Does Pi terminate or never?

[shadron]

That never made sense to me. It would not seem possible given the simple definitions necessary to derive Pi to somehow hide data within it.

That is, of course, Sagan's entire point. A being who has the ability to hide such a message within a universal math constant truely fulfills the requirement of omnipotent in the theological meaning of the term. Not something logically crazy like able to move an infinitely large stone or some such, but something that no one can fake, no one can mistake, no one can say it came through divine inspiration or any other intelligent agency, something no one has to trust another mortal in order to believe. It's a Turing test for godhood.

 

[nimzov]

That is, of course, Sagan's entire point. A being who has the ability to hide such a message within a universal math constant truely fulfills the requirement of omnipotent in the theological meaning of the term. Not something logically crazy like able to move an infinitely large stone or some such, but something that no one can fake, no one can mistake, no one can say it came through divine inspiration or any other intelligent agency, something no one has to trust another mortal in order to believe. It's a Turing test for godhood.

Nicely said.

 

[Lucky]

Anybody who's read Contact knows there's a buried message in pi...

That never made sense to me. It would not seem possible given the simple definitions necessary to derive Pi to somehow hide data within it.

That is, of course, Sagan's entire point. A being who has the ability to hide such a message within a universal math constant truely fulfills the requirement of omnipotent in the theological meaning of the term. Not something logically crazy like able to move an infinitely large stone or some such, but something that no one can fake, no one can mistake, no one can say it came through divine inspiration or any other intelligent agency, something no one has to trust another mortal in order to believe. It's a Turing test for godhood.

RussDill is right that it doesn't make sense.

Pi belongs in the logical universe, not the physical universe. A creator-god of our physical universe could (arguably) determine the values of physical constants, but could no more control the value of pi than the value of 1 + 1 ('logically crazy' is precisely what it would be!). Pi is completely determined by its definition, independent of the properties of any physical universe (in fact, independent of the existence of any physical universe).

 

[readme.txt]

I am neither a mathematician nor a physicist, but I cannot believe that any scientist in the world who is measuring things would ever need to use more than 200 digits of pi.

Earlier in this thread a claim was made that 39 digits was all that was needed to measure the circumference of the observable universe to a degree of gradation involving Plank lengths. A quick survey of the internet show estimations for the number of digits of pi for this feat range from 39 to 61.

Oops, my bad. Well, I don't know then. Maybe it's a new way to count sheep before going to bed.

 

[The Fool]

Yep.


Oooh, I opened the door to that, didn't I? I haven't thought about it myself until there was a thread on this a month or so ago in this very section in which it came up. I'll give you a very fragmentary reply.

Let's first see how a base-N system works for a positive integer N > 1. There are digits with values 0, 1, 2, ... N-1. The number representation
abc.de
(where a, b, c, d and e are digits) denotes the number
a * N^2 + b * N + c + d * N^-1 + e * N^-2

The nice thing about this is that you get a unique representation for each number. Well, for nearly each number. There's the obvious duplicate
1.000... = 0.999...
and likewise, all numbers with a finite fraction part can also be represented with an endless fraction of repeating 9's (or in the general case: repeating (N-1)-digits). This is very systematic so it doesn't bother much. We call the finite fraction representation the "standard representation" and we're done. Everybody from grade 4 upwards understands this.

Now let's turn to other possible bases.

0 is out because you'd have no digits. Even if you'd allow a digit 0, you could only represent the number 0.

1 is out because you'd only have digit 0. Even if you allow digit 1, you could only represent integer numbers. There's a thing called "unary numbers", but they're nothing else than tally marks.

Likewise, each base between 0 and 1 suffers from the lack of available digits.

Non-integer bases > 1: as an example, let's take the golden ratio phi as base. Phi is the solution of the equation
phi^2 - phi - 1 = 0
and has the value phi = (1+sqrt(5))/2, approximately 1.6. So you'd have as digits 0 and 1. Now, the representation of a number is not anymore unique. As a simple example, the number phi^2 has two finite representations in base-phi: 100 and 11, as follows from the equation above. The wiki page on this has an algorithm to determine a "standard representation", but as you see it's much more complicated. There's also a separate wiki page explicitly about the golden ratio as base.

