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.