I can imagine how this goes: Most of the world adopts Tau, the US refuses and stays with Pi and the UK starts to switch to Tau but stops half way through leaving somethings using Tau and others still using Pi.
Think of the top of pi/tau as a sin wave, representing circumference and the "legs" are radii under the fraction. Pi = C / (r+r) Tau = C / r The symbol for pi is two radii under the circumference, and the symbol for tau is one radius under the circumference.
actually, tau was originally proposed as a way to express pi/2, and tau looks like half of a pi symbol, but that never caught on. As to why proponents of a symbol for 2pi chose tau, -I imagine it's simply because of the visual similarity and nothing more.- I just thought of something: Tau is roughly equivalent to the English letter 't', which is the first letter of the word 'turn', and tau radians corresponds with one turn around a circle.
The other thing which came to mind while watching this again was that when saying a whole circle is pi radians, pi is defined in terms of diameter and radians in terms of radius. It's relating apples and oranges. A circle being tau radians relates apples with apples, as both tau and radians are defined in relation to radius. Tau then becomes a simple conversion constant between distance and angle, and is a much more powerful concept.
circle with diameter 1 has circumference 3.14159... =π apples to apples. when you mesure it, it's easier to get the diameter of a circle/ globe.. yet when you draw it, you use the radius then you have circumference = 2.π.radius here again you have apples with apples. then you go in angle / frequency/ period measurements and calculations that go more or less elegant depending on if you're using π or τ.. and that can reduce easier as well.... everybody say τ=2π.. it would be different if we were all saying π=τ/2
@@fockoff But when you construct a circle, you use a radius rotating around a point. The point comes first, then the line, then the circle. A natural progression of dimension. The diameter requires you to first draw a line, then find it's center, then rotate it around that center. Which is not a natural progression of dimension.
+Francesco Favro Well, i looked it up, and the name roughly translates as "navigator". If you're not gonna trust a navigator, whom or what should you trust? Besides, it's just a common Irish name. Perhaps you shouldn't trust any Irish person either just in case they have a Moriarty in their family tree.
It probably wouldn't catch on in America simply because we are not good at transitioning, we still refuse to transition to Metric even though it makes so much more sense.
Not saying metric is WORSE, but our system does have the advantage of being easy to divide. A yard can be divided into 2, 3, 4, 6, 8, 12, or 18 inches while a meter can be divided into 2, 4, 5, 10, 20, 25, or 50 centimeters. Same number of ways to divide, but 36 is much smaller than 100, making it roughly thrice as dense. Same goes for 12 and 10. Like pi v tau, it is not a trivial decision.
Jeremy Hoffman I know about that in terms of arguments that a base 12 system is better than base 10 for that reason, but how is a mile easily divisible into feet?
But right after you learn that Circumference = 2 PI R, you also learn that Area = PI R² so that would make Area = Tau R² / 2. Seems more confusing to me.
I think in fact it's not a problem. Try to separate equation like (Area) = (Tau) (R²/2). So why R²/2 ? You probably know the utility of dérivative in physics (derivative concern the variation of something in time). Firstly, when you derivate x²/2 it gives you x and if you dérivate x² (without the /2) you get 2 x (so a 2 appear). Well, I think x²/2 is more natural in a derivative problem, and the waves are derivative problems. But maybe it's not a way to explain to a student who ask your question why use tau instead of pi, cause pi/tau appear earlier than derivative ^^ But I don't know much more tings about Pi and derivative, so maybe I'm wrong.. or not ^^
just a random thought what about pi/2 so that you work with a right angel and you would have the important points of the sin cos... also complex numbers would be easier to handel if you use right angels.
Well, think about it like this: if you approximate it with triangles with infinitesimally small base and add them all up, the area will be Area = 0.5 * h * b by using the formula for triangles, whereas h = R and b = Tau * R in our case. Area = 0.5 * Tau R² makes sense that way, don't you think?
