Pi doesnt have reapting random digits

[master_coda]

My argument is simple: Pi is not finite, so you cannot base anything on its specific value. My original problem consisted of completing one single revolution of plotting a circle. On the Graph of r = 0.5, when theta = Pi, there could be no point, and the circle could not be drawn. If you plotted at a very close value of pi, there would be an indent in the curve. This all mental, of course. I am not suggesting a perfect circle could exist physically.

But saying "Pi is not finite, so you cannot base anything on its specific value" doesn't mean anything. Pi is finite, like all real numbers. And it's perfectly well defined, so we don't have any reason to assume that we can't base anything on its value.

 

[daveyp225]

But saying "Pi is not finite, so you cannot base anything on its specific value" doesn't mean anything. Pi is finite, like all real numbers. And it's perfectly well defined, so we don't have any reason to assume that we can't base anything on its value.

Pi is finite? Is there some new breakthrough I have not heard of?

And I said you cannot base anything on its specific value, since it does not have one. We base many ratios having to do with circles, most notably, its circumference, on an approximation of pi. In the real world, so far, it has been fine for engineers and physicists, so I don't see a problem with just using an approximation. It just interests me, that's all.

 

[master_coda]

Pi is finite? Is there some new breakthrough I have not heard of?

And I said you cannot base anything on its specific value, since it does not have one. We base many ratios having to do with circles, most notably, its circumference, on an approximation of pi. In the real world, so far, it has been fine for engineers and physicists, so I don't see a problem with just using an approximation. It just interests me, that's all.

All real numbers are finite.

Pi has a very specific value that's very specifically defined. In order to do this, you have to think at a level of abstraction higher than that of a pocket calculator, but that's not too much to ask. Math is not grade school arithmetic.

 

[Data]

I can define pi to be the arclength of the curve defined by the function f(x) = sqrt(1-x^2) from x=-1 to x=1. Is this any more imprecise than if I were to define 2 to be the area between the x-axis and the curve defined by the function g(x) = 2/x^2 for x>1? Why?

If you insist on measuring using atoms, then how would a measurement of 3.25 atoms be any more precise than a measure of pi atoms? You seem to be insisting that integers are the only numbers that are 'finite.'

Let's say that I define a new measurement system. We'll assume that we can measure down to the width of an atom to infinite precision. We then decided to write that one atom has a length equal to the sum of the sequence {2(3)^(1/2-k)(-1)^k/(2k+1)} for all integers k>=0. Then two atoms have a length equal to two times the value of this series in our measurement system, and so on. What is imprecise about this?

 

[NateTG]

Using exact, unrounded approximations of Pi, we have : 3.1, 3.14, 3.141, 3.1415, 3.14159, 3.141592, 3.1415926, 3.14159265, 3.141592653, 3.1415926535, 3.14159265358, 3.141592653589...

Um, what precisely is an 'exact, unrounded approximation of Pi'? All of the approximations I listed are 'exact, unrounded approximations' as well.

How about fractional approximations:
\frac{3}{1},\frac{13}{4},\frac{16}{5},\frac{19}{6},\frac{22}{7}...
are they 'exact unrounded approximations' of Pi?

The fact that the series of approximations you see is increasing is an artifact of the way we write down numbers and has nothing to do with the properties of \pi in particular.

For example, consider
9,9.9,9.99,9.999... \Rightarrow 10
That's a distance, and it should be 'exactly measurable', but all of the arguments that you make about \pi apply here.

 

[daveyp225]

I can define pi to be the arclength of the curve defined by the function f(x) = sqrt(1-x^2) from x=-1 to x=1. Is this any more imprecise than if I were to define 2 to be the area between the x-axis and the curve defined by the function g(x) = 2/x^2 for x>1? Why?

If you insist on measuring using atoms, then how would a measurement of 3.25 atoms be any more precise than a measure of pi atoms? You seem to be insisting that integers are the only numbers that are 'finite.'

