Wikipedia:Reference desk/Archives/Mathematics/2013 December 22

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December 22

Roots Calculating And Computing Roots

What are the mathematical differences between calculating an nth root different from calculating another nth root?174.3.125.23 (talk) 02:01, 22 December 2013 (UTC)

Your question is not clear to me. Can you rephrase it? -- Jack of Oz ^{[pleasantries]} 02:07, 22 December 2013 (UTC)

From [1]:

== Babylonian Method Of Calculating Roots ==

What is the history behind the Babylonian Method? Such as:

• Do we have an article on this topic?
• Was it published in a treatise?
• Was it compiled in a anthology of mathematical formulae?174.3.125.23 (talk) 01:11, 20 December 2013 (UTC)

See "Babylonian method".—Wavelength (talk) 01:37, 20 December 2013 (UTC)

How is calculating square roots any different from calculating cube roots?174.3.125.23 (talk) 01:46, 20 December 2013 (UTC)

Maybe you should ask this on the Math desk... AnonMoos (talk) 12:45, 20 December 2013 (UTC)

Because the first calculates square roots and the second calculates cube roots? I don't understand your confusion. If you read our article on the Babylonian method, it's clear that the method doesn't work for cube roots. --Bowlhover (talk) 21:35, 20 December 2013 (UTC)

My question is: What makes calculating a $n$ root different from others? Such as: What makes the-method-which-works-for-square-roots render the computation of a cubic root impossible?174.3.125.23 (talk) 02:44, 21 December 2013 (UTC)

Um, as Bowlhover already said, because it calculates square roots, not cube roots? Seriously, this question will receive better answers on the maths desk --109.189.65.217 (talk) 21:11, 21 December 2013 (UTC)

Computing the square root is different from computing the cube root just like dividing by 2 is different from dividing by 3. In the case of those divisions, we have the long division algorithm which enables you to divide by whatever you want by changing the process slightly. But the Babylonian method is specially constructed only to give the square root. Like it says at Methods of computing square roots, the Babylonian method works for the square root because it matches to first order with the Taylor series for the square root function. It won't work for cube roots because the Taylor series for the cube root is different. I imagine you could modify the Babylonian method to work for other roots by making it match with the Taylor series for other roots. If you're going to do that you may as well use more terms of the Taylor series. And if you're going to do that, then you may as well use other methods entirely, like using some fancy approximations for exponential and natural logarithms, and doing $x^{1/n}=e^{{\frac {1}{n}}\ln x}$, which is what typical calculators do. Staecker (talk) 13:09, 22 December 2013 (UTC)

Endomorphisms of Symm(n) for each integer n.

There are 2 for S2, 10 for S3. Is there a general formula for n>6 perhaps?(I bet any such formula would break down for S4 and S6, these have to be calculated separately) In any case, if you know where a table of the number per n is, or other information, I'd appreciate it. Thanks, Rich Peterson76.218.104.120 (talk) 05:01, 22 December 2013 (UTC)

Assuming n>6, if the endomorphism is onto then it's an automorphism, there are n! of them. The fact that Sn is nearly simple drastically reduces the remaining cases, namely the image must have order 1 or 2. This gives one for each element of order 2 in Sn. The number of elements of order 2 is a well known sequence, the exponential general function is exp(x+x²/2). There isn't a closed form expression but several recursive formulas and asymptotic expressions are known. See Telephone number (mathematics). So the exponential generating function of the sequence you want is (some polynomial) + 1/(1-x) + exp(x+x²/2). --RDBury (talk) 05:45, 22 December 2013 (UTC)

thanks, good answer.Rich (talk) 18:42, 24 December 2013 (UTC)

Things that you cannot see on a graph

I am not very knowledgeable regarding graphs (the ones with nodes and vertices) but a math geek guy a knew told me that there are things quite difficult to calculate, or even impossible to prove. It was something along the line of 'you can't be completely sure if you already got the shorted path.' So, what are some counter-intuitive things that you cannot do with a graph? OsmanRF34 (talk) 17:43, 22 December 2013 (UTC)

Maybe the traveling salesman problem, which is (probably) impossible to solve efficiently. Staecker (talk) 21:37, 22 December 2013 (UTC)

Another example is that it is thought to be very hard to determine if a graph contains a hamiltonian path, a path which visits each vertex once. (This is rather interesting because it is easy to determine whether a graph contains an Euler circut, a path which traverses each edge once.) Look at List of NP-complete problems for many more examples of these sorts of problems. Gutworth (talk) 01:31, 23 December 2013 (UTC)

Bingo Probability Question

Which Bingo card has a better chance of winning by standard bingo rules (ie one ball after another, first card to 5-in-a-row wins): A regular bingo card with 25 different numbers, or a card where each column is all the same number? Example:

2 - 25 - 39 - 52 - 70

2 - 25 - 39 - 52 - 70

2 - 25 - 39 - 52 - 70

2 - 25 - 39 - 52 - 70

2 - 25 - 39 - 52 - 70

70.211.7.169 (talk) 20:19, 22 December 2013 (UTC)

Unlike your example above, bingo cards shouldn't repeat the same number. If we also assume that numbers are truly drawn at random, then every card has an equal chance of winning.

However, if there were cards with repeated numbers like that, and we assume that getting one of the repeated numbers (as opposed to 5, in a system where the same number could be redrawn) is an instant bingo, then that card would have a much better chance of winning. It would also have a much better chance of not having any matches at all, and a zero chance of have some matches but not getting a bingo. StuRat (talk) 20:26, 22 December 2013 (UTC)

I know the second card couldn't exist. I meant that the second card is a nonexistant class of card, and the former is a "regular" bingo card. 66.74.10.59 (talk) 22:30, 22 December 2013 (UTC)

The special card with the constant columns has the better chance. The following is obviously wrong, but is correct in the limit: the probability of getting a particular ball on a single draw is 1/75, so the odds of getting a particular column, row or diagonal of a standard card in a single draw is (1/75)^5, or (1/75)^4 if it contains the free space. Since there are 8 ways to get bingo on a standard card without free space, and 4 ways with it, the probability of getting bingo in a single draw with a standard card is 8*(1/75)^5 + 4*(1/75)^4. Meanwhile, the probability of getting bingo in a single draw with the modified card is 5/75, which is significantly larger.--71.175.63.37 (talk) 21:36, 22 December 2013 (UTC)

Gotcha, thanks. What if the custom card was just a single number repeated 25 times? 98.176.56.95 (talk) 21:26, 23 December 2013 (UTC)

That would obviously lower your chances relative to the other special card. It will take some math to figure if it's better or worse than the normal cards. StuRat (talk) 17:15, 26 December 2013 (UTC)