But 1/2xy^2 is one of the most occuring quadratic expression. Distance fallen: y = 1/2 gt^2, Potention energy in a spring: U = 1/2 kr^2, Kinetic energy K = 1/2 mv^2. As you see this term is everywhere.
How about instead of radians for radius, we had dongers for diameter. Then a circle would have pi dongers as circumference. or pi/4, pi/2, 3pi/2, pi dongers at every 90 degrees !
Then the maths doesn't go together. Calculus and trigonometry builds on radians. Angles are defined as arc lengths of the unit circle. And the diameter of the unit circle is two.
+EpikCloiss37 Tau + Tau / 2 = (1 + 1/2)Tau = 1.5 * Tau Did I really need to teach you this? Just treat pi's as half tau's, and it should also make sense in the context, when you actually need a pi and not a 2pi.
I've been ignoring these notifications forever, but I think I'm going to clear something up. The fact that the video addresses both tau AND pi is why I chose tau + pi and not 3pi or 3tau/2.
this brings me back to learning pi for the first time at school i remember feeling the disconnect between using diameter to define pi but then as soon as you use pi to define other aspects of geometry (area of a circle etc.) you forget about diameter and use radius and it always struck me as weird and a bit redundant and this sorta explains why.
3:46 You "have no idea" as to why? I'm surprised to hear that coming from a mathematician. Think about it: if you were to measure angles clockwise, you'd start in the positive x-value, negative y-value quadrant (Quadrant IV)-rather than the positive x-value, positive y-value quadrant (quadrant I)! It's the same reason for which the quadrants are measured counterclockwise.
***** I left another comment on whatever video was talking about Zora's paradox, I think it helps with this comment. While the whole positive negative quadrant comment may very well be why we measure counter clockwise, it does not actually mean anything. Going around a circle clockwise is 360 degrees just the same as going around counter clockwise. Numbers do not exist in nature. Numbers and words are purely descriptive about existence and do not define it. That's why you cannot travel negative distance. Mathematically positive is forward and negative is backward but if you walk backwards for 10 miles you successfully looked like a fool for a positive 10 miles. If you walk forward for 5 and backwards for 5 you walked 10 miles and not 0. If you disagree I challenge you to find a mountain and walk up the mountain forwards and down the mountain backwards and then tell your legs you walked a total of zero mountains and see what they say. They won't say anything unless you're in serious need of mental evaluation but the point is your body confirms that terms in math cannot be applied to real life if you forget that math and words are describing and not defining existence.
The Real Flenuan You do realize that quadrants could be the other way around. The fact that they are what they are is what we don't have any idea about. It's just convention
+AlphaMineron But that's not what he was saying. Given the fact that the convention about how the quadrants are laid out is the way it is, the way angles are measured makes sense.
Wait why? Even with Fourier transforms the factor is 2pi. The only thing I can think of where pi comes up without 2 is the F transformation of Sin and cosine (with the delta impulses)
@@Imbeachedwhale Wouldn't mind a change there, an already difficult signals class was made worse with the terms t, T, and τ being used in the same equations
I'm more of a tau person, but I think Euler's identity with pi gets much more amazing. Think about it, you're investigating logarithms and circular functions, when you realize their relation by the simple and beautiful formula, but not only that, you find out by accident the logarithms of negative numbers! People usually don't pay attention to that.
Most graphing calculators allow you to program a value into a letter; I always have the letter T defined as 2*pi on my calculator. For some extra fun, you can of course also go ahead and plug in various useful values in all the other letters as well, like the gravitational constant into G and so on.
No gradual adoptions! Just look at the US sticking to their feets and pounds. You have to make a proclamation that from this day forward we will use this so "deal with it!"
one minor issue i can think of with tau is that as an engineering student, tau is very common notation for time constants, time delays and dimensionless time and stuff like that. whereas pi is pretty much unanimously the circle constant. using the symbol tau might confuse me as an engineer. i wonder if there is another symbol available that could possible be used that wouldn't have this problem?
