Does Pi terminate or never?

[Delvo]

Another way to look at the issue with alternative number bases is that any number that's either terminating or repeating in one base must be either terminating or repeating in all others, since they're all writable as fractions (with integers for the numerator and denominator) and those all either terminate or repeat.

Use Knuth's up-arrow notation to describe obscenely large numbers, such as Graham's number which has more digits than there are planck volumes in the visible Universe; give an upper bound.

The fun thing to ponder regarding those kinds of ludicrous numbers is that they still don't get us any closer to infinity. No matter how many ways you think of to try to build up and compoundify one super-duper notation scheme on top of another, you still end up with a number that's finite, and is thus still practically nothing compared to infinity.

 

[Soapy Sam]

Occasionally- generally after imbibing alcohol- I wonder in what sense numbers can be said to exist.
Clearly, when something like pi turns up not just in circle formulae, but in things like actuarial statistics, it must actually exist in some sense that does not require the existence of humans. Yet if there are no humans, it clearly cannot exist as a number, but must be an actual physical property of the universe, of which the number is merely a map.
If this is so, pi is, in some sense "wired into" each and every point of spacetime, which is bounded by the Planck length and (one imagines) by a Planck time. In which case, that point is finite in all possible dimensions, yet the map of one of it's properties is infinite.
Clearly something is amiss.
When we say " a number is infinite" we cannot mean that it extends infinitely in either space or time.
So what the hell do we mean?

 

[dasmiller]

If this is so, pi is, in some sense "wired into" each and every point of spacetime, which is bounded by the Planck length and (one imagines) by a Planck time. In which case, that point is finite in all possible dimensions, yet the map of one of it's properties is infinite.
Clearly something is amiss.

IMHO, pi is a purely mathematical relationship and, as such, is independent of spacetime.

Of course, it's a mathematical relationship that's turned out to be extremely useful in our particular spacetime. If there are sentient things in other spacetimes, it's probably useful to them, too.

Now . . . could there be "spacetimes" for which pi was not relevant? I think that "space" implies dimensionality which, in turn, brings pi back into the picture. But maybe in a 1D universe, pi wouldn't be relevant. Or if we imagined a multidimensional universe with in which the dimensions would always be treated independently, so "X^2+Y^2" would make no more sense than, say "$^2 + Temperature^2"

I'd like to think about this further, but it's much to early in the morning to start drinking.

 

[shadron]

Occasionally- generally after imbibing alcohol- I wonder in what sense numbers can be said to exist.
Clearly, when something like pi turns up not just in circle formulae, but in things like actuarial statistics, it must actually exist in some sense that does not require the existence of humans. Yet if there are no humans, it clearly cannot exist as a number, but must be an actual physical property of the universe, of which the number is merely a map.

Cool so far.

If this is so, pi is, in some sense "wired into" each and every point of spacetime, which is bounded by the Planck length and (one imagines) by a Planck time.

I don't think that is the way in which those constants are understood. The Plank length and time aren't bounds, but rather granularities. They form a lower limit under which finer gradations have no physical meaning.

In which case, that point is finite in all possible dimensions, yet the map of one of it's properties is infinite.
Clearly something is amiss.

But then again, pi is not a dimension. It doesn't align with any spatial or temporal axis, nor is it decomposable into a set of such dimensions. It is dimensionless, a ratio of values which is not bounded by any dimension. There is no limit on a ratio, even if measurement were used to invoke the ratio, for math has found better means to define it. Just the fact that it can be decomposed into an infinite series of non-zero values is a hint that it's exact value is not contained, just as the largest number is never the last word in large.

When we say " a number is infinite" we cannot mean that it extends infinitely in either space or time.
So what the hell do we mean?

Who makes statements like "Pi is infinite"? What pi is is transcendent, and there is a definite difference in the two statements.

Could this rather be a statement about drink givething the desire but takething away the ability?

