Hard Light Productions Forums
Off-Topic Discussion => General Discussion => Topic started by: Petrarch of the VBB on November 05, 2002, 01:40:15 pm
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For those of you with an unhealthy interest in maths (this means you, CP) i have a challenge, try and write Graham's number using conventional methods.
For those of you who dont know what graham's number is, here is a summarized description, the number 3^^^...^^^3 in which there are 3^^^^3 arrows, then take that number and make that the number of arrows, reapeat 63 times!
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It is theoretically possible to do this using the usual exponents since 3 is an integer but the problem is that I could use up all the webspace Gamespy has (and all the time in my life) in writing a huge post and only get a very small fraction of the required number. :p :D
Here are a couple I can think of that will use up less time and webspace: :D (the dots indicate that it goes on forever)
1: Prove the following:
z(2) = 1/1²+1/2²+1/3²+1/4²... = p²/6
2: Prove the following:
(1+1/1²)×(1+1/2²)×(1+1/3²)×(1+1/4²)... = 1/p × ( ep + e-p )
3: If y=xxxx..., prove that x=y1/y for convergent values of x. (or in terms of y, y = - plog( - log(x) ) / log(x) )
now this is what math is all about. ;7
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CP please explain me how a line of jibba-jabba can be better than hangin around with people, getting drunk, and having sex. mmh? give me one good reason.
*makes yoda face*
but it's actually nice too...in its own way.
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Originally posted by Petrarch of th VBB
try and write Graham's number using conventional methods.
"Graham's number cannot be expressed using the conventional notation of powers" (http://www-users.cs.york.ac.uk/~susan/cyc/g/graham.htm)
Ha!
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Originally posted by CP5670
1: Prove the following:
z(2) = 1/1²+1/2²+1/3²+1/4²... = p²/6
:lol:
I wrote a java program using a loop to test this out and then multipled it by 6 and took the square root of it to get pi.
where the formula goes:
From 1 to 1000000 the loop used x^-2 and added it to itself 1000000 times. The end result was 3.1415916986605086 (which is only accurate up to 5 places behind the decimal). This thing converges fricken slow! I prefer to use the inverse sine series to generate pi... lots faster. :p
Originally posted by CP5670
2: Prove the following:
(1+1/1²)×(1+1/2²)×(1+1/3²)×(1+1/4²)... = 1/p × ( ep + e-p )
btw... that looks a bit like the formula for hyperbolic cosine (where Cosh(x) = (e^x+e^-x)/2)... I've always wondered, what is the purpose of the hyperbolic functions anyway? :confused:
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Originally posted by CP5670
1: Prove the following:
z(2) = 1/1²+1/2²+1/3²+1/4²... = p²/6
2: Prove the following:
(1+1/1²)×(1+1/2²)×(1+1/3²)×(1+1/4²)... = 1/p × ( ep + e-p )
3: If y=xxxx..., prove that x=y1/y for convergent values of x. (or in terms of y, y = - plog( - log(x) ) / log(x) )
How we say it in good ol' YU,....the same to you. :yes:
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Originally posted by ZylonBane
"Graham's number cannot be expressed using the conventional notation of powers" (http://www-users.cs.york.ac.uk/~susan/cyc/g/graham.htm)
Ha!
Bah! From the Department of Don't-Believe-Everything-You-Read Department (http://www.brainzipper.com/grahm.htm). :rolleyes:
;)
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Prove the meaning of life mathmatically.
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e^(i*pi)+1=0, QED. :D
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Originally posted by Thor
Prove the meaning of life mathmatically.
:ha: lets see mister CP meet that challenge. But everyody knows thats impossible.
*hides and puts on flame retardant clothes*
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CP please explain me how a line of jibba-jabba can be better than hangin around with people, getting drunk, and having sex. mmh? give me one good reason.
*makes yoda face*
actually even if it was just "jibba-jabba," it would still be much better than the other stuff. :p (people in general are pretty boring unless you are talking about math, computer games or legos while the other two activities are equivalent to smashing your head against a wall)
but it's actually nice too...in its own way.
there you go... ;7
"Graham's number cannot be expressed using the conventional notation of powers"
Ha!
I guess that is because there is not enough ink in the universe, but it is still possible in theory. :D
I wrote a java program using a loop to test this out and then multipled it by 6 and took the square root of it to get pi.
where the formula goes:
From 1 to 1000000 the loop used x^-2 and added it to itself 1000000 times. The end result was 3.1415916986605086 (which is only accurate up to 5 places behind the decimal). This thing converges fricken slow! I prefer to use the inverse sine series to generate pi... lots faster. :p
lol yeah this formula is no good for numerically computing pi, but it is actually one of the better converging sums I have seen out there. :D (there is one for the Stieltjes constants that takes some 50 million terms to get one digit of accuracy. :rolleyes: )
btw... that looks a bit like the formula for hyperbolic cosine (where Cosh(x) = (e^x+e^-x)/2)... I've always wondered, what is the purpose of the hyperbolic functions anyway? :confused:
Yep, that's where that term comes from in this case. :D (this can be seen by evaluating and simplifying the product using gamma functions) I guess those are basically just real versions of the circular trig functions and have their own notation only because they come up often (same goes for the circular functions).
