Question

How do you simplify \[\left| {19i} \right|\]?

Answer 1

Hint: We are given a complex number here and asked to find its modulus. For this, we have to use the formula for finding the modulus of complex numbers. Therefore, we will first identify the real and imaginary part of the given complex number and then by using them, we will determine our answer.

Formula used:
If we have the complex number $z = x + yi$, where, $x$ is the real part of the complex number and $y$is the imaginary part of the complex number, then its modulus is given by $\left| z \right| = \sqrt {{x^2} + {y^2}} $.

Complete step by step answer:
We are given the complex number $z = 19i$.
We can rewrite it as $z = 0 + 19i$.
Now, we can identify the real and imaginary part of the given number.
We can clearly see that the given function has no real part which means that $x = 0$.
And the imaginary part is $y = 19$.
Now we will use the formula $\left| z \right| = \sqrt {{x^2} + {y^2}} $to find our final answer.
$\left| z \right| = \sqrt {{x^2} + {y^2}} $
Putting the values of real and imaginary part, we will get
\[\left| {19i} \right| = \sqrt {{0^2} + {{19}^2}} = 19\]
Thus, by simplifying \[\left| {19i} \right|\], we get our final answer as \[19\].

Note: In this question, we have determined the modulus of the given complex number. Modulus of the complex number is the distance of the point on the plane representing the complex number from the origin. It is also important to keep in mind that the modulus of any complex number is always greater than zero.