Does Pi terminate or never?

[CaveDave]

An acquaintance just told me that he had seen a newspaper article claiming an "ending to Pi had been discovered".:jaw-dropp

Now, as I have always understood it, Pi is a transcendental number that by definition can NEVER terminate. It, and others in that class, continue endlessly, adding precision but never ending.

Is it possible something changed and I missed it, or that I understood wrong?

PLEASE HELP! I know there are Bleeding Edge mathmeticians here that will know.:)

Cheers,

Dave

[Ladewig]

An acquaintance just told me that he had seen a newspaper article claiming an "ending to Pi had been discovered".:jaw-dropp


Was it an April's Fool joke by the newspaper or by your friend?

[ddt]

Was it an April's Fool joke by the newspaper or by your friend?

Or just a very badly informed journalist.
Your understanding is 100% correct, CaveDave.

[Zep]

...unless you are counting in base Pi. ;)

[Complexity]

Never 'terminates' - Pi is irrational.

A most excellent number.

[ynot]

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

[Complexity]

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

Why don't you simply tell us how many digits there are to the right of the decimal place before Pi starts to repeat?

Some things never change...

[Pi_314]

I am Pi, what is it exactly you would like to know?

[ynot]

Why don't you simply tell us how many digits there are to the right of the decimal place before Pi starts to repeat?

Some things never change...
I would but it would take forever to tell you ;)

[Complexity]

I would but it would take forever to tell you ;)

Actually, it would be very easy for you to describe if you actually had an answer.

Since the knowledge that you are claiming contradicts what is provable, full of beans you are.

[Damien Evans]

I suppose it's possible that Pi has an end, but if it does it certainly hasn't been discovered yet.

[ynot]

Actually, it would be very easy for you to describe if you actually had an answer.

Since the knowledge that you are claiming contradicts what is provable, full of beans you are.
Would it help for you to realise I'm merely joshin if I used alot of >>> :D:D:D:):):):cool::cool::eek::boggled::eye-poppi:jaw-dropp:blush::rolleyes::(:o:D:p

[ynot]

I suppose it's possible that Pi has an end, but if it does it certainly hasn't been discovered yet.
As Complexity is about to correctly tell you - NO IT ISN'T (possible). . . and it never will be.

[Loss Leader]

Outside of pure theoretical mathematics, Pi does terminate in a way. If, say, you are measuring Pi in meters, after a surprisingly small number of decimal places, you reach a measurement that is smaller than a planck length. If the 3 is 3 meters, about thirty five or so digits deep, the number becomes physically meaningless. The next digit would attempt divide a space that does not, for all practical purposes, exist.

[CaveDave]

Was it an April's Fool joke by the newspaper or by your friend?

Or just a very badly informed journalist.
Your understanding is 100% correct, CaveDave.

Much as I suspected. I could not see how that story could be true, but I have seen many wondrous things in my life...:D

Thank you all for your replies (including the humorous ones:))

I KNEW I would quickly discover the truth on this forum.:)

Cheers,

Dave

[CaveDave]

I am Pi, what is it exactly you would like to know?

But since you are truncated and have lost your decimal, why should I believe you?:D

Dave

[Wowbagger]

Perhaps some robots had been sent from the future to terminate Pi.

[Newtons Bit]

Pi terminates, just not in base 10 Cartesian :)

[Accidental Martyr]

Of course it terminates, but only in Alabama. ;)

http://www.snopes.com/religion/pi.asp

[Andrew Wiggin]

Outside of pure theoretical mathematics, Pi does terminate in a way. If, say, you are measuring Pi in meters, after a surprisingly small number of decimal places, you reach a measurement that is smaller than a planck length. If the 3 is 3 meters, about thirty five or so digits deep, the number becomes physically meaningless. The next digit would attempt divide a space that does not, for all practical purposes, exist.

Even with things like calculating the diameter of the observable universe, Pi reaches the limit of meaningful precision rather quickly. I don't have the figures at hand, but I remember Asimov calculating it in one of his essay series. Diameter of the observable universe calculated to less than the radius of a proton, or some such.

A

[Madalch]

Actually, it would be very easy for you to describe if you actually had an answer.

Since the knowledge that you are claiming contradicts what is provable, full of beans you are.
I think you're confusing ynot with yrreg.

[ynot]

Thank you all for your replies (including the humorous ones:))

Dave

There was humour!!!??? - DOH!

[ynot]

i think you're confusing ynot with yrreg.
YYUR with the Y's

[Ladewig]

Even with things like calculating the diameter of the observable universe, Pi reaches the limit of meaningful precision rather quickly. I don't have the figures at hand, but I remember Asimov calculating it in one of his essay series. Diameter of the observable universe calculated to less than the radius of a proton, or some such.

A

Wikipedia says 39 decimal places are enough for the scenario you describe; quite a small number considering that the current record is in the low trillions of decimal places.

[ynot]

Forget Planck, numbers speak only the truth.

[Complexity]

Would it help for you to realise I'm merely joshin if I used alot of >>> :D:D:D:):):):cool::cool::eek::boggled::eye-poppi:jaw-dropp:blush::rolleyes::(:o:D:p

Oops! I think I did confuse Ynot with Yrreg. Two years is a long time to be away...

My apologies. I'm glad that dueling is no longer in fashion, for I would surely die at dawn.

Yrreg would have held the position you adopted for our entertainment until the bloody end, believing it just to be contrary.

I'm wondering which other gaffs I'll make. Might as well get a bunch of them out of the way as quickly as possible. :o

[Complexity]

Engineering may only have use for a paltry handful of Pi's decimal places, but there is more to life than engineering (important though it is).

Pi is transcendental and far beyond mere usefulness.

[CapelDodger]

Never 'terminates' - Pi is irrational.

A most excellent number.

Exceedingly excellent.

[CapelDodger]

Or just a very badly informed journalist.

More likely just average, in my experience.

Your understanding is 100% correct, CaveDave.

Unlike the average journalist's. In my experience ...

[Chris Hegarty]

In Indiana, it was almost Pi = 4. (http://en.wikipedia.org/wiki/Indiana_Pi_Bill)

[dasmiller]

Even with things like calculating the diameter of the observable universe, Pi reaches the limit of meaningful precision rather quickly. I don't have the figures at hand, but I remember Asimov calculating it in one of his essay series. Diameter of the observable universe calculated to less than the radius of a proton, or some such.

A

Now, I'm making this up as I go, but I think the situation's even worse than that.

Suppose we postulate a universe that has gravity and massive objects (for example, our universe). Now, that means space isn't flat and I think that means that the relationship between radius and circumference for a particular patch of space isn't exactly pi.

Of course, to the extent that you know the mass distribution, then you can compensate for the curvature of space (assuming that you're far, far better at math than I), but I suspect that the uncertainty principle keeps you from ever perfectly knowing the distribution of masses in a volume of space, so you're never perfectly know the curvature, so pi's utility may run out even sooner.

ETA: Or maybe it's a really small effect. I don't know.

[shadron]

Hello, Complexity. Welcome back.

[CaveDave]

Thank you all, especially thanks for the April Fools hoax link to Snopes.

I can imagine there being a reporter getting hold of that "viral" story, second or more handed, then hearing the hoax had "come to an end", reporting that in a brief notation, and due to the wording, my acquaintance misreading that as an "end" of Pi.

Well, that is just a fantasy, but it might explain how he came to believe that...:D

------

On a related note, how many ways are there to calculate the digits of Pi?

I know that NO measurement of a circle can possibly yield more than a handful of digits, and I was once told of a method (semi-graphical) using inscribed and circumscribed polygons or some such, but I think I have heard there are better algorithms.

Is it possible to get a listing w/references to the method?

Thanks to all,

Dave

[Andrew Wiggin]

Now, I'm making this up as I go, but I think the situation's even worse than that.

Suppose we postulate a universe that has gravity and massive objects (for example, our universe). Now, that means space isn't flat and I think that means that the relationship between radius and circumference for a particular patch of space isn't exactly pi.

Of course, to the extent that you know the mass distribution, then you can compensate for the curvature of space (assuming that you're far, far better at math than I), but I suspect that the uncertainty principle keeps you from ever perfectly knowing the distribution of masses in a volume of space, so you're never perfectly know the curvature, so pi's utility may run out even sooner.

ETA: Or maybe it's a really small effect. I don't know.

Space isn't flat, but the rulers we use are imbedded in it so they flex and stretch along with it, so flat, newtonian space is a pretty good approximation. Pi would vary by space curvature assuming a rigid, non imbedded ruler so that would make it rather useless on a universal scale, but it also makes the eyes cross and the mind warp, so that's OK...

A

[gabeygoat]

...unless you are counting in base Pi. ;)



egads!

[stilicho]

Complete non-mathematical amateur here but doesn't the relationship among "e", the square root of -1, and the value of the ratio of the circumference of a circle to its diameter mandate that it doesn't terminate?

Can you also curve the circle so that pi is an integer? What would that do to "e" and the square root of -1?

[Floyt]

On a related note, how many ways are there to calculate the digits of Pi?

I know that NO measurement of a circle can possibly yield more than a handful of digits, and I was once told of a method (semi-graphical) using inscribed and circumscribed polygons or some such, but I think I have heard there are better algorithms.

Is it possible to get a listing w/references to the method?

Thanks to all,

Dave

This Wiki page (http://en.wikipedia.org/wiki/History_of_numerical_approximations_of_%CF%80) has a lovely collection of them. Some of these people must have had ... strange... minds... (I'm looking at you, Ramanujan)

I've been idly wondering - would physical reality be observably different in any way if pi was equal to 3, or any other rational number? I.e., is the irrational or transcendental character of pi in any way responsible for major characteristics of physical reality?

[erlando]

Pi is 3 for sufficiently large values of 3. There. Terminated.. ;-)

[Yuri Nalyssus]

Thanks to all,

Dave
Let's close with a song - http://www.youtube.com/watch?v=VqpWETqoD5Q

Excellent,

Yuri

[stilicho]

Pi is 3 for sufficiently large values of 3. There. Terminated.. ;-)

Is that possible? Is there such a thing as 3-ish or pi-ish? I could easily be falling for some kind of inside math joke but you'd have to explain it...which would ruin its humour value but might be educational in some way.

[CaveDave]

Is that possible? Is there such a thing as 3-ish or pi-ish? I could easily be falling for some kind of inside math joke but you'd have to explain it...which would ruin its humour value but might be educational in some way.

As I understand it, it IS a joke of sorts.
You can occasionally find, in various scientific and engineering writings, statements in the general form of:
"For sufficiently [comparative modifier {such as "large", "high", "long","narrow", etc.}] values (or ranges) of [quantity or property {such as "hardness", "mass", "lifetime", "voltage", "luminosity", "incremental change", etc.}], one can expect [some interesting or surprising result]".

It IS often twisted into a humorous form that may be difficult for non-insiders to catch on to. One of my favorites is "for sufficiently large values of "nil", you approach infinity"; other people like to make up their own.

I suspect the origin is from limit and infinitesimal theorems. (I hope I said that right.)

Technical people often have VERY quirky senses of humor.:D
(You may have seen the Intel TV ad where one researcher sneaks over to his co-worker"s calculations, adds a "plus" symbol to a spot, and then sits down and waits for the other guy to return. The other immediately notices, and they both have a long, hearty laugh. The tag line is "Our jokes aren't like your jokes".)

Cheers,

Dave

[erlando]

Dave is spot on.

My lecturer in calculus used to say "1 plus 1 equals 3 for sufficiently large values of 1".

[Ririon]

Dave is spot on.

My lecturer in calculus used to say "1 plus 1 equals 3 for sufficiently large values of 1".
1.3 is approximately 1. 1.3 + 1.3 = 2.6. Which is approximately 3. It does not have to be any deeper than that in the world of physics. :) I am sure mathematics-gurus have more complex explanations that are funny to them. :p

[Michael C]

This guy (http://www.correctpi.com/) has calculated the "correct value for Pi". He says it's 3.125 :jaw-dropp And also that it's 0.78125, 1.28, 0.64 and 0.21875 :confused:

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?

[Delvo]

I am Pi, what is it exactly you would like to know?When will you die?

(Or are you immortal?)

[3point14]

I am Pi, what is it exactly you would like to know?

Bloody imposters.

I must admit, I do like to stop and get my head down occasionally.

