Pi doesnt have reapting random digits

[fizzzzzzzzzzzy]

Supposedly Pi has infinite and random digits. Therefore pi cannot be a fraction unless the top numeber or the bottom number is a number with infinite and reapiting random digits. so therefore the diameter or the circumference must always be a infinite and random digited. BUT, this isnt true. if you measured all the way down to the very last atom. you would have the exact measurement. and that exact measurement would not have infinite and reapiting random digits. But if that is true, Pi must either be limited or reapiting

[Ba]

Measure a circle's circumference, a loci of points equadistant fom one foci on a plane, and you measure from a concept, atom's aren't small enough, quark's aren't small enough because a point has a definition of having no dimensions in any directions. The Pi we use is an approximation, approaching the asymptote that is pi. A circle is a concept that we have no way to measure acuratly, therefore we will not actually find a decimal.

[robert Ihnot]

BUT, this isnt true. if you measured all the way down to the very last atom. you would have the exact measurement. and that exact measurement would not have infinite and reapiting random digits. But if that is true, Pi must either be limited or reapiting

This is really a very interesting and even a deep observation, but it does not take in account the discovery of the Greeks, namely the existence of the "unutterables," i.e. numbers that are not the quotient of two integers.
(Previously it had been assumed that all numbers were such, we just had to find a large enough denominator, which is a way of talking about going all the way down to the atoms, etc.)

For example: \sqrt{2} is not the quotent of two integers. That is why some numbers are called, "irrational."

See http://www.mathacademy.com/pr/prime/articles/irr2/index.asp

[fizzzzzzzzzzzy]

you could measure it exactly. If you got a piece of string that was exactly 5 inches long(measured down to the very atom),and made a circle from it, the circumference could be measured exactly as 5 inches. then if you measured the diameter of that circle to the very last atom, it would have an absolute diameter and circumference

[zefram_c]

If you got a piece of string that was exactly 5 inches long(measured down to the very atom),and made a circle from it, the circumference could be measured exactly as 5 inches. then if you measured the diameter of that circle to the very last atom, it would have an absolute diameter and circumference You can not form a perfect circle from any physical object, since it requires an infinite amount of fine-graining. So yes, you could measure the perimeter and diameter down to the atom (in fact I believe the LIGO experiment is sensitive to displacements of a few atoms), but their ratio would not be exactly pi. pi is one of many real numbers that cannot be physically constructed nor calculated in its entirety (though it can be calculated to arbitrary precision)

[LastOneStanding]

you could measure it exactly. If you got a piece of string that was exactly 5 inches long(measured down to the very atom),and made a circle from it, the circumference could be measured exactly as 5 inches. then if you measured the diameter of that circle to the very last atom, it would have an absolute diameter and circumference

True, and at this point you've run into the limitation which is the reason a true circle, by it's defintion, cannot physically exist. As zefram c said, a true circle, expressing the true ratio of pi, would require an infinite amount of fine graining. The problem is, atoms are not infinitesimally small. They themselves have physical extent, so "zooming in" on the circle, to the point where you can see the individual atoms will show that your circle isn't a perfect circle, since its outer edge isn't perfectly smooth. The problem with your argument that pi cannot be irrational is that you take something that only can exist in abstract thought, namely a true circle, and try and place into the physical world.

[Felis]

you could measure it exactly. If you got a piece of string that was exactly 5 inches long(measured down to the very atom),and made a circle from it, the circumference could be measured exactly as 5 inches. then if you measured the diameter of that circle to the very last atom, it would have an absolute diameter and circumference

Yes. As the others mentioned, you cannot make a perfect circle from a physical object without infinite fine-graining.

Another way to look at it is that because the size of an atom limits you, you could basically have a physical circle with 25 miles of string for it's circumference, and would run into the size of the atom as your limit. Although your results would be more "accurate" because the relative size of the atom compared to that of the circle, it is still a limit and therefore is not the "true" pi.

[ZeAsYn51]

I think the important thing to remember is that all numbers have an infinate number of decimals. Even five exact inches is 5.00000000000000000000 where the 0's will continue forever. :smile: Besides no physical measurement can be absolutely exact. It is only in the mind where numbers are exact.

