Wikipedia:Reference desk/Archives/Mathematics/2011 June 20

= June 20 =

Elementary proof
Hi. I am sure I remember reading or hearing somewhere that an "elementary proof" is one that does not use Cauchy's integral theorem. But I can't find any evidence on the net that this is the case. Can anyone verify this? Robinh (talk) 03:09, 20 June 2011 (UTC)
 * Are you trying to verify whether this has been said somewhere, or verify if the claim is true? Because I'm pretty sure that in general what constitutes an elementary proof is going to depend entirely on context. Rckrone (talk) 04:48, 20 June 2011 (UTC)
 * thanks for this. I guess I'm asking whether anyone can find a reputable source that defines "elementary proof" as one that doesn't use Cauchy.  best wishes, Robinh (talk) 08:09, 20 June 2011 (UTC)
 * I think you're most likely to find this in accounts of proofs of the prime number theorem. As Robinh pointed out, it's a pretty context-dependent thing. Michael Hardy (talk) 17:40, 20 June 2011 (UTC)
 * thanks! Quote: "Newman's proof is arguably the simplest known proof of the theorem, although it is non-elementary in the sense that it uses Cauchy's integral theorem from complex analysis".  .  Robinh (talk) 20:23, 20 June 2011 (UTC)

help me railway in dire straits of runnaway train
HI need a bit of help with a right angle triangle a b c  ok looking to find the length of a-b    a-c equals 12 metres b-c angle 7 degrees what is the length from a-b this is not homework I am building a model railway and the engines can pull up to 7 degrees the trains come out of a curve  (b) and proceed to climb at 7 degrees for 12 meters and then go into a tunnel (c) and into another room so have got to have the right distance from a -b email me with the answer  thanks will send the person who answers my question a photo of the construction Ron — Preceding unsigned comment added by 124.184.8.94 (talk) 07:32, 20 June 2011 (UTC)


 * If 12 metres is the length of the sloping track then the horizontal distance will be 12 times the cosine of 7 degrees. If 12 metres is the horizontal distance then the length of the sloping track will be 12 divided by the cosine of 7 degrees. In either case, the cosine of 7 degrees is very close to 1 (just over 0.9925), so you get 12 metres +/- about 9 cm. That is a very long slope for a model railway - at a constant slope of 7 degrees the track rises nearly 1.5 metres in 12 metres distance. Gandalf61 (talk) 09:18, 20 June 2011 (UTC)

Proof of Aleph 0 Smallest Transfinite Cardinal
How do you prove aleph 0 is the smallest transfinite cardinal? Thanks.voidnature 11:41, 20 June 2011 (UTC)
 * For an arbitrary infinite set $$X$$, you need to prove the existence of an injection $$f: \mathbb{N}\to X$$. This can only be done assuming the Axiom of choice or a weaker version.


 * What you do is simply let $$f(0) \in X$$, then, having defined $$f(0), \cdots, f(n-1)$$, you define $$f(n) \in X - \{f(0), \cdots, f(n-1)\}$$. It's not too difficult to turn this into a fully formal proof. --COVIZAPIBETEFOKY (talk) 11:54, 20 June 2011 (UTC)
 * That was very helpful, but can you please elaborate it more, and which part of it depends of the axiom of choice? Thanks.voidnature 10:58, 22 June 2011 (UTC)
 * When you define $$f(n)$$ you need to pick an element in $$\{f(0), \cdots, f(n-1)\}$$. To make this proof formal we need a choice function on the set of all non empty subsets of X, selecting an element out of each such subset, and this needs the axiom of choice. Money is tight (talk) 07:17, 23 June 2011 (UTC)

Convert stddev from population to sample
I have a standard deviation calculated on the population of size n. I don't have the original data. The stddev was calculated using n in the denominator. Statistically, I can multiply by sqrt(n/(n-1)) to convert this to sample stddev, correct? -- k a i n a w &trade; 16:51, 20 June 2011 (UTC)
 * Yes, but note that for small n where this factor may be important, you would have a significant standard deviation of the population standard deviation anyway :) . Count Iblis (talk) 17:30, 20 June 2011 (UTC)
 * Was "n" the population size or the sample size?   D b f i r s   18:56, 20 June 2011 (UTC)


 * Sample size. The population size is unknown. -- k a i n a w &trade; 19:20, 20 June 2011 (UTC)
 * Thanks, I suppose that was obvious, but I was slightly confused. I also get confused between the terms "sample standard deviation" and "standard deviation of the sample", but  our article explains it (with careful reading).    D b f i r s   06:44, 21 June 2011 (UTC)