2020-11-25, 21:13 | #1 |
"Matthew Anderson"
Dec 2010
Oregon, USA
3·367 Posts |
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I am working through an unpublished number theory text book.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 |
2020-11-25, 21:49 | #2 |
"Curtis"
Feb 2005
Riverside, CA
5,279 Posts |
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 |
2020-11-26, 00:48 | #3 |
Feb 2017
Nowhere
53·109 Posts |
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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