A Mathematical Mystery Revealed

[CGUE]

A very interesting article for all.

http://www.math.utah.edu/~palais/pi.html

Quote from the article:
"What really worries me is that the first thing we broadcast to the cosmos to demonstrate our 'intelligence' is 3.14... I am a bit concerned about what the lifeforms who receive it will do after they stop laughing..."

It's saying e.g.

cos(x + π) = cos(x) ?

 

[cristo]

It's saying e.g.

cos(x + π) = cos(x) ?

No it's not; he's just defined some new symbol (a pi sign with three 'legs') to be equal to 2pi, and is then saying that cos(x+newpi)=cos(x).

This doesn't really change anything!

 

[LukeD]

While the article's name is terrible (and the article itself isn't all that well written), he does have a point. I can't think of anywhere I've found [itex]\pi[/itex] to be more useful than [itex]2\pi[/itex]. It would simplify a lot of things, and, if I did mathematics in a vacuum and never had to interact with anyone else, I'd strongly consider inventing a symbol for [itex]2\pi[/itex] and using that everywhere instead of [itex]\pi[/itex]

Of course the difference between them is always related by a factor of 2 (or some power), so pi itself isn't that clumsy. But it's similar to the way that physicists decided that [itex]\hbar[/itex] is slightly less clumsy than h

 

[arildno]

I have some sympathy for Palais' point, but then again, why bother overmuch?
Many formulae will become uglier, rather than prettier, with the new pi-symbol, not the least Euler's identity.

 

[LukeD]

I have some sympathy for Palais' point, but then again, why bother overmuch?
Many formulae will become uglier, rather than prettier, with the new pi-symbol, not the least Euler's identity.

Maybe I'm just being clouded by the earliness of the day and maybe I just haven't had enough advanced mathematics to appreciate the choice of pi over 2pi, but I can't think of any formulas that would be more ugly. To me, Euler's identity looks better as [itex]e^{is}=1[/itex] (where s = 2pi) and [itex]e^{\frac{1}{2}is}=-1[/itex] because it better mirrors how you use it. Euler's formula projects an angle onto the unit circle in the complex plane. [itex]e^{is}=1[/itex] expresses that a full turn is the same as doing nothing at all while [itex]e^{\frac{1}{2}is}=-1[/itex] expresses that a half turn is the same as turning around.

(Sorry to "argue" about this... It's not that I have anything invested in the conversation; I'm just bored and have nothing else to do at this time of day lol)

 

[HallsofIvy]

This has been discussed before. [itex]\pi[/itex] was originally defined as the ratio of circumference to diameter. Why not "circumference to radius"? Because it is much easier to actually measure the diameter of a circle- especially if the "circle" in question is a long tree trunk. Even with a mathematical "circle", determining the radius involves either first finding the diameter (and then dividing by 2) or first finding the center of the circle. Just finding the diameter is much easier.

 

[Astronuc]

I thought that new pi looked like pi overstruck with tau, or perhaps tau-pi.

There would also be the case of using 2[itex]\tau\!\pi[/itex] where one would use 4[itex]\pi[/itex]. So I don't see an advantage of introducing a new symbol.

 

[RonL]

No it's not; he's just defined some new symbol (a pi sign with three 'legs') to be equal to 2pi, and is then saying that cos(x+newpi)=cos(x).

This doesn't really change anything!

I have always been amused by how the picture of a square pie was such a great help in remembering how to determine the area of a circle, but now that my square pie might have three legs, is really a hoot!

 

[LukeD]

There would also be the case of using 2[itex]\tau\!\pi[/itex] where one would use 4[itex]\pi[/itex]. So I don't see an advantage of introducing a new symbol.

Sure... but how often does one reaaally talk about surface area?

HallsofIvy: I was thinking more from a mathematical aesthetic point of view, but there are definitely a lot more engineers than there are mathematicians!

 

[Hootenanny]

Sure... but how often does one reaaally talk about surface area?

Who mentioned surface area?

 

[Hurkyl]

I propose that we use the symbol "toopie" to represent this quantity. Toopie is, of course, the symbol [itex]2\pi[/itex]. It's apparent similarity to the product expression of 2 with [itex]\pi[/itex] is an added convenience.

 

[Hootenanny]

I propose that we use the symbol "toopie" to represent this quantity. Toopie is, of course, the symbol [itex]2\pi[/itex]. It's apparent similarity to the product expression of 2 with [itex]\pi[/itex] is an added convenience.

 

[Cyrus]

You nerds. Stop tagging this thing with pi with ever increasing significant digits or the number of tags will approach infinity!