say, of course, that those fields of mathematics which tau makes "easier" are the ones important in early math education (especially below the university level). Hence adopting tau would make them substantially friendlier and more intuitive to many people. While the idea of "fixing" math concept to make them more bearable to laymen should not be dismissed automatically, I would like to point out that the question of tau vs. 2pi is by no means the only issue of this kind. Indeed, there are a couple of more "warts" in everyday maths that could also warrant "fixing". Consider:

* The direction in which positive and negative angles on two-dimensional, Cartesian plane are measured [1]. Counter-intuitively, the measure increases when going counterclockwise, while going clockwise decreases it.

* The main diagonal [2] of a matrix goes from upper-left to lower-right corner, which coincides with the shape of backslash character.

* The established order of indices for matrix' elements is row-column, so that A_xy refers to element in x-th row and y-th column of matrix A. This goes against the habit of specifying the horizontal coordinate before the vertical coordinate when talking about XY planes [3].

* Definition of convex [4] and concave [5] functions (for R->R ones) do not agree with the intuitive associations based on plots of those functions. Clearly, the convex one looks like a valley, and the concave one resembles a hill or mountain.

I'm sure there are many more examples of such unreasonable, counter-intuitive conventions, so we really have a lot of work ahead of us. So, anyone fancies writing the Slash Diagonal Manifesto?...

[1] http://en.wikipedia.org/wiki/Angle#Positive_and_negative_ang... [2] http://en.wikipedia.org/wiki/Main_diagonal [3] http://en.wikipedia.org/wiki/Cartesian_coordinate_system#Car... [4] http://en.wikipedia.org/wiki/Convex_function [5] http://en.wikipedia.org/wiki/Concave_function

* Counter-clockwise angles comes from making charts and graphs with the independent variable going from left to right, and the dependent variable from bottom to top. This is far too ingrained in the way we teach and learn about mathematics to change now, much more than the choice of pi or tau (though we do make y go from top to bottom in many computer graphics contexts). There’s no reasonable way that we could integrate direction of reading on a clock with direction of reading typical charts with direction of reading text into a uniform system, and I don’t consider changing all the clocks, texts, and charts in the world to be a reasonable possibility.

* What does the shape of the slash character have to do with anything? A slash is a fraction bar. Fractions and matrix diagonals are entirely unrelated, though in both cases numbers are read from top left to bottom right, in accordance with our typical reading direction in western texts.

* Matrix index ordering is a pain in the butt and will be confusing whichever way they’re labeled. The logic behind the current system is to use the first index for the component that results when multiplying by a vector, and using the second index within that component. Picking the opposite convention would also end up confusing. Figuring out the proper ordering when dealing with non-commutative “number” systems in general is a pain, and I don’t think there’s any easy answer. We have matrices multiply column vectors on the left, because that’s typically how we notate operators acting on some input. But it means that composition is multiplication from left to right, which is a bit confusing. There’s no way to make the order be always left-to-right or always right-to-left. But much more importantly, matrices are a kind of painful abstraction to use in general. Mathematics education would be much improved in many ways if we used Geometric Algebra instead of matrix representations a lot more of the time. http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf

* It’s much easier to just call these “concave up” and “concave down”. Problem solved.

This brought me back to Khan where I previous did every math exercise, and I was a bit disappointed to see that not too much has changed with the instruction they offer. I keep wanting math exercises beyond elementary calc, but they just gave the same stuff a face lift.

I'm guessing they're heads down getting the Khan platform that we've heard about, but does anyone have any updates?

Right now we're focusing more on middle school–level math -- we're adding literally hundreds of new exercises in the topics that we are working on but haven't yet had time to work on calculus and beyond.

Hey, nice work. My daughters do Khan Academy at home in the mornings for a bit rather than their elementary school math class. They love it.

I wanted to tell you guys a story though, about your UI transition.

I walked in a few months ago to my 8 year old daughter slumped at the desk next to her computer, nonresponsive. When I was like "Oh my God, are you okay?" She told me, with many tears, that she was no longer to obtain I believe they were called "Master" badges in some future subjects, like Geometry.

