27-04-2002 13:55									#121
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For all you pi freaks, there is a link to a pi page http://mathforum.com/dr.math/faq/faq.pi.html
You'll find a proof for the irrationality of pi, based on showing that there are no integers p and q such that pi = p/q. It requires some calculus (integration, differentiation). I won't reproduce it here, because it requires more prerequisites and is less interesting than the infinity things.

I didn't find any proof on the web regarding "any finite sequence of digits can be found in pi". If it exists, it is probably rather difficult. Note that it is not equivalent to normality, it is a weaker statement (being a corollary of normality). Any mathematician here knows about this?

I'm a bit puzzled now, because I thought it already had been shown.

quote: Just like I can show you that it's probable that any number you show me is irrational...

Krazy Horse, you proof will need the assumption that SnowFire has no information about what is a rational and irrational number ...

All odd numbers are prime

	27-04-2002 14:01									#122
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quote: Any mathematician here knows about this?

I'm pretty damn certain it hasn't been shown. The proof for that and normality would probably fall together...

quote: Krazy Horse, you proof will need the assumption that SnowFire has no information about what is a rational and irrational number ...

That's the point. Snowfire said that Pi was probably normal, for the same reason that I can say "x" is probably irrational for any "x"

quote: All odd numbers are prime

Actually, very few odd numbers are prime. As you can imagine, the density of primes --> 0 as your sample size grows larger...

	27-04-2002 14:12									#123
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quote: All odd numbers are prime

15? 21?

	27-04-2002 14:15									#124
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There's a 62.5% chance that 15 is prime, and a 63.636% chance that 21 is prime.

	27-04-2002 14:53									#126
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At least someone has understood that I wanted to hint to an old joke. How to show by induction that all odd numbers are prime:

Mathematician:
1 - ok, doesn't count really
3 - prime
5 - prime
7 - prime
therefore shown by induction that all odd numbers are prime.

Physicist:1 - ok, doesn't count really
3 - prime
5 - prime
7 - prime
9 - error of measurement
11- prime
therefore shown by induction that all odd numbers are prime.

Economist:
1 - ok, doesn't count really
3 - prime
5 - prime
7 - prime
9 - prime
11- prime
therefore shown by induction that all odd numbers are prime.

	27-04-2002 15:03									#127
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Math jokes aren't really my thing.

Putting it that way though:

	27-04-2002 19:38									#128
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Lots of good point of course ure right KrazyH
and i was wrong about that being true for every irrationnal number
your 0.010010001 w (in decimal) hich IS Transcendental by the way does not contain a 7 of course
i checked about this today
for pi though it is it just cant be generalized to all irrational like i thought.
What aldbertus said was right
You can prove pi irrational and even transcendental (i can dig up the one on the irrationality somewhere)
And like he said, for infinite sets, the way to check which set is bigger is if you can make a bijection beetween them.
Its a know property of infinite sets that they can have the same cardinalty as a subset of themselves (integers and even numbers come to mind).

	27-04-2002 20:18									#129
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quote: If you think a bit over it you will make the stunning discovery that there are as many even numbers as integers (see below for something even stranger). This seems to be odd at first, but there is no better way to define the "number of elements" for infinite sets.

wouldn't that mean odd integers don't exist? which would mean 2 is the only prime number? hmm....

	27-04-2002 20:38									#130
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quote: wouldn't that mean odd integers don't exist? which would mean 2 is the only prime number? hmm....

No. Al LulThyme said, for infinite sets, a real subset can have the same cardinality. This is sensible, and most fun to understand the thing with the infinitely big hotel.
Atm I think that the better exercises are:
1) A bus with n guests arrives
2) A bus with an infinite (#N) number of guests arrives
3) An #N buses with #N guests each arrive.
You all get them into the hotel, even if it is full before the first bus arrives.

Again to integers: The set of all integers, all rational numbers, all even, and all odd integers, and the set of all primes each have the same cardinality. Even if some of them are subsets of some other.

If this seems all too strange for you: The mathematicians themselves didn't really understand this until about 1880 (mostly work of G. Cantor).

	27-04-2002 21:52									#131
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quote: for pi though it is it just cant be generalized to all irrational like i thought

I'm pretty sure that nobody's ever proved it true even for Pi. It certainly smells like it's true, but we aren't certain...

	28-04-2002 10:03									#132
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Yeah yeah, I meant normal. It seems highly unlikely that pi is rational, being that if it is, we have a whole lot of math to rewrite.

Don't forget other weird possibilities as well, folks: maybe pi is normal to all bases except base 7, base 320, and base 3 trillion.

Adal: That Economist version I've usually seen tagged as "English major," but indeed, 'tis always amusing.

KrazyHorse: Probable? Nay, certain, if I'm being fair and picking from the reals. But yeah, it still stands as just probable for pi.

	28-04-2002 12:00									#133
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quote: It seems highly unlikely that pi is rational

I hope you meant "not normal" instead of "rational". When you follow the link I provided a bit above and surf a bit you'll find a proof that pi is irrational.

	28-04-2002 18:12									#134
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"The proper word is frequency. Conner's phone number appears more frequently than your number.

Please note that both numbers appear an infinite number of times inside Pi."

I don't understand how a a pattern of numbers which repeats an infinite number of times can be said to appear 'more frequently' than another pattern which ALSO appears infinetly.

How many times does pattern A appear in pi? Answer: Infinitely.

How many times does pattern B appear? Infinitely.

On what basis do you say that one is more 'frequent'?

If you have a number like this: 0.1111111111.., but every millionth digit is a number 2-9, and the number was infinite, why is it that '1' is considered more frequent? Doesn't 1 appear at the same frequency, i.e. infinitely?

	28-04-2002 20:34									#135
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quote: Originally posted by Adalbertus I hope you meant "not normal" instead of "rational". When you follow the link I provided a bit above and surf a bit you'll find a proof that pi is irrational.

I think that was his point...

	28-04-2002 20:39									#136
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quote: Originally posted by Seeker "The proper word is frequency. Conner's phone number appears more frequently than your number. Please note that both numbers appear an infinite number of times inside Pi." I don't understand how a a pattern of numbers which repeats an infinite number of times can be said to appear 'more frequently' than another pattern which ALSO appears infinetly. How many times does pattern A appear in pi? Answer: Infinitely. How many times does pattern B appear? Infinitely. On what basis do you say that one is more 'frequent'? If you have a number like this: 0.1111111111.., but every millionth digit is a number 2-9, and the number was infinite, why is it that '1' is considered more frequent? Doesn't 1 appear at the same frequency, i.e. infinitely?

What he said is just a lie (we don't know if Pi has the property he's stated), but that doesn't mean you can't define frequency on an infinite sting of digits; it's simply:

frequency of string a = lim_n-->infinitya_n/n

Where a_n is the number of times string a appears in the first n digits.

	28-04-2002 21:07									#137
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Does anyone know how far PI has been calculated? Can someone post it if possible. I'm lost after 3.14

	29-04-2002 01:34									#138
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Sava, surfing around, I found that there are (probably) currently 206,158,430,000 digits of pi calculated. However, there is an algorithm with which you can calculate the n-th digit without knowing the previous digits, at least hexadecimal (don't know if there is a similar thing in decimal expansion). The first billion digits (only) is a 10 MB file, I hope you don't want me to post this.
3.1415926535897932 is about the accuracy you can achieve with an IEEE double variable.

	29-04-2002 02:39									#139
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*head explodes*