Pi doesnt have reapting random digits

[daveyp225]

No, we can't measure a side of length one. We can define something to be of length one, but that's not the same thing as measuring something. I could just as validly define a certain length to be equal to pi.

And pi is "finite" if you use base pi; it's just 10.

You misunderstood me.

First of all, Pi does not have a specific length! It never ends! How can you define something to be the length of Pi?

Pi can't be counted with anything. A rock, an atom, a ruler..etc. And you can't say a non-finite number is finite. How can you have a base number that is non-finite? You can't raise pi to a number, nor can you divide or multiply my pi. Only symbolicly. If you have ever taken a computer science class with conversion from Decimal to other bases, you'd see why this is not possible. Or even if you could use Pi as a base, that entire system would be useless, since NO numbers would have a finite value. Whats pi^2? Whats 1/Pi^6? Try converting 10 in base Pi to decimal.. Does it ever end?

However an integer can be counted. That is why a length of one can be defined as anything that is countable. I.E. 1 Foot = 12 Inches. 1 Meter = 100 Centimeters. For example, a single atom is countable, as is 10 atoms, 100 atoms..etc. This makes conversion possible. If I define a Unit as 100 atoms, I can find out how many atoms are in 6354 units. You can't have Pi of anything. You could have an extremely close amount of Pi things, but never exactly Pi.

You can't say 1 Unit = 3Pi, because that "1 Unit" would be just as infinite as Pi. You can't say that 1Pi = 3Units either, because One unit would be One third of Pi, which is also non-finite.

Dave

 

[master_coda]

You misunderstood me.

First of all, Pi does not have a specific length! It never ends! How can you define something to be the length of Pi?

Yes, I did misunderstand you. I made the mistake of assuming that you had some understanding of the concept of length that made sense.

It makes no sense to say "Pi has infinite length". Its decimal expansion has infinite length, but that has nothing to do with the size of a number, which is what we care about when we're measuring a physical length. The decimal expansion of 2 has the same length as the decimal expansion of 1, but we don't say that a 1m object and a 2m object have the same length. 1/3 has an infinite decimal expansion, but we don't say that a 1/3 m object has infinite length.

Decimal notation is used because it's a convenient notation for performing arithmetic. Do not attach too much significance to the fact that there are certain numbers that have inconvenient decimal expansions.

 

[Zurtex]

Pi can't be counted with anything. A rock, an atom, a ruler..etc.

Neither can 1, think about it. Find me something exactly 1 metre long.

 

[K.J.Healey]

Neither can 1, think about it. Find me something exactly 1 metre long.

Before 1906, I could have shown you a bar that was exactly, without a doubt, one meter long.

 

[strid]

I remember that some years ago I made a formula to calculate pi...
I don't remember the formula but I remember that the concept I used was that he circle is a polygon with infinite sides... så if x in the formula was number of sides, the higher value I put in the closer i got to pi...

and as the circle had infinite many sides I'äll never come close enough, and hence there are unlimited digits...

 

[Zurtex]

Before 1906, I could have shown you a bar that was exactly, without a doubt, one meter long.

Do you refer to the definition of a metre being:

"the length of a pendulum with a half-period of one second"

or:

"one ten-millionth of the length of the Earth's meridian along a quadrant"

or the modern version established in 1875 by the international treaty the "Metre Convention" which defines it as:

"the length of the path traveled by light in an absolute vacuum during a time interval of exactly 1/299’792’458 of a second"

Or some other definition? And which one of these could you show me exactly without a doubt (to the very infinitesimal) a bar exactly 1 metre long and why?

I remember that some years ago I made a formula to calculate pi...
I don't remember the formula but I remember that the concept I used was that he circle is a polygon with infinite sides... så if x in the formula was number of sides, the higher value I put in the closer i got to pi...

and as the circle had infinite many sides I'äll never come close enough, and hence there are unlimited digits...

