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Complex Conjugate

The complex conjugate of a Complex Number $z\equiv a+bi$ is defined to be $z^*\equiv a-bi$. The complex conjugate is Distributive over addition, $(z_1+z_2)^* = {z_1}^*+{z_2}^*$, since

{[}(a_1+ib_1)+(a_2+ib_2)]^* &=& [(a_1+a_2)\!+\!i(b_1+b_2)]^*\...
&=& (a_1-ib_1)+(a_2-ib_2)\\
&=& (a_1+ib_1)^*+(a_2+ib_2)^*,

and Distributive over multiplication, $(z_1z_2)^* = {z_1}^*{z_2}^*$, since
$[(a_1+b_1i)(a_2+b_2i)]^* = [(a_1a_2-b_1b_2)+i(a_1b_2+a_2b_1)]^*$
$\quad = (a_1a_2-b_1b_2)-i(a_1b_2+a_2b_1)$
$\quad = (a_1-ib_1)(a_2-ib_2)$
$\quad = (a_1+ib_1)^*(a_2+ib_2)^*.$


Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 16, 1972.

© 1996-9 Eric W. Weisstein