Chapter 1, Exercise 4, p 45
On page 7 we see that |z|2 = zz, and that z1 z2 = z1 z2
So
|z1 z2 |2 = z1 z2 z1 z2 = z1 z1 z2 z2
and so
|(a + ib)(c + id)|2 = (a + ib)(a − ib)(c + id)(c − id) = (a2 + b2 )(c2 + d2 ) = M N
Also
|(a + ib)(c + id)|2 = |ac − bd + i(bc + ad)|2 = |p + iq|2 ,
where, p = ac − bd and q = bc + ad
Since a, b, c, d are integers, p and q are also integers and so
|p + iq|2 = (p + iq)(p − iq) = p2 + q 2 = M N
QED
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