If z _ 1 and ¯ z 1 represents adjacent vertices of a regular...

Let ${z}_{1} = re^{i\theta}$

then, $\overset{¯}{z}1 = re^{- i\theta}$

$\dfrac{z1}{\overset{¯}{z}1} = e^{2i\theta} = e^{2\pii / n}$

$⇒ \theta = \dfrac{\pi}{n}$

Also, $\dfrac{{I}_{m}\left({z}_{1}\right)}{{R}_{e}\left({z}_{1}\right)} = tan\theta = tan\dfrac{\pi}{n} = \sqrt{2} - 1$

$⇒ tan\dfrac{\pi}{8} = \sqrt{2} - 1$

$∴ n = 8$