ETA: another drawback of a non-integer base is that you can't easily determine if a number is an integer or a rational number. Look at the above example of the golden ratio: phi is an irrational number, so 10 denotes an irrational number, and phi^2 is irrational too, so 100 denotes another irrational number. On the other hand, writing 2 in that base would require a fraction (and possibly an infinite fraction at that too).

Finally, with negative bases you'd first have to answer what the applicable digits would be. Would base (-4) have digits 0, -1, -2 and -3? Or somesuch? And then, negative numbers have the obnoxious property that their powers are alternating positive and negative. I'm getting headaches just thinking about it while I type this.

ddt...this post is most disturbing.
You really need to spend more time concerned about wine, women and song.

 

[readme.txt]

ddt...this post is most disturbing.
You really need to spend more time concerned about wine, women and song.

I think it's very interesting.

 

[Lucky]

To expand a bit on my previous post (that it's a logical impossibility for a hypothetical creator-god to control the value of pi), here are some algorithms for the expansion of pi.

Let's take, for example, the Gregory series: pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...

Several posters have defended Sagan's fantasy of an 'artist's signature' in pi as logically possible. So, how does this 'artist' prevent the value of pi from being what it actually is (ratio of a circle's circumference to diameter in euclidian space)?

Does 'it' alter the way calculus works when constructing the Gregory series? Or does 'it' alter the way arithmetic works when computing the expansion from the series?

It's just silly.

 

[dasmiller]

Several posters have defended Sagan's fantasy of an 'artist's signature' in pi as logically possible. So, how does this 'artist' prevent the value of pi from being what it actually is (ratio of a circle's circumference to diameter in euclidian space)?

Does 'it' alter the way calculus works when constructing the Gregory series? Or does 'it' alter the way arithmetic works when computing the expansion from the series?

So if I were an quasi-omnipotent being, I'd change the test a bit. I agree that pi is what it is, but couldn't an omnipotent being intervene whenever someone was calculating pi, perhaps starting at the 10,000th digit, so that whoever was calculating pi saw the message sequence? It would mean that we never got to see the 'real' pi beyond the 10,000th digit, but I could live with that.

 

[Complexity]

RussDill is right that it doesn't make sense.

Pi belongs in the logical universe, not the physical universe. A creator-god of our physical universe could (arguably) determine the values of physical constants, but could no more control the value of pi than the value of 1 + 1 ('logically crazy' is precisely what it would be!). Pi is completely determined by its definition, independent of the properties of any physical universe (in fact, independent of the existence of any physical universe).

See how silly the idea of an omnipotent 'god' is?

 

[Delvo]

Why is knowing the first two trillion digits of pi considered so much more worthwhile than knowing the first two trillion digits of radical 2 or e ?

If "radical 2" means the square root of 2, then that's no different in significance from any other exponential root of any other positive integer.

e is different, being a more special number than just some irrational exponential root, but it's too abstract to get most people's attention (and comprehension), and it isn't used as early (either in an individual's life or in human history) or as widely. Also, there might just not be any available simple methods of finding so many digits of e as there are with pi, particularly not methods that let you start where somebody else left off instead of back at the beginning.

 

[RussDill]

RussDill is right that it doesn't make sense.

Pi belongs in the logical universe, not the physical universe. A creator-god of our physical universe could (arguably) determine the values of physical constants, but could no more control the value of pi than the value of 1 + 1 ('logically crazy' is precisely what it would be!). Pi is completely determined by its definition, independent of the properties of any physical universe (in fact, independent of the existence of any physical universe).

Its all fiction, so its more about what makes an interesting story. I think the physical constant thing would make a much more interesting story with lots more possibilities of cool plot elements.

 

[3point14]

To expand a bit on my previous post (that it's a logical impossibility for a hypothetical creator-god to control the value of pi), here are some algorithms for the expansion of pi.

Let's take, for example, the Gregory series: pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...

Several posters have defended Sagan's fantasy of an 'artist's signature' in pi as logically possible. So, how does this 'artist' prevent the value of pi from being what it actually is (ratio of a circle's circumference to diameter in euclidian space)?

Does 'it' alter the way calculus works when constructing the Gregory series? Or does 'it' alter the way arithmetic works when computing the expansion from the series?

It's just silly.

