Is the Notation for Inverse Functions Ambiguous?

[ebits21]

Curious on people's thoughts. Especially since most people here are a lot better at math than me!

https://www.youtube.com/watch?v=jG7vhMMXagQ

 

[disregardthat]

I have read his blog, and I find this whole thing awkward and ridiculous. The practical "benefits" are hardly worth the argument, and compared to the historical significance and present popularity of the current definition it's worth nothing. Sure you can find a whole bunch of neat identities using tau instead of pi, but at the same time a lot of identities would lose that.

 

[Anonymous217]

I find celebrating pi to be pretty useless simply by going through logic, but then again, by going through the same logic, any other holiday such as Christmas is even more useless. So if you celebrate religious or recreational holidays, why not celebrate this one?

 

[Mensanator]

Curious on people's thoughts. Especially since most people here are a lot better at math than me!

Pi turns up in places that have nothing to do with circles.

 

[spamiam]

I was pretty convinced by this link that the video provided: http://tauday.com/ . Although it seems a pretty negligible change, for me at least it gives better geometric intuition, especially for e^{i\theta}.

 

[Anonymous217]

I have to admit that I'm more of a fan of e rather than pi. e is like pi's underrated and neglected younger brother.

 

[ebits21]

Curious on people's thoughts. Especially since most people here are a lot better at math than me!

Pi turns up in places that have nothing to do with circles.

Tau would be 2pi, so I don't really see your point. It just makes more intuitive sense for students trying to make sense of fractions and the unit circle. You could integrate tau into any equation that uses pi.

The link in the video shows modifications in many well known formulae: http://www.tauday.com"

@Anonymous217

The video isn't really about celebrating pi day so much as suggesting a clearer way (as in clearer and more consistent) to formulate trigonometry.

 

[micromass]

Well I certainly consider pi to be an unfortunate historic mistake. Math would be slightly more beautiful with tau. But this is the way it is and we need to stick with it.

Other unfortunate mistakes are the definition of the gamma function and the notation \subset to mean subset or equal. But there's nothing we can do about those now...

 

[myth_kill]

Well I certainly consider pi to be an unfortunate historic mistake. Math would be slightly more beautiful with tau. But this is the way it is and we need to stick with it.

Other unfortunate mistakes are the definition of the gamma function and the notation \subset to mean subset or equal. But there's nothing we can do about those now...

why do u consider it to be a mistake ?


you ought to read this :

http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Pi_through_the_ages.html

 

[jhae2.718]

...Other unfortunate mistakes are the definition of the gamma function...

For some reason I am reminded of the joke where a math professor is giving a lecture to freshman and brings up factorials (n! is of course N!), and when no one knows what they are exclaims "But it's only a special case of the gamma function!"

Back on topic, I say stick with \pi.

 

[micromass]

For some reason I am reminded of the joke where a math professor is giving a lecture to freshman and brings up factorials, and when no one knows what they are exclaims "But it's only a special case of the gamma function!"

That made my day

 

[jhae2.718]

I also like the joke where a visiting professor gives a lecture about Bessel functions to a group of students. When he realizes it's mostly undergrads, he asks if anyone hasn't seen a Bessel function before. When one student finally says no, the professor replies, "Well, you have now," and continues on with the lecture.

 

[micromass]

I also like the joke where a visiting professor gives a lecture about Bessel functions to a group of students. When he realizes it's mostly undergrads, he asks if anyone hasn't seen a Bessel function before. When one student finally says no, the professor replies, "Well, you have now," and continues on with the lecture.

That reminds me of an oral exam that I once had. The professor asks something very complicated using terms that I had never heard of before. So I say: "I'm pretty sure we have never seen those things." The professor just answered: "True, but I have seen them before." And I had to answer the question...

 

[jhae2.718]

Doesn't sound very fun. Did you get it right?

 

[micromass]

Nope, but the professor was incredibly rude and impossible to the entire class. So I ended up passing the class...

To one student he asked: "draw a circle on the board with radius 1." The student was quite confused, but did it anyway. Then he said: "this is your score, see you next year."

It wasn't funny then, but we can laugh with it right now...

 

[jhae2.718]

To one student he asked: "draw a circle on the board with radius 1." The student was quite confused, but did it anyway. Then he said: "this is your score, see you next year."

I would have drawn a circle of radius r and then said let r=1...

 

[Dickfore]

In Fourier series, the period should be 2l instead of l, because the cosine and sine series are the even and odd part of the function, respectively, which, in turn, must be defined on a symmetric interval. Thus, it makes more sense to have the period of the trigonometric functions equal twice a fundamental constant.

 

[Mensanator]

Tau would be 2pi, so I don't really see your point. It just makes more intuitive sense for students trying to make sense of fractions and the unit circle. You could integrate tau into any equation that uses pi.ometry.

And such integration automatically makes it less intuitive for those applications not involving unit circles. And as far as intuition goes, everyone knows that if your garden hose won't reach to your rose bed, you buy an extension and make it longer. But if your hose stetches 30
feet beyond, you don't cut off the excess and attach a new nozzle.

