Exploring the Infinity of Pi: A Scientific Perspective

In summary: it's still a conjecture. the only numbers we know for sure are normal are those whose definition is "randomly chosen."a number like 0.1101001000100001000001... is called "algorithmically random" or "chaitin random" or "kolmogorov random." these are all equivalent notions. a number like pi is "normal" in base 10, but not "random." in fact, pi is "algorithmically compressible" -- its digits can be generated by a short computer program.i don't think it's too hard to define "pattern" if we're talking about the digits of a number. maybe it's just a sequence of digits that
  • #36
If that is what you believe, Jug, then you are commiting several common mistakes. The first is presuming that there is something special about decimal expansions (or any other base). pi is as easily descibed, and as easily calculted as 1/3. That is to say, given a line with integer units marked on it pi and 1/3 are just as easily located. in fact, I'd say pi is more easilyl located; look at the proof of existence of the first transcendental number.

pi, 1/3 sqrt(2) are just symbols reprsenting some cuachy sequence of rational numbers, they are all as useful as each other, pi perhaps more so. How is 22/7 more accurate than pi?

pi is not infinite. Its decimal expansion is not eventually periodic, but that isn't important. I can more easily construct sqrt(2) than 1/3 given a ruler and compass.

The rest of your post is mostly uninterpretably to my mind.
 
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  • #37
My contention then is that the true ratio of pi cannot be arbitrarily determined but must in fact conform exactingly to some full set of ratios describing the finite condition. Just some thoughts on the thing...


If you mean that π is the limit of a sequence of fractions that approximate it, it may interest you to know that one model of the real numbers is just that; a real number is a sequence of 'converging' fractions.
 
  • #38
Pi is infinite

MG, we appear to be equally incoherent to one another. By example, you ask:

1) "How is 22/7 more accurate than pi?" It is not, regardless of what value you give to pi.

2) "Pi is not infinite." Then what is its finite value?
 
  • #39
"Pi is not infinite." Then what is its finite value?

Pi.


pi is of course infinite.

Then prove it's bigger than 4.
 
  • #40
Originally posted by Hurkyl
Pi.




Then prove it's bigger than 4.

Pi cannot possibly be bigger than 4. Prove that 4 is finite.
 
  • #41


Originally posted by Jug
MG, we appear to be equally incoherent to one another. By example, you ask:

1) "How is 22/7 more accurate than pi?" It is not, regardless of what value you give to pi.

2) "Pi is not infinite." Then what is its finite value?

I don't think 'equally' is the word you want.

First as pi is a number between 3 and 4, it is not infinte, is it? It is finite. If you think it is infinite, then you do not know what infinite means.

You were the person that suggested that some fraction was 'better' than pi.

I can tell you what pi is exactly: it is the ratio between a circle's diameter and its circumference. Just as sqrt(2) is the number (positive) which squares to two, just as 22/7 is the number which when multiplied by 7 gives the answer 22.

Look at Tim Gowers's discussions on what is solved when one solves an something.

Your replies to Hurkyl's post indicate that you ought to learn about things that you are speaking on.
 
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  • #42
The definition of finite for ordinal numbers is:
n is finite if and only if n is a natural number.
4 is a natural number, thus as an ordinal number, it is finite.


The definition of finite for the extended real numbers is:
n is finite if and only if n is a real number.
4 is a real number, thus as an extended real number, it is finite.


The definition of finite for hyperreal numbers is:
n is finite if and only if there is a natural number m with -m < n < m.
5 is a natural number, and -5 < 4 < 5.
Thus, as a hyperreal number, 4 is finite.


The definition of finite for sets is:
S is finite if and only if it is bijective to a natural number1.
4 is a natural number, and is bijective to itself.
Thus, as a set, 4 is finite.


Um... I can't think of any other (standard) mathematical terms labelled 'finite' or 'infinite' that apply to something labelled '4'. Is that good enough for you?


Hurkyl

1: In set theory, there is a "standard" model for the natural numbers in terms of sets. Since it is fairly common to label these sets by the symbol used to label the natural number it models, I figured I might as well include this definition too.
 