 

[gnome]

I think that the planck length doesn't kill the utility of pi to multiple places... sure, maybe for purely physical measurements. But what about formulas in which pi is a constant, but is heavily multiplied? If the formula is measuring something besides just size, greater precision may have applied value.

 

[Audible Click]

​

 

[roger]

This guy has calculated the "correct value for Pi". He says it's 3.125 And also that it's 0.78125, 1.28, 0.64 and 0.21875

He also says you can win 300,000 Swedish Crowns if you find a mistake in the theories in his book. Anybody want to try?

Sure. From the first page:

Also find a mistaken in the book and win 300,000 Swedish Crowns and a copy of
the book. For more information please see www.correctpi.com or write to
correctpi@hotmail.com

My bolding.

I'm going to guess that he will not give me 300,000 crowns.

 

[Bill Thompson]

What makes you think that has anything to do with it? Actually, Pi exhibits its transcendental nature no matter what integer you use as a base.

What makes me think what has anything to do with what?

I said:

It is based on a circle which cannot be calculated 100% accurately by base 10 numbers that we use.
​

And you said:

What makes you think that has anything to do with it? Actually, Pi exhibits its transcendental nature no matter what integer you use as a base.
​

So you think the fact that it is based on a circle has nothing to do with it or are you saying that the fact that we use integers has nothing to do with it?

Well, one person said they understand what I am saying. Are you suggesting this person is clueless?

PI is a number, not a function. You can represent curves by a polynomial or a function of a variable. You cannot represent a curve by a single number. That is not how math works.

On the other hand, you can calculate pi using a function. But that function's accuracy depends on how many iterations you use. You can never reach an end to this function where the number is 100% accurately determined.

 

[Bill Thompson]

I get the analogies, you explain it well.
But it's still curious to me what makes it an irrational (non-repeating) number.

But I'm relearning calculus, where limits allow you to throw around pi, infinity, etc, so I see it doesn't impede anything (...also as per the Planck length observation above).

If it repeats, it is the same sort of thing as saying it ends. It repeats because we use base 10 digits.

1/3 repeats in decimal because we use a base 10 numbering system. If we used a numbering system that divided by 3 without a remainder, it would not repeat.

===============

Base 10 is a poor system, by the way. You can only divide by 2 or 5 without getting repeating digits. Apes like us have 10 fingers. So we are stuck with this system.

 

[tsig]

Of course Pi terminates. I would write the terminal number down for you but I can’t find a piece of paper big enough.

I prove it ends and jotted down the answer in the margin.

 

[stilicho]

Another way to think of it. Suppose you had a ruler and a pencil. You could only make straght lines. How many little lines would it take to make a circle? First, with three lines, you would could make a triangle. With 4, a square.

I said before on this thread that I tried something similar with smaller triangle sections during a high school class where I was bored. I did realise that there didn't seem to be a way to predict the next increment in an orderly way. (Don't ask me to recreate the thing...it took up a whole 80 minute class and I wasn't finished once the bell rang).

I was asking about the hypothetical consequences of there being no transcendental numbers on other concepts such as SQRT (-1), which I thought were dependent on "e" and pi being transcendental.

Remember, I'm not a professional mathematician but I'm really curious about the way it would affect other things.

 

[ddt]

I was asking about the hypothetical consequences of there being no transcendental numbers on other concepts such as SQRT (-1), which I thought were dependent on "e" and pi being transcendental.

Remember, I'm not a professional mathematician but I'm really curious about the way it would affect other things.

No, the concept of SQRT(-1), commonly known as i, is not dependent on e or pi. In fact, you can do algebra without transcendental numbers.

Let's start with the natural numbers N: 0, 1, 2, 3, 4, ... That set is obviously closed under addition and multiplication - that is, if you take two natural numbers, their sum and their product are natural numbers too. However, it is not closed under subtraction: 3 - 5 is not a natural number.