[DavidS]

Outside of pure theoretical mathematics, Pi does terminate in a way. If, say, you are measuring Pi in meters, after a surprisingly small number of decimal places, you reach a measurement that is smaller than a planck length. If the 3 is 3 meters, about thirty five or so digits deep, the number becomes physically meaningless. The next digit would attempt divide a space that does not, for all practical purposes, exist.
If you're measuring pi, you're not doing it right.
If your pi is in meters, you've got the wrong one.

[readme.txt]

Pie terminates when I eat it

[skip]

I think the Texas School Board is working on this as we speak… They would like to give a more rounded view of mathematics. ;)

[AvalonXQ]

This guy (http://www.correctpi.com/) has calculated the "correct value for Pi". He says it's 3.125 :jaw-dropp And also that it's 0.78125, 1.28, 0.64 and 0.21875 :confused:

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?

I actually tried to get what he was saying, but I lost patience around chapter 2 part 2 of his "book".

[Raze]

Pie terminates when I eat it

Har har har, well said. Pie are round. :boxedin:

[Fnord]

Outside of pure theoretical mathematics, Pi does terminate in a way. If, say, you are measuring Pi in meters, after a surprisingly small number of decimal places, you reach a measurement that is smaller than a planck length. If the 3 is 3 meters, about thirty five or so digits deep, the number becomes physically meaningless. The next digit would attempt divide a space that does not, for all practical purposes, exist.

It is not that Pi eventually terminates in this way.

It is that space as we understand it is inadequate for a practical demonstration of Pi beyond a certain scale.

Wise you are.

[Speed of Light]

How is it possible that a finite formula can produce an infinite amount of information?

[AvalonXQ]

How is it possible that a finite formula can produce an infinite amount of information?

Assuming there's an infinite amount of information there, there's no reason it can't.
Finite formula: Start with one. Add one. Repeat.
... see how that works?

[paiute]

I am Pi, what is it exactly you would like to know?

I. M. Pei. How many stories you want?

[Speed of Light]

How is it possible that a finite formula can produce an infinite amount of information?

Assuming there's an infinite amount of information there, there's no reason it can't.
Finite formula: Start with one. Add one. Repeat.
... see how that works?

That's not the same, because there is an obvious pattern
With pi, there is no repetitive pattern, and the lack of predictability continues to infinity

[nimzov]

That's not the same, because there is an obvious pattern
With pi, there is no repetitive pattern, and the lack of predictability continues to infinity
Since there is an algorithm that generates the decimals, I would say they are predictable. But randomly distributed.

nimzo

[uk_dave]

Now, as I have always understood it, Pi is a transcendental number....

I shall have to meditate upon this for awhile...

[readme.txt]

How is it possible that a finite formula can produce an infinite amount of information?

Finite formula : 1/3 = 0.3333333... : infinite amount of information
Finite formula : sqrt(2) = 1.41421356... : infinite amount of information.

ETA: What's that "finite formula" you are talking about?

[Tarot_Is_A_Card_Game!]

Finite formula : 1/3 = 0.3333333... : infinite amount of information
Finite formula : sqrt(2) = 1.41421356... : infinite amount of information.

ETA: What's that "finite formula" you are talking about?

1/3 repeats in base 10 notation. In other bases, a base divisible by 3, this decimal expansion wouldn't repeat.

[readme.txt]

1/3 repeats in base 10 notation. In other bases, a base divisible by 3, this decimal expansion wouldn't repeat.

See under, as a response to "Pi does not terminate"

...unless you are counting in base Pi. ;)

Pi terminates, just not in base 10 Cartesian :)

[dasmiller]

Finite formula : 1/3; infinite amount of information
Finite formula : sqrt(2); infinite amount of information.

Just string the integers together, as a decimal:
0.123456789101112131415161718192021222324 etc.

It will never end, and it will never repeat. And there are an infinite number of variations on that (by 2s: 0.2468101214 etc. Start with 31: 0.31323334 etc. Add a 0 for each: 0.0102030405 etc). And there are many other approaches to generating long strings of digits.

It's simply the case that for some infinitely long sequences of digits, there are exact finite descriptions ("'0.' followed by the series of Base 10 digits created by appending the Base 10 digits of each successive integer, starting at 1"). Of course, for a particular "maximum description length" within any finite language, the fraction of infinitely long sequences that can be finitely described will be infinitesimal.

[Towlie]

Here's a question that I find more interesting:

Is it true that any arbitrary sequence of digits, no matter how long, will appear within the digits of pi, not merely once, but an infinite number of times?

[dasmiller]

Here's a question that I find more interesting:

Is it true that any arbitrary sequence of digits, no matter how long, will appear within the digits of pi, not merely once, but an infinite number of times?

Let's take this a step at a time.

If we start at the nth digit of pi, pi has an infinitely long sequence of digits after that. It follows that there's an infinitesimally small chance (effectively 0) that any particular sequence of digits will not appear in pi somewhere after the nth digit. So, your arbitrary sequence will appear at least once.

But if it appears at, say, the billionth digit, then starting at the billion+1'th digit, pi has an infinitely long sequence of digits after that, so your arbitrary sequence will appear a 2nd time, somewhere after the billionth digit. But pi has an infinite sequence of digits after that 2nd appearance, too . . .

and so ad infinitum.

[Monketey Ghost]

My wife's long, long tresses go on for as far as I can see, and I've never found the ends, which surely must be split...

that's right. My wife has a hair-pi.

[erlando]

Pie terminates when I eat it

There is no cake pie..

[stilicho]

It IS often twisted into a humorous form that may be difficult for non-insiders to catch on to. One of my favorites is "for sufficiently large values of "nil", you approach infinity"; other people like to make up their own.

I suspect the origin is from limit and infinitesimal theorems. (I hope I said that right.)

Cheers,

Dave

I get it now. Kind of.

I remember thinking there had to be an easy way to calculate pi in some high school class--probably one I wound up not doing very well in. I tried to figure it out by using progressively smaller triangles fitted within a circle and seeing how much longer the sides got. I hate to date myself but we didn't even have pocket calculators. I tried to work it out with a slide rule.

[Dave Rogers]

Is it true that any arbitrary sequence of digits, no matter how long, will appear within the digits of pi, not merely once, but an infinite number of times?

Yes, and therefore the following cartoon is factually correct (though somewhat truncated).

http://imgs.xkcd.com/comics/pi.jpg

As usual, XKCD is way ahead of you.

Dave

[GodMark2]

I've been idly wondering - would physical reality be observably different in any way if pi was equal to 3, or any other rational number? I.e., is the irrational or transcendental character of pi in any way responsible for major characteristics of physical reality?

If you use the geometric determination of pi (the ratio of a circle's circumference to it's diameter, where a 'circle' is the set of all points a fixed distance from a single, central, point), pi only takes on it's familiar value in perfect Euclidean space*. Other spaces have a different pi metric, and many have no fixed value for pi.

Example:

On the surface of the earth, make a circle one inch across, and measure it's circumference: You'll get a ratio of the two close to the familiar 3.14. Increase the diameter to 12,000 miles, and you'll find that the ratio is close to 2.000! Increase the diameter to 25,000 miles, and the ratio actually goes to zero! What happened? You weren't using a flat surface to draw on, but one that curved, and that curvature will change how geometry works.

In the real universe, it gets even weirder, as the curvature of space (and thus the 'radius' of a circle) would depend on how much matter and energy were inside the circle, and could change as that energy moved around!

Fortunately, at low enough energies for us to survive, things work pretty close to Euclid's ideal.

*Actually, there are other spaces for which pi has the same value, but they're not salient to the point being made

[Towlie]

Good point, GodMark2. In the same vein, the mysterious Bermuda Triangle has three angles that actually add to more than 180 degrees. :jaw-dropp

[Lucky]

Is it possible something changed and I missed it ... ?
I think the April Fool suggestion must be right. CaveDave, didn't it occur to you that "an ending to pi" would imply a proof of the non-existence of transcendental numbers, which (if true) would blow apart the foundations of (post-medieval) mathematics and logic? It would be huge orders of magnitude more world-shattering (and mind-boggling) than any conceivable scientific reversal (e.g. the speed of light is infinite, the age of the universe is 6000 years, most human disease and death is caused by vaccination).


... a transcendental number that by definition can NEVER terminate. It, and others in that class, continue endlessly, adding precision but never ending.
Your understanding is 100% correct, CaveDave.
Not 100%. First, the definition should exclude repeating expansions (and non-rational bases). Second, transcendental numbers are a subset of irrationals, and the corrected definition would apply to all irrationals. Third (though it's a subtle point), continuing the numeric expansion adds precision to the expansion only, not to the number itself.


On a related note, how many ways are there to calculate the digits of Pi?

...

Is it possible to get a listing w/references to the method?
Not a complete list, as there are infinitely many!

Here's a start. (http://mathworld.wolfram.com/PiFormulas.html)


How is it possible that a finite formula can produce an infinite amount of information?
It can't, for any useful definition of "finite formula".

So, here's a "finite formula": "Pick any number"! Or, if you prefer, "Write down all numbers"!

That's not what you meant, though, is it? Let's stipulate that a "finite formula" for some number (such as pi) has to define that number uniquely. Also, of course, that it can be expressed in a finite number of symbols (or steps, or definitions). In that case, it can't produce an infinite amount of information - though it can produce an infinite number of digits (repeating or not).

The numeric representation of pi, in any (non-transcendental!) base, is infinite (and non-repeating) - else it wouldn't be transcendental. But, by definition, this numeric representation is exactly equivalent to any formula for pi, so both contain the same information.


Assuming there's an infinite amount of information there, there's no reason it can't.
Finite formula: Start with one. Add one. Repeat.
... see how that works?
Not a useful answer, as your formula is of the "Pick any number" type. For sure, "assuming there's an infinite amount of information there" (all positive integers, in this case), a finite formula can produce this infinity, but the topic here is a formula for a unique number, pi.


Finite formula : 1/3 = 0.3333333... : infinite amount of information
Finite formula : sqrt(2) = 1.41421356... : infinite amount of information.

Neither of these numeric expansions contains an infinite amount of information (see above).


1/3 repeats in base 10 notation. In other bases, a base divisible by 3, this decimal expansion wouldn't repeat.
See under, as a response to "Pi does not terminate"

...unless you are counting in base Pi. ;)
Pi terminates, just not in base 10 Cartesian :)
Not sure whether you're being serious, but if so:
1. Note smilies!
2. Doesn't help your argument - a finite formula for a number can't produce an infinite amount of information. You can't increase the information content of a number by changing its base - even to a transcendental base.

[stilicho]

I think the April Fool suggestion must be right. CaveDave, didn't it occur to you that "an ending to pi" would imply a proof of the non-existence of transcendental numbers, which (if true) would blow apart the foundations of (post-medieval) mathematics and logic? It would be huge orders of magnitude more world-shattering (and mind-boggling) than any conceivable scientific reversal (e.g. the speed of light is infinite, the age of the universe is 6000 years, most human disease and death is caused by vaccination).

This is interesting to me. What would that proof imply? I imagine it would extend to "e" and SQRT(-1) too. Am I right or wrong about that?

Please bear with me. The whole topic interests me but I'm not a professional mathematician nor do I play one on TV.

[CaveDave]

I think the April Fool suggestion must be right. CaveDave, didn't it occur to you that "an ending to pi" would imply a proof of the non-existence of transcendental numbers, which (if true) would blow apart the foundations of (post-medieval) mathematics and logic? It would be huge orders of magnitude more world-shattering (and mind-boggling) than any conceivable scientific reversal (e.g. the speed of light is infinite, the age of the universe is 6000 years, most human disease and death is caused by vaccination).
I absolutely agree!
Actually, the way it occurred was:
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], and I immediately started shaking my head emphatically and said "no, never, impossible, can't happen, you're wrong, it was a joke or misprint or you dreamed it, not in this universe!", and he being stubborn as I, asked me to "go on the internet and check" so I promised I would.

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.:)


Not 100%. First, the definition should exclude repeating expansions (and non-rational bases). Second, transcendental numbers are a subset of irrationals, and the corrected definition would apply to all irrationals. Third (though it's a subtle point), continuing the numeric expansion adds precision to the expansion only, not to the number itself.
Thanks for the clarification.


Not a complete list, as there are infinitely many!

Here's a start. (http://mathworld.wolfram.com/PiFormulas.html)

Thanks for that link.