1/2 + 1/4 + 1/8 + ... + 1/(2^n) + ... = 1

That doesn't prove 1 is irrational and neither does your proof prove that Pi is irrational. Unfortunately it's not easy to prove the irrationality of certain numbers like Pi. If you get far into mathematics you’ll find that infinite sequence and series are quite common.

 

[Chronos]

Pi has been known to be irrational since the proof by Lambert in 1768.

 

[daveyp225]

Yes, I did misunderstand you. I made the mistake of assuming that you had some understanding of the concept of length that made sense.

It makes no sense to say "Pi has infinite length". Its decimal expansion has infinite length, but that has nothing to do with the size of a number, which is what we care about when we're measuring a physical length. The decimal expansion of 2 has the same length as the decimal expansion of 1, but we don't say that a 1m object and a 2m object have the same length. 1/3 has an infinite decimal expansion, but we don't say that a 1/3 m object has infinite length.

Decimal notation is used because it's a convenient notation for performing arithmetic. Do not attach too much significance to the fact that there are certain numbers that have inconvenient decimal expansions.

If you reread what I had typed, you'll see I never said Pi has an infinite length, just that the decimal never ends.

3.14, 3.141, 3.1415, 3.141592, 3.1415926.. etc

The closer you approximate Pi, the larger your approximation becomes! Not just in decimal length, but in numerical value as well. Even though Pi never reaches 3.15, it keeps getting closer and closer to it. Same with 3.142, 3.1416, 3.141593.. You can't measure a length that is always changing with no correleation to time. You can only say that it is about so-so long. If you meant to say that 1 atom cannot be measured exactly, so that many atoms cannot be measured exactly, that's very different. The only reason we can measure anything, is because we can define a length of 1 to be a certain number of something else. This way of thinking would lead you to believe that nothing can be measured, ever, but since the atom is so very small, it is a good starting point to define a system of measurement, and non-repeating, non-terminating decimals as lengths cannot be expressed by something even one trillionth the size of an atom.

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

 

[master_coda]

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.

1 can be counted. But counting is very different than attempting to measure a continuous value. You can produce exact counts, but you cannot use counting to perform every kind of measurement.

Even if I take some object and say "this thing has length 1", I still cannot measure things of length one exactly. Any measurement you make using your special object will still be an approximation.

And you're still continuing to make the mistake of thinking that measurements made with integers or rational numbers are somehow "true measurable lengths". When measuring a continuous value, all measurements are approximations. Finite decimal approximations are not the basis of measurement. It doesn't matter if the approximation is a terminating decimal or a repeating decimal or an irrational number.

 

[daveyp225]

1 can be counted. But counting is very different than attempting to measure a continuous value. You can produce exact counts, but you cannot use counting to perform every kind of measurement.

Even if I take some object and say "this thing has length 1", I still cannot measure things of length one exactly. Any measurement you make using your special object will still be an approximation.

And you're still continuing to make the mistake of thinking that measurements made with integers or rational numbers are somehow "true measurable lengths". When measuring a continuous value, all measurements are approximations. Finite decimal approximations are not the basis of measurement. It doesn't matter if the approximation is a terminating decimal or a repeating decimal or an irrational number.

What I typed above completely agreed with what you are saying. In fact, right before the paragraph you quoted.

I do, however, disagree with your notion that you cannot compare two things that are of the same units. 10 atoms arranged side by side will always have the same length as 10 atoms of the same type arraned in the same order. If one unit of another system is defined as those 10 atoms, then both segments would have a length of one in that unit system. What is the true "length" of those units? There is none. As I stated before, all measurements are based on finite, real, physical things. I can't give a specific example, but it makes sense to say you cannot measure a mile without something smaller than a mile. You cannot measure a milimeter unless you have something smaller than a milimeter. You cannot measure an atom unless you have something smaller than an atom. Same with Time. How long is a atto-second? Or do we use the attosecond as a basis for measuring time?

But if we base our measurement system on the "length" of the smallest known thing, then all other measurements can be considered approximations, like you said, or they can be said to be exact values in that measurement system, even though the true "length" can never be known. Pi (the number, not the symbol) cannot be measured in any measurement system, nor can you base a measurement system on it.

