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 isn't 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 isn't 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: [tex]\sqrt{2}[/tex] 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)

 

[VantagePoint72]

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. 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. 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]

A circle is not a physical object.

"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 infinite 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 written 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

[tex]x^2 + y^2 = 1[/tex]

 

[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 [itex]\sqrt{2}[/itex] 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 [itex]\pi[/itex] (or [itex]\sqrt{2}[/itex]). 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 [itex]\sqrt{2}[/itex]. It really doesn't matter how we represent numbers, it's just what makes it easier for us, for example in decimals:

[tex]\sqrt{2} = 1.41421356237310 \ldots[/tex]

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

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

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 [itex]\sqrt{2}[/itex]. It really doesn't matter how we represent numbers, it's just what makes it easier for us, for example in decimals:

[tex]\sqrt{2} = 1.41421356237310 \ldots[/tex]

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

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

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
[tex]\sqrt{2}[/tex]
as representing a farmily of arbitrarily precise measurements with rational values.

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

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 explanations, 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.