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Finitizing Pi
pi-based number system?

Finitizing Pi

jposamen
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Posted 03/17/08 - 08:52 PM: Subject: Finitizing Pi	#1
Right now we derive pi from circumference/diameter. This inevitably yields an irrational number. But what if we start backwards and deliberately make pi finite. Starting with pi forces us into a "base pi" system rather than a base 10 system, I would imagine. But I'm not sure if this is allowed in the math world. To play around with this, I'd suggest we use a simple circle that otherwise has a diameter of 1 and circumference of pi. I'm not quite sure how to manipulate numbers in a pi-based system. It would still be the case that c/d=pi. But now we have pi and the circumference as known veriables, and we're solving for the diameter, which should be easy to measure??? My math skills aren't that good. Thanks for humoring me. Jordan

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Posted 03/17/08 - 09:14 PM:	#2
Looks incredulously .... .... What the H... What the heck do you mean? This is completely random. How do you "finitize" (nonexistent term) pi? It won't be pi anymore. You just said a bunch of completely random nonsense. Sorry.

moonlight
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Posted 03/18/08 - 01:08 AM:	#3
If you changed Pi in this way, then the points of a circle's circumference would not anymore be all at the same distance from its center. In other words, those special ellipses where the two focuses coincide (which we call circles) wouldn't exist anymore in a system with a finite Pi. Your proposition can be reworded therefore as: why don't we ban circles? Not sensible in my opinion. But why do you want to make Pi rational anyway? Cordially, moonlight.
All are lunatics, but he who can analyze his delusion is called a philosopher. - Ambrose Bierce -

Cognitive Dissident
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Posted 03/18/08 - 02:36 AM:	#4
I suppose you can create positional number systems to any base you want. Just as the standard number system is base 10 (pardon the ugly notation): ...10^3 10^2 10^1 10^0 . 10^-1 10^-2 10^-3 .... You could have a number system in base pi ...pi^3 pi^2 pi^1 pi^0 . pi^-1 pi^-2 pi^-3 .... All other numbers would then be expressed as powers of pi. However I am not sure how to specify the numerals that you use in each of the positions.... For number systems to natural number base n there are n numerals which are the natural numbers 0 to n-1.... Heres's a link to a wiki article the number system based on the Golden ratio phi = 1.61803... : http://en.wikipedia.org/wiki/Golden_ratio_base It's interesting to note that every number has two decimal expansions in this system as well, similar to 1 = 0.999.... in base 10. Edited by Cognitive Dissident on 03/18/08 - 07:22 PM
All things are subject to interpretation. Whichever interpretation prevails at a given time is a function of power and not truth. - Friedrich Nietzsche

jposamen
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Posted 03/18/08 - 03:36 PM:	#5
Cognitive D, I think you see where I'm coming from. I'm having trouble with the units in a pi-based system, so I'll look at the link you provide. Moonlight, I think you're still thinking in a base 10 system. My system doesn't get rid of circles; it just measures them differently, i.e., in a pi-based system rather than a base 10 system. I am curious to "finitize" pi just to see what happens. Maybe other equations will get easier. Maybe new numbers will become infinite. I dunno. I just don't see any inherent reason to prefer our base 10 system. Jordan

moonlight
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Posted 03/18/08 - 06:57 PM:	#6
jposamen wrote: Moonlight, I think you're still thinking in a base 10 system. My system doesn't get rid of circles; it just measures them differently, i.e., in a pi-based system rather than a base 10 system. I am curious to "finitize" pi just to see what happens. Maybe other equations will get easier. Maybe new numbers will become infinite. I dunno. I just don't see any inherent reason to prefer our base 10 system. I see your point now. Sorry, I read too quickly the first time around. Well, in a base pi system a number x would require log_pi(x) 'rounded to the next integer' numerals, instead of the log_10(x) 'rounded to the next integer' numerals it would require in the usual base 10 system. But that's the only major difference I can see: no equation would become easier to solve I think. The reason I say this, is that the syntax will be different for the numbers, but the semantics would remain the same. So in a formal mathematical system of the first order, the problem would be just as hard, since there is an equivalence between syntax and semantics by Godel's completeness and Hilbert-Ackermann's soundness theorems. So you'd probably just get some rational numbers with infinite decimal expansion maybe, but I don't think it would make proofs any simpler. The decimal system is simply more popular because we're more used to it I think. I like the thought though... a pinary system. Nice thinking Cordially, moonlight.
All are lunatics, but he who can analyze his delusion is called a philosopher. - Ambrose Bierce -