  • #43
What y'all say sounds reasonable, standard and as curricula for an abstract, logarithmic progression of numbers for the defining of pi; yet, IMO, showing no relationship to any finite conclusion, particulrly as pertaining to a numbers set of such finality. How can you say that the pi ratio is finite when it merely represents one particular ratio of such finality?

I have to hold to the Pythagorean concept of mathematics as the science of describing exacting relationships. The concept, at least by my interpretation, allows for but one finite condition, all participles of which emanate from a finite fundamental and in return to which can only find final resolve. Furthermore, I would suggest that all participles emanating from the fundamental must demonstrate a repetitive numbers set of infinite progression. In short, whatever its value the pi ratio can only be infinite.
 
  • #44
"Furthermore, I would suggest that all participles emanating from the fundamental must demonstrate a repetitive numbers set of infinite progression."

Isn't it remarkable how one can put together a sentence that sounds like it actually means something!

There are several problems I have with that sentence. First is the fact that "fundamental" is an adjective, not a noun so I cannot find a subject in it. On the other hand "numbers" is a noun rather than an adjective so I have no idea what "a repetitive numbers set" is. I guess it would be too much to point out that a "set", by definition, cannot be "repetitive".

Finally, I can't see how a "conclusion" can be either "finite" or infinite. What I would like is for Jug to tell us explicitely what definitions of "finite" and "infinite" he is using. They are clearly not the standard ones.
 
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  • #45
I know the first digit in the decimal representation of pi is 3. Therefore, I know -4 < pi < 4. Thus, pi is finite.


If you would like to supply an alternative definition of "infinite" (either by saying what "infinite" is, or properties "infinite" has), we can discuss whether pi is finite or infinite with respect to that definition.

However, even if pi is infinite according to your definition, that does not change the fact that pi is finite according to the definitions I quoted.


(And for the record, your post made very little sense)
 
  • #46
Originally posted by HallsofIvy
"Furthermore, I would suggest that all participles emanating from the fundamental must demonstrate a repetitive numbers set of infinite progression."

Isn't it remarkable how one can put together a sentence that sounds like it actually means something!

There are several problems I have with that sentence. First is the fact that "fundamental" is an adjective, not a noun so I cannot find a subject in it. On the other hand "numbers" is a noun rather than an adjective so I have no idea what "a repetitive numbers set" is. I guess it would be too much to point out that a "set", by definition, cannot be "repetitive".

Finally, I can't see how a "conclusion" can be either "finite" or infinite. What I would like is for Jug to tell us explicitely what definitions of "finite" and "infinite" he is using. They are clearly not the standard ones.

I was hoping we might avoid the pedantry. By "fundamental" is of course meant fundamental wavelength. Do we really need to engage in a discussion over ordinary physics terms?

As to your claim that a "set" cannot be repetitive, what is 1.185185185...ad infinitum, if not a repetitive set?
 
  • #47
I was hoping we might avoid the pedantry. By "fundamental" is of course meant fundamental wavelength. Do we really need to engage in a discussion over ordinary physics terms?

Oh, of course, it was so obvious.


As to your claim that a "set" cannot be repetitive, what is 1.185185185...ad infinitum, if not a repetitive set?

A decimal expansion? A sequence of digits, perhaps?
 
  • #48
the solution of 1000(x-1)-185=x-1?

I'm still also at a loss to understand how a ratio is infinite, in whatever sense you use.As yoy adopt the pythagorean attitude of exactitude, whatever that might be, can you get more precise than the statement pi is the ratio of the circumference to the diameter of a circle?
 
  • #49
Is pi finite?

Originally posted by matt grime
the solution of 1000(x-1)-185=x-1?

I'm still also at a loss to understand how a ratio is infinite, in whatever sense you use.


As yoy adopt the pythagorean attitude of exactitude, whatever that might be, can you get more precise than the statement pi is the ratio of the circumference to the diameter of a circle?

If any number is the ratio to some fundamental that is itself finite, how can that number be said to also be finite? Y'all may be perfectly right in your assertion but, with all due respect, I'm not going to take your word for it. The academia is ever being corrected.
 
