Pi doesnt have reapting random digits

[daveyp225]

Does the fact that it was not obvious to Zeno that the sum over k from 1 to n of (1/2)^k converges to 1 as n goes to infinity make it the case that this series does not converge to 1? Just because you can't tell what something is converging to by looking at it doesn't mean that it doesn't converge.



I am not agreeing with or repeating your statements. In fact I explicitly disagree with the statements in the preceding quotation. A circle of radius 1 has circumference equal to exactly 2pi. The real number "pi" is just as "exact" as the real number "1." If I like, I can choose to express numbers in terms of sums of powers of pi with coefficients chosen to be smaller than pi, in which case pi = 10, 2pi = 20, 2 pi^2 = 200, and 1=1. Of course, in this representation, what is 4? Not pretty!

Pi cannot be used as a base for any useful purpose. Whenever you convert ANY number in base Pi to decimal (basically the only useful base aside from computer science applications) you will get a number just as random as Pi!

I am also explicitly disagreeing with you. Pi is NOT exact! Therefore a circumference of 2Pi is NOT exact! Where would you plot the point when Theta=Pi on your graph of the circle? Wherever you plot it, it is wrong!

Dave

 

[HallsofIvy]

I am also explicitly disagreeing with you. Pi is NOT exact! Therefore a circumference of 2Pi is NOT exact! Where would you plot the point when Theta=Pi on your graph of the circle? Wherever you plot it, it is wrong!

Dave

You seem to have trouble distinguishing between mathematics and applications of mathematics (applications are, of necessity, approximate as opposed to the exact definitions of mathematics).
Pi is a specific number. It is every bit as "exact" as 1 or 3 or 1/2. It's value does NOT depend upon actually measuring some physical object approximating a circle.'

Even if we were to look at a circle made up atoms of a specific type (which, I've just said is irrelevant to the mathematical value of pi), it seems to me we would have difficulty (remembering the quantum nature of such things) determining exactly where one atom ends and another begins- so the nature of such a circle is not as clear as you might think.

 

[daveyp225]

You seem to have trouble distinguishing between mathematics and applications of mathematics (applications are, of necessity, approximate as opposed to the exact definitions of mathematics).
Pi is a specific number. It is every bit as "exact" as 1 or 3 or 1/2. It's value does NOT depend upon actually measuring some physical object approximating a circle.'

Even if we were to look at a circle made up atoms of a specific type (which, I've just said is irrelevant to the mathematical value of pi), it seems to me we would have difficulty (remembering the quantum nature of such things) determining exactly where one atom ends and another begins- so the nature of such a circle is not as clear as you might think.

Just because I am arguing something doesn't mean I believe it. Did Zeno believe motion was impossible? Of course I can picture a perfect circle in my head, and there are no gaps in its graph. I was just having a bit of fun with the terminology. There's nothing wrong with that. Many discoveries came from just challenging what was, at the time, an unchallengeable idea.

Dave

 

[HallsofIvy]

On the other hand if Zeno had said "Motion is impossible because the sky is blue" no one would have paid any attention to him. "Challenging" something with patently invalid arguments isn't helpful.

 

[daveyp225]

On the other hand if Zeno had said "Motion is impossible because the sky is blue" no one would have paid any attention to him. "Challenging" something with patently invalid arguments isn't helpful.

What is so invalid about Pi not being exact? What is so invalid about saying you cannot plot an exact point on a polar graph when Theta = Pi? And for those of you who believe it is exact and finite, where is your proof? Just to make clear, I mean finite in the sense that it has a definite value, not an unbounded one.

 

[Hurkyl]

Your problem, I believe, is that you have your own pet meaning of the word "exact".

In any case, this seems to have gone on quite far enough.

 

[HallsofIvy]

What kind of proof would you accept? pi is a specific value. It is, among other things, half the fundamental period of f(x)= sin(x) which can be defined and calculated without reference to geometry. pi can be shown to be equal to the sum of certain infinite series- and it is well known that if a series has a sum, then it is unique. There's nothing more precise than that!