The next step is the set of integers, named Z: it consist of all (negative, positive or zero) whole numbers. It is closed under subtraction. Every integer has an inverse w.r.t. addition (i.e., its negative).

Another step further is the set Q of rational numbers, consisting of all fractions. It is closed under division, and every rational (non-zero) number has an inverse w.r.t. multiplication (i.e., its reciprocal).

The next step, algebraically speaking, is to look at polynomials. The polynomial
X^2 - 2
has no zero within the rational numbers. And there are many more. So the next set we define is that of the algebraic numbers, A. Those are all numbers that are the zero of some polynomial with integer (or rational) coefficients. This brings with it that you also have to define a number i which is the square root of -1. Algebraically, you can define these without resort to the full real or complex numbers.

Transcendental numbers are simply those real/complex numbers which are not algebraic.

The remarkable thing is that all these sets have the same cardinality ("size") as the natural numbers, whereas the real numbers have a bigger cardinality. On the other hand, the rational numbers (and therefore the algebraic numbers too) are dense in the real numbers, i.e., how small a distance you take, you can always find a rational number that close.

 

[Lucky]

My acquaintance and I and a couple other guys were shooting the breeze after work, and the conversation came to a point somehow where he announced that he had read that factoid in [local paper]

I'd still like to know whether it was April 1st!

When I am fulfilling an honest request for knowledge, my standard is to go, not where I will have my preconceptions confirmed, but to go where I can expect many answers and trust that many of those will be will be good and likely to be including a couple at least from REAL experts, which is why I usually choose JREF.
In composing my title post, I make every attempt to present the question in an honest and fair way so that I don't bias the responses toward my point of view, or to leave out important detail to the same end. Even though I don't try to conceal my opinion, I am loath to unduly slant the responses in my favor, and whatever the results, I report back to the person with as much accuracy as my (aging) memory will permit.

I like your approach! It does bother me when we get requests here for the expert skeptics to provide the (skeptical) facts.

Your points are, of course, right. But it's quite nitpicking, and not all correct criticism on what CaveDave wrote.
As to the first, the issue whether the expansion is repeating or not is not relevant to CD's statement that "pi is transcendental and therefore has an infinite expansion". It is true regardless whether the expansion is repeating or not.
Likewise, the second is also irrelevant to the truth of that statement.
The third is just a bit sloppy wording, of which everyone with a modicum of math knowledge knows what is actually meant.

From the wording of the OP, CaveDave could have been saying that a non-terminating expansion is the definition of a transcendental number. It may well have been just some slightly careless wording, but it's not nitpicking to try and clear up a potential confusion - especially as there will be people viewing this thread who will read it that way and not know it's wrong.

A number with a repeating expansion is neither transcendental nor irrational, so it's clearly relevant to point that out. The distinction between irrational and transcendental numbers is also worth pointing out (transcendentals being a subset of irrationals) - it makes pi a more interesting number than a mere irrational.

Also, the distinction between a number itself and its possible representations is very relevant to this topic, and I disagree that it's clear to "everyone with a modicum of math knowledge". I think many people do have some idea that generating further digits of pi is generating information about the number itself - they see it as akin to measuring some physical constant to greater precision. (This thread is evidence that some people believe the termination or otherwise of pi to be an empirical question.)

 

[Complexity]

Well, one person said they understand what I am saying. Are you suggesting this person is clueless?

That was me. I was hungry. I just wanted some cherry pie. I would have said anything.

I am such a cherry-pie slut.

You said to say I understand, I said I understand.

Where's my pie?

 

[Towlie]

What makes me think what has anything to do with what?

I began composing an answer to that question but quickly realized that what I meant has to be patently obvious to you. You made that clear in your post through the words you quoted from me. There's really nothing left for me to add. It seems like you're being deliberately obtuse here.

So you think the fact that it is based on a circle has nothing to do with it or are you saying that the fact that we use integers has nothing to do with it?

No, I'm saying the "base 10 numbers that we use" has nothing to do with it, as you implied it did, but I think you already know that.