And if you multiply pi by two, carry the tau, go around, divide again by pi, divide by two (the very same two as previously), to the power of the given radius, times pi, divided by tau, you can never go back home again.
Well, I completely disagree. Why? In my math classes, the first time we came across Pi was circumferrence = Pi * diameter. At that point we also defined r as one half of d, which caused a lot of confusion for many kids. (Why are we measuring from the middle?) By the time we got the part of trigonometry where you start using Sin and Cos (and ...) or rotations in physics, everyone was used to the number Pi. Admittedly, one revolution being 2 Pi caused some confusion again, but that was trivial in comparison with the other problems people had with trigonometry. So, in conclusion I believe that using Tau instead of Pi wouldn't have caused less confusion, but rather caused the confusion earlier on - which I think is worse. It makes sense to me now, but I don't think that it would have helped the kids that struggled at mathematics anyway - and those who didn't struggle never had a problem with Pi in the first place... PS: I love your accent!
I disagree; radii are much more intuitive. We measure the distances between celestial objects as the radius (more technically, 1/2 the major/minor axis). Heck, the tangent of an angle is defined as the slope of the radius of the unit circle. We teach kids that Circumference=pi*diameter. I think we might want to start transitioning to Circumference=tau*radius
Neil Dey All the screws and pipes I work with are defined by diameter, because it's much easier to measure diameter given an arbitrary circle. For celestial purposes, maybe tau is more convenient. But for earthly applications, I would argue for pi.
Everest314 I can't possibly agree that diameters are less confusing or more intuitive. After all, what is the definition of a circle? A circle is the set of all points that are a specific distance (radius) from a specific point (center). If you are actually teaching the definition of a circle, there is NO question about why one would measure a circle by the radius. The issue that is causing confusion, then, is even introducing the idea of a diameter, which is a much more confusing and less intuitive topic than the radius. And, AMGwtfBBQsauce I would agree that it is easier to *estimate* a diameter, but disagree that it is easier to *measure* one. In order to measure a diameter correctly, you still need to know where the center is. Otherwise, you are just measuring a chord. So, sure, if you don't need perfect accuracy, use diameters. But, if you are trying to do any kind of logical, mathematical reasoning or arguing (i.e. the whole point of mathematical education), the radius is superior.
zanJoKyR I am only speaking from my own experience in school. Nowadays, I know that a circle is more logically defined via its radius and I see no real difference between 2 Pi and Tau - there are formulas that are "simpler" with either one (are we honestly arguing about an additional 2 or 1/2?). However, we knew what a circle looks like long before we could describe it mathematically and from that perspective i still think that the diameter is easier to grasp than the radius because at that point you have never thought of the centre of a circle (at least most people haven't). I have also experienced the same when I did math tutoring. The "all points with the same distance from the center" never got through, even if they had used it numerous times with their compasses. Okay, those were not the brightest kids... And has others have pointed out, by far the most praticable way to measure the size of a circular object is by using a calliper which gives you the diameter. I just don't think that the advantages of Tau are enough to justify overturning the convention. (Not that i don't see the advantages...)
JustWatchingVideo56 pi is also used in many places in math, pi can represent different functions such as capital pi of x = x / ln(x), this is a function that gives a rough approximation of how many prime numbers there are that are less than x. All Greek letters are used in several places: theta is used to represent angles, also it’s a constant where (theta)^3^n rounded down will give you a prime number.
I do not think there is a problem. At 7:25 : "omega = 2 pi * f". Now, if you use tau for a time constant, you would commonly have "omega * tau", so, if tau also used for "2 * pi", it would turn into "f * tau^2". Now you can cancel one of the "tau"s without breaking anything. Easy!
Not really. There's no reason we have to eradicate the use of pi to introduce tau, we could use tau in equations where tau is better, and pi in equations where pi is better.
I agree it is cleaner and more intuitive, however it also seems relatively trivial issue to suggest changing all the books forever. I also think for any serious student of math, this will represent a very minor issue in the grand scheme of things. If anyone is actually being held back because of this, they probably shouldn't be studying math anyway.
