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Axiverse

Pi in other bases

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Wolfram says:

An irrational number is a number that cannot be expressed as a fraction (p/q) for any integers p and q.

So, no. 0.333333333..... is rational. (which is a misconception I think you might have.)

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Quote:
Original post by Axiverse
I know tha, but some numbers which are rational in base 10 are irrational in other bases. Right?


right

but finding a base in which pi is rational is impossible (or very very hard)

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Quote:
Original post by Axiverse
I know tha, but some numbers which are rational in base 10 are irrational in other bases. Right?
I'm pretty sure that's wrong. Give an example.

Actually, I'm sure that's wrong. Because base has nothing to do with rationality.

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Guest Anonymous Poster
No. I repeating decimal can be non-repeating in other bases, but a repeating decimal is still rational. PI is not a fraction, so is irrational.

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Quote:
Original post by cowsarenotevil
Where as e and phi, for instance, have very obvious patterns as continued fractions.


I'm not sure about phi, but I thought e only seemed to have a pattern for the first few digits, but the digits after that have no pattern. Surely if those numbers do indeed have patterns they would then be rational, which e is not.

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Quote:
Original post by Doc
Quote:
Original post by cowsarenotevil
Where as e and phi, for instance, have very obvious patterns as continued fractions.


I'm not sure about phi, but I thought e only seemed to have a pattern for the first few digits, but the digits after that have no pattern. Surely if those numbers do indeed have patterns they would then be rational, which e is not.

e is transcendental - e.g. not rational. No patterns whatsoever.

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You're all ignoring the fact that I said "as a continued fraction" this is a continued fraction, such that phi is obvious, and e, less obvious still has a patern.

Also, irrational numbers can have patterns. Take 0.121122111222111122221111122222... that's clearly irrational, but has a very distinct pattern.

Pi posseses none of these characteristics (that we know of) in any of it's forms.

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the important part has already been said at least twice, but to bring it on home for those who don't always make the connections:

1. A "rational" number is any number which can be expressed as a "ratio" ... and I believe specifically a ratio of integers ...

2. All whole number bases have exactly the same expresses power as all other whole number bases when it comes to whole numbers / integers ... every single base can express 3, and 1201313, and 52 just as well as every other base ...

3. Therefore, any ratio expressed in any base (whole number), is guaranteed to also be expressable as a ratio in any other base (whole number). And in fact are 100% as factorable and reducable in any base as in any other base - whole number bases provide no numerical benifit over each other in the whole number realm I.E. 17/3 is a fully reduced decimal ratio, and would be 11/3 in hex, 10001/11 in binary, 122/1 in base three (and this shows the only case when a different base "looks" different, if the base equals the denominator, then the denominator becomes 1 and can be elided), aka 122/1 == 122.

4. Trying to express numbers WITHOUT ratios causes the base to make a major difference. IE. 1/2 (as decimal ratio) becomes 0.5 (as decimal non-ration), and 0.1 (as binary non-ratio) - but 7/3 as decimal ratio cannot be correctly expressed as decimal non-ration (without repeating bar), but becomes 2.1 in trinary non-decimal.

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That's really funny that this post should come up now, because just yesterday I was wondering if pi was a non-repeating decimal in some other base and 1. how we could find that base, and 2. if we did, what kind of uses would that base serve?

I'm retaking calculus now though =) (it's been years) so it'll be a while before I have any hope of conceiving of a solution. =)

I'm curious to see if anyone else does though.

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Quote:
Original post by serratemplar
...I was wondering if pi was a non-repeating decimal in some other base...


Well, it's already a non-repeating decimal, no? It's not terminating either, of course. But I'm pretty sure that no irrational number can turn rational based on base, and, as I mentioned above, certainly not pi.

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the rationality of a number has nothing to do with in what base it is expressed. pi can not be expressed as a ratio so it is not rational, but there is a patern to the digits, just for fun here is an explicit hexadecimal representation of the digits of pi. the bailey-borwein-plouffe series:
pi = sum from n=0 to infinity of
1/16^n * (4/(8n+1) -2/(8n+4) -1/(8n+5) -1/(8n+6))
omar hijab "introduction to calculus and classical analysis" (1991)

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Pi can only not be expressed as a rational number in a rational base, you can't make a widesweeping comment that Pi is not rational in any base. That is easy enough to prove wrong anyway:

Pi in base Pi = 10

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Base is only a way of representing a number, a symbol for it. The numbers properties are separate from it's base. IE The number represented by 10 (in base 10) is divisible by the number resented by 5 (in base 10), in any base (or roman numerals or words or what ever symbols). I hope I explained that well enough.

And c is a rational number if and only if c = a/b where a and b are both integers. And pi cannot be written as a/b, so it is irrational in any base since base is irrelevant. Although a/b can be arbitrarily close to pi.

