Find the conjugates of the following complex numbers: (i) 4 – 5i (ii) 1 / (3 + 5i)

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Find the conjugates of the following complex numbers:
(i) 4 – 5i
(ii) 1 / (3 + 5i)

Find the conjugates of the following complex numbers:
(i) 4 – 5i
(ii) 1 / (3 + 5i)

Solution:

(i) 4 – 5i
It is known that the conjugate of a complex number $(a + ib)$ is $(a - ib)$

$\therefore$ $(4 + 5i)$ is the conjugate of $(4 - 5i)$

(ii) 1 / (3 + 5i)

As the given complex no. is not in the standard form of $(a + ib)$

Convert it to the standard form by multiplying and dividing with (3 – 5i)

We obtain,
$\begin{aligned} \frac{1}{3+5 i} =\frac{1}{3+5 i} \times \frac{3-5 i}{3-5 i} \\ =\frac{3-5 i}{3^{2}-(5 i)^{2}} \\ =\frac{3-5 i}{9-25 i^{2}} \\ =\frac{3-5 i}{9-25(-1)}\left[\text { Since }, i^{2}=-1\right] \\ =\frac{3-5 i}{34} \end{aligned}$
It is known that the conjugate of a complex number $(a+i b)$ is $(a - ib)$
$\mathrm{So}$ ,
As a result, the conjugate of $(3-5 i) / 34$ is $(3+5 i) / 34$