Question

For z = 2 + 3i verify the following:

${\displaystyle \text{z}\overline{\text{z}}}$ = |z|²

Answer 1

z = 2 + 3i

∴ ${\displaystyle \overline{\text{z}}}$ = 2 – 3i

and |z| = ${\displaystyle \sqrt{2^2+3^2}}$

= ${\displaystyle \sqrt{4+9}}$

= ${\displaystyle \sqrt{13}}$

Also, ${\displaystyle \text{z}.\overline{\text{z}}=(2+3\text{i}).(2-3\text{i})}$

= 4 – 9i²

= 4 + 9 ...[∵ i² = – 1]

∴ ${\displaystyle \text{z}.\overline{\text{z}}}$ = 13 = |z|²