Timothy
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Posted 03/19/08 - 01:48 PM:	#7
Certainly, no matter how you choose to express a relation, its properties should remain the same. A "pi" base should be isomorphic to a 10 base as to the properties of the relations it can express; if it falls short, then we have a good pragmatic reason not to use it. Relations ought not to change as we switch from a form of expression to the other (following a quasi-platonism as that of Frege, Russell and Gödel). Yet the more expressive your system gets, the less decidability it accomplishes. A Pi base number theory isomorphic to elemental number theory, base 10 expressed, has to meet the same faith. Edited by Timothy on 03/19/08 - 06:55 PM
"I think I'll stop here." -- Andrew Wiles immediately after presenting the proof of Fermat's Last Theorem, Cambridge, 23 June 1993

jposamen
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Posted 03/21/08 - 09:03 PM:	#8
Timothy and Moonlight, I think your both making the point that "semantics" or "relations" will be isomorphic as between a pi-based system and a base 10 system. Given that most numbers are irrational, I wonder if a pi-based system would flip this around and allow for a majority of non-irrational numbers. Then, I also wonder if there's any advantage to having non-irrational numbers as the majority. Probably not, eh? Jordan

moonlight
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Posted 03/22/08 - 02:22 AM:	#9
Hi jposamen, Yes I was arguying that the semantics remain the same. I'm not sure I understood your question, therefore. But a pi-based system, would just change the syntax for the numbers, not their nature or inter-relations. For example the number written as "5" in decimal (base-10), will become in a unary system (base-1) "11111", in a binary system (base-2) it will be "101", in a ternary system (base-3) it will be "12" and in a pinary based system (base-PI) it will look something like: "11,2201220211211103010...". The syntax changes, but the number, semantically remains the same. It just looks different than in decimal when it is strictly superior to the base, in this case PI. You can check this with simple addition in different bases, using the number 'one' and you will see that it's properties do not change. 'One' has the same syntax in bases 1, 2, PI, 10, etc. because it is always inferior or equal to the base... it is always written "1". So just do the additions to see: In decimal: 1+1+1+1+1 = 5 In unary: 1+1+1+1+1 = 11111 In binary: 1+1+1+1+1 = 101 In ternary: 1+1+1+1+1 = 12 In pinary: 1+1+1+1+1 = 11,2201220211211103010... The syntax has changed, but the addition operation is relating one to itself in the same way semantically. The irrational numbers are still not definable by means of a division of 2 integers, i.e. numbers that are sums of 1. In other words, in decimal you have, if X = Y/Z, where Y and Z are sums of ones, then X is not irrational. So: PI = 3.1415... = Y/Z is not solvable in decimal (base-10) system. PI = 10 = Y/Z is still not solvable in pinary (base-PI) system. Cordially, moonlight.
All are lunatics, but he who can analyze his delusion is called a philosopher. - Ambrose Bierce -

Timothy
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Posted 03/22/08 - 06:23 PM:	#10
jposamen wrote: Given that most numbers are irrational... What do you mean by this? That there are infinitely many irrational numbers? Or that the infinitely many irrational numbers constitute a non-denumerable set? jposamen wrote: I wonder if a pi-based system would flip this around and allow for a majority of non-irrational numbers. I don't see how this could be the case. The definition of a rational number does not requires that the base notation is 10; it doesn't mention any base notation at all. Thus, the definition ought to apply to any couple of numbers, whatever the base notation is. No flipping should take place, as moonlight shows. jposamen wrote: , I also wonder if there's any advantage to having non-irrational numbers as the majority. Probably not, eh? I don't see any theoretical advantage, assuming that the flipping can actually take place. But then again, I'm no mathematician.
"I think I'll stop here." -- Andrew Wiles immediately after presenting the proof of Fermat's Last Theorem, Cambridge, 23 June 1993

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Posted 03/23/08 - 07:38 AM:	#11
jposamen wrote: Timothy and Moonlight, I think your both making the point that "semantics" or "relations" will be isomorphic as between a pi-based system and a base 10 system. Given that most numbers are irrational, I wonder if a pi-based system would flip this around and allow for a majority of non-irrational numbers. Then, I also wonder if there's any advantage to having non-irrational numbers as the majority. Probably not, eh? A number's rationality or irrationality does not change with a change of representation. After all, you do not become a different person by changing your tee-shirt. A rational number is one which is the ratio of two integers. Also relative cardinalities of the sets of repeating/terminating or irregular-appearing digit strings does not change. What changes is that in an integer base all rational numbers and only rational numbers have repeating or terminating expansions. In a pi-based system most rational numbers will have irregular-appearing expansions. But most irrational numbers will continue to have irregular-appearing expansions also.

