Question

Find the modulus and principal argument of -4?

Answer 1

Hint: Modulus of a complex number z = x + iy is the magnitude of complex number z and is represented as $\left| z \right|=\sqrt{{{x}^{2}}+{{y}^{2}}}$and principal argument of a complex number z is $\arg \left( z \right)=\arctan \left( \dfrac{y}{x} \right)$. So, here we are going to use these formulae.

Complete step-by-step answer:

We can write -4 in the form of complex number representation as:
z = -4 + i(0)
Here, the imaginary part of the complex number is 0.
We can find the modulus of -4 using the formula$\left| z \right|=\sqrt{{{x}^{2}}+{{y}^{2}}}$. We have written -4 in the form of a complex number as z = -4 + i(0). So, x = -4 and y=0. Substituting the values of x and y in modulus formula we get,
$\begin{align}
  & \left| z \right|=\sqrt{{{\left( -4 \right)}^{2}}+{{\left( 0 \right)}^{2}}} \\
 & \Rightarrow \left| z \right|=\sqrt{16} \\
 & \Rightarrow \left| z \right|=4 \\
\end{align}$
Hence, the modulus of -4 equals 4.
Now, for evaluating the principal argument of -4 we are using the formula $\arg \left( z \right)=\arctan \left( \dfrac{y}{x} \right)$. So, substituting the values of x and y in principal argument formula, we get:
$\begin{align}
  & \arg \left( -4 \right)=\arctan \left( \dfrac{0}{-4} \right) \\
 & \Rightarrow \arg (-4)=\arctan (0) \\
 & \Rightarrow \arg (-4)=0 \\
\end{align}$
As we know from trigonometry that arctan(0) is 0 in value.
Hence, the modulus of -4 is 4 and the principal argument of -4 is 0.

Note: You can directly answer the modulus of -4 without using the formula. As -4 is a real number and modulus of any negative real number is always positive so the answer is +4. The expression of -4 in terms of complex numbers is z = -4 and as you can see the imaginary part is 0 so it looks like a line parallel to x axis so the argument ( which is an angle made by a complex number with x axis in argand plane) is 0.