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Old 2002-12-09, 14:13   #1
xtreme2k
 
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Default Would finding a definate Pi value easier if...

We use a floating point base other than 10? Has this being thought before? What I mean by a floating point base is, for example, 3.1235 base 7. (not only the 3 is base 7, but the floating point value is also based on 7). Is there such thing as floating point base or are we limiting ourselves to whole numbers for no apparent reasons?

Just because we have 10 fingers doesnt mean all the maths in the universe revolves around base 10. Would it be a new challange if we start using this approach? I am sure we can express a lot of values a lot easier if we also use a prime base floating point. EG, base 3,5, 7, 11, 13...

I wanted to ask this in a math specific forum but couldnt find one. If you know some give me the URL of one. I am not a maths specific student though.
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Old 2002-12-13, 11:41   #2
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Default Pi is transcendental in all number bases.

Pi is irrational and transcendental in all integer number bases. Generally, ease of computing its value in a particular base depends only on the computer's ease of calculation in that base, not on how Pi's value is expressed in that base. This applies both to floating-point arithmetic and fixed-point arithmetic.

Most recent Pi calculations by digital computer have been performed in base 2 or base 16, not base 10, because those computers operate more efficiently in base 2 or 16 than in base 10.

A famous Bailey-Borwein-Plouffe algorithm for Pi features the number 16 prominently, making its use in base 16 particularly efficient. But similar algorithms can be derived that feature integers other than 16 and would be efficiently computed in number bases other than base 16.
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Old 2002-12-13, 14:57   #3
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The floating point standard set by IEEE (Institute of Electrical and Electronics Engineers) is actually in a base other than 10. It is in binary. And there have been other floating point standards which have been in Hex and Octal. The octal standard that comes to my mind was made by IBM, who certainly like to go across the grain at times
xtreme2k, when you talk about finding a value for pi, are you talking about the ratio of the circumference of a circle to its diameter, or are you talking about the number of prime numbers less than or equal to a given number?
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Old 2002-12-14, 09:39   #4
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If you can express a value in base 10, you can express it in any base 2,4 6, 8, 12, 14, 16 just as easy if you allow floating point in the original value. What I am saying is, have we tried a prime base and allow for floating point values as well? EG, base 3, 5, 7, 11, 13... EG, 2.56925653758445 base 17

I am talking about the Pi as in relation to circumference of a circle. But in stead of stating 3.1415..... (base 10), there MIGHT be a way to express Pi in a base other than 10, allowing floating point values (not floating pt base). Would there be a chance if we find a figure say, for example.... 3.0352868285384963 (base 653) as a definate value of Pi instead of the millions of figures...

The reason why base 10 is not accurate enough for such value because you only have 10 intervals between a value. If you increase that intervals to a very high value (base) wouldnt it have a much higher chance of reaching pi without millions of digits.
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Old 2002-12-14, 15:05   #5
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Default Re: Pi is transcendental in all number bases.

Quote:
Originally Posted by cheesehead
Pi is irrational and transcendental in all integer number bases.
Let me put it another way. No matter what base you choose, the representation of pi will allways be an infinite number of digits long. There is no base in which pi can be represented by a fixed number of digits. Any fixed number of digits in any base is an approximation only.
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Old 2002-12-14, 17:12   #6
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well... that's not TOTALLY true... if you found pi base pi now, then it'd be an integer right? Perhaps even unity?!?!? :(
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Old 2002-12-14, 22:24   #7
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Quote:
Originally Posted by Deamiter
well... that's not TOTALLY true... if you found pi base pi now, then it'd be an integer right? Perhaps even unity?!?!? :(
... which is why I specified integer number bases. :)
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Old 2002-12-14, 22:49   #8
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Quote:
Originally Posted by xtreme2k
If you can express a value in base 10, you can express it in any base 2,4 6, 8, 12, 14, 16 just as easy if you allow floating point in the original value. What I am saying is, have we tried a prime base and allow for floating point values as well? EG, base 3, 5, 7, 11, 13... EG, 2.56925653758445 base 17
But Pi could not be exactly expressed in a finite number of digits in any integral (or even rational) base, be it 17, another prime, or any other integer (or rational number).

Quote:
I am talking about the Pi as in relation to circumference of a circle. But in stead of stating 3.1415..... (base 10), there MIGHT be a way to express Pi in a base other than 10, allowing floating point values (not floating pt base). Would there be a chance if we find a figure say, for example.... 3.0352868285384963 (base 653) as a definate value of Pi instead of the millions of figures...
No, not in any integral base, be it 653 or 92843719421843275321322.

Quote:
The reason why base 10 is not accurate enough for such value because you only have 10 intervals between a value.
No, it's not a matter of accuracy. Between 3.1(base 10) and 3.2(base 10) lie an infinite number of intervals, not just ten of them. E.g., 3.14, 3.141, 3.1415, 3.14159, and 3.141592 all lie between 3.1 and 3.2. You can construct an approximate value of Pi in base 10 that meets any finite accuracy standard you specify, no matter how precise, but never the exact value of Pi.

Quote:
Originally Posted by xtreme2k
If you increase that intervals to a very high value (base) wouldnt it have a much higher chance of reaching pi without millions of digits.
Nope. It's a fundamental property of all irrational numbers that they cannot be exactly expressed in a finite number of digits in a rational number base.
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Old 2002-12-15, 04:59   #9
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Just out of curiosity... I understand well that Pi is irrational and has an infinite number of digits, but where can I find an analysis of the proof (if there is one) Seems to me that it'd be pretty hard to prove that a number is prime, even if you can find no pattern in the first Million or so digits...
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Old 2002-12-16, 00:14   #10
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I think i know one way pi was shown to be irrational. Through some type of work with continued fractions, it can be shown that 'If x is a rational number other than zero, then tan(x) cannot be rational'. The corrolary to this is 'If tan(x) is rational, x must be irrational or zero'. tan(pi/4)=1 -> 1 is an integer -> pi/4 is irrational -> pi is irrational.
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Old 2002-12-16, 06:50   #11
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Quote:
Originally Posted by Deamiter
Just out of curiosity... I understand well that Pi is irrational and has an infinite number of digits, but where can I find an analysis of the proof (if there is one)
The first three results of a Google search on "pi irrational proof" I just did are:
http://pi314.at/math/irrational.html ,
http://www.math.clemson.edu/~rsimms/neat/math/piproof.html , and
www.lrz-muenchen.de/~hr/numb/pi-irr.html
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