Dave

 

[NateTG]

If you reread what I had typed, you'll see I never said Pi has an infinite length, just that the decimal never ends.

3.14, 3.141, 3.1415, 3.141592, 3.1415926.. etc

The closer you approximate Pi, the larger your approximation becomes! Not just in decimal length, but in numerical value as well. Even though Pi never reaches 3.15, it keeps getting closer and closer to it. Same with 3.142, 3.1416, 3.141593..

Interesting I have a series of approximations:
4, 3.2, 3.15, 3.142, 3.1416, 3.14160...
that gets closer and closer to 3, and gets *smaller* as it gets closer to pi. Just because you chose an increasing bunch of approximations doesn't mean that any family of approximations has to be increasing. Moreover, pi does not get closer and closer to 3, althought the approximations that you're using may.

 

[master_coda]

But if we base our measurement system on the "length" of the smallest known thing, then all other measurements can be considered approximations, like you said, or they can be said to be exact values in that measurement system, even though the true "length" can never be known. Pi (the number, not the symbol) cannot be measured in any measurement system, nor can you base a measurement system on it.

That's an interesting approach, but it still doesn't support the idea that the measurement must use 1 as the base unit of length. You can't just suggest a technique for measuring and then grandly assert something about all systems of measurement. Especially when you're so inconsistent in your arguments; you agree that we can't make exact measurements of the "true values" of things, but then you assert that we can just arbitrarily declare integer based measurements to be exact, and then immediately afterwards you assert that we can't do the same thing using the value of pi without giving any actual reason for this restriction.

 

[daveyp225]

Interesting I have a series of approximations:
4, 3.2, 3.15, 3.142, 3.1416, 3.14160...
that gets closer and closer to 3, and gets *smaller* as it gets closer to pi. Just because you chose an increasing bunch of approximations doesn't mean that any family of approximations has to be increasing. Moreover, pi does not get closer and closer to 3, althought the approximations that you're using may.

Maybe I don't always say what I mean to say.

I meant that as the amount of Pi's decimal digits increase, its numerical value increases also.

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......

Each number in that list is greater than the number that precedes it in both decimal digits and numerical value, just as 1.11 is greater than 1.1. I was not trying to imply that there is one method of approximating Pi, as there are several.

Dave

 

[daveyp225]

That's an interesting approach, but it still doesn't support the idea that the measurement must use 1 as the base unit of length. You can't just suggest a technique for measuring and then grandly assert something about all systems of measurement. Especially when you're so inconsistent in your arguments; you agree that we can't make exact measurements of the "true values" of things, but then you assert that we can just arbitrarily declare integer based measurements to be exact, and then immediately afterwards you assert that we can't do the same thing using the value of pi without giving any actual reason for this restriction.

I don't see my arguments as being inconsistent. I simply tried to use your way of thinking to explain what I was talking about.

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.

That whole tangent on defining Pi to be a length was just a supplement to my original question. Trying to use your way of thinking, I attempted to show how nothing can truly be measured, but you can use integer values (or even finite decimal values) to represent the exact length of real physical things. For instance, a piece of wood conatining exactly all the same types of atoms, has 1 billion atoms from one end to the other. If there is a unit defined as one WoodMeter = 20,000,000 atoms, then the piece of Wood would be exactly 50 WoodMeters long. But although this is an "exact" measurement in WoodMeters, what WoodMeters are based on (atoms) may not be able to be measured because there is nothing smaller to measure them with. Unlike my WoodMeter, Pi can never be a number to represent "how many" of something we have, or how many Pis we can make from X amount of units.

It is very hard (impossible) to approximate how small is infinitely small, so therefore, there will always be a size smaller than the smallest size. That is why a true "length" is impossible to have, since there is nothing to base it on. Using something to base your measurements can give an exact value in those units.

Dave

 

[DeadWolfe]

Isn't this topic really more about physics than math? The topic creator appers to be questioning how well the real numbers match the physics world, which is not at all related discovering properties of them.

 

[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!