Let's say that I define a new measurement system. We'll assume that we can measure down to the width of an atom to infinite precision. We then decided to write that one atom has a length equal to the sum of the sequence {2(3)^(1/2-k)(-1)^k/(2k+1)} for all integers k>=0. Then two atoms have a length equal to two times the value of this series in our measurement system, and so on. What is imprecise about this?

Maybe if you didn't just read my first post, you'd see you are repeating what I had been saying. Are you disagreeing that the length of 2 atoms of the same type are of different sizes? No? Well then you are agreeing with me. The problem with Pi is that it can not be measured with anything physical, where as an atom can measure anything that made up of the same atoms. That is, any natural number multiple of a unit can be measured with the same unit.

 

[daveyp225]

Um, what precisely is an 'exact, unrounded approximation of Pi'? All of the approximations I listed are 'exact, unrounded approximations' as well.

How about fractional approximations:
\frac{3}{1},\frac{13}{4},\frac{16}{5},\frac{19}{6},\frac{22}{7}...
are they 'exact unrounded approximations' of Pi?

The fact that the series of approximations you see is increasing is an artifact of the way we write down numbers and has nothing to do with the properties of \pi in particular.

For example, consider
9,9.9,9.99,9.999... \Rightarrow 10
That's a distance, and it should be 'exactly measurable', but all of the arguments that you make about \pi apply here.

You are saying that 22/7 is an exact, unrounded approximation for Pi?
3.142857... You're right, this is not rounded, but I would hardly call this an approximation. Maybe in 4th grade it was.

I see what you are saying, but you just don't understand what I was trying to make a point of, that 3.14 is an "Exact" unrounded approximation for Pi, but 3.142 is not. 3.141 is, 3.141X (x not being 5) is not. Maybe what I said wasn't the proper way of saying it. I won't try to repeat myself again, as I am growing a bit bored of this topic.

And in your example, you are sort of proving my problem. It is true that 0.9999... is equal to one, but 0.9999 is not. It is obvious where 0.9999... is heading to. But where is 3.1415926535898 et cetera heading to? You can't use the ... notation in this case because there is no pattern. For example, 0.49999999... is obvious as a pattern going to 0.5. 0.49597993 et cetera cannot be precieved as having a definite limit.

Dave

 

[Zurtex]

And Zurtex.. 1 can be counted and measured. Maybe not one meter, but one atom. If you define a unit to be 10 atoms, you can measure something that is 100 atoms long, having an EXACT length of 10 in our defined units. So one tenth of that length would have a length of exactly one. My whole point here was that you cannot say pi has a finite length, just that its length can be approximated by a true measureable length of some other form.

Dave

Counted and Measured are 2 very very different things. If you define a length to be 10 atoms long then which atoms do you choose? You assume all atoms are exactly spherical, never change length and are all exactly the same length? It would certainly be in an interesting universe were that true. Also where would you consider the boundaries of an atom? What model of an atom are you basing this length on? How would you measure such atom? How would you ever be able to exactly measure a length of 10 atoms as atoms don't exactly like be right next to each other? etc.. etc..

 

[Data]

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.

And I said you cannot base anything on its specific value, since it does not have one. We base many ratios having to do with circles, most notably, its circumference, on an approximation of pi.

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!

 

[daveyp225]

Counted and Measured are 2 very very different things. If you define a length to be 10 atoms long then which atoms do you choose? You assume all atoms are exactly spherical, never change length and are all exactly the same length? It would certainly be in an interesting universe were that true. Also where would you consider the boundaries of an atom? What model of an atom are you basing this length on? How would you measure such atom? How would you ever be able to exactly measure a length of 10 atoms as atoms don't exactly like be right next to each other? etc.. etc..

In this case, counting IS measuring! I am using logic, not a ruler! Two atoms of the same type should be the same size. My argument here is asuming mentally the atom's are right next to each other, just as you assume mentally that Pi is an exact value! Although, in any case, assuming anything can be bad.

This thread is turing more into a philosophy topic than a math one.

dave

 

[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!