It's not rational in base pi, since 1 2 3 4 5 etc don't represent integers in base pi.

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Isn't that definition of rationality highly arbitrary? For example, if we started with the irrational numbers instead of the rationals, there would e no reason that the entire system wouldn't just be flipped around.

On the other hand, if you say that 1basePi is an Integer, then rationality depends on the base. I guess what it boils down to is that I don't see any reason that 5basePi is not an Integer.

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That is the definitation of rational in standard mathemtics.
[ulr]http://mathworld.wolfram.com/RationalNumber.html[/url]

As far as standard math is concerned the integers are [...-2,-1,0,1,2...] thats all of them (there is a concrete definition in standard mathematics). If you want to make up new "integers" then we'll not playing in standard mathematics. Now convert those to basepi and you'll have the integers in basepi, I think -2 to 2 go fine but then you get problems. You have to remmeber that operations on the numbers are independent of bases. IE if m basex = n basey. Then
m basex*m basex = n basey*n basey
m basex/m basex = n basey/n basey
m basex+m basex = n basey+n basey
m basex-m basex = n basey-n basey

For fun.

Using my best guess about how a irrational base would work (I thought about it a bit, it gets really ugly)

0 1 2 3 are representable in base pi.

0basepi=0base10
1basepi=1base10
2basepi=2base10
3basepi=3base10
10basepi=pibase10
11basepi=pi+1base10
321basepi=3*pi^2+2pi+1 base10

There all kinds of wierd things with this 2basepi+2basepi=2base10+2base10=4base10=?basepi.
4basepi is as meaningless as 3base2 I think it's about 10.2201....basepi. So 2basepi+2basepi=10.2201....basepi. If you want to go with multiplication 2basepi*2basepi=10.2201....basepi.

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at the end of the day 10 is just a base we picked, and whether we use base 11,121,12321 ...
3base76 = 3base21
its just the symbols(representation) we use to mean the different numbers;

therefore the basic mathematical operations still stay the same.
using base pi would be weird but the numbers will still stay the same
e.g
1 , 2 , 3 ... 10.858407346

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i don't think some of you get a few subtle details still ...

the only "normal" bases, meaning bases where the standard assumptions of positional mathmatics ALL hold true are the natural number bases greater than 1 ... think about it for a minutes ...

Natural numbers = 1, 2, ...
Whole numbers = 0, 1, 2, ...
Integers = ..., -2, -1, 0, 1, 2, ...

for ALL natural number bases (and only those)

this is the system of positional representation

11.1 = 1 * base ^ 1 + 1 * base ^ 0 + 1 * base ^ -1

so this number, 11.1 means different things in different bases, but for all whole number bases you can do all normal math operations upon it, including representing all integers, etc ...

now how many symbols must you have to use a base ... the same number as the value of the base ... base 3 requires 3 symbols, base 10 uses 10 symbols, base 16 uses 16 symbols ...

and doing addition you say stuff like:

123
+ 42

if this is in base 5 ...
3 + 2 == 5, so carry a 1, remainder 0
1+2+4 == 7, so carry a 1, remainder 2
1+1 == 2, so carry nothing, remainder 2

so 123 + 42 = 220 in base 5 ...

now imagine a pi based number system ... which sounds perfectly reasonable and logical at first ...

how many symbols do we need?

3.14159...

well, already I see a problem, but lets try to just ignore it for a minute (I'll use 4 symbols for now, representing 0,1,2,3 and ignoring pi ..., just like base 4 has 0,1,2,3 and no 4).

so we have a number, 123. What does this number mean (in decimal terms)?

1 * pi^2, + 2 * pi^1 + 3 * pi ^0

simplified

pi^2 + 2pi + 3

so far, not great, but not obviously broken ...

now try to ADD, 123 + 22 (22 == 2pi + 2)

3 + 2 == 5, so how many pis come out of 5 (1) and leave how many 1s? I can't do it ... can you?
2 + 2 == 2, so how many pi^2 come out of that (1) and leave how many pis? I still can't do it ...

so you can't ADD in base pi ... because the ONES digit is not 1 pi ... it's ONE ... the real one, as in "If there is only 1 creature of a kind, that creature is unique" ... the number 1 has the same meaning in EVERY positional base that makes since ...

But to invent a thing LIKE a base (but not quite the same) for uing pi, you would have to change the fundamental rules ... you would have to say 1 == 1 pi, not normal 1 ... and in that system, what does 12.1 mean? does it mean 1*pi^2 + 2*pi^1 + 1*pi^-1?

That might be usefull in many circles, but it is definately exhibits different behaviors than the whole number bases ...

just as many ancient cultures had no formal 0, and had many hard times expressing certain important ideals, can you imagine trying to use a number system in which the concept of 1 is incompatible with your symbols?

hope this post at least makes you think :)

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