Prove the meaning of life mathmatically.
Here is one way of putting it: :D
sin( x³ )=0
(in other words, life has an infinite number of meanings, but not all possible meanings)
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I now sacrifice myself on the alter of bad humour in the hopes of saving all who come after me:
42
*dies*
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"deatomizing the entire universe to write down a simple number"
I think I've found CP's purpose of exsistance
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Originally posted by Bobboau
"deatomizing the entire universe to write down a simple number"
I think I've found CP's purpose of exsistance
Heh - I was beginning to wonder if anyone read that page... ;)
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I read it. Heavy stuff. I'm still not sure I completely understand ^^^^^ notation.
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Originally posted by Goober5000
e^(i*pi)+1=0, QED. :D
Now THAT is beauty. :D
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That result is possibly the most beautiful in all of mathematics. :D :nod:
Here are some other neat ones:
(http://www.3dap.com/hlp/hosted/procyon/misc/image2.gif)
And then there is also Khinchin's law for continued fractions, which is one of the most astounding things I have ever seen, but I don't know enough about that field to be able to prove it.
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I think that e^(i*pi)+1=0 is God's signature on the mathematical continuum of the universe. :)
Some of us less enlightened (but no less interested in mathematics) haven't heard of some of these things. Links are A-1 SUPAR. :) For example, what is G, the gamma function, the beta function, and Khinchin's Law?
On that note, thanks for describing K(x). :) What is it used for?
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Originally posted by CP5670
2: Prove the following:
(1+1/1²)×(1+1/2²)×(1+1/3²)×(1+1/4²)... = 1/p × ( ep + e-p )
I have to disagree with this. I went to term 10000000 in the generation of this series and I ended up with 3.6760775427785917... whereas (e^pi+e^-pi)/pi = 7.37966665555814939701766565277332......
Are you sure you have the series correct? :wtf:
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*Looks at this entire thread, then throws up*
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Ah crap. Math sucks. If I were like CP, and did all this math crap, I would have gone crazy 2 days later. :shaking:
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*picks up shotgun from under the seat and tries to shoot down this thread*
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Guys, quit the math/CP bashing - now. :mad: Like Venom said in his Anime render thread - if you don't like it, don't go there. Personally, while I don't understand a single thing here, I find it fascinating that people are applying their minds to something actually useful outside of render and 3D modelling techniques. I can understand if this was a thread about something else, and maths was brought into the discussion just out of spite, but it isn't, and it wasn't, so cut it out.
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Thanks, Sandwich, although I don't really mind it too much. ;) :D
Some of us less enlightened (but no less interested in mathematics) haven't heard of some of these things. Links are A-1 SUPAR. :) For example, what is G, the gamma function, the beta function, and Khinchin's Law?
On that note, thanks for describing K(x). :) What is it used for?
Sure; I don't have much time at the moment, but I will point you to some links.
The gamma function is basically an extension of the usual factorial to noninteger arguments, and the two are related by G(x+1) = x! . There is a bunch of information about this one available here:
http://mathworld.wolfram.com/GammaFunction.html
There exist several formulas and identities for this function as well, some of which can be used to prove that product result I gave earlier.
G is called Glaisher's constant and is actually defined by that limit; I think it is not known whether the constant is irrational, (although it is likely to be). It can be written in terms of derivatives of the zeta function, but not in terms of any of the usual functions or constants. I would post a link for this as well but the site I usually go to for constant-related information is currently down.
Here's one neat identity I can remember at the moment:
(http://www.3dap.com/hlp/hosted/procyon/misc/image3.gif)
(pi and e you know, G is Glaisher's constant, and g is Euler's constant)
As for Khinchin's Law, it has to do with continued fractions. A CF is an "infinite fraction" of the following form:
(http://mathworld.wolfram.com/s1img2031.gif)
Just like there is a decimal representation for every number, there is also a CF representation. For rational numbers, the CF does not go on forever but terminates. It turns out that, while irrational numbers have no periodic repeats in their decimal form, such patterns do frequently exist in CF form, and many common roots and constants can be expressed as CFs with some pattern in them. What Khinchin found is that, for almost all real numbers, taking the geometric mean of the numbers in the CF (a1, a2, a3, etc.) actually yields a special constant that is independent of the original number. In fact, if you select a real number completely randomly, the probability that it will satisfy Khinchin's law is actually 100%. (there are numbers that do not satisfy it, but there are infinitely more that do satisfy it) The constant is called Khinchin's constant and is given by the following infinite product:
(http://www.3dap.com/hlp/hosted/procyon/misc/image4.gif)
It is not known to be expressible in terms of any other functions or constants, although there are a few really messy integrals and sums that give it.