[CRGreathouse]

you could measure it exactly. If you got a piece of string that was exactly 5 inches long(measured down to the very atom),and made a circle from it, the circumference could be measured exactly as 5 inches. then if you measured the diameter of that circle to the very last atom, it would have an absolute diameter and circumference

You're assuming that circles can be drawn exactly with atoms, that atoms are all of the same width, that they are all at exactly the same angle, and that the angle gives rise to a rational measure. These are mostly, if not entirely, incorrect.

[robert Ihnot]

I think the important thing to remember is that all numbers have an infinate number of decimals. Even five exact inches is 5.00000000000000000000 where the 0's will continue forever. :smile: Besides no physical measurement can be absolutely exact. It is only in the mind where numbers are exact.

It seems that part of the confusion about this problem is the difference between a practical measurement and a theoretical result, AS ZeASYn51 has brought up. Pi has been calculated to billions of decimals, but this has absolutely no practical value for manufacturing milk bottles. Similarly to a carpenter, probably a 5 foot plank is no different from one add or subtract a 1/32 of a inch. All measurements are a question of approximation.

[Integral]

you could measure it exactly. If you got a piece of string that was exactly 5 inches long(measured down to the very atom),and made a circle from it, the circumference could be measured exactly as 5 inches. then if you measured the diameter of that circle to the very last atom, it would have an absolute diameter and circumference
How do you know this? Are you positive? Is there not the slightest chance you are wrong? how did you cut the string? Perhaps you had to cut it at the atom NEAREST the actual diameter.

Your physical arguments are meaningless in a mathematical universe.

[CRGreathouse]

Alternately, a circle with physical circ. of 5 inches, with inches defined as a certain integral number of atoms, wouldn't have a diameter measurable in whole atoms -- the last atoms wouldn't quite tough, or wouldn't quite close.

[cogito²]

you could measure it exactly.
Not according to Quantum Mechanics.

[geraldmcgarvey]

"if you measured all the way down to the very last atom"

A circle is not a physical object that can be measured, it is a mathematical object. There's a distinct difference between mathematics, which is used to model the physical world, and the physical world itself.

Pi has been proven to be irrational, it has also been shown to be transcendental, which means it is not algebraic, i.e. it is not the root of any polynomial with rational coefficients. An example of an algebraic number which is irrational is the square root of 2, which is a solution to x^2 - 2 = 0.

[DaveC426913]

"Supposedly Pi has infinite and random digits."

False. They may look random, they may have no definable pattern, but that does not make them random. If I measure a plank of wood as 5.641 feet long, can I say that the number is random?



"All measurements are a question of approximation."

True. Because you said "measurement".


"...all numbers have an infinate number of decimals. Even five exact inches is 5.00000000000000000000 where the 0's will continue forever...."

False

A foot is 12 inches. Not 12.0 inches, not 12.000000000000000000>infinity inches. It is - by definition - 12 inches.

[matt grime]

Actually the best guess is that the digits of pi are indeed random, though you ought to know what the means in this context (Conjecture: pi is normal).

The last thing you wrote probably deserves a reply, too.

The primes are randomly distributed in some sense too, if you're interested - that is if you look at the density of primes it behaves as if they were picked randomly. This is a powerful observation used in Number theory to prove quite difficult theorems.

[ohwilleke]

"Supposedly Pi has infinite and random digits."

False. They may look random, they may have no definable pattern, but that does not make them random. If I measure a plank of wood as 5.641 feet long, can I say that the number is random?

You are correct that Pi's digits are not sochastic (indeed, IIRC, it is possible to determine a digit of Pi in some formulas, without determining all the intervening digits) which is one layman's definition of random. Rather the digits of Pi can be determined deterministically. But, it is also true that they are random in other definitions. E.g. they do not contain repeating sequences and appear in statistically equal proportions independent of the length of the string of numbers you examine.

In contrast, output from certain quantum mechanical processes is both sochastic (to the best of our theoretical knowlege) and has a pattern of digits which meets statistical tests of randomness.

