Re: pi & 2 pi
- From: "Freiddy" <fei.yuanbw@xxxxxxxxx>
- Date: 24 Jan 2007 11:57:15 -0800
I understand that "pi" is just a label made by mathematicians to
indicate a certain constant in the "Idealized Platonic world" thus any
expression of pi (e.g. 2*pi) is equally valid to define its value. But
I'm not sure whether "pi" & "tau" & other expressions of pi (e.g.
pi^0.5) are in equal footing (that is, equally important, after all,
some are more frequent and useful than others) in math. Nonetheless, it
is often preferred that such things are to be left alone, and I do
accept the current "pi", since such matters are of little importance.
However, I do believe that in certain situations, notation does play a
role (even if not a mathematical role). Personally I find the math
notations in this world is very messy. There are so many conflicts in
the current system (e.g. tensor notation & powers). I think they should
be solved someday. (Like the language analog, a universal language
would be better than thousands of different ones)
Freiddy
On Jan 24, 3:00 am, "Jon Slaughter" <Jon_Slaugh...@xxxxxxxxxxx> wrote:
"Freiddy" <fei.yua...@xxxxxxxxx> wrote in messagenews:1169585167.901363.170340@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
I just want to raise a simple question, why is the constant Pi definedgets confused or frustrated by having to write a few extra multiplicative
as the ratio between the circumference and diameter not radius?
If, say, pi, was never defined this way, instead, a constant "tau"
(it's just a name tag) is defined to be the ratio between the
circumference and the radius, then how would the world of maths be?
Personally I think it <might> be more natural, since then a full period
of the y=sin(x) would be "tau" and not the unnatural-looking "2*pi". It
would also make a complete revolution in radians one "tau" rather than
two "pi".
I'm aware that if so, the formula, e^(i*pi)+1=0 would then have to be
e^(0.5*i*tau)+1=0, and the area of a circle would be 0.5 tau r^2 (but
then again, area of a triangle is 0.5 b h).
Are there any other benefits of using "tau" / or troubles?Should any serious mathematician being worrying about such things? If he
constants here and there then what's he doing working with numbers in the
first place?
Your problem is analogous to many others, One being the fourier transform
pair. You can put your constant factor in front of the transform, in front
of the inverse transform, or in both(ofcourse not the same as the previous
factors). In all cases your results will be identical, they just look
different.
Its like saying 2/2 is different than 1. Ok, sure it uses different symbols
but they represent the same mathematical concept. This is what math is all
about. finding out that 2/2 is the same is 1 is no big deal but the same
thing applied some some advanced relation is much more satisfying. This is
why math has all those cool identities.
For example, the abstract object zeta(x) is "idential"(Well, maybe not
completely but they overlap) to sum(1/k^x,k=1..oo).
Now the first time this function was defined it might have been defined that
way but then later on someone found a new way to express it. e.g., zeta(x) =
1/prod((1 - p^(-x)),p = prime). There are hundreds of these expressions for
the same abstract concept.
Now suppose that we defined zeta(x) := Q*sum(1/k^x,k=1..oo).
then the other identify would be
zeta(x) = Q/prod((1 - p^(-x)),p = prime).
Q could be anything. It could be another complicated expression. Pretty much
all of mathematics is based on finding these relationships. The relationship
is what is important and not how you symbolically notate it. Sure the
symbolism can make things easier to remember or write but that should be of
very little concern to a true mathematician unless there specically
intersted in symbolism.
The fact of the matter is that you won't ever get away from things like
this. Its sorta like conservation of energy. You might be able to hide some
energy to make something else easier to do but that energy will pop up in
some other place. You'll never be able to get away from it in the grand
scheme of things.
If I were you I'd worry about more important things. Knowing that C =
2*Pi*r is the exact same thing as C = Pi*d. Its also pretty obvious since d
= 2*r. did you know that d = cos(34)*zeta(34)*q so that C =
Pi*cos(34)*zeta(34)*q?
Whats q? well its q = d/cos(34)*zeta(34). It might also be something else
like a special point on a line intersecting the center of the circle that
represents some odd ball property.
You can think of the problem in a way similar to natural languages. We can
write/speak about the same concepts in just about any language and even
though in some languages its easier to do than others utlimately the
abstract thing we are talking aout is not dependent on the language(sure
there are some issues with this analogy but lets not complicate things).
Jon
.
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