Question

How do you find the absolute value of $1 + 3i?$

Answer 1

Hint : In general the absolute value of any number is its distance from zero. In the case of a complex number,$a + bi$, its absolute value will be the distance from the zero complex number, i.e. $0 + 0i$to the number $a + bi$.

Complete step-by-step answer :
We know that the distance between any points can be found out with the help of distance formula which is given by,
$d = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2}}$
Where, ( ${x_1},{y_1}$ ) and ( ${x_2},{y_2}$ ) are the coordinates of two points
And as per the question if we have to calculate the absolute value of a point, then we have to calculate the distance of the point from the origin which will be given by,
$$ $d = \sqrt {{{({x_2} - 0)}^2} + {{({y_2} - 0)}^2}}$
Where ( ${x_2},{y_2}$ ) are the coordinates of the given point and cane be written as $d = \sqrt {{{({x_2})}^2} + {{({y_2})}^2}}$
Now, for the absolute value of $1 + 3i$ , our equation will become
$d = \sqrt {{{(1)}^2} + {{(3)}^2}}$ , As the coordinate is ( $1,3$ )
$d = \sqrt {1 + 9}$
Solving this will give us,
$d = \sqrt {10}$
And $d = \sqrt {10}$ cannot be simplified further, so we can say that the absolute value of $1 + 3i$ is $\sqrt {10}$
So, the correct answer is “ $\sqrt {10}$ ”.

Note : If you have learned how to plot complex numbers on a coordinate system you can plot the two points and see that the distance between $a + bi$ and $0 + 0i$. Here we can see that the origin itself is expressed in the form of a complex number.
Additional information: A complex number is a number that can be expressed in the form $a + bi$, where a and b are real numbers, and i is a symbol called the imaginary unit, satisfying the equation $i^2 = -1$. Because no "real" number satisfies this equation, it is called an imaginary number.