It can't, for any useful definition of "finite formula".

So, here's a "finite formula": "Pick any number"! Or, if you prefer, "Write down all numbers"!

That's not what you meant, though, is it? Let's stipulate that a "finite formula" for some number (such as pi) has to define that number uniquely. Also, of course, that it can be expressed in a finite number of symbols (or steps, or definitions). In that case, it can't produce an infinite amount of information - though it can produce an infinite number of digits (repeating or not).

The numeric representation of pi, in any (non-transcendental!) base, is infinite (and non-repeating) - else it wouldn't be transcendental. But, by definition, this numeric representation is exactly equivalent to any formula for pi, so both contain the same information.
Fascinating.


Not a useful answer, as your formula is of the "Pick any number" type. For sure, "assuming there's an infinite amount of information there" (all positive integers, in this case), a finite formula can produce this infinity, but the topic here is a formula for a unique number, pi.

Neither of these numeric expansions contains an infinite amount of information (see above).

Not sure whether you're being serious, but if so:
1. Note smilies!
2. Doesn't help your argument - a finite formula for a number can't produce an infinite amount of information. You can't increase the information content of a number by changing its base - even to a transcendental base.
Cool!

Thank you (and the others) for the informative replies.

Cheers,

Dave

[Mark6]

Never 'terminates' - Pi is irrational.

A most excellent number.
Transcendingly excellent.

[readme.txt]

There is no cake pie..
The cake is a lie
or
The pie is a lie
or
The pie is a lake.


Neither of these numeric expansions contains an infinite amount of information (see above).

I was referring to his concept of "infinite amount of information". Of course, it's not infinite, but I tried to interpret what he said by giving other examples that fit his definition of "infinite amount of information".



Not sure whether you're being serious, but if so:
1. Note smilies!
2. Doesn't help your argument - a finite formula for a number can't produce an infinite amount of information. You can't increase the information content of a number by changing its base - even to a transcendental base.

Ibidem. I know a finite formula for a number can only THAT number, but I was still using his "definition". In base Pi, Pi is 10. That isn't useful at all, but it opposes his notion of "infinite amount of information".

I think this is just a misunderstanding of positions. I didn't agree with him, so I used arguments and examples using his "suppositions" to prove he was wrong and, therefore, that his definitions were wrong. That also why I added : "ETA: What's that "finite formula" you are talking about? "
:o

[quarky]

It bothers me how very little this ratio would need to be nudged to terminate.

It should prove the existence or non-existence of god.

It makes me so angry sometimes.

[Ladewig]

It bothers me how very little this ratio would need to be nudged to terminate.

It should prove the existence or non-existence of god.

It makes me so angry sometimes.

What are you talking about?

[Towlie]

This is interesting to me. What would that (hypothetical) proof (of the non-existence of transcendental numbers) imply? I imagine it would extend to "e" and SQRT(-1) too. Am I right or wrong about that?"e", yes, but not the square root of minus 1.

[Tarot_Is_A_Card_Game!]

See under, as a response to "Pi does not terminate"

How can there be a base pi? The base of a number system is the number of digits used. Wouldn't this require that the base be an integer?

[ddt]

Not 100%. First, the definition should exclude repeating expansions (and non-rational bases). Second, transcendental numbers are a subset of irrationals, and the corrected definition would apply to all irrationals. Third (though it's a subtle point), continuing the numeric expansion adds precision to the expansion only, not to the number itself.
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.

[Tarot_Is_A_Card_Game!]

How can there be a base pi? The base of a number system is the number of digits used. Wouldn't this require that the base be an integer?

Oops! Should have looked at Wikipedia more closely. It does appear to be possible at least in theory to have a non-integer base for a number system.

http://en.wikipedia.org/wiki/Positional_notation

http://en.wikipedia.org/wiki/Base_(mathematics)

[readme.txt]

Oops! Should have looked at Wikipedia more closely. It does appear to be possible at least in theory to have a non-integer base for a number system.

http://en.wikipedia.org/wiki/Positional_notation

http://en.wikipedia.org/wiki/Base_(mathematics))


Hehe :D

There's a little mistake in one of your links. I fixed it in the quote (see above)

You can also read this small, but interesting article : http://www.dwheeler.com/essays/bases.html

[Towlie]

Too bad that Wikipedia article (http://en.wikipedia.org/wiki/Base_%28mathematics%29) doesn't explain how Halloween = Christmas.

[stilicho]

"e", yes, but not the square root of minus 1.

I thought the evaluation of SQRT(-1) was partly dependent on the existent of transcendental numbers. Or vice versa.

Or is it that the result of SQRT(-1) wouldn't be SQRT(-1) but something "close-ish"?

[Towlie]

Although the square root of minus one is not a transcendental number, it is an imaginary number, and therefore, together with π and e, it belongs in the general category of "geek numbers".

[Complexity]

Of course, the ultimate geek numbers are 1 and 0, in that order.

Not a binary thing. Two friggin important ideas.

[quarky]

What are you talking about?

Just goofing. Not angry.
But i find it amusing that theologists and sometimes mathematicians prefer tidy sums. Pi is quite bizarre. Perfectly so. It almost deserves a religion. That sequence of numbers could be a Bible! My old girl-friend's phone number is in there, for christ's sake.

[Orphia Nay]

My brain just exploded:

http://i176.photobucket.com/albums/w194/orphia/Pi.jpg

http://www.chicagonow.com/blogs/redeye-puzzler/2010/04/get-ready-to-have-your-mind-blown.html

[Bill Thompson]

{reminder to self. Post lots of lauging gifs}

No, Pi does not terminate. It also never repeats. It is based on a circle which cannot be calculated 100% accurately by base 10 numbers that we use.

Think of it this way. If you wanted to represent a perfect sphere by using a box and you could only make angles to the shape of the box, how many angles would you have to make to form it into a perfect sphere.

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.

With each digit we add to PI, we are increasing the accuracy of representing a circle using numbers by 10 times. But we cannot ever got there and say "all done". Impossible.

I wonder if people here can get it. Please tell me you understand.

[Complexity]

I understand.

Now I want some pie. Cherry pie.

[Towlie]

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

[Furcifer]

Although the square root of minus one is not a transcendental number, it is an imaginary number, and therefore, together with π and e, it belongs in the general category of "geek numbers".


\sqrt{-69} :D

[Towlie]

Pi is transcendental no matter what integer base is used. For example, in base 16, Pi is approximately 3.243F6A8885A22. (Online Base Converter (http://www.easysurf.cc/cnver17.htm))

In my post #63 I asked, "Is it true that any arbitrary sequence of digits, no matter how long, will appear within the digits of Pi?"

The answer appears to be "yes."

Hexadecimal (base 16) numbers are a convenient way of expressing the contents of a computer file. Other bases that are equally appropriate are base 8 (octal) and even base 2 (binary).

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.

Can anyone seriously claim to own the rights to any portion of the digits of Pi?

Of course not!

That settles that! Now to see if Avatar is available on BitTorrent yet.

[orange31]

With each digit we add to PI, we are increasing the accuracy of representing a circle using numbers by 10 times. But we cannot ever got there and say "all done". Impossible.

I wonder if people here can get it. ..

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).

[dropzone]

Outside of pure theoretical mathematics, Pi does terminate in a way.I'm a CAD-oid and find, in everyday calculations OR in converting between metric and imperial, or vice versa, five decimal points usually suffice. HOWEVER, when aiming ones laser cutter, "good enough is NOT good enough," despite my Latin-trained brother's claims, such unseemly language leaves end and start points far enough away to cause the software to fail.

Which is pretty good, since in reality several MILLIONTHS will cause it to fail.

These days people ask for dimensions rounded off to the nearest inch. Between mentally asking when they will go metric and making the conversion between decimal and fractional, I ask why they bother, but give the answers. BOTH answers.

FTR, I thought I was the last defender of the Imperial scales. I was wrong.

[dasmiller]

I'm a CAD-oid and find, in everyday calculations OR in converting between metric and imperial, or vice versa, five decimal points usually suffice.

You'd think that, wouldn't you? And yet, I was associated with an aerospace project, maybe 15 years ago, for which the weight database was maintained in lb and the customer worked in kg. The software would work in any units, so this wasn't a problem . . . until a customer checked our math and discovered that we were using 0.453592 as a conversion factor. Of course, the true conversion is 0.45359237.

Yes, they were upset that the weight of our several-thousand-pound thing was being mis-reported by 0.001 kg, which was a full 1000X better than our weighing accuracy.

Okay, um, back to pi . . . back in my aspiring-SF-writer days, I had a civilization that discovered that it was in a computer-generated universe. The thing that first tipped them off was when they were doing precise measurements of space, and they discovered that sometimes their calculations would come closer to the measured values if they used a 64-bit approximation for pi rather than more exact solutions.

[soylent]

I would but it would take forever to tell you ;)

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.

[soylent]

Is it possible that pi is a rational numbers if you express it in some unconventional integer base like base 2927?

[Towlie]

Is it possible that pi is a rational numbers if you express it in some unconventional integer base like base 2927?No. A rational number is a number that can be expressed as a ratio of two integers, and it doesn't make any difference what base you use to write those two integers. In fact, you could just line up two rows of apples to represent the two integers and not use any particular base at all. Therefore, any rational number that can be written in one base can be written in any other base, including base 10.

The bottom line is that there's no number of apples that you could line up in two rows such that the number of apples in one row would exactly represent the circumference of a circle, and the number of apples in the other row would exactly represent the diameter of the same circle.

[Sherman Bay]

Is it possible that pi is a rational numbers if you express it in some unconventional integer base like base 2927?No. If that were true, then it would be possible to establish a 1:1 relationship with the {other base} number and base 10. That would require a fixed (rational) number for both. By definition, not possible.

[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? :D

[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]

;)

http://forums.randi.org/imagehosting/thum_266624bc22bb062294.jpg (http://forums.randi.org/vbimghost.php?do=displayimg&imgid=19686)

[roger]

This guy (http://www.correctpi.com/) has calculated the "correct value for Pi". He says it's 3.125 :jaw-dropp And also that it's 0.78125, 1.28, 0.64 and 0.21875 :confused:

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.comMy 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 (http://forums.randi.org/showpost.php?p=5810477&postcount=89), 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 (http://forums.randi.org/showpost.php?p=5813424&postcount=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? :D

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]

;)

http://forums.randi.org/imagehosting/thum_266624bc22bb062294.jpg (http://forums.randi.org/vbimghost.php?do=displayimg&imgid=19686)



I laughed at that. :D Then I read the caption. :o

[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.

[ddt]

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.

For what value of "excellent"? Theoretically, you're right. Practically, both the compression part and the decompression part of this algorithm just cost too much time.

[Modified]

No, I'm saying the "base 10 numbers that we use" has nothing to do with it, as you implied it did (http://forums.randi.org/showpost.php?p=5810477&postcount=89), but I think you already know that.

His other point seems to be that a circle is a curved shape, which also has nothing to do with it.

[MetalPig]

For what value of "excellent"? Theoretically, you're right. Practically, both the compression part and the decompression part of this algorithm just cost too much time.
Throw more hardware at it :D

[Jack by the hedge]

Throw more hardware at it :D

Could you train an infinite number of monkeys to search pi for stuff?

Every book, every song, every movie that ever was or ever could be is in pi somewhere.

There's an infinite number of copies of every movie, in fact. Plus an infinite number of each of the infinite number of variations on the original. Of course an infinite number of these would be extremely pornographic. (Won't somebody think of the monkeys?)

So - which version of which movie shall we set them to find first?:D

[Mikemcc]

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

[69dodge]

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.

An interesting idea. But the integers could be very large. Then, the compression ratio would be poor, because the compressed version would consist of all the digits of those large integers.

[ddt]

Throw more hardware at it :D

Are you trying to imitate Jan Sloot (http://en.wikipedia.org/wiki/Jan_Sloot) and his alleged super compression? :D
Romke Jan Bernhard Sloot (27 August 1945, Groningen—11 July 1999[citation needed], Nieuwegein) was a Dutch electronics technician, who claimed to have developed a revolutionary data compression technique, the Sloot Digital Coding System, which could compress a complete movie down to 8 kilobytes of data— this is orders of magnitude greater compression than the best currently available technology.
(wiki)

[dasmiller]

For what value of "excellent"? Theoretically, you're right. Practically, both the compression part and the decompression part of this algorithm just cost too much time.