 

[Redbelly98]

Well I certainly consider pi to be an unfortunate historic mistake. Math would be slightly more beautiful with tau. But this is the way it is and we need to stick with it.

That sums up my thoughts pretty well. In math, it is the radius of the circle that turns up in many equations, more so than the diameter. So why not define the ratio circumerence/radius as a fundamental mathematical constant, and voila! You have 6.283185...

That being said, we might as well stick with the system we have. If we Americans aren't willing to change over to metric units, I can't imagine people adopting τ over π.

 

[Dickfore]

So, it's simpler to remember:

<br /> A = 2 \tau r^{2}<br />

and

<br /> V = \frac{2 \tau}{3} r^{3}<br />

for surface area and volume of a sphere?!

I think the author of the video should start memorizing formulas instead of 'thinking critically'.

Also:

Imagine you have a pipe with a circular opening or a ball. Is it easier to measure the radius or the diameter?

 

[Redbelly98]

You might want to double check that area formula.

Imagine you have a pipe with a circular opening or a ball. Is it easier to measure the radius or the diameter?

Since when does easier-to-measure matter in mathematics?

 

[jhae2.718]

Imagine you have a pipe with a circular opening or a ball. Is it easier to measure the radius or the diameter?

Whoa! This is mathematics. You can't just start bringing up practical examples like that!

 

[ebits21]

Whoa! This is mathematics. You can't just start bringing up practical examples like that!

I think the point is that the radius is the fundamental unit. Diameter can always be broken down into radius.

Plus.. in real life you can't just take a tape measure and hope to find the absolute value of the diameter. This is because you can't be sure you're going through the center of the pipe exactly. You would have to know the center exactly (and the radius, by definition, relates to the center).

If you use a metal wedge with a known angle to find the diameter of a pipe, you geometrically find the radius of the pipe, then multiply by two to find the diameter.

Edit: Okay, I forgot about calipers. :P

 

[Dickfore]

You might want to double check that area formula.

I thought the whole point of the vid was that \tau = 2\pi.

Since when does easier-to-measure matter in mathematics?

Since when is not being able to remember formulas considered being a mathematician?

 

[jhae2.718]

I personally say stick with \pi. Far too late to change anything.

 

[ebits21]

So, it's simpler to remember:

<br /> A = 2 \tau r^{2}<br />

and

<br /> V = \frac{2 \tau}{3} r^{3}<br />

for surface area and volume of a sphere?!

Seems equally easy to remember as with pi to me. That is, if you were a person learning from scratch.

 

[spamiam]

To one student he asked: "draw a circle on the board with radius 1." The student was quite confused, but did it anyway. Then he said: "this is your score, see you next year."

All right you jokesters, enough of that!

But seriously, the section of the manifesto on quadratic forms was quite persuasive. It compared
\frac{1}{2}gt^2, \frac{1}{2}kx^2, and \frac{1}{2}mv^2, concluding with \frac{1}{2}\tau r^2 for the area of a circle. I was pretty impressed.

 

[HallsofIvy]

Of course, the reason the Greek's chose to give a symbol for \pi rather than 2\pi is that the cirumference of a circle is 2\pi r or \pi d and it is much easier to measure the diameter of a log than the radius.

 

[Redbelly98]

I thought the whole point of the vid was that \tau = 2\pi.

My mistake, I was thinking of the area of a circle, but you did say in the post you meant the surface area of a sphere.

 

[micromass]

Just imagine that advanced civilizations in space have missed us because we've been sending pi into space, and not tau

 

[Dickfore]

Just imagine that advanced civilizations in space have missed us because we've been sending pi into space, and not tau

Well, they wouldn't have been so advanced to not recognize we were sending \tau/2.

 

[Redbelly98]

Especially if we send in binary, you'd really have to be an idiot not to recognize it.

 

[TylerH]

Of course, the reason the Greek's chose to give a symbol for \pi rather than 2\pi is that the cirumference of a circle is 2\pi r or \pi d and it is much easier to measure the diameter of a log than the radius.

Since when are logs circular?

 

[Dickfore]

Since when are logs circular?

By right about the same time cows became spherical.

 

[jhae2.718]

By right about the same time cows became spherical.

Wait...cows aren't really spherical?

 

[TylerH]

Theoretically...

 

[Dickfore]

Wait...cows aren't really spherical?

Just in case you haven't been aware:

http://en.wikipedia.org/wiki/Spherical_cow"

 

[dimension10]

Well I certainly consider pi to be an unfortunate historic mistake. Math would be slightly more beautiful with tau. But this is the way it is and we need to stick with it.

Other unfortunate mistakes are the definition of the gamma function and the notation \subset to mean subset or equal. But there's nothing we can do about those now...

Yes. Notations like sin^-1 are a bit ambiguous quite often.

 

[TylerH]

Yes. Notations like sin^-1 are a bit ambiguous quite often.

IMHO, it's the notation sin^n(x)=\left( sin(x) \right)^n that is ambiguous, rather than inverse notation. In general, f^{-1} is ubiquitously the inverse of f.