Wikipedia:Reference desk/Archives/Mathematics/2009 January 31

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January 31

formula to calculate pi

this formula which has been given by the great indian mathematician srinivasa ramanujam, is the cause of my headache. i am an indian and without understanding this formula, i won't be a true indian. please explain this for a ninth grade student ${\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}\!$ and this also ${\frac {426880{\sqrt {10005}}}{\pi }}=\sum _{k=0}^{\infty }{\frac {(6k)!(13591409+545140134k)}{(3k)!(k!)^{3}(-640320)^{3k}}}\!$

Do you just want an explanation of the meaning of the terms in the formulae, or do you want an explanation of why the formulae are actually true? The first is substantially easier than the second. Algebraist 01:49, 31 January 2009 (UTC)

Even if I did understand Ramanujan's formulas I would probably not be a true indian. :-) . The formula components are the summation sign Σ , the square root sign √ , the factorial sign ! , and the exponentiation notation, and the Pi symbol π. Click on the colored words to find explanations. You may like to check the formulas numerically, for instance using the J (programming language). The left hand side is

  %o.1
0.31831

and five terms of the right hand side is

  ((2*%:2)%9801)*+/(!4*k)*(1103+26390*k)%(!k)*296^4*k=.i.5
0.31831

the terms of the sum decreases rapidly:

  ((2*%:2)%9801)*(!4*k)*(1103+26390*k)%(!k)*296^4*k=.i.5
0.31831 2.48051e_8 5.31965e_15 4.08817e_21 7.72733e_27

Bo Jacoby (talk) 08:00, 31 January 2009 (UTC).

Some further remarks: The first formula was discovered by Ramanujan in 1910 and published in 1914, with no proof, as his abitude. The first proof appeared in a 1987 book by Jonathan and Peter Borwein, where they reconstructed Ramanujan work on the subject. The other formula is not due to Ramanujan, but to the Chudnovski brothers, following Ramanujan's method. The point of both formulae is that they converge very quickly, as Bo Jacobi shows here above. For instance 4 terms of the second series already give pi with more that 50 correct decimal digits. It was used by its authors in the '80s to compute 4 billions digits of pi. (If you ask why, it seems that computing digits of pi is a sort of competition, I do not know what is the current record; it was over 1 trillion around 2000 by the japanese mathematician Kanada and his team). pma (talk) 17:26, 31 January 2009 (UTC)

Euler's proof that god exists

This is an excerpt from wiki's page on Leonhard Euler:

There is a famous anecdote inspired by Euler's arguments with secular philosophers over religion, which is set during Euler's second stint at the St. Petersburg academy. The French philosopher Denis Diderot was visiting Russia on Catherine the Great's invitation. However, the Empress was alarmed that the philosopher's arguments for atheism were influencing members of her court, and so Euler was asked to confront the Frenchman. Diderot was later informed that a learned mathematician had produced a proof of the existence of God: he agreed to view the proof as it was presented in court. Euler appeared, advanced toward Diderot, and in a tone of perfect conviction announced, "Sir, ${\frac {a+b^{n}}{z}}=x$, hence God exists—reply!". Diderot, to whom (says the story) all mathematics was gibberish, stood dumbstruck as peals of laughter erupted from the court. Embarrassed, he asked to leave Russia, a request that was graciously granted by the Empress. However amusing the anecdote may be, it is apocryphal, given that Diderot was a capable mathematician who had published mathematical treatises.

"Sir, ${\frac {a+b^{n}}{z}}=x$, hence God exists."

What does that mean? Was it just a joke by Euler to humiliate Diderot, and has no meaning, or does it mean something else? Because no matter how many times I try to interpret it, the sentence makes no sense to me. Could anyone clarify this for me?? Johnnyboi7 (talk) 05:36, 31 January 2009 (UTC)

Isn't this anectode taken from Bell's booklet "Men in mathematics"? I've been also puzzled with its meaning, but first I'd like to check the sources. If it's just the author's version, as I suspect, things could be much different, and easier. I'll have a look to the article you quoted...--pma (talk) 07:34, 31 January 2009 (UTC) PS: Bell quotes the anecdote from De Morgan's book "A budget of paradoxes".PPS: I have read the interesting Gillings' article you mentioned [1], where it is reported the tradition of the anecdote, going back to Thiébault's version (1804) that is probably the origin. (Note that initially the denominator had n instead of z, not that this makes less nonsensical the use of the formula). --pma (talk) 09:03, 31 January 2009 (UTC)

Now that you have convinced me that God exists. I'm going to confront Richard Dawkins and say to his face.

"Sir, ${\frac {a+b^{n}}{z}}=x$, hence God exists."