Here's a variation on it - rather than pi, start with a string of digits that simply represents the integers smashed together: 01234567891011121314 etc

Now you can trivially compute where your digital book would lie on that number string, so there's no time wasted searching the string, and no more messy computations to come up with zillions of digits of pi. Simply give someone the index of the starting point and the length and - voila!

(yes, I know the critical flaw in this plan. That's what makes it funny for me)

[Ziggurat]

Is that possible? Is there such a thing as 3-ish or pi-ish? I could easily be falling for some kind of inside math joke but you'd have to explain it...which would ruin its humour value but might be educational in some way.

It's a joke. In math (and physics), many functions have limiting behavior, meaning that they approach some particular value for large enough (or small enough) values of the variable you plug into the function. So it's common to talk about, say, large values of 'x', or 'y', or whatever your variable is. The 'joke' (which isn't 'ha-ha' funny so much as absurd funny) is in treating a number like a variable. There's no such thing as "large values" of a number: a number has a specific value. Treating it like a variable has a sort of bizarre logic to it, but it produces nonsense. Like 2+2=5 for sufficiently large values of 2.

[Ziggurat]

Could you train an infinite number of monkeys to search pi for stuff?

Every book, every song, every movie that ever was or ever could be is in pi somewhere.

There's a classic problem in probability where you calculate the probability that some vast number of monkeys randomly hitting keys on a typewriter would type out some particular book. That's essentially equivalent to what you're asking for. It turns out that even for large numbers of monkeys (say, 10 billion), fast typing speeds (10 characters/sec), and long time scales (the current age of the universe), the probability of any of them typing even a few lines of Hamlet is incredibly small.

Variants of this problem are often assigned in statistical mechanics classes to give students some idea of how improbable certain outcomes can be, and why for macroscopic systems we can treat the improbable outcomes as being impossible even though technically they aren't.

In other words, regardless of what might be in pi, you won't find much. Also, there's no guarantee that every book, or even any book, is in pi. pi doesn't have to contain every possible finite number string. In fact, it's trivially easy to construct irrational numbers which are guaranteed to NOT have every finite number string.

[Towlie]

Also, there's no guarantee that every book, or even any book, is in pi. pi doesn't have to contain every possible finite number string. In fact, it's trivially easy to construct irrational numbers which are guaranteed to NOT have every finite number string.I think your first two sentences are definitely wrong and your last sentence is probably wrong too. Let's see what the others have to say about it.

[69dodge]

Here's an irrational number: 1.101001000100001000001...

(The pattern is that each 1 is followed by a string of 0s consisting of one more 0 than the previous string of 0s.)

There are lots of finite strings it doesn't contain: any string containing even a single 2, for instance.

[Ziggurat]

I think your first two sentences are definitely wrong and your last sentence is probably wrong too. Let's see what the others have to say about it.

My last sentence is rather easy to prove. Take the binary expression for pi. It's non-repeating and infinite, but it's only 0's and 1's. Now make a number whose base 10 expression is equal to pi's binary expression. It's infinite and non-repeating, so this new number is also irrational. But it's guaranteed to not have any strings containing the decimals 2, 3, 4, 5, 6, 7, 8, or 9, even though it's a base-10 number. So clearly, it's possible for irrational numbers to not contain every possible finite digit sequence. I just made one.

As for my former statement, well, that's really just a claim of a LACK of proof that pi does contain every string. You may suspect that pi contains every finite string, and maybe it does, but if you haven't proven that it does, then you only have a suspicion.

[readme.txt]

I think your first two sentences are definitely wrong and your last sentence is probably wrong too. Let's see what the others have to say about it.

His last sentence is not wrong. You can build a transcendental number that only has zeros, ones and twos and it will never contain the sequence 6578.

1,012210210101210210202010101020120210210210210210 10201201021210120201202222211202100001212...

[readme.txt]

My last sentence is rather easy to prove. Take the binary expression for pi. It's non-repeating and infinite, but it's only 0's and 1's. Now make a number whose base 10 expression is equal to pi's binary expression. It's infinite and non-repeating, so this new number is also irrational. But it's guaranteed to not have any strings containing the decimals 2, 3, 4, 5, 6, 7, 8, or 9, even though it's a base-10 number. So clearly, it's possible for irrational numbers to not contain every possible finite digit sequence. I just made one.

As for my former statement, well, that's really just a claim of a LACK of proof that pi does contain every string. You may suspect that pi contains every finite string, and maybe it does, but if you haven't proven that it does, then you only have a suspicion.

I agree. Maybe there's a proof, but I don't know it. Does Pi contain its own first billion decimals somewhere else after the billionth decimal?

[ddt]

Also, there's no guarantee that every book, or even any book, is in pi. pi doesn't have to contain every possible finite number string. In fact, it's trivially easy to construct irrational numbers which are guaranteed to NOT have every finite number string.

The notion you're looking for is normal number (http://en.wikipedia.org/wiki/Normal_number):
In mathematics, a normal number is a real number whose digits in every base follow a uniform distribution: all digits being equally likely, all pairs of digits equally likely, all triplets of digits equally likely, etc.

While a general proof can be given that almost all numbers are normal, this proof is not constructive and only very few concrete numbers have been shown to be normal. It is for instance widely believed that the numbers √2, π, and e are normal, but a proof remains elusive.

So it's an unproven, but widely believed, conjecture that this property holds for pi.

[Bill Thompson]

we are debating this?

that is the most amaizing thing I have ever seen.

If it repeated, it would also end using a different numeric base other than base 10. Repeating is the same thing as ending. Any number that does not divide into 10 without a remainder repeats just like 1/3rd.

Pi never repeats.

People are arguing mathimatical truths here. I find that astounding.

[dasmiller]

I agree. Maybe there's a proof, but I don't know it. Does Pi contain its own first billion decimals somewhere else after the billionth decimal?

Almost certainly, but of course, "almost certainly" isn't sufficient for a mathematical proof.

But there's a fair chance (50%?) that a repeat of the first billion digits occurs somewhere in the first 10^billion digits.

Of course, 1E1,000,000,000 is a large number.

[Bill Thompson]

There are 10 kinds of people in the world: the people who understand binary and the people who do not.



No, I'm saying the "base 10 numbers that we use" has nothing to do with it, as you implied it did (http://forums.randi.org/showpost.php?p=5810477&postcount=89), but I think you already know that.


It does.

if we used a numeric base that you could divide by 3 without a remainder, then lots of numbers that repeat under base 10 would not.

What numeric base you use determines what numbers repeat (like 0.3333333333...) and what numbers do not.

Maybe I do not understand what your point is. If you are saying it does not matter what numeric base we use to count and that Pi will never end or repeat, then you are agreeing with me.

(There are 10 kinds of people in the world: the people who understand binary and the people who do not.)


The point is, Pi never repeats. It does not matter what sort of counting system you use. It never repeats. Ever. IF PI REPEATED IN BASE 10 NUMBERS IT WOULD ALSO END IF WE USED A DIFFERENT COUNTING SYSTEM. Pi is the length it takes to draw a circle divided by its diameter. The diameter can be an easily determined number 1, 2, 3... but use that straight line as a unit to represent the circumference and you will get a number that can never be accurately 100% represented because you will always be using what amounts to straight line sebments to represent a curve.

Maybe some people just can't get it. Maybe it is just one of those things you can see or you cannot see (like Fermi's Paradox or what year the millennium began).

The fact that people will debate and argue mathematics is proof that we are not an intelligent species. That is what I am getting from this discussion thread.

[readme.txt]

if we used a numeric base that you could divide by 3 without a remainder, then lots of numbers that repeat under base 10 would not.

What numeric base you use determines what numbers repeat (like 0.3333333333...) and what numbers do not.

Maybe I do not understand what your point is. If you are saying it does not matter what numeric base we use to count and that Pi will never end or repeat, then you are agreeing with me.

(There are 10 kinds of people in the world: the people who understand binary and the people who do not.)


The point is, Pi never repeats. It does not matter what sort of counting system you use. It never repeats. Ever. IF PI REPEATED IN BASE 10 NUMBERS IT WOULD ALSO END IF WE USED A DIFFERENT COUNTING SYSTEM. Pi is the length it takes to draw a circle divided by its diameter. The diameter can be an easily determined number 1, 2, 3... but use that straight line as a unit to represent the circumference and you will get a number that can never be accurately 100% represented because you will always be using what amounts to straight line sebments to represent a curve.

Maybe some people just can't get it. Maybe it is just one of those things you can see or you cannot see (like Fermi's Paradox or what year the millennium began).

Wow, are you angry at someone or what?

The fact that people will debate and argue mathematics is proof that we are not an intelligent species. That is what I am getting from this discussion thread.

Isn't that a bit far-fetched?

[Towlie]

Here's an irrational number: 1.101001000100001000001...You and the people who posted after you are arguing that if you rule out certain digits or combinations in your definition of a number then those digits won't appear in the number. That seems petty and trivial. I could just as easily define an irrational number that contains all of the same digits as the square root of two except that whenever the sequence "69" appears it is replaced by "42". That would also be an irrational number that's guaranteed to NOT have every finite number string.

But enough of that. I did, after all, say "probably". I still dispute this statement by Ziggurat:

Also, there's no guarantee that every book, or even any book, is in pi. pi doesn't have to contain every possible finite number string.

[Towlie]

We've discussed decimal numbers that go on forever, irrational numbers, and transcendental numbers here. I wonder if Bill Thompson knows the difference.

[eeyore1954]

Does Pi terminate or never?
My high school math teacher had a cat named Pi and seeing as that was close to 40 years ago I would bet that Pi did terminate.

[readme.txt]

I still dispute this statement by Ziggurat:

<quote by Ziggurat>

As ddt said, it depends on its normality. I suggest you to read this brief article, which explains this well-known math issue : http://www.askamathematician.com/?p=177

It was never proven, therefore it is still a belief. So Ziggurat might not be wrong.

Does Pi terminate or never?
My high school math teacher had a cat named Pi and seeing as that was close to 40 years ago I would bet that Pi did terminate.
:D

[readme.txt]

Almost certainly, but of course, "almost certainly" isn't sufficient for a mathematical proof.

http://en.wikipedia.org/wiki/Almost_surely :D

[dasmiller]

http://en.wikipedia.org/wiki/Almost_surely :D

I had no idea that there was a formal definition! Thanks

[orange31]

The point is, Pi never repeats. It does not matter what sort of counting system you use. It never repeats. Ever....
Pi is the length it takes to draw a circle divided by its diameter. The diameter can be an easily determined number 1, 2, 3... but use that straight line as a unit to represent the circumference and you will get a number that can never be accurately 100% represented because you will always be using what amounts to straight line sebments to represent a curve.



No problem that Pi never repeats. But I'm curious why it doesn't repeat. If the reason relates to a difference between measurement on a straight line versus measurement on a curved line, it isn't obvious to me why a repeating decimal is ruled out in that situation. Is there something in calculus that gives insight into that? Long ago a math teacher mentioned that you can actually prove -using calculus- the shortest distance between two points is a straight line.

[W.D.Clinger]

No problem that Pi never repeats. But I'm curious why it doesn't repeat. If the reason relates to a difference between measurement on a straight line versus measurement on a curved line, it isn't obvious to me why a repeating decimal is ruled out in that situation. Is there something in calculus that gives insight into that? Long ago a math teacher mentioned that you can actually prove -using calculus- the shortest distance between two points is a straight line.
Any finite sequence of digits in the decimal expansion of pi might be repeated somewhere else in the decimal expansion of pi.

What never happens is that the digits of pi settle down into some finite sequence of digits that repeats infinitely, that same sequence followed immediately by that same sequence, and so on. The reason we know this can't happen is that pi is irrational (http://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational).

[RussDill]

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.

Now, a far more interesting plot element would be embedded information in unitless physical constants. The more accurately determined the number, the more data that could be seen.

(ETA: with instructions embedded early on in the sequence for building machines that determine the number with greater accuracy)

[gnome]

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.

I'm not sure why that would be an obstacle to the kind of creator Sagan envisioned. You're looking at data that OUGHT to be random and determined by the nature of the very universe itself... but clearly isn't random... a dead giveaway isn't it? It proves you exist and so therefore you don't.

Oh dear, I hadn't thought of that *puff of logic*.