Well, one person said they understand what I am saying. Are you suggesting this person is clueless?

I never said I don't understand what you're saying, I only said you're wrong.

PI is a number, not a function.

And I never implied otherwise. In fact, I never mentioned functions. Now you seem to be wandering completely off the track as if you're responding to some other post and not mine.

For further reading of why our base 10 number system has nothing to do with the the trancendental nature of Pi, please see my post #99.

 

[Soapy Sam]

Cool so far.

I don't think that is the way in which those constants are understood. The Plank length and time aren't bounds, but rather granularities. They form a lower limit under which finer gradations have no physical meaning.

But then again, pi is not a dimension. It doesn't align with any spatial or temporal axis, nor is it decomposable into a set of such dimensions. It is dimensionless, a ratio of values which is not bounded by any dimension. There is no limit on a ratio, even if measurement were used to invoke the ratio, for math has found better means to define it. Just the fact that it can be decomposed into an infinite series of non-zero values is a hint that it's exact value is not contained, just as the largest number is never the last word in large.

Who makes statements like "Pi is infinite"? What pi is is transcendent, and there is a definite difference in the two statements.

Could this rather be a statement about drink givething the desire but takething away the ability?

Hah! No. The ability probably ain't there to start with.

I do appreciate that pi has no spaciotemporal extent and yet pi must, in some way, be a property of spacetime. What else is there?
Of course this is equally true of the square root of 2 and any other number.
Or we make them up. They are all imaginary, pure mindstuff.
But while that may be a sufficient explanation of number, it is no explanation of why the ratio of C/d is pi for a circle.

I suppose it's the same question as one I have asked elsewhere- Why is it that to model a 3 body problem requires immense computing resources, yet the bodies themselves seem to do it with no need for computation whatever? Mathematics is cool and occasionally elegant, but rarely so elegant as reality.

 

[shadron]

Hah! No. The ability probably ain't there to start with.

I do appreciate that pi has no spaciotemporal extent and yet pi must, in some way, be a property of spacetime. What else is there?
Of course this is equally true of the square root of 2 and any other number.
Or we make them up. They are all imaginary, pure mindstuff.
But while that may be a sufficient explanation of number, it is no explanation of why the ratio of C/d is pi for a circle.

I suppose it's the same question as one I have asked elsewhere- Why is it that to model a 3 body problem requires immense computing resources, yet the bodies themselves seem to do it with no need for computation whatever? Mathematics is cool and occasionally elegant, but rarely so elegant as reality.

Well, you're way to metamathecal for me, young feller. Warp on.

On your second, it is well known that analog computing, in some problem spaces, is much better at getting an answer than digital. But that's not math; that's modeling. As a set of differential equations it is very elegant; just impossible to solve in a closed form.

 

[a_unique_person]

​

I laughed at that. Then I read the caption.

 

[a_unique_person]

Occasionally- generally after imbibing alcohol- I wonder in what sense numbers can be said to exist.
Clearly, when something like pi turns up not just in circle formulae, but in things like actuarial statistics, it must actually exist in some sense that does not require the existence of humans. Yet if there are no humans, it clearly cannot exist as a number, but must be an actual physical property of the universe, of which the number is merely a map.
If this is so, pi is, in some sense "wired into" each and every point of spacetime, which is bounded by the Planck length and (one imagines) by a Planck time. In which case, that point is finite in all possible dimensions, yet the map of one of it's properties is infinite.
Clearly something is amiss.
When we say " a number is infinite" we cannot mean that it extends infinitely in either space or time.
So what the hell do we mean?

Alcohol does weird things to your brain?

 

[MetalPig]

Therefore, any file that exists on any computer or digital storage medium, whether it be music, a movie, a computer program, a picture, or whatever, can be found within the digits of Pi.

Which makes for an excellent compression algorithm. Any movie, mp3, application or what have you can be reduced to 2 integers; starting position and length.