  • #50
With least due respect, are you trying to be a crank or a troll?

You perhaps want to change some of the prepositions in that last post. You at least want to rewrite it so it makes sense, anyway.

1 is finite, I think, though it's rather unclear what you think finite means. It is also a ratio, of 1/1, and I think it's fundamental, in some sense...
 
  • #51
If any number is a ratio to some fundamental that is itself finite, how can that number be said to be finite? Y'all might be right in your assertions but, with all due respect, I'm not going to take your word for it. Saying that pi is finite is like saying that the sky is finitely blue.
 
  • #52
Originally posted by Donde (let's stop kidding ourselves, shall we?
Saying that pi is finite is like saying that the sky is finitely blue.

What are you talking about? Earlier in this very thread, you acknowledged that p is less than 4. Of course it is finite, by your own admission!

p is simply the ratio of the circumference of a circle to its own diameter. Consider any given circle. Since the circumference is not infinite and the diameter is not zero, p is finite.

This question was answered in the first two posts. Why on Earth is this silly debate still going on?
 
  • #53
Crackpot rule #1: when you make up your own language, no one can ever prove you wrong!
 
  • #54
If any number is a ratio to some fundamental that is itself finite, how can that number be said to be finite?

By proving that number satisfies the definition of finite.


Y'all might be right in your assertions but, with all due respect, I'm not going to take your word for it.

We're not asking you to take our word for it... there's a reason we use something called a "proof".


Saying that pi is finite is like saying that the sky is finitely blue.

Is it? In what way?
 
  • #55


Originally posted by selfAdjoint
Physics assumes space is a continuum and that the values it discusses can take on all real numbers. The exception is action, which can only take on integer multiples of h.

i know that in Bohr-Sommerfeld quantization, this was the approach, but does that have any resemblance to modern quantum theory? i have not heard said about modern quantum theory that the action must be discrete. i would be surprised to find this is true.
 
  • #56
The word "infinite" has a very specific mathematical meaning. No number that falls between 3 and 4 can ever satisfy that meaning. What else is there to talk about?

I mean really, even from a common sense viewpoint, if you asked any person with an IQ over 90 if "infinite" was bigger than 4, what do you think the right answer would be?
 
  • #57
Is it your contention that a circle with a radius of 1 meter contains within it an infinite amount of area? Can a silo that is 4 Meters in diameter and 10 meters high hold an infinite amount of grain? How much coffee can I fit in my cup? Is the volume of my cup equal to the volume of the silo? How large is a baseball?

If this discussion started pi hours ago and ends pi hours from now, does that mean that we are half-way done? Or will it go on forever...?
 
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  • #58


Originally posted by Tom
What are you talking about? Earlier in this very thread, you acknowledged that p is less than 4. Of course it is finite, by your own admission!

p is simply the ratio of the circumference of a circle to its own diameter. Consider any given circle. Since the circumference is not infinite and the diameter is not zero, p is finite.

This question was answered in the first two posts. Why on Earth is this silly debate still going on?

Tom, 1) Please do not put words in my mouth. I have never admitted to any such (ridiculous) thing as pi being finite.

2) If pi were the "finite" value for describing ratio of circumference to diameter, are you then saying that there is no other value that is capable of describing that same relationship? For if there were, then the pi ratio could not possibly be finite, could it?
 
  • #59
1) Please do not put words in my mouth. I have never admitted to any such (ridiculous) thing as pi being finite.

You have admitted that pi satisfies a condition which is (one) definition of finite.


If pi were the "finite" value for describing ratio of circumference to diameter, are you then saying that there is no other value that is capable of describing that same relationship?

There is no other value that describes the relationship, but that has nothing to do with the finiteness of pi.


By Finite we mean that both ends of some interval are reachable.

No, Organic, that is what you mean, not what we mean.



Anyways, Jug: If you want to talk math, then do so. If you want to continue talking about your own personal ideas and theories, I suggest moving over to an appropriate forum, such as Theory Development or one of the Philosophy forums.
 
