Complex Numbers
4.0 Properties of conjugate
4.0 Properties of conjugate
$\mathop z\limits^ - $ is the mirror image of $z$ in the real axis.
1. $\left| z \right| = \left| {\mathop z\limits^ - } \right|$
2. $z + \mathop z\limits^ - = 2\operatorname{Re} (z)$
3. $z - \mathop z\limits^ - = 2i\operatorname{Im} (z)$
4. $z\mathop z\limits^ - = {\left| z \right|^2}$
5. $\overline {{z_1} + {z_2}} = \overline {{z_1}} + \overline {{z_2}} $
$ \Rightarrow \overline {{z_1} + {z_2} + {z_3} + ...{z_n}} = \overline {{z_1}} + \overline {{z_2}} + \overline {{z_3}} + ... + \overline {{z_n}} $
6. $\overline {{z_1} - {z_2}} = \overline {{z_1}} - \overline {{z_2}} $
$ \Rightarrow \overline {{z_1} - {z_2} - {z_3} - ...{z_n}} = \overline {{z_1}} - \overline {{z_2}} - \overline {{z_3}} - ... - \overline {{z_n}} $
7. $\overline {{z_1}{z_2}} = \overline {{z_1}} .\overline {{z_2}} $
8. $\overline {\left( {\frac{{{z_1}}}{{{z_2}}}} \right)} = \left( {\frac{{\overline {{z_1}} }}{{\overline {{z_2}} }}} \right)$
9. $\overline {{z^n}} = {\left( {\overline z } \right)^n}$
10. $\overline {\left( {\overline z } \right)} = z$
11. ${z_1}\overline {{z_2}} + \overline {{z_1}} {z_2} = 2\operatorname{Re} (\overline {{z_1}} {z_2}) = 2\operatorname{Re} ({z_1}\overline {{z_2}} )$
12. If $z = f({z_1})$, then $\overline z = f(\overline {{z_1}} )$
Question 2. Given $$x + iy = \sqrt {\frac{{a + ib}}{{c + id}}} $$ Prove that $${\left( {{x^2} + {y^2}} \right)^2} = \frac{{{a^2} + {b^2}}}{{{c^2} + {d^2}}}$$
Solution: Replace $i$ by $-i$ in given equation, we get $$x - iy = \sqrt {\frac{{a - ib}}{{c - id}}} $$
Multiply both the equations
$$\begin{equation} \begin{aligned} (x + iy)(x - iy) = \sqrt {\frac{{a + ib}}{{c + id}}} \sqrt {\frac{{a - ib}}{{c - id}}} \\ {x^2} + {y^2} = \sqrt {\frac{{(a + ib)(a - ib)}}{{(c + id)(c - id)}}} = \sqrt {\frac{{{a^2} + {b^2}}}{{{c^2} + {d^2}}}} \\\end{aligned} \end{equation} $$
Squaring both sides, we get $${\left( {{x^2} + {y^2}} \right)^2} = \frac{{{a^2} + {b^2}}}{{{c^2} + {d^2}}}$$
