Question

Find the modulus and argument of the complex number z = -7+24i​

Answer 1

Given :-

• Complex number $z = -7 +24i$

To find :-

• Modulus and argument of $z$

Solution :-

Let's assume,

• $x = -7\: (\sf{Real\:part)}$
• $y = 24 \:(\sf{Imaginary\:part)}$

Modulus of a complex number is given by:

$\displaystyle \rm{:\implies |z|=\sqrt{Re(z)^2+Im(z)^2}}$

$\displaystyle {:\implies |z|=\sqrt{(x)^2+(y)^2}}$

$\displaystyle {:\implies |z|=\sqrt{( - 7)^2+(24)^2}}$

$\displaystyle {:\implies |z|=\sqrt{49+576}}$

$\displaystyle {:\implies |z|=\sqrt{625}}$

$\displaystyle {:\implies |z|=25}$

Hence the modulus of z is 25.

$\displaystyle {:\implies \alpha = \tan^{ - } \left | \dfrac{y}{x}\right |}$

$\displaystyle {:\implies \alpha = \tan^{ - } \left | - \dfrac{ 24}{7}\right |}$

$\displaystyle {:\implies \alpha = \tan^{ - } \dfrac{ 24}{7}}$

Since x is -ve and y is positive, the given complex number will lie in 2nd quadrant on argànd plane.

Angles of 2nd quadrant are given by, θ = π - α

$\displaystyle {:\implies \theta = \pi - \alpha }$

$\displaystyle {:\implies \theta = \pi - \tan^{ - } \dfrac{ 24}{7}}$

This is the required argument θ.