The look on Richard Dawkins face when I say this will be priceless. 122.107.205.162 (talk) 10:24, 31 January 2009 (UTC)

The difficult thing is to organize everything as a public event, like Catherina the II was able to do, according to the anecdote. Of course, she was particularly interested, as it was God who gave her the kingdom of all the Russias (according to her and to her colligues kings all around Europe). Euler himself was interested, as he got a salary from Catherina (and not by the French Republic, which paid for Diderot, of course). Just to say that nothing has really changed so much. For instance, today a mathematician could gain some founds proving with mathematical authority that Economy is governed by suitable mathematical rules (is it so different from (a+bⁿ)/z=x?). Personally, I suspect that it's always been the same story, in paleolithic times, in Catherina times, and today: just brute force proofs, so to speak. --pma (talk) 16:37, 31 January 2009 (UTC)

To answer the original question, Euler's alleged quote is indeed nonsense. The mathematical sentence $(a+b^{n})/z=x$ is a statement regarding the relationship between a, b, n, z, and x, but in this context these variables have no meaning, so the sentence is meaningless. It would be as if I had said "I am writing a novel where Jane's second cousin is named Richard" -- I've told you absolutely nothing about my novel, because you don't know anything about Jane and Richard, so knowing that they are second cousins is useless. Eric. 131.215.45.82 (talk) 23:32, 31 January 2009 (UTC)

Indeed and, since it is just nonsense, there is no possible logical retort (other than pointing out that it's just nonsense, which is never very impressive). --Tango (talk) 00:42, 1 February 2009 (UTC)

Well we can't reject a proof as nonsense just because you do not understand it. Let say: "the author shoud fill some passage that is not completely clear, or provide a refernce for it. Some variables are not defined. The existence result is in any case quite poor as it is, and demands for further properties of the found solution". pma (talk) 01:44, 1 February 2009 (UTC)

So the ruling monarch comes to you and says - "The guy Diderot is a pain in the ass - please prove to him that God exists so he'll shut up about the atheism thing."...Well, when the ruling monarch tells you to do something - it's generally a good idea to put other matters aside and attend to it right away. Euler knows he can't prove the existence of God - but he knows that Diderot doesn't know squat about math - so he writes down any old mathematical-looking gibberish that's sufficiently complicated that nobody is going to argue about it and challenges Diderot to prove that it's NOT true. With the onus suddenly on him - and with no knowledge whatever about math - poor Diderot can neither challenge nor disprove Euler's assertions. This is nothing to do with math or religion - it's a piece of clever social engineering. SteveBaker (talk) 05:36, 1 February 2009 (UTC)

We have to rely on mathematicians when we do not understand a mathematical proof. So it is sad if Euler really testified that he had a proof of God's existence. I do not understand Andrew Wiles' proof of Fermat's last theorem, and perhaps some day it will be published that an error in Wiles' proof is found and that Fermat's theorem is false after all. SteveBaker's argument applies to Wiles as well: more is earned by providing a proof than by trying in vain. Bo Jacoby (talk) 18:12, 1 February 2009 (UTC).

The crucial point here is that Euler basically posed a question to Diderot that he couldn't answer. Euler, likewise, probably believed that it was impossible to prove God's existence, and ALSO believed it impossible to disprove. The question of God's existence is an unaswerable question, based on the extent of knowledge and intelligence given to humans. This is ironic, because Diderot had no problem advocating atheism, which faces that same problem. Euler's little gag exposed his hypocrisy.

Localization Morphism

Resolved

 – Shahab (talk) 17:09, 31 January 2009 (UTC)

Hello all. I am currently reading a book on rings. Unfortunately the book is assuming a lot of basic knowledge which I possess only in bits and parts. In an example for showing that a morphism between local rings is not necessarily a local morphism (i.e. doesn't map the maximal ideal into the maximal ideal) the book says:

Let A be a local ring with a prime ideal P, such that $P\subset M\subset A$ where M denotes the unique maximal ideal. If we denote by $\phi$ the localization morphism with respect to the multiplicative system S=A\P, then $\phi :A\to A_{P}$ is not local.

My problem is that I do not understand what is a localization morphism (how it is defined, and what idea it conveys). I believe that $A_{P}$ has S^-1P as its unique maximal ideal but thats about all I understand here. Any help will be appreciated.--Shahab (talk) 14:35, 31 January 2009 (UTC)

A_P is the ring formed from A by adding multiplicative inverses for every element not in P. Thus the elements are of the form a/b for a in A and b in A\P. There is a natural map from A into A_P sending a to a/1. This is the localization map. The canonical example is to take A to be the integers and P={0}. Then A_P is the rationals and the localization map is the natural embedding of Z into Q. Algebraist 15:22, 31 January 2009 (UTC)

Thanks. I don't quite understand your first sentence though. Shouldn't it be: A_P is the ring formed from A by multipliying A by multiplicative inverses for every element not in P. Also if m is in P it is going to be mapped to m/1. In that case it is not a unit and so possibly in the maximal ideal of A_P. How can I conclude that the morphism isn't local?--Shahab (talk) 15:45, 31 January 2009 (UTC)

Perhaps 'adjoining' would have been better than 'adding'. On your last point, yes of course P is mapped into the maximal ideal of A_P. The point is the M is not (we're assuming here that M≠P). Algebraist 15:50, 31 January 2009 (UTC)

An explicit example is A = { a/b : b odd, a,b in Z }, P = 0, M = 2A = { a/b : a even, b odd, a,b, in Z }, A_P = Q = { a/b : b nonzero, a,b in Z }, and φ : A → A_P : x → x. The maximal ideal of Q is 0, and the image of M under φ is much larger than 0. JackSchmidt (talk) 16:14, 31 January 2009 (UTC)