:D

[sol invictus]

The fact that people will debate and argue mathematics is proof that we are not an intelligent species. That is what I am getting from this discussion thread.

That makes no sense whatsoever. If anything, it is proof that we are an intelligent species.

That said, who's arguing about anything?

[Perpetual Student]

No problem that Pi never repeats. But I'm curious why it doesn't repeat. If the reason relates to a difference between measurement on a straight line versus measurement on a curved line, it isn't obvious to me why a repeating decimal is ruled out in that situation. Is there something in calculus that gives insight into that? Long ago a math teacher mentioned that you can actually prove -using calculus- the shortest distance between two points is a straight line.

This might help: LINK (http://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational)

and (regarding the shortest distance proof): LINK (http://www.instant-analysis.com/Principles/straightline.htm)

[Soapy Sam]

Given the curvature of spacetime is scarcely zero this close to a star, is it actually true that C/d=pi anyway? Or is this just an approximation?

[Audible Click]

I laughed at that. :D Then I read the caption. :o

I laughed at it and ignored the caption. :)

[drkitten]

But enough of that. I did, after all, say "probably". I still dispute this statement by Ziggurat:

You shouldn't. As has been already pointed out, Ziggurat's statement is simply that pi has not been proven normal (to base 10, technically), which is true.

[Soapy Sam]

What does "normal" mean in that post drk?
This is not a philosophical query, merely one about definition.

[JoeyDonuts]

Put it in Pascal's Triangle!

See if it does something!

[ddt]

What does "normal" mean in that post drk?
This is not a philosophical query, merely one about definition.

A "normal number (http://en.wikipedia.org/wiki/Normal_number)" is a number in whose decimal expansion the digits are uniformly distributed (i.e., they all occur with equal probability), as well as all series of two digits, of three digits, etc.; and not only in decimal expansion, but in expansion in an arbitrary (integer) base.

Nearly all numbers are normal.

[Soapy Sam]

Ta.

[Modified]

Pi is the length it takes to draw a circle divided by its diameter. The diameter can be an easily determined number 1, 2, 3... but use that straight line as a unit to represent the circumference and you will get a number that can never be accurately 100% represented because you will always be using what amounts to straight line sebments to represent a curve.

That is not an explanation, it is some sort of (incorrect) intuition. Any number of closed curves have integer relationships between circumference and some property represented by a straight distance.

[drkitten]

A "normal number (http://en.wikipedia.org/wiki/Normal_number)" is a number in whose decimal expansion the digits are uniformly distributed (i.e., they all occur with equal probability), as well as all series of two digits, of three digits, etc.; and not only in decimal expansion, but in expansion in an arbitrary (integer) base.

Nearly all numbers are normal.

I suppose technically that what Ziggy was saying was not that pi has not been proven "normal," but that pi has not been proven "disjunctive (http://en.wikipedia.org/wiki/Disjunctive_sequence)", which is a weaker property. But since pi is almost universally believed to be normal, and all normal numbers are disjunctive, the distinction is quite weak.

Oh, and since nearly all numbers are normal, nearly all numbers are disjunctive, too.... (Duh.)

[readme.txt]

I suppose technically that what Ziggy was saying was not that pi has not been proven "normal," but that pi has not been proven "disjunctive (http://en.wikipedia.org/wiki/Disjunctive_sequence)", which is a weaker property. But since pi is almost universally believed to be normal, and all normal numbers are disjunctive, the distinction is quite weak.

Oh, and since nearly all numbers are normal, nearly all numbers are disjunctive, too.... (Duh.)

That's exactly it. Normality (therefore "disjunctivity") on every irrational number is strongly suggested, but still unproven, thus what Ziggy said.

[stilicho]

The point is, Pi never repeats. It does not matter what sort of counting system you use. It never repeats. Ever. IF PI REPEATED IN BASE 10 NUMBERS IT WOULD ALSO END IF WE USED A DIFFERENT COUNTING SYSTEM. Pi is the length it takes to draw a circle divided by its diameter. The diameter can be an easily determined number 1, 2, 3... but use that straight line as a unit to represent the circumference and you will get a number that can never be accurately 100% represented because you will always be using what amounts to straight line sebments to represent a curve.

The fact that people will debate and argue mathematics is proof that we are not an intelligent species. That is what I am getting from this discussion thread.

I disagree. Some of us aren't professional mathematicians. (I'd gauge rather more than a few). Curiosity is actually one sign of possible intelligence.

But I can absolutely see the logic in what you're saying above. I trust you're saying that the ratio of the circumference to the radius of a circle has to be transcendental in all numeric systems if it transcendental in any of them. That makes logical sense to me since any number would have to have the same characteristics in all systems to have them in any one of them. There's nothing magical about our base ten system AFAIK.

Is this true of all number "types"? Irrational numbers? Imaginary numbers? Negative numbers? Whole numbers? If "type" isn't the right word then tell us what it is.

Are there different rules that apply to ratios?

[Ladewig]

I have a question about pi that is not worth its own thread, so I'll ask it here.

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 ?






ETA: changed billions to trillions.

[readme.txt]

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 ?


Well, there's some kind of unexplainable metaphysical adoration for Pi. It might be because of its "mysteriousness"... I dunno really.

There's also another explanation that has a lot to do with physics, since Pi is absolutely useful in angle determination : the more digits you know, the more accurate you will be, for a higher distance (i.e. in astrophysics and all related fields, Pi might be more useful than "e" or sqrt(2) ). I don't want to go further in my "explanation", since I'm more of a mathematician than a physicist.

[ddt]

But I can absolutely see the logic in what you're saying above. I trust you're saying that the ratio of the circumference to the radius of a circle has to be transcendental in all numeric systems if it transcendental in any of them. That makes logical sense to me since any number would have to have the same characteristics in all systems to have them in any one of them. There's nothing magical about our base ten system AFAIK.

Is this true of all number "types"? Irrational numbers? Imaginary numbers? Negative numbers? Whole numbers? If "type" isn't the right word then tell us what it is.

Are there different rules that apply to ratios?

What you call "numeric system" is actually only the representation of the number as a string of digits. The inherent properties of a number - if it's a whole number, or if it's rational, or algebraic, or transcendental - do not change whether you write out the number in base-10 or in base-2 or in base-37 or whatever positive integer base you might choose. Note, I restrict myself to positive integer bases, as they're the only ones commonly regarded.

How the representation looks like is simple.

Whole (=integer) numbers have always a representation without fraction, irrespective of the base.

Rational numbers have a representation with a finite fraction part, or a repeating fraction. There is at least one base in which a rational number has a finite fraction: a rational number is the division of two whole numbers, so just take the denominator as the base.

Irrational numbers have in every base a representation with an infinite, non-repeating fraction part.

Transcendental numbers are a subset of the irrational numbers, so they always have an infinite, non-repeating fraction part.

[Ziggurat]

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 ?

Well, pi and e both have a special place in mathematics in a way that roots of integers just don't. Plus, circles are cooler than right triangles. And e has no simple corresponding geometric shape. So people think pi is just more interesting. On some level it's a subjective evaluation, but that's OK.

[stilicho]

What you call "numeric system" is actually only the representation of the number as a string of digits. The inherent properties of a number - if it's a whole number, or if it's rational, or algebraic, or transcendental - do not change whether you write out the number in base-10 or in base-2 or in base-37 or whatever positive integer base you might choose. Note, I restrict myself to positive integer bases, as they're the only ones commonly regarded.

Perfect. That's what I wanted to know. It's the property of the number that doesn't change.

What's this about non-positive and/or non-integer bases now?

[3point14]

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.

That first integer might be remarkably big on occasion.

[Ziggurat]

That first integer might be remarkably big on occasion.

In fact, that integer should be about the same size as the file, on average. So... not really compression at all, actually.

[Towlie]

In fact, that integer should be about the same size as the file, on average.Why?

[3point14]

In fact, that integer should be about the same size as the file, on average. So... not really compression at all, actually.

The same? Or bigger?

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.

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

I'm very bad at maths.

[dasmiller]

Why?

Let's do a quick estimate of how far down the string of digits your movie is likely to be.

Let's say you're encoding a tiny picture that requires 1000 digits. A particular random sequence of 1000 digits has 1 in 10^1000 chance of being identical to the one you want. Thus, if you had 10^1000 sequences of 1000 digits, you'd have a reasonably good chance of having the one you want.

As you travel down pi, every new digit is the beginning of a 1000 digit sequence, so you'll likely have to go on the order of 10^1000 digits down the line until you find the sequence you want. And how many base-10 digits does it take to specify a number on the order of 10^1000? Why, around 1000, of course.

[orange31]

re the attention to Pi,

as many useful functions involve cycles or repetitive functions, the unit circle is widely distributed throughout classical and quantum physics (ie, think how often you see sine, tangent, etc), therefore pi is all over the place.

Slightly different, but I always loved how Feynman in his undergraduate physics course lecture, once as an aside (I think the topic was vectors) told the students in his NYC accent- "anytime ya see somethin' with a bunch of square roots in it, ya got a^2 + b^2 = c^2 in dere somewhere....the Pythagorean theorem."

[Ladewig]

Well, there's some kind of unexplainable metaphysical adoration for Pi. It might be because of its "mysteriousness"... I dunno really.

I can buy that

There's also another explanation that has a lot to do with physics, since Pi is absolutely useful in angle determination : the more digits you know, the more accurate you will be, for a higher distance (i.e. in astrophysics and all related fields, Pi might be more useful than "e" or sqrt(2) ). I don't want to go further in my "explanation", since I'm more of a mathematician than a physicist.

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.

[Ladewig]

re the attention to Pi,

as many useful functions involve cycles or repetitive functions, the unit circle is widely distributed throughout classical and quantum physics (ie, think how often you see sine, tangent, etc), therefore pi is all over the place.


I will agree that pi is all over the place, but that does not account for the publicity surrounding and the desire to know more than the first trillion digits.

[Complexity]

Will this never end?

[Ladewig]

Will this never end?

I thought we already established the irrationality of this thread.

[Complexity]

I thought we already established the irrationality of this thread.


Thank you for not suggesting it is transcendent as well.

[ddt]

Perfect. That's what I wanted to know. It's the property of the number that doesn't change.
Yep.


What's this about non-positive and/or non-integer bases now?
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 (http://en.wikipedia.org/wiki/Non-integer_representation) 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 (http://en.wikipedia.org/wiki/Golden_ratio_base) 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.

[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. :D

[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 (http://en.wikipedia.org/wiki/Non-integer_representation) 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 (http://en.wikipedia.org/wiki/Golden_ratio_base) 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 (http://mathworld.wolfram.com/PiFormulas.html) are some algorithms for the expansion of pi.

Let's take, for example, the Gregory series (http://mathworld.wolfram.com/GregorySeries.html): 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 (http://mathworld.wolfram.com/PiFormulas.html) are some algorithms for the expansion of pi.

Let's take, for example, the Gregory series (http://mathworld.wolfram.com/GregorySeries.html): 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 (http://en.wikipedia.org/wiki/Non-integer_representation) 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 (http://en.wikipedia.org/wiki/Golden_ratio_base) 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:

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

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 25x1010 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 101.5x1010, 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.

[sol invictus]

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.

What I like about Sagan's idea is that it's logically possible, actually - just extremely unlikely if there's no god. That is, suppose it is the case (as everyone seems to believe) that pi is normal. That means that every finite sequence occurs in it infinitely many times.

So whatever the message is, it's in there somewhere, and all the creator has to do is make sure it occurs near enough to the beginning that we have a chance of noticing it by computing enough digits of pi. Since any sequence can be a significant message in some code, it's just a matter of creating life that will see some relatively early sequence as significant.

Off the top of my head, one example would be a string of a million 1's in the base 10 decimal expansion of pi. The frequency with which that would occur is something like 10-1,000,000 if pi is normal, so if we find it it would be pretty good proof of the existence of god. But if we had 6 fingers, or 2, or 12, and used a number system with a different base, we might not notice it.

So here's a question for the mathematicians - if the base ten decimal expansion of pi has a string of a million 1s starting, say, at the trillionth digit, would we be likely to notice if we used a base 7 number system (and had computed pi to well past that point)?

[Robin]

The question becomes, can an ultimate creator fashion a universe where the laws of mathematics operate differently? Philosophically speaking I think so.
Philosophically speaking, I would say definitely no. Mathematics would be the same in any possible universe or any possible mode of existence.

