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-   -   Pythagorean Theorem in Complex Numbers (https://www.mersenneforum.org/showthread.php?t=26230)

MattcAnderson 2020-11-25 21:13

Pythagorean Theorem in Complex Numbers
 
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Hi All,

I am working through an unpublished number theory text book. Some university professors who are my friends wrote it. I want to share.

I have found an example of complex numbers a, b and c such that
a^2 + b^2 = c^2

But here the solution does not represent a length.

Regards,
Matt

VBCurtis 2020-11-25 21:49

I have a set of complex a, b, c also:
3 + 0i, 4 + 0i, 5 + 0i.

Edit: or 0 + 3i, 0 + 4i, 0 + 5i.

Dr Sardonicus 2020-11-26 00:48

There are algebraic formulas, e.g.

a = k*(p^2 - q^2), b = k*(2*p*q), c = k*(p^2 + q^2)

for which a^2 + b^2 = c^2 is a polynomial identity.

So k, p, and q can be rational integers, Gaussian integers, Eisenstein integers, arbitrary complex numbers, or just variables -- it's all good.

There is a book entitled [u]Mathematics, Its Magic And Mastery[/u] by Aaron Bakst. One of its chapters is entitled [b]Algebra, Boss of Arithmetic[/b].


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