  • #60
Jug,

Learn to read, will you please? I said that you admitted to pi being less than 4. That automatically means that it is finite.

Here is where you said it:

Originally posted by Jug
Pi cannot possibly be bigger than 4. Prove that 4 is finite.
 
  • #61
No, Hurkyl, I have not described any finite condition and I don't think you can either. And yes, there ARE other values to describe the same relationship as does pi. Because you (et al) do not believe that does not make it untrue.

As to your dictum, I was referred to this site as being a premier science forum...and I do find it to be that...trusting it may not go the way of so many others by the elitist attitude of a few. I'm here only to learn and hopefully to find some way to contribute.
 
  • #62
If you are here to learn, then why don't you tell us what you think it means for a real number to be finite and/or infinite, then we can all debate the issue, and perhaps we can all learn something. There are several good mathematicians who've posted in this thread all expressing surprise at the notion that pi is infinite, which means not finite. In what way is pi not a finite number, whatever that might mean? Perhaps we cna then explain to you what the more sensible name for this property is.

And why is pi a fundamental (wavelength)? pi is dimensionless and wavelengths aren't, if we're going to think of physics.
 
  • #63
Originally posted by Jug
No, Hurkyl, I have not described any finite condition and I don't think you can either. And yes, there ARE other values to describe the same relationship as does pi. Because you (et al) do not believe that does not make it untrue.

As to your dictum, I was referred to this site as being a premier science forum...and I do find it to be that...trusting it may not go the way of so many others by the elitist attitude of a few. I'm here only to learn and hopefully to find some way to contribute.
That doesn't make any sense. Are you actually claiming that pi and some other value BOTH define the same ratio? If so, what it is?
 
  • #64
Jug,
Is it your position that you are offering rigorous proof that infinity lies between +3 and +4 on a real number line?
 
  • #65
Sorry if I'm covering ground already well and truly stomped, but... Someone mentioned something along the lines of "Pi isn't infinite because it's less than 4". Now, I'm truly crap at maths, but how the heck is being less than four proving that something isn't infinite? There are infinite numbers between 3 and 4. There are infinite numbers between 3.99999999999 and 4. It depends entirely on how much time you want to waste going down to smaller and smaller numbers.
 
  • #66
Originally posted by Adam
Now, I'm truly crap at maths, but how the heck is being less than four proving that something isn't infinite? There are infinite numbers between 3 and 4. There are infinite numbers between 3.99999999999 and 4. It depends entirely on how much time you want to waste going down to smaller and smaller numbers.

Because a real number a is said to be finite if there exists another real number x so that a < x. It is a definition, not something that is out in the open or in need of debating!

Also, "There are infinitely many numbers between 3 and 4". What does that proof? Just because there are an infinite number of numbers does not mean that every (or any) single number is infinite.

Discussions such as these could be avoided if people took the trouble to read an elementary math book before arguing about this stuff...
 
  • #67
adam,
what number lies between 3.999... and 4?
 
  • #68
I went out and bought lunch, paid with a ten dollar bill, and I got $3.14 cents back...I didn't realize I was so close to having infinite money...who knew?
 
  • #69
Originally posted by Dimitri Terryn
Discussions such as these could be avoided if people took the trouble to read an elementary math book before arguing about this stuff...

I agree.

I think they also need to understand the role of definitions. Many people try to "challenge" definitions because they think there "should be more to it". That in itself shows that they don't understand what the concept of "definition" means in math (and science).
 
  • #70
No, Hurkyl, I have not described any finite condition and I don't think you can either. And yes, there ARE other values to describe the same relationship as does pi. Because you (et al) do not believe that does not make it untrue.

As to your dictum, I was referred to this site as being a premier science forum...and I do find it to be that...trusting it may not go the way of so many others by the elitist attitude of a few. I'm here only to learn and hopefully to find some way to contribute.


Which is why I suggested https://www.physicsforums.com/forumdisplay.php?s=&forumid=95. Part of what makes Physicsforums work is that we have different subforums for different topics and methods of inquiry.
 

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