I am not too experienced with this stuff myself so I do not know exactly why this law is true, but you can find some more information here: http://mathworld.wolfram.com/KhinchinsConstant.html
If you find this stuff interesting, let me know, as I can go on about this all day. :D
Are you sure you have the series correct?
:o looks like I wrote that one in all wrong; it should be ( ep - e-p ) / (2p), or sinh(p)/p. sorry about that; I think those errors got buried in all the html/vb tags. :p
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Originally posted by CP5670
:o looks like I wrote that one in all wrong; it should be ( ep - e-p ) / (2p), or sinh(p)/p. sorry about that; I think those errors got buried in all the html/vb tags. :p
Ah, ok, that works. :)
I went to the 100000000th term in the series and came up with 3.1415926460901864... after applying the inverse hyperbolic sine to it.
How do you generate Gamma(x) in terms of a series and whatnot?
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Yeah, this is cool stuff. :)
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Originally posted by CP5670
G is called Glaisher's constant and is actually defined by that limit; I think it is not known whether the constant is irrational, (although it is likely to be). It can be written in terms of derivatives of the zeta function, but not in terms of any of the usual functions or constants. I would post a link for this as well but the site I usually go to for constant-related information is currently down.
Y'know, you should offer your services as a tech description writer - you certainly have a handle on all the techno-jargon. ;)
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How do you generate Gamma(x) in terms of a series and whatnot?
For numerical computations, there is something called Stirling's asymptotic series that is valid for about z>1/2, and gets very accurate for higher values: (this is actually based on the limit I posted with the factorial; it converges on order x-5)
(http://mathworld.wolfram.com/s3img1520.gif)
To get negative values using that, use the following relation: (this one is an exact formula)
G(-z) = - p csc( pz ) / G(z+1)
Y'know, you should offer your services as a tech description writer - you certainly have a handle on all the techno-jargon.
lol thanks, maybe I should try writing bits of this stuff in the weapon descriptions... :D
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Originally posted by CP5670
For numerical computations, there is something called Stirling's asymptotic series that is valid for about z>1/2, and gets very accurate for higher values: (this is actually based on the limit I posted with the factorial; it converges on order x-5)
(http://mathworld.wolfram.com/s3img1518.gif)
To get negative values using that, use the following relation: (this one is an exact formula)
G(-z) = - p csc( px ) / G(z+1)
Would you mind posting a link to where I could find this series? :p
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:wtf: it looks like all the images on that site were switched around; I better edit my posts... :p
one moment...
[edit] okay, I updated the links; let's hope they stay like that now... :p
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hmm how exactly do you get
(1+1/(12z)+1/(288z²)-139/(51840z³)-571/(2488320z4))
aka how are those values computed so i can find the nth term?
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There are both recursive and explicit evaluations of the coefficients, but they are extremely complicated and themselves involve several gamma functions. :p However, there is a fairly simple series for the logarithm of the gamma function, of which taking the exponential yields direct gamma function values. (this can be derived directly from Binet's second formula) I also added in the above series with a few more terms on it:
(http://www.3dap.com/hlp/hosted/procyon/misc/image5.gif)
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Bleh, forget these stupid expressions. Here's a simple challenge: write out googolplex (10^(10^100)) :p
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You fool, Graham's number is much higher than a googolplex!
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triple integrals anyone?
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just use the dirichlet multiple integral reduction on those; it works well for most of the common regions and integrands. :nod:
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Originally posted by Petrarch of th VBB
You fool, Graham's number is much higher than a googolplex!
I never said it was higher. I said it was a "simple challenge". And if it's so much smaller, write it out :p
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still big enough to eat up all of gamespy's webspace though... :p
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< nevermind > this is too risky for my processor :D
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That would be 100*100*100 = 1,000,000. A lot less than a googolplex.
A googol is
10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
And a googolplex is 10 raised to that. :D
EDIT: ARGH! Why is the number all chopped up?
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Originally posted by Goober5000
That would be 100*100*100 = 1,000,000. A lot less than a googolplex.
Actually, no.... what my loops do is add zeros onto the end of the number... no multiplication. For example:
[ first loop here ]
[ second loop here ]
00000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000 generated here
[ second loop "adds" those zeros onto the end of the one 100 times ]
[ first loop loops through all of that 100 times ]