Defining what "random" means is non-obvious. How you define the term "random", depends on why you care whether something is random or not. If you are using digits of Pi as a seed for an encyption program, one definition might be appropriate. If you are using digits of Pi as an example in a discussion about whether we live in a deterministic or sochastic universe, a different defintion would be useful.

[houserichichi]

A foot is 12 inches. Not 12.0 inches, not 12.000000000000000000>infinity inches. It is - by definition - 12 inches.

That's fine and dandy, but if you want to start getting technical with the semantics, how do you define 12? It's equally valid to say 12, 12.0, or 12 with as many zeros after the decimal point. Just the more zeros you have after that decimal place the more exact you are.

In REAL LIFE the number 12 is pretty inexact and we SHOULD be writing things with as many decimal places as measurable. If a foot is 12 inches (no more, no less) by definition then really we should write 12.0000... ad infinitum so we don't get confused with another infinitely close approximation.

[Zurtex]

That's fine and dandy, but if you want to start getting technical with the semantics, how do you define 12? It's equally valid to say 12, 12.0, or 12 with as many zeros after the decimal point. Just the more zeros you have after that decimal place the more exact you are.

In REAL LIFE the number 12 is pretty inexact and we SHOULD be writing things with as many decimal places as measurable. If a foot is 12 inches (no more, no less) by definition then really we should write 12.0000... ad infinitum so we don't get confused with another infinitely close approximation.
Not really, if you are working in integers than 12.0 is a meaningless statement.

[matt grime]

In REAL LIFE

What's real life got to do with a post in a thread in a forum on Number Theory?

[damoclark]

In response to the randomness of Pi. I think we should consider Pi only appears random in the base we choose to define it in. It may not appear random in another number system we choose to construct.

Firstly, a decimal representation of a number is merely just an infinte series, call it a decimal series.
For example, the number 33.333333...

= ....0*10^n +... 3*10^1 + 3*10^0 + 3/10^1 + 3*/10^2 + ..... etc

= 0+0.. 30 + 3 + .3 + .03 + etc

= ...0033.3333333...........

So any number can be expressed as a decimal series. If the number is irrational then the co-efficients in the decimal series appear random and never ending.

The co-efficients of the number "e" when writen in a decimal series appear random, however when we modify the series in which we construct the number, then the co-efficients may not appear random.....

Consider our series to which we want to represent our number has the form..

N = an*n! + a(n-1)*(n-1)! ...... a2*2! + a1*1! + a0*0!
+ b1/1! + b2/2! + b3/3! +..........

Call this an exponential series

Any number can be expressed in this exponential series, just like any number can be expressed in the decimal series.

For N= "e" then our co-efficients become

an=0,... a2=0, a1 = 0, a0 = 1

b1=1, b2=1, b3=1, b4=1, b5=1, b6=1
...... bk=1..... etc.

So in the exponential series we could write..

e = 1.11111111111111111111111........

These co-efficients are not random.

but in the decimal series ...

e = 2.71828.......... (random)

However the number 3.333, which when wriiten in the decimal series has a quite respectable form of 3.333, when written in the exponential series might not appear so pretty.

You can also find a series representation for Pi, where the co-efficients are not random, it just depends on the series you choose to express the number in.


Pardon my notation, I don't know how to write subscipts, if anyone can tell me i'd appreciate it.

[daveyp225]

What makes me have problems concepualizing that Pi has an infinite amount of decimal digits, is that a circle with a Diameter of 1 has a circumference of Pi. If Pi wasn't exact, then this CANNOT be a circle, because the starting point and ending point of the circle would be different (considering one single revolution).

Since the definition of a circle is the locus of point that are all equidistant from a single point, even the infinite amount of points that make up the circle's perimeter, can't draw the entire circumfernece of the circle with diameter Pi. There would be no point at (r, Pi) on a polar representation of the graph, breaking the graph of the circle.

For example, graphing r = 0.5, when theta = pi, the point would be (0.5, pi). Which doesn't exists if Pi is not finite.

Dave

[Zurtex]

daveyp225 could you give me an example of a real life perfect circle?