Or maybe students should be expected to handle a factor of two with little difficulty? The people who want to introduce tau to replace 2pi are hipsters wanting change for the sake of change and nothing else. Forget the fact that pi is rarely used as a symbol in other contexts since everyone knows what pi means to most people. Let's use tau which already has multiple meanings (proper time, decay time constant, volume in some cases where V is used for voltage or potential, torque!).
+TheWindWaker333 Pi is used in several areas as well. Chemists have Pi-bonds and the osmotic pressure is denoted as pi. Basically at any point you wish to distinguish something otherwise denoted with a p as a special type of property you'd use pi, just like you would use tau instead of t. Frankly there are not enough letters to denote every single property of something uniquely. The reason why pi even is called pi is from the greek word for periphery. Likewise tau could imply "turn" or something similar.
As far as I know, the story of pi says that Euler used pi to describe the perimeter of a circle. It just happens that the page they looked at looking for the definition of pi had a semicircle not a full circle. But in the same book, in a different page, he has pi = 6.28... So if they would have seen that page first, today we might be using the better way. I don't know if this story is true, but I don't think Euler would have defined pi the way we use it today.
This guy makes it seem like using 2pi is something that is holding a lot of people back or something that people really struggle with when they are first introduced to it. I haven't come across anybody that has been even remotely confused by this concept. It's really not that confusing at all.
How about all the calculators that already have PI programmed in. If we switch to Tau we have to either remake every calculator. For example. Ti-83, Ti-84, Ti-89 etc. Or every time we see tau we have to times it by two to get the PI units.
I just opened my first year physics textbook to the chapter on rotational motion: first thing I see is it saying that 1 revolution is 2 pi. Earlier than that it mentions 2 pi as 6.28 before ever saying 3.14 or a lone pi. The only lone pi in the entire chapter is equated to 180 degrees. (after saying that 2 pi = 6.28 is 360) The section on circular motion mentions radial acceleration as 4pi^2R/T^2 (aka tau^2R/T^2)
I'm an engineer, and I think this whole discussion of pi vs tau is silly, I don't get why mathematicians take sides so radically instead of just using both at convenience. That being said, when the time comes I'll be teaching my kids trigonometry with tau and substitute it with 2*pi if necessary, it's far better for learning because it's way more intuitive, there will be dickhead close-minded teachers that won't like it, I'm sure, but all arguments are invalid if the procedures and results are correct.
(@3:46): ...gotta love the irony in the fact that, in Prof. Moriarty's argument in support of those who wish to abandon the convention of pi in favor of tau, he mentions references the convention of measuring angles counterclockwise rather than clockwise!
Having the square root of two in the problem promotes the correspondence for an integer solution among conceptual square triangles used for measurement. The square root of two is useful for mediating between polar and Cartesian considerations.
The reason we originally used pi instead of tau is because it is much more natural to measure the *diameter* of a circle or cylinder with something like calipers than it is natural to measure the *radius* of the same.
Counter-clockwise kinda makes sense, because most of us would say that the direction pointing away from the paper (as opposed to into the paper) should be considered "positive", because when you are working with a sheet of paper at your desk, away from the paper is the same direction as gravitational "up". That, combined with the right hand rule (kind of an arbitrary choice, I suppose, as opposed to left hand) tells us that the direction of "positive" rotation should be counter-clockwise.
What is peculiar is pi often turns in elegantly in formulas instead of 2 pi. The area of a circle is r pi², that's more elegant than r tau²/4. The series 1/1²+1/2²+1/3²+1/4²+.... = pi²/6 instead of tau²/24. The area of the graph under e^x² is the root of pi instead of the root of tau/2. What about e^(i pi) = -1 that would turn into e^(i tau/2) = -1 and spoil the most beautiful formula in maths!!!! I mean although the definition of Tau may seem more natural, still pi turns out to position itself more elegantly in formulas especially in higher/intermediate mathematics. Tau only wins in elegance in basic mathematics, that's why I root for pi!
dekippiesip The are of circle is π r², but really is 1/2 τ r², just like the kinetic energy 1/2 m v², and right, we make mistakes, every variable means something, τ means a circle while π means the half of a circle, It is not practical.
