Wikipedia:Reference desk/Archives/Mathematics/2008 June 15

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June 15

solution of a Diophantine equation

Given integer n, what are solutions in integers x and y of the Diophantine equation x^2-y^2=n? Bubba73 (talk), 06:11, 15 June 2008 (UTC)

x = √n*sec(t) y = √n*tan(t)

for some parameter t. —Preceding unsigned comment added by 124.191.114.87 (talk) 06:45, 15 June 2008 (UTC)

x²−y²=(x+y)(x−y). So factor n=ab and chose x=(a+b)/2, y=(a−b)/2. If n is odd then x and y are integers. If n is even but not divisible by 4 then a+b and a−b are odd and there are no integer solutions. If n is divisible by 4 then chose a and b even. Bo Jacoby (talk) 08:12, 15 June 2008 (UTC).

Thanks, I had thought of factoring n, but I didn't know if that was the best way. I need all solutions, so that will do it. Bubba73 (talk), 14:28, 15 June 2008 (UTC)

question about inequalities

Hi Why does the inequality sigh change when both sides are multiplied or divided by a negative number?Does this happen with equations? —Preceding unsigned comment added by Lighteyes22003 (talk • contribs) 08:37, 15 June 2008 (UTC)

Is the "inequality sigh" the sound you make when confronted with an inequality ? :-) StuRat (talk) 05:08, 16 June 2008 (UTC)

Hi, your questions sound very much like homework, and we will not do your homework for you. We may give hints if you have at least shown you have tried to do it. I suggest you look at the article on inequality. -mattbuck (Talk) 14:13, 15 June 2008 (UTC)

The questions don't sound like homework to me, but rather requests for clarification of topics that were not clearly explained in the OP's class. They are very difficult to answer without knowing more about the OP's background. -- Meni Rosenfeld (talk) 14:18, 15 June 2008 (UTC)

Look what happens with this simple inequality: $3<4$. You see that multiplying both sides by minus 2 gives $-6>-8$. That demonstrates the inequality sign turning around. Equations don't contain inequality signs, instead they have an equals sign that doesn't change, as long as you multiply or divide both sides by the same (any) number.Cuddlyable3 (talk) 14:55, 15 June 2008 (UTC)

(This doesn't sound like homework to me, it's a general question.) When you multiply by a negative number, the number line is basically flipped over, it's reflected about zero. This reverses all the inequalities. For example 1 < 2, but -1 > -2. It's pretty obvious that this works with numbers, but it works in exactly the same way with more complicated expressions, since once you evaluate them at a particular point, they are just numbers. Diving works in the same way as multiplying, since it's just multiplying by one over the number. Does that help at all? --Tango (talk) 14:58, 15 June 2008 (UTC)

Oh, and no, it doesn't happen with equations. You can't flip an equals sign - if two things are equal, you can do anything you like to them and as long as you do the same thing to both, they will still be equal. --Tango (talk) 14:59, 15 June 2008 (UTC)

This doesn't look anything like homework to me either. The "flipping the inequality sign" bit also applies to taking a negative root. Thus X² > 4 has solutions of X > 2 and X < -2. With some practice you should have flippin' inequality signs mastered in no time. StuRat (talk) 05:12, 16 June 2008 (UTC)

Inequality

How do you know if a value is a solution to an inequality?How is this difference from determining if a value is a solution to an equation?Lets say you replace the equal sigh with the inequality sign,is there ever a time when you have the same value solution for both an inequality and the equation? —Preceding unsigned comment added by Lighteyes22003 (talk • contribs) 08:44, 15 June 2008 (UTC)

Hi, your questions sound very much like homework, and we will not do your homework for you. We may give hints if you have at least shown you have tried to do it. I suggest you look at the article on inequality. -mattbuck (Talk) 14:13, 15 June 2008 (UTC)

See my comment above. -- Meni Rosenfeld (talk) 14:19, 15 June 2008 (UTC)

Put the value you think is the solution back into the equation or inequality concerned and see whether it works. For example, $5x+2=12$ is an equation and its solution is $x=2$. You know this solution is right because $5$ times $2+2$ indeed equals 12. Note that many different numbers will satisfy (not "solve") most inequalities. Cuddlyable3 (talk) 15:08, 15 June 2008 (UTC)

THIS WAS HOMEWORK.......I HAVE THE SAME CLASS for AXIA COLLEGE!!!!