Of course the symbol system used to work out maths could be different - that would not even require a different universe. We could work out a complete mathematics based on a different set of axioms. But those different systems would also work identically in all possible universes.

[Robin]

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.
But then that wouldn't be Pi, that would be something else.

Pi is defined for a plane - a cartesian product of reals.

[orange31]

So here's a question for the mathematicians - if the base ten decimal expansion of pi has a string of a million 1s starting, say, at the trillionth digit, would we be likely to notice if we used a base 7 number system (and had computed pi to well past that point)?

Generally yes, and specifically the answer would be "42".

Thanks Dr K for the math lesson-
----Helpful thought for the math invalids like me, the expression "rational" number simply means ratio , as in the number can be expressed as the ratio of two integers. "Irrational number" has nothing to do with rational/ logical, or being far out like 'imaginary' numbers, etc.
A 'duh' point, but I needed it.

[ddt]

Am I able to practice any of these non-standard bases in Excel?
I don't know, I don't use Excel. I tried in OpenOffice Calc to use a non-integer base for the BASE function, and it didn't work - that is, I tried using 2.5 as base and it gave the answer as if I had given it 2 as base. You could program the algorithm on that wiki page into a VBScript function, or whatever it's called.


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.
What do you mean with "some of its dimensions"? phi is a number, full stop.

It's irrational because sqrt(5) is irrational. It's obvious that if sqrt(5) were rational, then phi = (1+sqrt(5))/2 were also rational; and conversely, if phi were rational, than sqrt(5) were also rational.

Proof that sqrt(5) is not rational is pretty easy, and goes by contradiction.
Suppose sqrt(5) is rational, and it is equal to p/q where p and q are both (positive) integers. Then simplify this fraction to p'/q' where p' and q' have no (prime) factors in common.
Then we have: sqrt(5) = p' / q' and when we square that equation, and move q' to the other side, we get
5 * q' * q' = p' * p'
The main theorem of number theory states that factorization into prime numbers is unique. So p' * p' must contain a factor 5, and therefore p' must contain a factor 5 and therefore, p' * p' must contain at least two factors 5. That, in its turn, implies that q' * q' must contain at least one factor 5, and therefore q' contains a factor 5. However, we started out with saying that p' and q' had no common factors, and we have now established they both have a factor 5. Contradiction.
Ergo, sqrt(5) is not rational.


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.
:jaw-dropp The idea of a system of weights and measures in base-phi is even less practical than imperial weights and measures with its haphazard factors of 3, 12, 14, 16 and what have you.

[Robin]

What I like about Sagan's idea is that it's logically possible, actually - just extremely unlikely if there's no god. That is, suppose it is the case (as everyone seems to believe) that pi is normal. That means that every finite sequence occurs in it infinitely many times.

So whatever the message is, it's in there somewhere, and all the creator has to do is make sure it occurs near enough to the beginning that we have a chance of noticing it by computing enough digits of pi.
I don't think it is logically possible. If some intelligence could alter the digits of pi to make this sequence close to the beginning then it could make the sequence right at the beginning, thus this intelligence could make a pi that was 2.1419 for example.

Then the immediate operations that produce these digits would also have to be different and the definitions upon which pi was based would have to be different and going right back the axioms would have to be different - it would end up not being pi at all and not the ratio between the diameter and circumference of a plane circle.

[sol invictus]

I don't think it is logically possible.

It obviously is.


If some intelligence could alter the digits of pi to make this sequence close to the beginning then it could make the sequence right at the beginning, thus this intelligence could make a pi that was 2.1419 for example.

That's not what I suggested. We know that isn't the case, so you're discussing some sort of bizarre countermathfactual.

Then the immediate operations that produce these digits would also have to be different and the definitions upon which pi was based would have to be different and going right back the axioms would have to be different - it would end up not being pi at all and not the ratio between the diameter and circumference of a plane circle.

But that isn't the case, and it's not what I suggested (or Sagan for that matter).

Again - as far as we know, to the best of our knowledge, pi may contain a sequence of 10^6 1's starting from the trillionth digit of its base 10 decimal expansion.

There is nothing logically inconsistent about that; it may be true and we may discover it to be true in a decade. However, the odds of it being true - assuming pi is normal - are something like 10^{-1,000,000}. Therefore if it does turn out to be true, we might try to interpret it as a proof of the existence of god.

Why? Because god, having noticed that the transcendental number we call pi has this interesting feature in its base 10 decimal expansion, could have created us with 10 digits on our hands and in a world where the ratio of circumference to diameter of a Euclidean circle has special significance, so that we would notice this incredible coincidence and worship her.

This was just an example; you can think of others (and Sagan's was different as far as I can recall). But the basic idea is simply that god created us so that we would find something we interpret as unusual in the expansion of pi. That's clearly not logically inconsistent.

[Robin]

It obviously is.
Nothing obvious about it, the way you put it.
But that isn't the case, and it's not what I suggested (or Sagan for that matter).

Again - as far as we know, to the best of our knowledge, pi may contain a sequence of 10^6 1's starting from the trillionth digit of its base 10 decimal expansion.

There is nothing logically inconsistent about that; it may be true and we may discover it to be true in a decade. However, the odds of it being true - assuming pi is normal - are something like 10^{-1,000,000}. Therefore if it does turn out to be true, we might try to interpret it as a proof of the existence of god.

Why? Because god, having noticed that the transcendental number we call pi has this interesting feature in its base 10 decimal expansion, could have created us with 10 digits on our hands and in a world where the ratio of circumference to diameter of a Euclidean circle has special significance, so that we would notice this incredible coincidence and worship her.

This was just an example; you can think of others (and Sagan's was different as far as I can recall). But the basic idea is simply that god created us so that we would find something we interpret as unusual in the expansion of pi. That's clearly not logically inconsistent.
Well here is what you said:
So whatever the message is, it's in there somewhere, and all the creator has to do is make sure it occurs near enough to the beginning that we have a chance of noticing it by computing enough digits of pi.

So you cannot blame me for misinterpreting this to mean that you were saying that the creator could make the digits of pi different to what they are.

[Towlie]

Here's something I've always wondered about Pi: Why do we use the circumference divided by the diameter and not the circumference divided by the radius? I began to wonder about that when I noticed how many times I encountered the expression "2π", such as the fact that there are 2π radians in a circle.

[Robin]

As I recall Ellie finds a rasterised circle made of 1's and 0's in the first 10^20 digits of pi.

But she is trying various different bases and she can arrange the 1's and 0's found in any row/column combination.

So I wonder what the odds really are that for some base, for some combination of row/column there might be a rasterised circle of some diameter in the first 10^20 digit expansion.

[sol invictus]

So you cannot blame me for misinterpreting this to mean that you were saying that the creator could make the digits of pi different to what they are.

"Make the digits of pi different to what they are": as far as I can parse that it's an oxymoron, so I can indeed blame you for misinterpreting what I said that way.

[sol invictus]

As I recall Ellie finds a rasterised circle made of 1's and 0's in the first 10^20 digits of pi.

But she is trying various different bases and she can arrange the 1's and 0's found in any row/column combination.

So I wonder what the odds really are that for some base, for some combination of row/column there might be a rasterised circle of some diameter in the first 10^20 digit expansion.

Essentially 1 - all you'd need to do is find

...010...
...101...
...010...

somewhere. But presumably she found a really big one, or something.

[gnome]

But she is trying various different bases and she can arrange the 1's and 0's found in any row/column combination.

Not quite--she was looking for a sequence that had a length that was the product of two primes... it lends itself to only one way of lining them up. This (we hope) is the same way a sequential binary signal could self-identify as a two-dimensional image for signals we send for aliens to receive... one that was the product of three primes could be examined as a three-dimensional diagram or (more likely) as a two-dimensional movie.

[gnome]

Here's something I've always wondered about Pi: Why do we use the circumference divided by the diameter and not the circumference divided by the radius? I began to wonder about that when I noticed how many times I encountered the expression "2π", such as the fact that there are 2π radians in a circle.

Honestly it's just a convention I think.

Plus using the diameter we get (e^(pi*i))+1=0, which is pretty nifty.

[gnome]

This was just an example; you can think of others (and Sagan's was different as far as I can recall). But the basic idea is simply that god created us so that we would find something we interpret as unusual in the expansion of pi. That's clearly not logically inconsistent.

It isn't, but it is different from what was presented in the book, which implied heavily that Sagan's "God" set the value of PI so that the "message" would be in it. In fact, it appeared in base eleven according to the book.

he was implying intelligence in the creation of the universe, not a personal creator for humans.

[gnome]

I don't think it is logically possible. If some intelligence could alter the digits of pi to make this sequence close to the beginning then it could make the sequence right at the beginning, thus this intelligence could make a pi that was 2.1419 for example.

Then the immediate operations that produce these digits would also have to be different and the definitions upon which pi was based would have to be different and going right back the axioms would have to be different - it would end up not being pi at all and not the ratio between the diameter and circumference of a plane circle.

Here's the beauty... by putting it umpteen thousands of digits down the line, so that in any practical scale it would not prevent circles from existing properly... the creator ensures that the physics of it works out the same way--and that we need a certain degree of development (computers) before we can discover the message.

[Ron Webb]

The value of pi is what it is, and not even God can change it. It would be extremely unlikely to find a string of a million 1s or a rasterized circle or any other such simple sequence or pattern in the first trillion or so digits, in any base. If we found such a thing I'm not sure what we would make of it, but I wouldn't call it a message from God. I can only say that God would probably be as astonished as us. (And yes, I know it should probably be "as astonished as we", but that just sounds silly.)

However, if we found a more complex coded message, such as the phrase "I am the Lord your God", spelled out in ASCII characters, then I think that would be very good evidence of God. Not that God could have put that particular string there either, but He certainly could have influenced the historical development of the English language and the choice of ASCII character representations so that the string, when we eventually found it, would have the meaning that it does.

[Myriad]

So far, the analyses of the compression scheme seem to agree that the number of bits in the file to be compressed would be (as an expected value) comparable to the number of bits needed to express the position of the first chance occurrence of a sequence of digits identical to the file within the digits of pi. Thus, there is no compression.

However, if a lossy compression stream is acceptable, then we could do much better. For instance, suppose in our 6 gigabyte movie we allow a mere 20 single bit errors. The density of possible digit sequences to point increases by a factor of about 48,000,000,000 ^ 60, or 5.7 * 10^640 which means the expected digit offset to find the first one is only 1 / 5.7e640 the distance from the start... giving a compression factor of...

... oops, that only cuts about 2,135 bits off the length of the number giving the offset. Out of 48 billion. A compression factor of a few millionths of a percent. Darn those counterintuitive big numbers.

(Not to mention the minor "takes much longer than the age of the universe to decompress" problem.)

Respectfully,
Myriad

[Robin]

"Make the digits of pi different to what they are": as far as I can parse that it's an oxymoron...
Indeed it is, that is my whole point.

When you talked of the making sure the message appears close to the beginning of the digits of pi it sounds like you are talking about altering the digits of pi.
... so I can indeed blame you for misinterpreting what I said that way.
Sure - when somebody misunderstands them immediately assume the blame does not lie with the clarity of your writing.

[Robin]

It isn't, but it is different from what was presented in the book, which implied heavily that Sagan's "God" set the value of PI so that the "message" would be in it. In fact, it appeared in base eleven according to the book.

he was implying intelligence in the creation of the universe, not a personal creator for humans.
But it would not matter how a God created the Universe, it would not change the value of Pi by a single digit.

No God could alter Pi.

[shadron]

The value of pi is what it is, and not even God can change it. It would be extremely unlikely to find a string of a million 1s or a rasterized circle or any other such simple sequence or pattern in the first trillion or so digits, in any base. If we found such a thing I'm not sure what we would make of it, but I wouldn't call it a message from God. I can only say that God would probably be as astonished as us. (And yes, I know it should probably be "as astonished as we", but that just sounds silly.)

However, if we found a more complex coded message, such as the phrase "I am the Lord your God", spelled out in ASCII characters, then I think that would be very good evidence of God. Not that God could have put that particular string there either, but He certainly could have influenced the historical development of the English language and the choice of ASCII character representations so that the string, when we eventually found it, would have the meaning that it does.

