An Elementary Proof of Wallis’ Product Formula for Pi (2005)
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An Elementary Proof of Wallis’ Product Formula for Pi (2005)
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Re: An Elementary Proof of Wallis’ Product Formula for Pi (2005)
#2I appreciate that this is likely trivial for mathematicians. For me, I was quite lost. Something I found a bit more obvious was constantly dividing a squares sides into two. When done infinite amount of times, getting the perimeter of the shape would equal 2xpixr
Re: An Elementary Proof of Wallis’ Product Formula for Pi (2005)
#3I appreciate that this is likely trivial for mathematicians. For me, I was quite lost. Something I found a bit more obvious was constantly dividing a squares sides into two. When done infinite amount of times, getting the perimeter of the shape would equal 2xpixr
I don't think anyone would call it trivial; it might be that the submitter erroneously translates elementary as trivial. Here elementary means that it doesn't use higher math (calculus, analysis, etc), it doesn't say anything about ease or complication, and indeed the elementary proof of the prime number theorem is hella complicated.
Re: An Elementary Proof of Wallis’ Product Formula for Pi (2005)
#4I don’t like the title change. The proof is elementary but non-trivial.
Re: An Elementary Proof of Wallis’ Product Formula for Pi (2005)
#5I had never heard of this product formula before, so I whipped up a quick python program: http://pastebin.com/FBR8rWxv
All I can say is wow, that formula slow to converge on pi. The product after 100 million terms is: 3.141592637878503
You can see that 100M terms only gets you 8 decimals of precision.
Re: An Elementary Proof of Wallis’ Product Formula for Pi (2005)
#6As others are pointing out: the paper's title is "elementary" (not "trivial" a in the current link title). Both of these have different and accepted meanings in mathematics.
"Elementary" roughly means doesn't bring in results from different fields (such as using calculus to prove a theorem about a sum or product). "Elementary" proofs can be quite difficult, and it can be quite a feat (well worth a publication) to find one for a standard result.
"Trivial" usually means something closer to easy, or means there is some linkage already implying the result (just in disguise).
Re: An Elementary Proof of Wallis’ Product Formula for Pi (2005)
#7Please fix the title change. 'Trivial' and 'elementary' have specific meaning in mathematics and are not interchangeable. Roughly speaking, 'trivial' means easy or obvious, while 'elementary' means a proof that does not use complex analysis or higher techniques. This proof is elementary but definitely not trivial.
Re: An Elementary Proof of Wallis’ Product Formula for Pi (2005)
#8As others are pointing out: the paper's title is "elementary" (not "trivial" a in the current link title). Both of these have different and accepted meanings in mathematics. "Elementary" roughly means doesn't bring in results from different fields (such as using calculus to prove a theorem about a sum or product). "Elementary" proofs can be quite difficult, and it can be quite a feat (well worth a publication) to fin…
In fact, there is a negative correlation between a proof being elementary and a proof being trivial. The reason people like abstraction is that it makes difficult proofs trivial. The reason people don't like abstraction is that it makes elementary proofs non-elementary.
Re: An Elementary Proof of Wallis’ Product Formula for Pi (2005)
#9I had never heard of this product formula before, so I whipped up a quick python program: http://pastebin.com/FBR8rWxv All I can say is wow, that formula slow to converge on pi. The product after 100 million terms is: 3.141592637878503 You can see that 100M terms only gets you 8 decimals of precision.
If speed of convergence is what you seek, i've yet to discover a more impressive case than a surprisingly simple method attributed to Gauss using the geometric and arithmetic means of two numbers. Three times around the loop and it's already correct to five decimal places: http://pastebin.com/674YJ9VM
Re: An Elementary Proof of Wallis’ Product Formula for Pi (2005)
#10I had never heard of this product formula before, so I whipped up a quick python program: http://pastebin.com/FBR8rWxv All I can say is wow, that formula slow to converge on pi. The product after 100 million terms is: 3.141592637878503 You can see that 100M terms only gets you 8 decimals of precision.
For another famous, but also very slow series converging to pi, try
pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...