[DaveC426913]

That's fine and dandy, but if you want to start getting technical with the semantics, how do you define 12? It's equally valid to say 12, 12.0, or 12 with as many zeros after the decimal point. Just the more zeros you have after that decimal place the more exact you are.No - that's the point. It is not a measurement. It is not more or less exact, depending on the number of zeros.

In REAL LIFE the number 12 is pretty inexact and we SHOULD be writing things with as many decimal places as measurable. If a foot is 12 inches (no more, no less) by definition then really we should write 12.0000... ad infinitum so we don't get confused with another infinitely close approximation.If we give it zeros, we are defining it to an arbitrary level of precision. Precision is a property of measurment. This is not a measurment, it is a definition.

[hypermorphism]

No - that's the point. It is not a measurement. It is not more or less exact, depending on the number of zeros.

You misunderstood. Consider: Is there a difference between adding more zeros to the decimal expansion of a measurement or adding more nonzero digits to a decimal expansion of a measurement ?
By looking at the decimal expansion as a measure of exactness, you are considering a measurement. By your analysis, every number is suspect as being "exact" or not. For example, say we had a measure we know to have the value 12. You may then be arguing that 12 is inexact because it requires an infinite number of decimal digits to place the point (See 12.000. All we know then is that the point is between 11.999 and 12.001. So we try to place one at 12.0000. Again the problem results). The problem is not with knowing what the digits are, anyone can get the nth digit of pi as much as the nth digit of 12.
In order to alleviate this "zero bias", let's get out of the arbitrarily chosen decimal expansions of numbers like 1/2 and look at them in other bases, such as the ternary expansion 1/2 = 0.111... ad infinitum. 0.5 doesn't seem so "exact" anymore! The problem you have may be with the construction of the real numbers and their relation (or lack thereof) to the physical world.

[daveyp225]

daveyp225 could you give me an example of a real life perfect circle? I was not talking about in the real, physical world.

I am trying to understand how a perfect circle could exist, even in our minds, since a circle diameter of one would have a circumference of pi, and there would be a discontinuation where theta = pi. Sure you could mentally visualize a radius extending 0.5 units from a certain point, but the circumfernece would me unmeasureble, having no finite length.

Dave

[CrankFan]

I am trying to understand how a perfect circle could exist, even in our minds


x^2 + y^2 = 1

[NateTG]

I am trying to understand how a perfect circle could exist, even in our minds.

Ah, but it doesn't have to. Just like a line segment of length exactly \sqrt{2} doesn't have to.

But we know that as we improve our measurements the ratio of circumference to diameter (or of one side of a square to the diagonal) is going to get closer and closer to \pi (or \sqrt{2}). Therefore, we could talk about a perfect ratio without actually having a 'perfect circle' (in our minds or physically).

In practice, mathematical objects like the mathematical notion of circle are 'perfect' and represent things similar to this 'perfect ratio' they're useful abstractions.

[Zurtex]

I was not talking about in the real, physical world.

I am trying to understand how a perfect circle could exist, even in our minds, since a circle diameter of one would have a circumference of pi, and there would be a discontinuation where theta = pi. Sure you could mentally visualize a radius extending 0.5 units from a certain point, but the circumfernece would me unmeasureble, having no finite length.

Dave
But the circumference does have a finite length. Do not get too caught up in the way we represent numbers, for example if you draw a right angled triangle with the two adjacent sides to the right angle of length 1 unit, you will get the hypotenuse being of length \sqrt{2}. It really doesn't matter how we represent numbers, it's just what makes it easier for us, for example in decimals:

\sqrt{2} = 1.41421356237310 \ldots

Where as it can be much more easily represented as the continued fraction:

\sqrt{2} = [1 ; \overline{2,2}]

But try coming up with a ruler that nicely represents continued fractions.

[daveyp225]

But the circumference does have a finite length. Do not get too caught up in the way we represent numbers, for example if you draw a right angled triangle with the two adjacent sides to the right angle of length 1 unit, you will get the hypotenuse being of length \sqrt{2}. It really doesn't matter how we represent numbers, it's just what makes it easier for us, for example in decimals:

\sqrt{2} = 1.41421356237310 \ldots

Where as it can be much more easily represented as the continued fraction:

\sqrt{2} = [1 ; \overline{2,2}]

But try coming up with a ruler that nicely represents continued fractions.

