# Write down the definition of the complex conjugate, bar{z} ,if z=x+iy where z,y are real numbers. Hence prove that, for any complex numbers w and z

Write down the definition of the complex conjugate, where z,y are real numbers. Hence prove that, for any complex numbers w and z,
$\stackrel{―}{w\stackrel{―}{z}}=\stackrel{―}{w}z$
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berggansS

Step 1
Complex consugate
Let z be a complex number defined by $z=x+iy$ , whre x and y are real and imaginary part.
complex consugate of z is the sum change of imaginary part.
$\stackrel{―}{z}=x-iy$
let two complex number
$\stackrel{―}{z}={x}_{2}-i{y}_{2}$
$w\stackrel{―}{z}=\left({x}_{1}+i{y}_{1}\right)\cdot \left({x}_{2}-i{y}_{2}\right)$
$={x}_{1}{x}_{2}-i{x}_{1}{y}_{2}+i{y}_{1}{x}_{2}-{i}^{2}{y}_{1}{y}_{2}$
$={x}_{1}{x}_{2}+{y}_{1}{y}_{2}+i\left({y}_{1}{x}_{2}-{x}_{1}{y}_{2}\right)$
Step 2
$\stackrel{―}{w\stackrel{―}{z}}=\left({x}_{1}{x}_{2}+{y}_{1}{y}_{2}\right)-i\left({y}_{1}{x}_{2}-{x}_{1}{y}_{2}\right)$
$\stackrel{―}{w\stackrel{―}{z}}=\left({x}_{1}{x}_{2}+{y}_{1}{y}_{2}\right)+i\left({x}_{1}{y}_{2}-{y}_{1}{x}_{2}\right)$
$\stackrel{―}{w}={x}_{1}-i{y}_{1}$
$\stackrel{―}{w}z=\left({x}_{1}-i{y}_{1}\right)\left({x}_{2}+i{y}_{2}\right)$
$\stackrel{―}{w}z=\left({x}_{1}{x}_{2}+{y}_{1}{y}_{2}\right)+i\left({x}_{1}{y}_{2}-{y}_{1}{x}_{2}\right)$
Form equation 1 and equation 2
$\stackrel{―}{w\stackrel{―}{z}}=\stackrel{―}{w}z$