Question

C=2πr

Answer 1

how come?

Answer 2

is this purely an experimental result?

Answer 3

circumferamce of a circle.

Answer 4

in the sense that 1+1=2 is an experimental result, then yes I guess
but I wouldn't call it experimental

Answer 5

Well. Pi is the ratio of circumference and diameter shared by EVERY circle.

\( \color{Black}{\Rightarrow \pi = {C \over d } }\)

Now, solve for C.

\( \color{Black}{\Rightarrow \pi d = C }\)

\(d = 2r\) by definition.

\( \color{Black}{\Rightarrow \pi (2r) = C }\)

Rearranging:

\( \color{Black}{\Rightarrow 2\pi r = C}\)

Answer 6

how do you know that @ParthKohli /

Answer 7

Know what, Unkle?

Answer 8

\[\pi=\frac Cd\]

Answer 9

That is supposed to be the definition of \(\pi\).

Answer 10

well is it?

Answer 11

here are many perspectives on what, how, and why pi is what it is
http://www2.fiu.edu/~rudomine/pi.pdf

Answer 12

hmm

Answer 13

pi, like e, comes from the assumption that there exists some constant that will solve some particular problem. For e, it is defined as the exponential base for which the derivative at x=1 equals itself. The value of e can be derived from that.
similarly if you read the link, you will see that the logic that circles are directly proportional to each other lead us to search that that constant of proportionality. Finding its actual value took a long time obviously...

Answer 14

so it is the result of experiment rather than deduction?

Answer 15

well if it was \(just\) the result of experiment then we wouldn't know that pi is a trancendental number since our instruments aren't precise enough to measure the exact ratio of two circles to the degree of specificity that would be required to find the full value of pi (it is an infinitely long decimal number, so it would take an infinitely precise instrument to measure, right?)

Answer 16

could we know pi is transcendental without experiment?

Answer 17

By experiment we can deduce roughly the value of pi, but the special properties it has like the fact that it is transcendental can be found only through mathematical analysis. How could we experimentally prove that the value of pi is what it is beyond the limits of the uncertainty principle? QM would get in our way if we tried it that way. How exactly pi is proven to be trancendental I do not know, you may have to ask one of the smart guys like @Zarkon @asnaseer about that

Answer 18

what are transcendental numbers anyway/?

Answer 19

numbers that like \(\sqrt2\) are non-rational and non-repeating decimals, but further cannot be expressed as irrational either. i.e. numbers for which we can only write a formula to represent the exact value, like pi and e.

Answer 20

so \(\phi\) the golden ratio is irrational
by not transcendental because it can be written with √5 's in it ,

what are any other trancendtals ?

Answer 21

good question, let's google it :)
I only use pi and e... but I bet there are more

Answer 22

but yes, the golden ratio is irrational, not transcendental (I'm pretty sure...)

Answer 23

hm, looks like \(2^{\sqrt2}\) is also transcendental. interesting...

Answer 24

hmm

Answer 25

"In mathematics, a transcendental number is a (possibly complex) number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients." -Wikipedia
so my definition of a transcendental number was not quite right I guess
I can't find a proof that \(\pi\) is transcendental, but the one for e makes me hesitant to try to understand this whole concept of proving a number as transcendental

http://en.wikipedia.org/wiki/Transcendental_number#Sketch_of_a_proof_that_e_is_transcendental

http://en.wikipedia.org/wiki/Transcendental_number