# Complex numbers

• Sep 12th 2012, 06:14 PM
franios
Complex 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 PM
MaxJasper
Re: Complex numbers
z = x+i y
z(with line on top) = x-i y
• Sep 13th 2012, 08:00 AM
Deveno
Re: 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 AM
Plato
Re: Complex numbers
Quote:

Originally Posted by MaxJasper
z = x+i y
z(with line on top) = x-i y

In LaTeX code [TEX] \overline{a+bi}[/TEX] gives $\displaystyle \overline{a+bi}$
• Sep 13th 2012, 08:17 AM
Deveno
Re: Complex numbers
if you look closely, i used that code at least once in my post :P