Garen Crownguard Dude, tau means a whole circle, meanwhile pi means a half of a circle. When we have a quarter of a circle, What we say? pi/2 or tau/4? evidently pi/2, that make sense? EVIDENTLY NOT! That is the reason why pi is wrong, pi usurp the place of tau, but you can use the pi number in your equations, this change just begun....
Um, you can't exactly call pi^2/6 more natural than tau^2/24. Several issues you stated here were addressed in the Tau Manifesto, so I think it would help if you looked more into the area of a circle and Euler's Identity. Often the Gaussian integrand is written as e^-(1/2 x^2), and one way to think of this is that d/dx 1/2 x^2 = x. You cannot get a much simpler expression in mathematics than x, thus making the expression 1/2 x^2 more differentially natural than x^2. (Also, the area of a circle in your comment somehow got messed up. Not r pi^2 = r tau^2 / 4, pi r^2 = 1/2 tau r^2. Here again you see an expression of the form 1/2 x^2.)
Yes, but what you're forgetting is that by the introduction of pi (when dealing with a circle, sphere etc.) you are already conflicting with the radius! Where pi=c/d, why not just have tau=c/r? It's senseless to build a foundation on the diameter (2*r, already arguably less fundamental), and have diameters and radii in the same equations? Euler's identity is without question prettier: One turn, one tau. Michael Hartl sums it up well: If you really need the zero, then e^i*tau = 1 + 0. Lastly, the unit circle is so much more intuitive to explain and understand. Half turn, tau/2 etc. I've loved and pondered pi since I was a kid, but please be wary the logical fallacy of appealing to tradition. Just because we were raised that way, it doesn't make it better, right etc. Best of luck, Chris
What I found surprising in this video was learning that I was never taught was a radian was until now -- the angle of an arc along the circumference with a length equal to the radius. I think I would have done better in math 10 years ago if they taught that better in my school.
To any scientist or engineer who doesn't get why tau is more natural and worth pushing for, let me make a comparison. You're often interested in oscillations, right? And what's the key thing about an oscillation? Its period. But suppose long ago, someone had instead put the focus on the half-period. Suppose they'd given _that_ a name - say the 'heriod' - and never bothered naming the concept of the period at all. So, the heriod notion takes root, and years later it pervades the literature. Then someone points out that actually, it would be better to be using _twice_ that - which we could call the 'period'. Well, basically that's what tau is to pi. It's about fixing a poor choice made long ago, to smooth out a wrinkle. Tau is the natural thing to focus on, not half-tau.
The choice to base pi on the ratio between the circumference and the diameter, then use half of the diameter for all the previous and subsequent definitions other than pi it's a weird blip in the convention.
As an electrical engineer I'll retort: using tau would simply cause a bunch of fractions in the indices of my equations, because often there are just as many pis as 2pis in complex system equations. Bottom line, if it doesn't help me I won't use it.
Sure, you'll have some of that. I don't think the main thrust is about conciseness though. It's more about making things potentially clearer and easier to understand (I forget how much the video goes into that). So, the fractions you speak of are not necessarily 'bad'; they could be usefully instructive. That said, you may of course value conciseness more, particularly if experienced and just viewing an equation as a tool.
The use of radians is confusing to anyone ,why not just use degrees? Nobody can get a mental visual of radians but they can in degrees. Also the use of radians is unnatural because they can't be constructed ,where as degrees can be .
Radians make more sense when you start doing calculus and beyond. They simplify a lot of the math. Degrees are great for everyday use, though! Especially because 360 has so many divisors.
