Letzj=xj+ij, where xj,yj are real number=1,2,3.
(a) We have
z1+z2=(x1+x2)+i(y1+y2)=(x2+x1)+i(y2+y1)=z2+z1. z1z2=(x1x2−y1y2)+i(x1y2+x2y1)=(x2x1−y2y1)+i(y2x1+y1x2)=z2z1.
is valid for real numbers and so it holds for the real and imaginary parts ofz1z2,z3.
Consequently we have
(z1+z2)+z3=z1+(z2+z3).
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