When Diameter is 7 units Circumference is always 22 units And that is basis of Pi But for that to be true, one has to do measurements in nanometers or even more finer. So who and how did he do that and when ? Or is Pi a HOAX ?
Basically whatever pi you use, it must be rational... To say pi is irrational AND transcendental in theory is beyond the accuracy of the math techniques used to quantify it..
You're confusing mathematics with physics again. There is no issue of accuracy in mathematics, because there are no measurements. That is the realm of physics. What you should be saying is that any physics experiment devised to check the value of pi will result in a rational approximation, due to finite precision. It follows that no finite set of experiments can definitively tell us whether the mathematics describing the geometry of circles corresponds exactly to physical reality. But that pi is irrational and transcendental is not an issue of accuracy, nor of serious doubt. Indeed, there do not seem to be any good reasons to question the validity of there basic geometric results, which have withstood thousands of years of scrutiny. Do you really think that you wouldn't get an irrational result if you were able to measure the diameter and circumference of a sphere with perfect accuracy? Do you not think that the result of any finite-precision experiment would grow closer and closer to pi as the accuracy improved?
Well argued Quadraphonics. Thanks all for a nice discussion. I 'wish' pi was (4/3)^4..... it would make physics so much more consistent. eg [(4/3)^5]*r^3 volume of a sphere (4/3)^-9 Newton's G N * (4/3)^-8n gravity planetary power wave etc..... life would be so consistent, LOL... why the 4 quarters(whole...cube) divided by 3 (???? dimensions) must be a reason... but I shall continue to use the sumbol 'pi' instead of any value, I suppose you can't have everything. One day I may understand the deeper secrets of spherics.
I'm not really sure why people think it's such a big deal that we cannot compute pi to an exact number... We've calculated to a billion (I think this is the best number so far) decimals.... Expansions of pi are exact values if we had an infinite amount of time. So if we can calculate pi to within whatever amount of precision you want, what is the problem?
Pi is irrational. Also sqrt(2) is irrational. However 2 is rational, so could pi² or pi³ or so on be a rational number ?
No. Pi is transcendental over the rationals. If Pi^n was rational, i.e. Pi^n=a/b where a and b are both integers, then Pi would be a solution to bX^n - a = 0 over the rationals. Which is impossible since Pi is transcendental.
Well, as Human001 explained, pi^n isn't going to be rational, but there are still some open questions. Since bother e and pi are transcendental, there's the question of whether certain combinations of them are transcendental, irrational, etc. I think e/pi is one of the unknown cases? The proof that pi is transcendental basically starts with the equation e^(i*pi) = -1, notes that e is transcendental, and concludes that pi must be also in order to get e^(i*pi) to be an integer.
I am not sure. 10^(log2) = 2 , log base 10 sqrt(3) ^2 = 3 Please Register or Log in to view the hidden image!
I have a problem with that proof. e^(a*i) = cos a + i sin a is a defined equation, not a derived one. You can't just define some relation and then use it to prove a result in a field that doesn't rely on that definition, such as real analysis.
Err, not clear on what you mean by "defined" as opposed to "derived", but how can a (real) number be transcendental in complex analysis but not in real analysis? I haven't seen a proof of pi's trescendence that doesn't use complex numbers; anybody know if there is one?
I mean that the complex extension of e^x (i.e. e^z) was defined to be e^x(cos y + i sin y). It wasn't a derived result. Sure, it was defined that way to be analogous to e^x in real analysis, but the fact remains that you can't get to e^pi*i = -1 without assuming a somewhat arbitrary definition for e^z.
http://mathworld.wolfram.com/BBPFormula.html Please Register or Log in to view the hidden image! >>The BBP (named after Bailey-Borwein-Plouffe) is a formula for calculating pi discovered by Simon Plouffe in 1995,>>> looks like the "mathematical pi" is a manufactured item... fancy math, redefinitions etc LOL this "mathematical pi" does not even show any evidence of it being 'integrated' (no not calculus)=[cross-linking] with other spherical constants.... Long live pi = ( 4/3 )^4 4/3 rules for some reason, IMO it rocks.
GMontag: The extension of the exponential function to complex numbers is usually defined as the standard power series: exp(z) = <sup>inf</sup>Σ<sub>n=0</sub> z<sup>n</sup> / n! where z can be any complex number, instead of any real number. This is chosen because the power series has infinite convergence radius, and is it's own derivative, and furthermore behaves very nicely. Euler's formulas are derived from this, not the other way round. e is nothing more than exp(1), but it is standard notation to write e<sup>z</sup> when what is meant is exp(z).
Right, I see what you mean. All of the proofs of transcendence for pi (and e) that I've seen make use of Euler's Identity; I don't know that there are any that don't involve complex numbers. I'm not sure this is such a big deal though. In what setting would you want to talk about transcendence but exclude complex analysis?
Yes, you're right. No, because all the other mathematical definitions produce the same value for pi, whereas yours does not. So, yours will never become popular. Bad luck.
