Find the modulus of each of the following complex numbers and hence

Solution:

Let Z = 4 = r(cosθ + isinθ)

Now, separating real and complex part, we get

4 = rcosθ……….eq.1

0 = rsinθ…………eq.2

Squaring and adding eq.1 and eq.2, we get

$16=r^{2}$

Since r is always a positive no., therefore,

r = 4,

Hence its modulus is 4.

Now, dividing eq.2 by eq.1, we get,

$\frac{r \sin \theta}{r \cos \theta}=\frac{0}{4}$

Tanθ = 0

Since cosθ = 1, sinθ = 0 and tanθ = 0. Therefore the θ lies in first quadrant.

Tanθ = 0, therefore θ = 0°

Representing the complex no. in its polar form will be

$Z=4\left(\cos 0^{\circ}+i \sin 0^{\circ}\right)$