Odd/stupid/interesting question

For the discussion of math. Duh.

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Odd/stupid/interesting question

Postby keenin » Fri Apr 20, 2012 12:34 am UTC

This might be a dumb question but I can't seem to come up with definite answer. Is it possible to have a single equation with two distinct unknown variables, that only has only one solution? I don't think it is but my mathematical abilities are very limited. The reason I think it is not possible I don't see how it would be possible to come up with an equation that when plotted gives you single point.

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Re: Odd/stupid/interesting question

Postby gfauxpas » Fri Apr 20, 2012 12:55 am UTC

x2 + y2 = 0 for real x, y only has the solution x = y = 0

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Re: Odd/stupid/interesting question

Postby MartianInvader » Fri Apr 20, 2012 1:01 am UTC

In the same vein, |x| + |y| = 0 for complex variables. Of course, the question comes down to what you consider an "equation".
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!

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Re: Odd/stupid/interesting question

Postby gorcee » Fri Apr 20, 2012 3:21 pm UTC

Sure, it's easy to construct such an example.

Take some function f : R x R -> R such that it has a global maximum m in its image. Then, there is a single tuple (x',y') such that f(x',y') = m. Thus, the equation f(x,y)-m = 0 has only a single point (x,y) as a solution.

As an example, cut a tennis ball in half. Put it on a table. The point on the surface of the tennis ball where the normal plane is parallel to the table is a single point.

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Re: Odd/stupid/interesting question

Postby Yesila » Sun Apr 22, 2012 6:23 am UTC

Time to search through that old collection of (functions with) negative definite hessian's for some global maxes.

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