Question

For z = 2 + 3i verify the following:
i) $\overline{\bar{z}} = z$
ii) $z\bar{z} = |z|^{2}$

Answer 1

Solution:

If a complex number is represented by z then it's complex conjugate is
$\bar{z}$
and it's mode
$z = a + ib\\\\ |z| = \sqrt{ {a}^{2} + {b}^{2} } \\\\ {|z|}^{2}={a}^{2} + {b}^{2}$

i) $\overline{\bar{z}} = z$

$z = 2 + 3i \\ \\ \bar{z} = 2 - 3i \\ \\ \overline{\bar{z}} = 2 + 3i = z$

hence proved

ii) $z\bar{z} = |z|^{2}$

$z = 2 + 3i \\ \\ \bar{z} = 2 - 3i \\ \\ z\bar{z} = (2 + 3i)(2 - 3i) \\ \\ = ( {2)}^{2} - ( { - 3i)}^{2} \\ \\ = 4 + 9 \\ \\ z\bar{z} = 13 = { |z| }^{2}$
Hope it helps you