Question

Show that 1+2i / 3+4i × 1-2i / 3-4i is real​

Answer 1

Step-by-step explanation:

We know that, the Complex Number product is real only if z = z'

Here z' is the Conjugate of z

$z = \frac{1 + 2i}{3 + 4i} \times \frac{1 - 2i}{3 - 4i}$

$z = \frac{1 + 4}{9 + 16} = \frac{5}{25}$

$z = \frac{1}{5}$

Now,

$z’ = \frac{1 - 2i}{3 - 4i} \times \frac{1 + 2i}{3 + 4i}$

$z’ = \frac{1 + 4}{9 + 16} = \frac{5}{25}$

$z’ = \frac{1}{5}$

z = z'

So, z is real