Thanks--Shahab (talk) 17:09, 31 January 2009 (UTC)

elementary graph theory

(This isn't the question... just background...) I was looking at a problem that introduced dropping a node from a connected graph and ensuring that the connected graph is still connected. My first thought was to have the dropped node add connections between all neighboring nodes, but that will be nasty if there are, say 100 neighbors. So, I thought about the minimum number of edges required to ensure connectivity. For 1 neighbor, no edges. For 2 neighbors, 1 edge. For 3 neighbors, 2 edges. For four neighbors, 3 edges (wrong - I noticed it is 4 later). So, it is n-1. Well, I then thought about five neighbors. It takes 7 edges to ensure connectivity. Then, I realized that I don't need this at all and went on to the next step.

(This is the question...) Is there a common proof for the minimum number of edges required to connect n nodes? I don't need it, but now the idea is stuck in my head and I have a lot more pressing things to work on. -- kainaw™ 22:23, 31 January 2009 (UTC)

There are many ways of showing that n-1 edges are required to connect n vertices. I don't know which, if any, is most common. The only textbook I have to hand is Bollobás's Modern Graph Theory, which does it by observing that the two algorithms for finding spanning trees he's given obviously make a graph with n-1 edges. Algebraist 22:33, 31 January 2009 (UTC)

I know that n-1 edges can create connectivity, but that doesn't ensure connectivity. With 4 nodes, ABCD, I can have vectors AB, BC, AC. That is n-1, but it is not connected. I must have 4 vectors to ensure connectivity. With 5 nodes, I must have 7 vectors. I was wondering about ensuring connectivity. -- kainaw™ 00:52, 1 February 2009 (UTC)

Oh, I see. We had that question here a little while ago. The answer is that with the disconnected n-vertex graph with most edges is a complete graph on n-1 of the vertices with an extra isolated vertex. Thus (n-1)(n-2)/2+1 edges are required to ensure connectivity: not much of an improvement on having all edges. Algebraist 00:57, 1 February 2009 (UTC)

Previous discussion. Algebraist 00:58, 1 February 2009 (UTC)

Thanks. Luckily, all the nodes in the program I was writing have unique IDs. So, when dropping a node with n neighbors, I only need n-1 vectors. If I consider the neighbors a line of unconnected nodes and put a vector between each pair along the line, I've ensured connectivity. The unique IDs makes it very easy to do that. -- kainaw™ 01:10, 1 February 2009 (UTC)

For completeness, it should be noted that you are assuming the graph is simple - that is, contains no self-loops or duplicated edges. --Tango (talk) 20:56, 1 February 2009 (UTC)

Modular arithmetic/number theory

Hi. For some work I'm doing, I need to work with PSL(3,2ⁿ). Trying to find the centre of SL(3,2ⁿ) involves solving the equation

$x^{3}\equiv 1{\pmod {2^{n}}}$

which some straight calculations for low n show to be only true for x=1. Is this true for any n? I was never any good at number theory...SetaLyas (talk) 23:28, 31 January 2009 (UTC)

Yes. The group of units of Z mod 2ⁿ has 2^n-1 elements, so every element has order a power of 2. If x³=1, then x must have order dividing 3, so the order must be 1, so x is 1. Algebraist 23:30, 31 January 2009 (UTC)

Wow, thanks for the speedy answer! SetaLyas (talk) 00:21, 1 February 2009 (UTC)

Vectors & Potentials

A particle at position vector r experiences a force (a(R⁻³) + b(R⁻⁴))r, where R=|r|. How would one find the function (say V(r)) of the potential in this case? I know with 1-D work done it's just the integral of F with respect to x, but how does one approach it in 3 dimensions? Can you simply integrate the function before the r with respect to R and ignore the r to get -(a/2(R⁻²) + b/3(R⁻³))r? It seems horribly wrong to me to just ignore the position vector in the integral but I'm not honestly sure where to go otherwise - what's the correct method?

On a similar vector-y calculus-y note, how would one differentiate 1/R with respect to time, when again R=r for the position vector?

Thanks for the help,

131.111.8.104 (talk) 23:40, 31 January 2009 (UTC)Zant

While the force is a vector, the potential is a scalar. As the force is in the radial direction and numerically depend on the distance only and not on the direction, it is the negative gradient of a potential which also depend on the distance only. The potential −a/(2R²)−b/(3R³) will do. Bo Jacoby (talk) 08:47, 2 February 2009 (UTC).

Sorry about the terse response. Unless I am confused (a strong possibility), the first question has no solution for nonzero b. See conservative vector field and scalar potential for information. See scalar potential#Integrability conditions for how to calculate the potential -- it requires a line integral.

For the second question, use the chain rule. You may find $R=|r|={\sqrt {\mathbf {r} \cdot \mathbf {r} }}$ helpful. (See dot product#Derivative for taking the derivative of a dot product.) Eric. 131.215.158.184 (talk) 09:38, 2 February 2009 (UTC)