Pythagorean Theorem in Complex Numbers

MattcAnderson · 2020-11-25, 21:13

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I am working through an unpublished number theory text book.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 Mathematics, Its Magic And Mastery by Aaron Bakst. One of its chapters is entitled Algebra, Boss of Arithmetic.

2020-11-25, 21:49	#2
VBCurtis "Curtis" Feb 2005 Riverside, CA 1240₁₆ 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

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2020-11-26, 00:48	#3
Dr Sardonicus Feb 2017 Nowhere 10E0₁₆ Posts	There are algebraic formulas, e.g. a = k(p^2 - q^2), b = k(2pq), 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*.