Not so. The point of the rasterized circle in base 11 is that this could be found by any civilization, any intelligence in the universe from a simple series expansion and a computer (or their own brain given time and paper enough). No need to know ascii, no need to know english, no need to be air breathers or carbon based life. Just know what a circle is, enough math to have developed a series expansion for pi, and time. Your message in terms of information is much smaller than hers was; it was something like 51x51 base-11 digits long.

[MetalPig]

Nothing to add, just a 'thank you' to those who picked up the 'compression' idea and analyzed it beyond the point where I would have given up :D

[Ziggurat]

Here's something I've always wondered about Pi: Why do we use the circumference divided by the diameter and not the circumference divided by the radius? I began to wonder about that when I noticed how many times I encountered the expression "2π", such as the fact that there are 2π radians in a circle.

With 2pi radians in a circle, the power series expansion for sin(x) and cos(x) become nice and tidy, the derivatives of each just jump back and forth between them without introducing any prefactors, and they connect nicely to ex. If you make it so there are pi radians in a circle, these relationships get cluttered.

[ddt]

Here's something I've always wondered about Pi: Why do we use the circumference divided by the diameter and not the circumference divided by the radius? I began to wonder about that when I noticed how many times I encountered the expression "2π", such as the fact that there are 2π radians in a circle.

Honestly it's just a convention I think.

Plus using the diameter we get (e^(pi*i))+1=0, which is pretty nifty.

With 2pi radians in a circle, the power series expansion for sin(x) and cos(x) become nice and tidy, the derivatives of each just jump back and forth between them without introducing any prefactors, and they connect nicely to ex. If you make it so there are pi radians in a circle, these relationships get cluttered.

However, approximations of pi are known already from the ancient Egyptians and the ancient Greeks, among many others. They didn't yet do power series or imaginary numbers. Now, it can be of course they really made approximations of 2*pi - I don't know, I didn't read the ancient sources.

But there's another equation - about the area of a circle - which I think is very old:
A = pi * r^2
and that one would get a fraction of 1/4 in it when you'd use 2*pi as the constant.

[ddt]

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.

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.

As late as Gauss - start 19th Century and contemporary of Lobachevsky - people regarded indeed (Euclidean) geometry as a description of reality. We now know better (think GR), and regard mathematics as "just" mind games, and it's up to the physicists to pick the right mathematical model. :)

Yes, you could think of a world where Lobachevskian geometry were the first type of geometry invented. That makes you think how that would influence the development of, e.g., calculus, or complex numbers. After all, pi is present all over the place: Euler's formula, Fourier series, etc.

In envisaging another world with different mathematics, you can go one step further. What about a different logic underlying what constitutes mathematical proof? There is such one, intuitionistic logic, which lacks the Law of excluded middle. That implies that you can't make a proof that "there is a number with property P" without, in fact, giving an algorithm to calculate that number. It also implies that reasoning about negatives becomes much more tedious. Take my above proof of sqrt(5) being irrational as an example. If you define "irrational" as "not rational", that proof is allowed; however, proving a number rational by contradiction, starting out with the assumption that the number is irrational, would not, as you'd at most get that the number is "not not rational" and you couldn't get away the double negative. Nevertheless, intuitionists have been able to recreate various parts of mathematics in an intuitionistic logical framework.

[3point14]

<SNIP>
No, it's easy to do.,SNIP>



You missed out the "for me" after that. Easy for you. For me, not so much, but I'm quite flattered you thought I might be able to understand all that squiggly stuff. :)

[W.D.Clinger]

Nevertheless, intuitionists have been able to recreate various parts of mathematics in an intuitionistic logical framework.
Indeed, all of the mathematics needed to formulate the laws of science and engineering appears to be available within a constructive (http://en.wikipedia.org/wiki/Mathematical_constructivism) framework.

Which is kind of fortunate, because we'd like to calculate with those laws.

[Lucky]

What I like about Sagan's idea is that it's logically possible, actually - just extremely unlikely if there's no god. That is, suppose it is the case (as everyone seems to believe) that pi is normal. That means that every finite sequence occurs in it infinitely many times.

So whatever the message is, it's in there somewhere, and all the creator has to do is make sure it occurs near enough to the beginning that we have a chance of noticing it by computing enough digits of pi. Since any sequence can be a significant message in some code, it's just a matter of creating life that will see some relatively early sequence as significant.

Off the top of my head, one example would be a string of a million 1's in the base 10 decimal expansion of pi. The frequency with which that would occur is something like 10-1,000,000 if pi is normal, so if we find it it would be pretty good proof of the existence of god. But if we had 6 fingers, or 2, or 12, and used a number system with a different base, we might not notice it.
But that's not at all what Sagan suggested in Contact (you seem not to have read the book). For a start, the 'message' is found in the base 11 expansion of pi - which removes any possibility that Sagan had your idea in mind.

He refers to the pattern in pi as the 'artist's signature', and it's quite clear that he really is implying an intelligence that created pi (and presumably mathematics and logic as a whole), along with the physical universe. Even worse, he implies that pi is dependent on the properties of space in our universe, and that we discover its value by measuring it!

Do you seriously want to argue that there's any other way to interpret this line (from the book's conclusion):

"The universe was made on purpose, the circle said. In whatever galaxy you happen to find yourself, you take the circumference of a circle, divide it by its diameter, measure closely enough, and uncover a miracle — another circle, drawn kilometers downstream of the decimal point. In the fabric of space and in the nature of matter, as in a great work of art, there is, written small, the artist’s signature. Standing over humans, gods, and demons, subsuming Caretakers and Tunnel builders, there is an intelligence that antedates the universe."

I and the couple of others here who are trying to explain that pi is not a property of the physical universe seem to be getting nowhere.

[gnome]

I accept that it's not a property of the physical universe. What I'm exploring (and not convinced one way or the other) is of the universality of the concept of pi -- starting as a thought experiment in universes where it is not physically relevant.

[Robin]

However, approximations of pi are known already from the ancient Egyptians and the ancient Greeks, among many others. They didn't yet do power series or imaginary numbers. Now, it can be of course they really made approximations of 2*pi - I don't know, I didn't read the ancient sources.

But there's another equation - about the area of a circle - which I think is very old:
A = pi * r^2
and that one would get a fraction of 1/4 in it when you'd use 2*pi as the constant.
It could just be that the ratio of the circumference to the diameter was the most practically useful ratio in ancient times.

For example early approximations of pi might have been used by builders or artisans to estimate the amount of material needed to build a round structure or object of a certain width.

[ddt]

Indeed, all of the mathematics needed to formulate the laws of science and engineering appears to be available within a constructive (http://en.wikipedia.org/wiki/Mathematical_constructivism) framework.

Which is kind of fortunate, because we'd like to calculate with those laws.

Not all. For one example: Quantum Mechanics (http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics) relies on functional analysis, in particular Hilbert spaces, and for that you need the Axiom of Choice (http://en.wikipedia.org/wiki/Axiom_of_choice), which is universally rejected by intuitionists/constructivists because it is non-constructive. In fact, it would re-introduce the law of the excluded middle through the backdoor.

[sol invictus]

But that's not at all what Sagan suggested in Contact (you seem not to have read the book).

I read it many years ago. What I recall is a hidden pattern in pi, not necessarily what it was. But I disagree that what I'm suggesting is significantly different.

For a start, the 'message' is found in the base 11 expansion of pi - which removes any possibility that Sagan had your idea in mind.

Umm, no. That was a simple example. Sagan's is more sophisticated.

He refers to the pattern in pi as the 'artist's signature', and it's quite clear that he really is implying an intelligence that created pi (and presumably mathematics and logic as a whole), along with the physical universe. Even worse, he implies that pi is dependent on the properties of space in our universe, and that we discover its value by measuring it!

No, I don't think so. Not necessarily at least.

Do you seriously want to argue that there's any other way to interpret this line (from the book's conclusion):

"The universe was made on purpose, the circle said. In whatever galaxy you happen to find yourself, you take the circumference of a circle, divide it by its diameter, measure closely enough, and uncover a miracle — another circle, drawn kilometers downstream of the decimal point. In the fabric of space and in the nature of matter, as in a great work of art, there is, written small, the artist’s signature. Standing over humans, gods, and demons, subsuming Caretakers and Tunnel builders, there is an intelligence that antedates the universe."

First off, I don't really have a problem with "measure" there. "Calculate" is more accurate, but he's being poetic. And surely that's made clear in the novel?

More importantly, I disagree that this suggestion is so different from the one I made, or necessarily at odds with any form of logic. Why do we focus on pi at all, and not its third root, or e^pi, or Riemann zeta(23)? Presumably because we live in a universe that's approximately flat, and we perceive circles as special, and we think it a certain sort of way, etc. I can easily imagine a universe in which the inhabitants would never bother to calculate pi and instead focus on some other number, or perhaps on some entirely different type of endeavour.

So perhaps, being omniscient, our putative creator noticed that the quantity we call pi had this special feature in its decimal (or base 11 or whatever) expansion, and created us so that we would eventually notice and appreciate the significance.

[W.D.Clinger]

Not all. For one example: Quantum Mechanics (http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics) relies on functional analysis, in particular Hilbert spaces, and for that you need the Axiom of Choice (http://en.wikipedia.org/wiki/Axiom_of_choice), which is universally rejected by intuitionists/constructivists because it is non-constructive. In fact, it would re-introduce the law of the excluded middle through the backdoor.
No, you're confusing the nature of Hilbert spaces with the way they're usually taught.

There are lots of constructive Hilbert spaces that don't need the axiom of choice. To support your claim, you'd have to identify a scientific law that requires a Hilbert space that

isn't constructive, and
doesn't have a constructive analogue that would do just as well.

That would be hard to do. Errett Bishop, in his 1967 book on Foundations of Constructive Analysis, presented a constructive development of Hilbert spaces and commutative Banach algebras. In his review of that book (http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183531480), Gabe Stolzenberg wrote:
He is not joking when he suggests that classical mathematics, as presently practiced, will probably cease to exist as an independent discipline once the implications and advantages of the constructivist program are realized. After more than two years of grappling with this mathematics, comparing it with the classical system, and looking back into the historical origins of each, I fully agree with this prediction.
Constructive Analysis, by Errett Bishop and Douglas Bridges, is an updated version published in 1985. I have read only the original version, but the updated version might be easier to find in university libraries.

[sol invictus]

That would be hard to do. Errett Bishop, in his 1967 book on Foundations of Constructive Analysis, presented a constructive development of Hilbert spaces and commutative Banach algebras. In his review of that book (http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183531480), Gabe Stolzenberg wrote:He is not joking when he suggests that classical mathematics, as presently practiced, will probably cease to exist as an independent discipline once the implications and advantages of the constructivist program are realized. After more than two years of grappling with this mathematics, comparing it with the classical system, and looking back into the historical origins of each, I fully agree with this prediction.


So why hasn't it happened?

[W.D.Clinger]

So why hasn't it happened?
Patience, grasshopper.
:)

Conservatism and laziness, mostly.

Mathematicians were taught to use non-constructive methods, and it is often easier to prove something non-constructively.

Constructive proofs yield stronger theorems, because the theorem doesn't need non-constructive assumptions, but mathematical tradition has implicitly made non-constructive assumptions. That means the statement of the constructive theorem doesn't look any different from the traditional statement of the weaker non-constructive theorem, which means there has been little incentive to find constructive proofs.

In some areas, notably computer science and logic (http://plato.stanford.edu/entries/mathematics-constructive/), mathematicians are beginning to understand that constructive proofs yield stronger and more useful theorems, but this is going to be a long slow process, much as the shift to machine-assisted or machine-verified proofs will be long and slow.

Part of the problem is that many departments of mathematics have regarded logic as the province of the philosophy department and computer science as the province of the computer science department. At MIT, for example, Marvin Minsky had to move from the math department to the department of electrical engineering despite his impeccable mathematical pedigree (http://www.genealogy.ams.org/id.php?id=6869). That particular math department eventually realized its mistake and made amends; for example, Michael Sipser has been a recent chair of the department.

[drkitten]

Not quite--she was looking for a sequence that had a length that was the product of two primes...

Er, 9 (3x3) is the product of two primes.....

