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How to check equations/identities?

asked 2017-05-05 05:47:32 -0500

screened00 gravatar image

updated 2017-05-05 05:48:01 -0500

So, i got a different cubic formula for the depressed equation than that was there in the standard formula from a book. $y^3+fy+g=0 $ has the solution

$ ({-g+\sqrt {g^2/4+f^3/27} })^{1/3} + ({-g-\sqrt {g^2/4+f^3/27} })^{1/3} $

$ -\frac{f}{3 (-\frac{1}{2}g + \frac{1}{6} \sqrt{\frac{4}{3}f^3+9g^2}) ^{1/3} } == ({-g+\sqrt {g^2/4+f^3/27} })^{1/3} $

How do I check. I want to use sage,not math. In particular the bool() function.

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answered 2017-05-05 09:59:28 -0500

dan_fulea gravatar image

updated 2017-05-05 10:03:13 -0500

Code:

sage: var( 'f,g' );
sage: LHS = - f/3/ ( -g/2 + 1/6*sqrt( 4/3*f^3 + 9*g^2) )^(1/3)
sage: RHS =        ( -g/2 - 1/6*sqrt( 4/3*f^3 + 9*g^2) )^(1/3)
sage: bool( LHS == RHS )
False
sage: bool( LHS^3 == RHS^3 )
True

Comments:

If a bool on a to be equality is answered by sage with False, this means that the equality is mathematically either false (for instance if we mistype a g/2 as g or a plus as a minus in the formulas for LHS or RHS above), or true, but sage could not prove it in the latter case. In case sage gives us the True, then it is really true in mathematics. In our case, sage is giving us the True only after passing to the third power. (And i would do the same mathematically, since without doing this it depends on the implicit choice of the third root for the two places to make the choice. If one correlates the roots, this is ok. But where should sage or i know this?)

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answered 2017-05-13 08:47:49 -0500

updated 2017-06-18 15:14:29 -0500

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Asked: 2017-05-05 05:47:32 -0500

Seen: 62 times

Last updated: Jun 18