360 is an easily divisible number, and you can construct equal dhapes. For instance, if I want to construct a circle with three equal parts, all I have to do is make 120 degree angles from the center of the circle, as 360/3 is 120
Tau is like the metric system. Sure it's a hell lot more convenient, but there will always be that one group that claims it's not worth it, *cough* America *cough*.
The argument to keep 2pi is the same reason the USA won't give up the mile. Or the silly Month, Day, Year date system that gets everyone all worked about pi in mid-March. Silly and wrongheaded.
+donfolstar τ is much easier to get used to than a whole new system of units. People who are used to inches, feet, miles etc will have to start developing a feel for a bunch of completely new lengths, like millimeters, meters, kilometers etc. τ is much simpler, it is essentially the exactly same thing as "period" or "revolution" - τ perfectly corresponds to one period, so you can just replace "period" and "revolution" with τ everywhere. It can't get much simpler than that.
The argument is actually not a silly and wrongheaded one. It is a financial one. So much signage would have to be changed, and so many books / calendars reprinted. Sure, the USA should switch, but it'd be too expensive.
+donfolstar I have to say, you form a really constructive argument. Just say what you said the first time, without supporting it with any new evidence. That's how discussions should go.
Bailis Cremey My state recently replaced every road sign because they wanted them to be more reflective. Sorry to tell you, but cost isn't the issue. Or are you still on about calendars because you are struggling with that concept?
Almost every time I deal with the sinus function in programming I end up needing to do pi*2 because often the environment has a pi constant but not a tau constant. With a tau constant you wouldn't have to do these things all the time.
It's because 2pi isn't very useful in programming since you're usually using it to do things like calculate trigonometric functions, which are 2pi periodic. And realistically 2*pi isn't a problem...
There's that joke on XKCD about choosing the shortest route past rectangular blocks in a cityscape. Then there's the myth that I heard that taking a right turn is always faster than taking a left turn (perhaps due to the flow of traffic). I guess that this is one of those things that are not apparent to people yet. I got the reminder when I was talking about PI being the amount of radians you would have to go in order to get past a watchtower. The direction causes a short crisis, then you decide that it has no importance. In a similar way, perhaps TAU or PI is just like that watchtower: Left or right... no matter. But perhaps it actually does matter because of circumstances.
The one thing I wish you would have addressed but didn't when you brought up springs was that using Tau in place of Pi would make the equations for the spring match the physical equations for other forms of physical motion, so they'd all look the same with just the particular symbols changing. I can't remember the details and am too tired to go hunting right now, but I remember seeing that...
"Natural Unit" is turn to opposite direction. Not turn to opposite direction then back again. Why should the negative cure underneath the surface be added to the measure? It is just exactly the same over again.
The reason angles roll counterclockwise is because of the way the plane is oriented. In order for the definitions cos(x) = Re(exp(x)) and sin(x) = Im(exp(x)) to be consistent with cos(x) = adjacent/hypotenuse and sin(x) = opposite/hypotenuse, we have to define the the angle x where 0 is the start from the positive real axis and a quarter turn is the positive imaginary axis. By convention, the positive real direction is right and positive imaginary up, so angles go in counterclockwise direction.
The natural logarithm when dealing with complex numbers is many-valued. So e^(i*tau) = 1 doesn't mean that i*tau = 1 (taking the ln of both sides), in the same way that x^2 = (-x)^2 doesn't mean that x = (-x) (taking the square root of both sides).
i think the best way to explain why tau is amazing for circle theory is using it via a graph of sin theta, cos theta, and tan theta. you will see a full map of where each goes through one circular rotation from 1 to -1 over the course of tau, and it presents a more clear image of what happens to all 3 when you go through one rotation. its just a nice thing. "don't forget to bring a tau!" -taulie
Positive angles are counter-clockwise because +y is upward on most graphs. If +y is downward as on a computer display, then your positive angle is now clockwise.
Unfortunately, Tau is used to represent the time it takes a capacitor to charge in electrical engineering, so even more textbooks would have to change. I like the idea of using tau to ease education, though.