[stilicho]

Proof that sqrt(5) is not rational is pretty easy, and goes by contradiction.
Suppose sqrt(5) is rational, and it is equal to p/q where p and q are both (positive) integers. Then simplify this fraction to p'/q' where p' and q' have no (prime) factors in common.
Then we have: sqrt(5) = p' / q' and when we square that equation, and move q' to the other side, we get
5 * q' * q' = p' * p'
The main theorem of number theory states that factorization into prime numbers is unique. So p' * p' must contain a factor 5, and therefore p' must contain a factor 5 and therefore, p' * p' must contain at least two factors 5. That, in its turn, implies that q' * q' must contain at least one factor 5, and therefore q' contains a factor 5. However, we started out with saying that p' and q' had no common factors, and we have now established they both have a factor 5. Contradiction.
Ergo, sqrt(5) is not rational.

I actually followed that. Are you a math teacher? If you aren't then you should be. Mine all sucked.


:jaw-dropp The idea of a system of weights and measures in base-phi is even less practical than imperial weights and measures with its haphazard factors of 3, 12, 14, 16 and what have you.

I have several impractical ideas. Perhaps you'd like to subscribe to my newsletter.

[jsfisher]

Constructive proofs yield stronger theorems


A semantic quibble: A theorem is no stronger nor weaker because of how it is proven. However, a constructive proof for a theorem can provide a useful technique for exploiting the theorem. Given an existence proof, for example....

[AvalonXQ]

A semantic quibble: A theorem is no stronger nor weaker because of how it is proven.

I don't think he means that Theorem A is "stronger" if it's produced by one proof as opposed to another. I think he means that with a constructive proof we can often get Theorem B instead, which implies Theorem A but not vice versa -- hence B is formally a stronger theorem than A.

[jsfisher]

I don't think he means that Theorem A is "stronger" if it's produced by one proof as opposed to another. I think he means that with a constructive proof we can often get Theorem B instead, which implies Theorem A but not vice versa -- hence B is formally a stronger theorem than A.

Perhaps, but that isn't what he said. On the other hand, had he stated it as you did, well, then, that would have been fodder for a separate thread and debate.

[AvalonXQ]

Perhaps, but that isn't what he said.

I really think that it was what he said. "Constructive proofs yield stronger and more useful theorems" -- I read his statements as explaining that you can prove more using constructive proofs, not that coming to the exact same conclusions is subjectively "better" if you do it by one method as opposed to the other.

[W.D.Clinger]

I don't think he means that Theorem A is "stronger" if it's produced by one proof as opposed to another. I think he means that with a constructive proof we can often get Theorem B instead, which implies Theorem A but not vice versa -- hence B is formally a stronger theorem than A.
Right. What I had in mind is the situation in which Theorem A is of the form "AC implies Theorem B."

Because it has been traditional for mathematicians to assume the axiom of choice implicitly, the traditional statement of Theorem A is exactly the same as the correct statement of Theorem B, even though Theorem B is stronger than Theorem A. That's confusing.

When a mathematician uses the axiom of choice to "prove" Theorem B, the mathematician is really proving the weaker Theorem A. A constructive proof of Theorem B proves a stronger theorem, even though the traditional statement of Theorem A looks exactly like Theorem B.

The situation has been improving, albeit slowly. Nowadays, a mathematician who intends to assume the axiom of choice throughout a paper or book is more likely to state that assumption up front. In a few areas, such as set theory or set theoretic topology, the axiom of choice may be explicit in the statement of every theorem that needs it; in most areas of mathematics, however, that degree of rigor would still be considered unusual.

[jsfisher]

Right....


So, then you are talking about constructivism and not merely constructive proofs.

[rsaavedra]

Of course it terminates, shortly after 3.14 and before 3.15 ;)

[Complexity]

A semantic quibble: A theorem is no stronger nor weaker because of how it is proven. However, a constructive proof for a theorem can provide a useful technique for exploiting the theorem. Given an existence proof, for example....


I think some confusion can be avoided by quoting a bit more of what W.D.Clinger said: "Constructive proofs yield stronger theorems, because the theorem doesn't need non-constructive assumptions..."

Let's play for a moment.

Let's say that P1 and P2 are valid proofs of theorem T.

P1 assumes X, Y, and Z. P1 is a proof of T if and only if X, Y,and Z are true.

P2 assumes Y and Z. P2 is a proof of T if and only if Y and Z are true.

So what's the big deal about assumption X?

Much depends upon what sort of assumption X is. X could be simple or deep, well accepted or controversial, rarely used in other proofs or often used in other proofs.

One the fun things about mathematics is that you get to pick which things you'll assume are true and work within the framework generated by those assumptions. As long as the set of assumptions is consistent, you're free to pick as you like.

While mathematics has been a remarkably useful tool for modeling reality, mathematics is much larger and more interesting than that, and a mathematician needn't care one whit about the results of his/her work conforming to reality. (Which makes it all the more fun when that work is used to illuminate some aspect of reality anyway.)


Some of the considerations that are taken into account when judging the quality of a proof include:

How original is the proof?
How unexpected is the proof?
How much much does the proof have to assume?
How well are the assumptions used in the proof?
How big a step (from prior mathematics) was taken in the proof?
What is the value of the new ideas (if any) introduced in the proof?
What is the value of the new techniques (if any) introduced in the proof?
How much new math does the proof open up and what is the estimated value of that new math?
What is the effect that this proof may have on mathematics (e.g. a yawn, a ripple, a tsunami)?
How elegant is the proof?
How beautiful is the proof?
Some proofs are plodding and others are gems.

It isn't as easy as saying that a proof of T that assumes X is better than one that doesn't.

If X is widely accepted, the decision to assume X for P1 doesn't appear to be a problem. However if the validity of X comes into question, or if X can be proven not to be true, P1 becomes suspect or useless.

Basing the correctness of P1 on X makes P1 vulnerable in a way that P2 is not.

In this sense only, P2 is a 'stronger' proof than P1.

A somewhat different play theme:

What if an assumption R is so useful that it gets spread around a lot and is used as an assumpation in a great many proofs.

What if R is an open question, a very active area of research, but mathematicians find R likely enough to be true (i.e. low risk as an assumption) and useful enough that they keep basing proofs on it even though they know R may not be true.

What would happen if R were proven not to be true?

Each proof that has R as an assumption would immediately be junked. A panic and mad dash (chaos is opportunity) would start to salvage what was possible by trying to modify the junked proofs so they don't have to assume R. Mathematicians would be thinking, amidst the wreckage, "Cool! Now that we know R is false..."

A prime example (pun not originally intended) of such an assumption R is the Riemann Hypothesis.

This is but one of the reasons why the Riemann Hypothesis is thought to be so important - much of post-Riemann mathematics rests upon it and would be undermined if the Riemann Hypothesis were ever to be proven false.

Long answer, but I like this stuff.

[W.D.Clinger]

So, then you are talking about constructivism and not merely constructive proofs.
I thought I was talking about increasing the precision, rigor, and usefulness of mathematical discourse by aligning the statements of theorems more closely with the hypotheses assumed by their proofs.

Constructivism is certainly relevant, but it is a philosophy of mathematics that sometimes verges on ideology. I prefer to speak of constructive or computational mathematics. Here's a nicely ironic summary of the situation (http://www.nuprl.org/Intro/ConstrMath/constrmath.html) from the PRL project (http://www.nuprl.org/Intro/intro.html):

Now that the dust has settled and the controversy dimmed by the passing of the main protagonists, we can see that there remain these two traditions: the old computational and the modern (so called "classical" or "ideal"). The computational tradition has received renewed attention because of the advent of modern digital computers. Fields like numerical analysis, symbolic algebra, computational geometry, computational number theory, and automated deduction have arisen fresh in the last fifty years to employ large numbers of mathematicians, many of whom are not working in mathematics departments. These fields have become vital to modern science and industry. Some of them are part of the subject of computer science.

Computing digits of pi is a recreational activity that also serves as a benchmark for progress in computational (constructive) mathematics.

[jsfisher]

Long answer, but I like this stuff.


Me, too.

Nonetheless, we seem to have a disconnect. With this I would agree: Constructive proofs are generally stronger than existence proofs. With this I would not agree: Theorems proven by constructive proofs are stronger than theorems proven by existence proofs.

I see people asserting the latter, and then justifying it by observing the former.

[Ron Webb]

(Sorry for the delaying responding, shadron. I'm still getting to know my way around this forum.)
Not so. The point of the rasterized circle in base 11 is that this could be found by any civilization, any intelligence in the universe from a simple series expansion and a computer (or their own brain given time and paper enough). No need to know ascii, no need to know english, no need to be air breathers or carbon based life. Just know what a circle is, enough math to have developed a series expansion for pi, and time. Your message in terms of information is much smaller than hers was; it was something like 51x51 base-11 digits long.

Let me repeat: the value of pi is what it is. God Himself cannot make it otherwise, any more than He can decide that 2 + 2 should equal 5 instead of 4.

Whatever you find in the expansion of pi, God did not put it there. The best He could do would be to influence the development of our language and our choice of the symbols we use to represent it, such that a particular sequence would appear to contain a message. If I found such a message, I would consider it strong evidence that some guiding Intelligence had been working throughout our history to make such a miracle appear to us.

The case of a hypothetical discovery of a rasterized circle, or any similar mathematical pattern, is somewhat different. I still could not conclude that God had adjusted the value of pi, because that is logically impossible. However, if the sequence is independent of language or culture or anything that God might have control over, I would be left without any explanation at all, beyond a mind-numbingly astonishing coincidence.

[Complexity]

Me, too.

Nonetheless, we seem to have a disconnect. With this I would agree: Constructive proofs are generally stronger than existence proofs. With this I would not agree: Theorems proven by constructive proofs are stronger than theorems proven by existence proofs.

I see people asserting the latter, and then justifying it by observing the former.


I don't think I've addressed the question of the relative strength or value of constructive and existence proofs.

Suppose that theorem T states that 'If set A has the jabberwocky property then a subset of A having the bandersnatch property exists.'

A proof that proves that T is true by actually constructing a subset that has the bandersnatch property is a constructive proof.

A proof that proves that T is true by showing that the non-existence of a subset that has the bandersnatch property results in a contradiction is an existence proof. It proves the theorem by showing that such a subset must exist without constructing a subset that has the bandersnatch property.

The remarks in my recent post had nothing to do with the proof technique - they touched on the notion of strength as related to proof assumptions.

I'm not at all sure what I think about the strength question and constructive vs. existence proofs. Fun to ponder upon sometime.

I freely intermix constructive and existence proofs in my work and enjoy and appreciate both. Constructive proofs may on occasion be more useful than existence proofs if one is trying to use the result of the construction, but existence proofs can be more fun by tantilizing one by promising the existence of something without showing how to get one.

[jsfisher]

I don't think I've addressed the question of the relative strength or value of constructive and existence proofs.

Suppose that theorem T states that 'If set A has the jabberwocky property then a subset of A having the bandersnatch property exists.'

Ok, we have a (perhaps unproven) theorem T.

A proof that proves that T is true by actually constructing a subset that has the bandersnatch property is a constructive proof.

Yes, and we now have a theorem T and a proof C.

A proof that proves that T is true by showing that the non-existence of a subset that has the bandersnatch property results in a contradiction is an existence proof. It proves the theorem by showing that such a subset must exist without constructing a subset that has the bandersnatch property.

No argument so far. Now we have a theorem T, a proof C, and a proof E.

The remarks in my recent post had nothing to do with the proof technique - they touched on the notion of strength as related to proof assumptions.

Proof assumptions. Not theorem. The theorem T remains unmorphed in any way by the methods used to proof it. We don't have a special versions of T, TC and TE, dependent upon proof method.

We do, however, have different theorems if we start from different foundations. A theorem T developed within ZF (one possible constructivism system) vs a similar in appearance theorem T' developed within ZFC, then sure, we could find differences in theorem "strength" and maybe declare T stronger than T', but we would be discussing different theorems.

I'm not at all sure what I think about the strength question and constructive vs. existence proofs. Fun to ponder upon sometime.

They [constructive proofs] can be more utilitarian, especially in applied mathematics (and I think that was one of W.D.Clinger's main points). You are right, though, about the pondering.

I freely intermix constructive and existence proofs in my work and enjoy and appreciate both. Constructive proofs may on occasion be more useful than existence proofs if one is trying to use the result of the construction, but existence proofs can be more fun by tantilizing one by promising the existence of something without showing how to get one.

As someone who always had far more talent for proof by contradiction -- seldom a constructive method -- I come with my own biases on what's to be preferred. Besides, I like having a Law of the Excluded Middle. :)