Prove the following properties of any complex number z:

Re[z]=(z+z(with line on top))/2

Im[z]=(z+z(with line on top))/2i

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- Sep 12th 2012, 06:14 PMfraniosComplex numbers
Prove the following properties of any complex number z:

Re[z]=(z+z(with line on top))/2

Im[z]=(z+z(with line on top))/2i - Sep 12th 2012, 06:24 PMMaxJasperRe: Complex numbers
z = x+i y

z(with line on top) = x-i y - Sep 13th 2012, 08:00 AMDevenoRe: Complex numbers
let z = a+bi

then z* (i am writing z* for z-conjugate, rather than $\displaystyle \overline{z}$) to avoid extra typing. so i'm lazy. sue me.) is a-bi.

now z+z* = (a+bi) + (a-bi) = (a+a) + (b-b)i = 2a + 0i = 2a, so (z+z*)/2 = a = Re(z).

and: z-z* = (a+bi) - (a-bi) = (a-a) + (b-(-b))i = 0 + 2bi = 2bi, so (z-z*)/(2i) = 2bi/(2i) = b = Im(z). - Sep 13th 2012, 08:08 AMPlatoRe: Complex numbers
- Sep 13th 2012, 08:17 AMDevenoRe: Complex numbers
if you look closely, i used that code at least once in my post :P