Yes, a "circle" in the real world will have a finite circumference. As someone stated in these posts somewhere, even if you estimated to the last atom, the number would be incredibly close to pi, but not equal to pi. So a circle with a circumference of pi could not be measured physically, only assumed mentally to be finite, even though pi is not.

I like the analogies of the right triangle with components of one and one. But similarly, even though we can represent the length with sqrt{2}, it can never be physically measured exactly (only symbolized), so how can we be sure that the hypotenuse truly does meet with the end of either one of the segments.

I am not a mathametician (though I may want to be someday), so if anything I am saying is just wrong, excuse my ignorance. I like to play with theories when I am bored.

Dave

[NateTG]

I like the analogies of the right triangle with components of one and one. But similarly, even though we can represent the length with sqrt{2}, it can never be physically measured exactly (only symbolized), so how can we be sure that the hypotenuse truly does meet with the end of either one of the segments.


We can't - or we can't be sure that the angle us 'perfectly square', but it doesn't matter much.

Even with conventional views, it's quite possible consider expressions such as
\sqrt{2}
as representing a farmily of arbitrarily precise measurements with rational values.

For example, the set of fractions
\frac{x_n}{n} \forall n \in \mathbb{N}
where x_n is chosen so that the fraction is as close as possible to
\sqrt{2}.

Clearly this could correspond to a series of ever more precise measurements. If you order them in increasing precision, these measurements represent a Cauchy sequence (http://mathworld.wolfram.com/CauchySequence.html). Cauchy sequences are, as you can see from the link, closely related to the notion of real numbers.

[master_coda]

I like the analogies of the right triangle with components of one and one. But similarly, even though we can represent the length with sqrt{2}, it can never be physically measured exactly (only symbolized), so how can we be sure that the hypotenuse truly does meet with the end of either one of the segments.

But we can't measure the sides of length one exactly either. We can write out a finite decimal representation of 1, but that doesn't have anything to do with measurement.

[hypermorphism]

So a circle with a circumference of pi could not be measured physically, only assumed mentally to be finite, even though pi is not.

Why do you think pi is not finite ? Do you think of 1/3 as being infinite as well ? If so, why ? I can write it in base 3 as 0.1, whereas your arbitrary base ten representation of 1/10 becomes strung out. What if we marked our rulers at multiples of pi instead of multiples of 1 ? Do you find the number 1 to be more "exact" than the number pi ? If so, why or why not ?

[daveyp225]

But we can't measure the sides of length one exactly either. We can write out a finite decimal representation of 1, but that doesn't have anything to do with measurement.

We can measure a side of length one, in any type of units. If the units are atoms, then one atom can be a length of one, or 5000 atoms can be a length of one because 1 is a finite number and so is 5000. Most of the world uses centimeters, which is already a defined, finite number system. The problem with pi is that no matter what base of number you use, no matter what type of scale you use, it is never finite.

And NateGT, I appreciate all the explainations, but unfortunately my mathematical knowedge is only at calc 3, and I have never gotten into number theory except for computing pi and trig functions. That link basically said "blah blah blah sequence" to me. Thanks anyway.

If anyone knows a good place (online) to start learning more about number theory, let me know.

Dave

[master_coda]

We can measure a side of length one, in any type of units. If the units are atoms, then one atom can be a length of one, or 5000 atoms can be a length of one because 1 is a finite number and so is 5000. Most of the world uses centimeters, which is already a defined, finite number system. The problem with pi is that no matter what base of number you use, no matter what type of scale you use, it is never finite.


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.

[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 cant have Pi of anything. You could have an extremely close amount of Pi things, but never exactly Pi.

You cant 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 dont remember the formula but I remember that the concept I used was that he circle is a polygon with infinte 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 infinte 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 travelled 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 dont remember the formula but I remember that the concept I used was that he circle is a polygon with infinte 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 infinte 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 in to 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, thats 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 dont 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 peice 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 peice 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 physic 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, thats 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, thats 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 eachother, 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!