Do the measurement on a surface that's not flat. By adjusting the amount of curve to the surface you adjust the ratio. Spacetime is curved. All the omnipotent being needs to do is make sure that spacetime is curved to exactly the right degree to make the ratio required.

I am guessing.

 

[dasmiller]

Do the measurement on a surface that's not flat. By adjusting the amount of curve to the surface you adjust the ratio. Spacetime is curved. All the omnipotent being needs to do is make sure that spacetime is curved to exactly the right degree to make the ratio required.

I am guessing.

That's an interesting approach. The big practical problem would be precision; and encoded message is going to take many, many digits and it's hard for me to imagine us ever measuring the curvature of space to more than a few dozen digits (not counting the leading 0s).

And the other practical problem, of course, is that spacetime isn't uniformly curved, what with all those lumps of mass lying around. That one may be manageable with some creative definitions.

Maybe we could combine the two problems to come up with a solution. Measuring a curved-space pi (well, measuring the curvature of space sufficiently) in one location to 5000 digits is probably impossible; determining 500 curved-space pis to 10 digits may be merely tedious. Someone would have to figure out the 'obvious' sequence of 500 places (and orientations?) that the curvature should be measured . . . hmmm . . ..

 

[Ladewig]

and it isn't used as early (either in an individual's life or in human history) or as widely.

I had never thought of that. I consider this a very insightful answer.

 

[orange31]

Thanks for the Wiki links to irrational numbers.

They describe irrational numbers, and null proofs, well.
My math is minimal, but I didn't catch if they described why the numbers will never repeat (as opposed to repeating decimals like 1/3).
Is there an explanation, or is it a property like quantum double split experiments, that simply 'is' ?

 

[drkitten]

Thanks for the Wiki links to irrational numbers.

They describe irrational numbers, and null proofs, well.
My math is minimal, but I didn't catch if they described why the numbers will never repeat (as opposed to repeating decimals like 1/3).
Is there an explanation, or is it a property like quantum double split experiments, that simply 'is' ?

Because if they repeated (or terminated, which is just a repeating zero), they would be rational numbers.

It's not that irrational numbers don't repeat, but that rational numbers do.

Specifically, if I've got a number that stops after the first umpteen zillion digits, if you multiply it by ten-to-the-umpteen-zillionth, you will get an integer. Which means it's the ratio of two integers.

If you have a number with a repeating block of umpteen zillion digits, then if you multiply it by ten to the umpteen zillionth minus one (i.e. umpteen zillion nines) you will get an integer(*). Which, again, means that it's the ratio of two integers.

For example, 0.2323232323.... is simply 23/99. 0.234234234234234... is 234/999. 0.234523452345.... is 2345/9999. And so it goes.

In either case this means it's rational.


(*) Minor technical point. You might have a number that's got some arglebargle and THEN starts repeating. In this case, you need to do it in two steps.

E.g. 0.66652323232323.... is 6665.232323232323..... divided by 10000.

We already know that 0.23232323....is 23/99. So 6665.2323232323.... is 6665 and 23/99. Which is
659,858 / 99.

So the original number is 659,858 / 99 * 10,000 or 659,858 / 9,900,000.

It's still a rational number.

 

[stilicho]

Let's first see how a base-N system works for a positive integer N > 1. There are digits with values 0, 1, 2, ... N-1. The number representation
abc.de
(where a, b, c, d and e are digits) denotes the number
a * N^2 + b * N + c + d * N^-1 + e * N^-2

The nice thing about this is that you get a unique representation for each number. Well, for nearly each number. There's the obvious duplicate
1.000... = 0.999...
and likewise, all numbers with a finite fraction part can also be represented with an endless fraction of repeating 9's (or in the general case: repeating (N-1)-digits). This is very systematic so it doesn't bother much. We call the finite fraction representation the "standard representation" and we're done. Everybody from grade 4 upwards understands this.

Now let's turn to other possible bases.

0 is out because you'd have no digits. Even if you'd allow a digit 0, you could only represent the number 0.

1 is out because you'd only have digit 0. Even if you allow digit 1, you could only represent integer numbers. There's a thing called "unary numbers", but they're nothing else than tally marks.

Likewise, each base between 0 and 1 suffers from the lack of available digits.