jposamen
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Posted 04/02/08 - 08:07 PM:	#12
Moonlight, I appreciate the symantics argument. But aren't some base systems relatively more useful in some circumstances? If not, why bother with other base systems? Timothy, What do you mean by this? That there are infinitely many irrational numbers? Or that the infinitely many irrational numbers constitute a non-denumerable set? I meant the cardinality of infinity for irrational numbers is greater than that for rational numbers. bobo, But most irrational numbers will continue to have irregular-appearing expansions also. See that's what I don't know. To reiterate, I wonder if there's any advantage of using rational numbers over using irrational numbers for anything. I think it's time for me to bow out. Thank you for the responses. Jordan

moonlight
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Posted 04/03/08 - 06:11 AM:	#13
jposamen wrote: Moonlight, I appreciate the symantics argument. But aren't some base systems relatively more useful in some circumstances? If not, why bother with other base systems? I think they're more useful for practical reasons not mathematical ones. Depends on who's dealing with the numbers. For machines, the binary digit (bit) is used because historically Shannon noticed in 1936 a symmetry between the 0 and 1 and low & high, or abscence & presence of electrical current. Later on, for progammers, as the code was getting complicated in binary, the Hex base was used to shorten it all. It was for practical reasons more than anything else, I guess. Usually humans deal with the decimal base for historical reasons too. Cordially, moonlight.
All are lunatics, but he who can analyze his delusion is called a philosopher. - Ambrose Bierce -

bobg0
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Posted 04/03/08 - 07:29 AM:	#14
jposamen wrote: I appreciate the symantics argument. But aren't some base systems relatively more useful in some circumstances? If not, why bother with other base systems? Use different tools for different jobs. Bakers and egg suppliers use base 12 (a dozen) probably because 12 can be divided into half, third, quarters or sixths evenly. That may also be the reason there are 360 degrees in a circle (divides by 2, 3, 4, 5, 6, 8, 9, 10, ...) and 5280 feet in a mile (divides by 2, 3, 4, 5, 6, 8, 10, 11, 12, ...). Base ten is "natural" for humans. Base 2 and 16 are natural for digital computers, but also look up Balanced Ternary: http://en.wikipedia.org/wiki/Balanced_ternary Primes are good bases for many mathematical problems like cryptography. I don't know of any application for an irrational base but that doesn't mean there are none. Best I can come up with off the top of my head (very rough idea): We have a clock with one hand 10 cm long. The tip of the hand sweeps out 20 cm of arc per second. We need to use your base pi to figure out the time.

rabeldin
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Posted 04/03/08 - 09:48 AM:	#15
Look into non-Euclidean geometries. We can have a system with a rational ratio of circumference to radius by distorting the geometry. It seems like a lot of maneuvering to satisfy one aesthetic.
Leave no assumption unquestioned.

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Posted 05/24/08 - 07:41 PM: Subject: reply to pi	#16
What if pi is resembling the infinite and therefore can't be put into a finite perspective. I don't understand how you could shrink infinite to finite. Thats like subtracting infinity by 1. What do you end up with? I don't think that the amount of numbers in pi or their sequence in any numeral system is important. From that perspective you are always dealing with quantities, quotients, products, etc. The fundamentals remain the same, (as stated by moonlight). How can somebody look at pi as something with a fundamental purpose other than mathematics when it's infinite proportion does not resemble anything in the universe other than the universe itself. Please reply if you think I need serious help with this

Gort
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Posted 05/25/08 - 03:36 AM:	#17
jposamen wrote: Right now we derive pi from circumference/diameter. This inevitably yields an irrational number. But what if we start backwards and deliberately make pi finite. Starting with pi forces us into a "base pi" system rather than a base 10 system, I would imagine. But I'm not sure if this is allowed in the math world. To play around with this, I'd suggest we use a simple circle that otherwise has a diameter of 1 and circumference of pi. I'm not quite sure how to manipulate numbers in a pi-based system. It would still be the case that c/d=pi. But now we have pi and the circumference as known veriables, and we're solving for the diameter, which should be easy to measure??? My math skills aren't that good. Thanks for humoring me. Jordan If you finitize pi you will infinitize a dozen. Bob Kolker

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Posted 05/26/08 - 11:09 PM:	#18
I don't get the effort, either. Pi is defined as a quotient of circumference and diameter. If we instantly assigned this number some value as a real number, then we've undone our reasons for caring about pi in the first place because we could not sensibly work the equation backwards to determine the circumference or diameter of any figure. I'm stealing someone else's question and starting a new thread with it, though.
"If it were not for the laughter, the Way would not be what it is." -- Laozi

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