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Old 2020-11-25, 21:13   #1
MattcAnderson
 
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"Matthew Anderson"
Dec 2010
Oregon, USA

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Smile Pythagorean Theorem in Complex Numbers

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I am working through an unpublished number theory text book.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
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Old 2020-11-25, 21:49   #2
VBCurtis
 
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"Curtis"
Feb 2005
Riverside, CA

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

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

Last fiddled with by VBCurtis on 2020-11-25 at 21:50
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Old 2020-11-26, 00:48   #3
Dr Sardonicus
 
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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 Mathematics, Its Magic And Mastery by Aaron Bakst. One of its chapters is entitled Algebra, Boss of Arithmetic.
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