Determine the absolute value of each of the following complex numbers:a. z = 5 + 12i b. 2 - 3i.

Determine the absolute value of each of the following complex numbers:
a. $$z = 5 + 12i$$
b. $$2 - 3i.$$

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Usamah Prosser

a. The absolute value of a complex number is its distance from the origin. If $$z=a+ bi$$, then $$\displaystyle{\left|{z}\right|}={\left|{a}+{b}{i}\right|}=\sqrt{{{a}^{{2}}+{b}^{{2}}}}$$
The absolute value of a complex number $$z=5+12i$$ is,
$$\displaystyle{\left|{z}\right|}={\left|{5}+{12}{i}\right|}=\sqrt{{{5}^{{2}}+{12}^{{2}}}}=\sqrt{{{25}+{144}}}=\sqrt{{169}}=\sqrt{{13}}$$
b. The absolute value of a complex number $$z=2−3i$$ is,
$$\displaystyle{\left|{Z}\right|}={\left|{2}-{3}{i}\right|}=\sqrt{{{2}^{{2}}+{\left(-{3}\right)}^{{2}}}}=\sqrt{{{4}+{9}}}=\sqrt{{13}}$$