Euler-Lagrange equations - The inverse problem

Is there a way to numerically compute the Lagrangian of a system, given a path that the system followed? 212.143.139.133 (talk) 11:08, 15 June 2008 (UTC)

What a good question! Historically, the path of planet Mars was observed and recorded numerically by Tycho Brahe. Then the data were reduced into physical laws by Johannes Kepler. Further reductions by Isaac Newton and Joseph-Louis Lagrange lead to the Lagrangian. So yes, there is a way! Usually the induction of a physical law from experimental observations is more tricky than the deduction of an experimental result from the physical laws. Bo Jacoby (talk) 10:25, 16 June 2008 (UTC).

Does a numerical algorithm for solving this kind of problems exist? I am particularly interested in a one dimensional system. Will I be needing more than one path in order to solve the problem? 212.143.139.133 (talk) 11:18, 16 June 2008 (UTC)

If the motion is periodic, then the energy is conserved. If the kinetic energy as a function of velocity is (1/2)mv^2, then you can numerically compute kinetic energy as a function of position, and from that a table of potential energy as a function of position. Then you need to compute an interpolating function of the table. As several interpolating functions are possible, your solution is not unique. Several paths might improve certainty. Bo Jacoby (talk) 12:20, 16 June 2008 (UTC).

Unfortunately, the motion is not periodic, it is very general. Furthermore, the potential itself changes all the time and I am not even sure that L=T-U. I guess it is quite a problem!
I am looking for a numerical algorithm which will help me approximate a family of Lagrangians based on paths of the one-dimensional system. 212.143.139.133 (talk) 07:41, 17 June 2008 (UTC)

Will they ever give up on this pi thing?

I mean, what the heck are the practical implications of finding the last number (if there is one)? Billions of places? Why?--Sam Science (talk) 13:28, 15 June 2008 (UTC)

There is no last digit, and it has no practical implications (except possibly the testing of new computers). I doubt everyone's going to give up on it any time soon, though: many people seem to enjoy it. Me, I prefer e. Algebraist 13:35, 15 June 2008 (UTC)

I guess it could help a little in answering the question of whether or not pi is normal. Obviously, an actual proof is needed to answer the question completely, but calculating enough digits could provide strong evidence against normality (I don't think it can really provide much evidence in favour of it). At least then, people would know what to try and prove. --Tango (talk) 14:53, 15 June 2008 (UTC)

And when they've finished with that, they should have one of thier interns take atom inventory.Sam Science (talk) 14:25, 15 June 2008 (UTC)

Question: I see from the article that e has been calculated up to 100 billion decimal places. Has it been done with only one algorithm or piece of software? If so, how do we know it's correct? Zain Ebrahim (talk) 14:35, 15 June 2008 (UTC)

I don't know about e, but when people calculate digits of pi, they will use two algorithms and compare the answers. See here for one of the most recent announcements, explaining the process. I expect the same applies to e. --Tango (talk) 14:53, 15 June 2008 (UTC)

The constant e and its computation Cuddlyable3 (talk) 15:32, 15 June 2008 (UTC)

If you had an idea for an algorithm that could compute the digits of pi that ran fast enough to find more digits than anyone has ever found before, how could you resist the urge the write and run it? GromXXVII (talk) 17:37, 15 June 2008 (UTC)

<joke> calculating enough digits of pi might also uncover secret messages. </joke> Oliphaunt (talk) 21:24, 15 June 2008 (UTC)

If pi is normal, you will uncover all possible secret messages... eventually. JohnAspinall (talk) 20:53, 16 June 2008 (UTC)