She had earned one of the Master badges at great effort, and when the UI changed, you guys retired them for more granular badges. Overall, I'd say it was the right decision to make; gaining subject mastery is done on a smaller slice of content now, and feels more achievable.

That said, she was totally devastated that she wouldn't be able to earn those future badges. Oh man, it was tough. She recovered nicely and loves the new interface, but I was thinking what an impact a UI change had, and thinking that a) probably many adults feel the same way with a change, but don't communicate it as well, and b) some sort of way to notify / slowly introduce / help transition kids who use the tool intimately as changes happen would be pretty awesome.

Thanks for all the work! I wish I'd taken some video for your UI guys, she was really bummed, the sort of response you can't get out of a focus group. :)

Right, we retired the four "challenge" badges (I believe they were arithmetic, pre-algebra, algebra, and trig) in favor of the new "topic" badges which replace what was there previously.

I hadn't heard of anyone particularly upset about those badges, but we had many dozens of people who complained about the disappearance of the streak bar. We try our hardest not to hurt users' feelings but sometimes it's unavoidable as we make changes (hopefully for the better!).

Email me your daughter's email address used on Khan and tell her to keep her eye out for a special badge.

ben@khanacademy.org

Aww, thanks!

Yea, it makes total sense to focus there - I'm just pissed that no one else taking on harder math in the Khan-style. MIT/Stanford style lectures are brutal and the practice is nothing like the great Khan system.

What exercises are being added now? I'm not seeing them if so. Good luck to you guys!

In the last month, we've added:

http://www.khanacademy.org/math/root/logical-reasoning/e/con...

http://www.khanacademy.org/math/arithmetic/factors-multiples...

http://www.khanacademy.org/math/geometry/basic-geometry/e/co...

http://www.khanacademy.org/math/geometry/basic-geometry/e/or...

http://www.khanacademy.org/math/geometry/basic-geometry/e/pe...

http://www.khanacademy.org/math/geometry/angles/e/exploring_...

http://www.khanacademy.org/math/geometry/angles/e/congruent_...

as well as made some a handful of improvements to the existing exercises.

(Oops, the first two basic-geometry exercises aren't available publicly yet.)

I still find it hard to see any practical, hard benefits of switching to tau. Perhaps you could argue that many things become "more natural" but at this point, pi is ingrained in mathematics, and it works perfectly well. I just can't imagine that switching to tau would allow us to reap any benefits that were previously inaccessible. I don't think there's any reason to fix what ain't broke.

I think the strongest argument is pedagogical. If tau makes the underlying concepts clearer (and I believe it does), then it makes sense to use it when teaching beginners.

This is a dangerous argument—sure, if (I did say 'if'!) it's a win for the beginners, then why not expose them to it when there's no baggage dragging them back?; but, on the other hand, in that case, in exchange for short-term gains you're giving them a long-term loss of ability to communicate with others who weren't taught that way.

(Essentially the same argument can apply to innovation; it doesn't mean that innovation is bad, just that it is rarely without cost.)

> a long-term loss of ability to communicate with others who weren't taught that way.

I'm not sure that the difference between pi and tau is severe enough to cause a communication problem. Conversion is trivial and anyone familiar with one can learn to convert to the other in about ten seconds.

Existing problems with conversion from older systems into metric make a pi<->tau conversion insignificant - and even then people manage to deal with it just fine.

I think their main point is not to switch to tau but to allow it in answers. I like the attitude.

Just for answers? Lame. It would be noteworthy if they had an option to replace instances of pi with tau in all the practice problems.

That would be unbelievably confusing...

I'd rather they concentrated on supporting openid so you don't need a google or Facebook account...

As we've seen with Stack Exchange, OpenID is confusing for most users, but good news: you don't need a Google or Facebook account anymore! You can sign up for a Khan Academy account with any email address here:

http://www.khanacademy.org/signup

Cool - when did that happen?

Thanks for pointing out :)

I don't remember exactly, but it was a few months ago. About a month ago we also launched the ability to create "child" accounts for under-13 users who aren't old enough for Google or Facebook accounts.

It would be cool to be able to transition accounts to the new child accounts; is that possible?

