Question

How do you find $\left| -1-2i \right|$?

Answer 1

We explain the term absolute value of a complex number and how the modulus function always remains positive. We expand the function and breaks into two parts. We take the general form of complex number $z=a+ib$ and the modulus value being $\left| z \right|=\sqrt{{{a}^{2}}+{{b}^{2}}}$. This is the direct distance of the complex value from origin.

Complete step by step solution:
Modulus function $f\left( x \right)=\left| z \right|$ works as the distance of the complex number from origin. The complex number can be anything but the distance of that number will always be positive. Distance can never be negative.
In mathematical notation we express it with modulus value. Let a complex number be $z=a+ib$ whose sign is not mentioned. The absolute value of that number will be $\left| z \right|=\sqrt{{{a}^{2}}+{{b}^{2}}}$.
For our given function $z=a+ib=-1-2i$.
We equate the given complex number with its general form and get $a=-1,b=-2$.
We place the value of $a=-1,b=-2$ in the formula of $\left| z \right|=\sqrt{{{a}^{2}}+{{b}^{2}}}$.
So, $\left| z \right|=\sqrt{{{\left( -1 \right)}^{2}}+{{\left( -2 \right)}^{2}}}=\sqrt{1+4}=\sqrt{5}$

Therefore, the modulus value of $z=-1-2i$ is $\left| z \right|=\sqrt{5}$.

Note: Modulus of the complex number is the distance of the point on the argand plane representing the complex number z from the origin. The modulus and argument are fairly simple to calculate using trigonometry. Use of Pythagoras’ formula gives the formula. When the complex number lies in the first quadrant, calculation of the modulus and argument is straightforward. For complex numbers outside the first quadrant, we need to be a little bit more careful.