Correct me if I'm wrong, but I always thought that we conventionally go counterclockwise around a circle (when talking about angles) because of the Quadrant System. We start in the upper right hand corner with Quadrant I, and move counterclockwise. So it makes sense that angles would do the same. As to why the Quadrant system goes this direction however, is another question.
The ease of understanding for newcomers is indeed very important, for example, my physics teacher was able to make e=mc^2 much easier for us to understand by rearranging it to m=e/c^2, thus encouraging us to focus not on the energy but on the question of "what is mass?" which was the more important concept for us to wrap our heads around in the beginning.
The ironic bit is that the whole thing starts from using both the radius and diameter as fundamental measurements of the circle. It just so happens that d=2r (look familiar?). If we're comfortable using both of those, it only makes sense to use BOTH pi and tau.
It's easier to measure the diameter of something accurately than its radius with standard measurement devices. You take that diameter and multiple it by (an approximation of) pi to find the circumference. To find the area, Tau would force you to divide the radius squared by two which may or may not be as handy a calculation to do on the fly. Pi works better for the unwashed masses, and the erudite can easily multiply by two when the need arises (for things like harmonic motion or radians).
I did not learn what radians were until I was in 9th grade, But I was just teaching my 4th grade sister the relationship between the circumference and diameter of a circle.I think that instead (π) * d it was (τ)/2 * d. Every one will by the time they are taking their first trig class know that 2π radians is a full revolution. It is really about keeping things simple when you start geometry with circles, and it's usually before kids learn algebra now.
this is just my opinion, but we should've started with teaching tao because if we try implementing it now, students would have to be taught both since pi is already so widely used. So it would be even more for them to remember. Maybe just the concept of what tao is would be useful, but forcing them to use tao in certain situations and pi in others would be less effective.
Ironically, the thing that convinced me of the superiority of Tau, was the area of a circle. The general equation of 1/2xy^2 pops up everywhere in science. Velocity is a*t. Change in distance is the integral of velocity which happens to be 1/2at^2. Momentum is m*v. Kinetic energy is the integral of momentum, 1/2mv^2. When you compare a circle's circumference and area, you could use 2pi*r and it's integral pi*r^2, or you could follow the same formula as everything else and use tau*r and 1/2tau*r^2
On the otherhand, getting your head around the concept of 2pi is kind of a milestone that opens the door to understanding the deeper equations. I see this as a shift of that issue: making it easier to understand initially but setting students up for a hurdle later on.
I thought about it and I think that that statement is faulty as a radian is based on the radius and "removing" the 2pi would make it harder to understand what a radian is and why it is so. A better solution would, in the case stated above, to remove the radian as a measurement for angles ...
2*pi radians is not the problem. The problem is radians. The most natural unit of angle is "turn". For example, the angle made by two adjacent sides of a rectangle is one quarter turn (0.25 turn), as opposed to pi / 2 radians (which I totally agree is utter brain damage!). The elegance of turn as a unit of angle results from the fact that it does not unnecessarily drag radius or diameter into the issue; turns involves exactly the one thing we care about: angle. Besides being unnatural, another problem with radians is that a complete turn is an irrational (in fact, transcendental) number of radians. This, of course, is a direct consequence of the unnaturalness of radians as a unit of angle. The turn has no such problem. Similarly, under the radian convention, the period of sin is irrational (again, 2*pi); whereas, if we were to adopt the turn, then the period would most elegantly be 1. What could be more satisfying?? Also for people who prefer to work with degrees, if the number of degrees you are dealing with is rational (e.g. the angle made by adjacent sides of a rectangle or regular triangle), then the equivalent number of turns is also rational (and vise versa). Whereas, switching back and forth between degrees and radians inevitably involves irrational calculations, which means that nobody is going to bother with hand calculations. One could probably go on at length about the evils of radians, but I think ya'll can see where this is going: radians are very silly, and turns just make sense.