Sorry, it's not possible, but the main reason to create a child account is to get around the Google/Facebook requirement -- since you've already surmounted that hurdle, a child account probably wouldn't be of any more use to you.

Off topic, but can you add the ability to check off completed videos instead of having to watch them within the website? I usually play KA videos at 1.5x or 2.0x on youtube, but then I don't receive any points :(

Thanks for everything by the way.

OpenID is dead. It was flawed from the beginning, they botched the deployment, and it's useless outside of the you-sitting-in-front-of-browser context. Mobile apps, delegating access, exchanging profile data... not supported.

So what is a better solution? Genuine question.

The only thing that works well for lots of people is Facebook, twitter and google login. Mozilla's browserID might change things, but I'm neither holding my breath or expecting it to work well anytime soon. This is Mozilla after all.

Quite possibly because they hired Vi Hart :)

That's not the only reason, though it might have been a small factor. Many of us here at KA are tau believers. :)

speak for yourself, infidel!

If tau means 2*pi, how do I represent torsion?

The Tau Manifesto discusses this (perhaps not to your satisfaction, but it's there): http://tauday.com/tau-manifesto#sec:ambiguous_notation

Consider the Frenet-Serret apparatus. We use tau to represent torsion. The Tau Manifesto proposes we replace it with N - that not only breaks the "scalars are lower-case Greek, vectors are capital Latin" pattern, but N is already used to represent the normal vector. And this is somewhere the circle constant (in whichever form) comes up frequently.

Some people say we should use e for the base of natural logarithms, but that's crazy. The letter e is already used for the charge on an electron, for the eccentricity of an ellipse, and for orthonormal basis vectors. Oh, and it's the identity element in group theory. Exponentials and logarithms show up in all those contexts, so the risk of confusion is too great. Using e also breaks the convention of using lower-case Greek letters for fundamental mathematical constants (such as the circle constant π, the golden ratio φ, the Euler–Mascheroni constant γ, and the Feigenbaum constants α and δ).

I'll keep using β = 0.367879..., thank you very much. Then the base of natural logarithms is 1/β, and 1/β^(iπ) = -1. How beautiful is that? Now, some people say that writing exponential growth as β^(-x) is confusing, but I say, come on, it's only a minus sign!

If that was already the convention, I'd be on board.

overloading variables is nothing new in math/science/engineering. Usual methods include picking a different variable to represent torsion, or appending a subscipt or superscript of some kind to differentiate between the two taus. you could also use a bar (like hbar and h in physics) or an apostrophe.

In macroeconomics, pi is commonly used to represent the rate of inflation. And, as in many fields, i is used as an iterator.

Which means that in your math, if you're assuming constant inflation, you need to deflate prices in year i by a factor of e^(i*pi).

beautiful, just beautiful.

APOSTROPHE MEANS DERIVATIVE!!!!1!!!11!!!

The tau deal breaker for me is EE sinewaves, where the equation naturally uses pi, and tau is conventionally used to parameterize the function.

> APOSTROPHE MEANS DERIVATIVE!!!!1!!!11!!!

Prime means derivative; fʹ is the derivative of f, whereas either of f' or f’ is an abomination. (TeX permits the former abomination by, essentially, translating f' into f^\prime.)

I stand corrected. Perhaps we should pick a different compromise, say, tau with Unicode combining ring above: τ̊.

Edit: On Mac. Trying to do that on a Kindle Fire was ... unproductive.

If we're going to be decorating the tau anyway, let's do it from the outset so I don't have to figure out which tau they chose to decorate in this book/paper/whatever.

So let's see. There's pi, used for ~2300 years, or tau, some niche US-only fad (check for the tau wikipedia page in every language BUT english). It makes absolutely no fscking sense. Now could we move on?

I'm still on the fence as to which is more correct, though I do find that tau eliminates a lot of the silly brainfarts I used to make at school.

But I do think this is actually quite important. Mathematics has a few 'special' values including; 0, 1, i, pi/tau and e, special because they turn up everywhere, so it'd be nice to know which of pi and tau is the 'correct' special number.

Tau is a fad.

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