Non-integer bases > 1: as an example, let's take the golden ratio phi as base. Phi is the solution of the equation
phi^2 - phi - 1 = 0
and has the value phi = (1+sqrt(5))/2, approximately 1.6. So you'd have as digits 0 and 1. Now, the representation of a number is not anymore unique. As a simple example, the number phi^2 has two finite representations in base-phi: 100 and 11, as follows from the equation above. The wiki page on this has an algorithm to determine a "standard representation", but as you see it's much more complicated. There's also a separate wiki page explicitly about the golden ratio as base.

ETA: another drawback of a non-integer base is that you can't easily determine if a number is an integer or a rational number. Look at the above example of the golden ratio: phi is an irrational number, so 10 denotes an irrational number, and phi^2 is irrational too, so 100 denotes another irrational number. On the other hand, writing 2 in that base would require a fraction (and possibly an infinite fraction at that too).

Finally, with negative bases you'd first have to answer what the applicable digits would be. Would base (-4) have digits 0, -1, -2 and -3? Or somesuch? And then, negative numbers have the obnoxious property that their powers are alternating positive and negative. I'm getting headaches just thinking about it while I type this.

I've had to bookmark those pages because those ideas interest me. Am I able to practice any of these non-standard bases in Excel? By the way, phi is irrational because of the use of a circle's radius to determine some of its dimensions, if I read that correctly. A base-phi system might be just what I'm looking for since I have planned out many parts of a new nation-state. The flag is proportioned according to the golden ratio already. So trying a system of weights and measures in base-phi might be the logical next step.

 

[gnome]

Do the measurement on a surface that's not flat. By adjusting the amount of curve to the surface you adjust the ratio. Spacetime is curved. All the omnipotent being needs to do is make sure that spacetime is curved to exactly the right degree to make the ratio required.

I am guessing.

I like the Sagan example in Contact myself, but this doesn't work... Arroway wasn't measuring pi, she was computing pi mathematically. If she didn't assume a curved space, it wouldn't matter if the real universe was curved.

The question becomes, can an ultimate creator fashion a universe where the laws of mathematics operate differently? Philosophically speaking I think so. I could probably describe a few myself. They wouldn't be very interesting universes, but that's not necessary for the point. On the other hand, a universe where pi works out mathematically the same except for a carefully crafted variation in the number sequence would function nicely.

 

[W.D.Clinger]

The question becomes, can an ultimate creator fashion a universe where the laws of mathematics operate differently? Philosophically speaking I think so. I could probably describe a few myself. They wouldn't be very interesting universes, but that's not necessary for the point.

She might fashion a universe in which the usual axioms were different. In some universes, for example, Lobachevskian geometry might be taught in junior high schools instead of Euclidean geometry, and pi might be regarded as an obscure transcendental number that comes up in the theoretical world of Euclidean geometry but not in the universe she has fashioned.

On the other hand, there might not be any junior high schools in that universe, so it's hard to say.

 

[Ziggurat]

The same? Or bigger?

On average (meaning averaged over all possible files of a given size), about the same. Assuming, of course, that pi is normal.

The first integer is going to be somewhere between 1 and infinity. I'm not sure how you project an average for that. Is it possible? I suspect if you do the average is going to be a very, very, very big number.

No, it's easy to do. Mathematically, it's essentially equivalent to this integral:

[latex]$\int\limits_{x=0}^{\infty}\frac{x}{\lambda}e^{-x/\lambda}dx = \lambda$[/latex]

The probability distribution may extend to infinity, but the average remains quite finite.

Actually, how long is the string of numbers that represent, say, a movie? Surely that's got to be the starting point...?

Depends on a lot of stuff, but it's big. Let's consider a DVD, which is, say, a bit over 6 Gigabytes, or about 5x10¹⁰ bits. Now we treat it like a big binary number, meaning it's one of 2^5x1010 possible numbers of up to that length (we include leading zeros). If you want to represent such a number in base-10, then we can figure out its length by noting that 2¹⁰ is approximately 10³. So the number of possibilities is about 10^1.5x1010, so we need around 1.5x10¹⁰ digits. Note that this doesn't mean 10 digits, it means 15,000,000,000 digits. I can't post a message with that many digits on this message board.