

Absolute Value of a Complex NumberHow to calculate a complex number's absolute value
The absolute value of a real number is its distance from 0 on the number line. A complex number's absolute value is also a measure of its distance from zero. However, instead of measuring this distance on the number line, a complex number's absolute value is measured from zero on the complex number plane.
In general
Illustrated Example
Practice Problems
To find the absolute value of the complex number, 3 + 4i, we find the distance from zero to that number on the plane.
Calculate 5+ 12i
Problem 1) What is $$ 6 + 8i $$ ? Problem 2) calculate the value of $$ 3 + 2i$$
$$
3 + 2i = \sqrt{3^2 + (2)^2}
\\
= \sqrt{9 + 4}
\\
= \sqrt{13}
$$
Problem 3) Calculate the value of $$ 3  2i $$
The only difference between this one and problem 2 is that the $$2i$$ has now become $$2i$$.
However, since we are squaring that term, the negative sign has no effect and you end up with the exact same answer$$ 3  2i = \sqrt{3^2 + (\color{Red}{}2)^2} \\ = \sqrt{9 + \color{Red}{4}} \\ = \sqrt{13} $$ Problem 4) calculate the value of $$  5i  3$$
$$
5 3i = \sqrt{(3)^2 + (5)^2}
\\
= \sqrt{9 + 25}
\\
= \sqrt{34}
$$
Problem 5) What is $$  x  ci$$
$$
 x  ci = \sqrt{(x)^2 + (c)^2}
\\
= \sqrt{x^2 + c^2}
$$
