Question

solve (with explanation), if u don't know then don't solve!!!!! ​

Answer 1

Given :-

$Z = 2 + 3i$

To verify :-

$\overline{\overline{Z}} = Z \\ Z \overline{Z} = |Z|^2$

Verification:-

Conjugate of a complex number :-

Let Z = a + ib be a complex number.

Then, the conjugate of z is denoted by $\mathsf{ z^{\overline}}$ and is equal to a- ib.

→$(\overline {\overline{ Z}})$

→$\overline {\overline{ (a+ib) }}$

→$\overline{(a-ib)}$

→$a+ ib$

→$Z$

hence,

$\overline{ \overline {Z}}= Z$

Modulus of a complex number :-

The modulus of a complex number Z is denoted by | Z |.

→$| Z | = \sqrt{Re(Z)^2 + Im(Z) ^2}$

• where Re (Z) is real part of complex number.
• Im(Z) is imaginary part of complex number.

Let,

Re(Z) = 2 and Re (Z) = 3

→$|Z|^2 = Z. \overline{Z}$

→$( \sqrt{2^2 + 3^2})^2 = 2 + 3i . \overline{(2+3i)}$

→$(\sqrt{4 + 9})^2 = (2+3i) (2-3i)$

→$(\sqrt{13})^2 = (2) ^2-(3i) ^2$

→$13 = 4 - 9i^2$

→$13 = 4 + 9$

→$13 = 13$

hence,

$Z. \overline{(Z) }= |Z|^2$

verified..