How do you find the absolute value of 10-7i?

Dec 25, 2016

$\left\mid \left(10 - 7 i\right) \right\mid = \sqrt{149} \approx 12.21$.

Explanation:

The absolute value of a number is better thought of as the distance that number is from the origin. For numbers in $\mathbb{R}$, this definition simplifies to "just take off the negative sign if there is one". But for complex numbers and multi-dimensional vectors, we have to do a little more calculation.

Since we use a 2-D plane to illustrate complex numbers, we will use the formula for the 2-D distance $d$ between two points $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$:

$d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$

This can be simplified further for the use of absolute values, since the first point will always be the origin $\left(0 , 0\right)$ and only the second point will be allowed to vary. Thus:

$d = \sqrt{{\left({x}_{2} - 0\right)}^{2} + {\left({y}_{2} - 0\right)}^{2}}$
$\textcolor{w h i t e}{d} = \sqrt{{x}_{2}^{2} + {y}_{2}^{2}}$

or simply

abs[(x, y)$= \sqrt{{x}^{2} + {y}^{2}}$

A complex number $a + b i$ is drawn in the complex plane exactly the same way an $\left(x , y\right)$ point is plotted in the $x \text{-} y$ plane: $a$ (or $x$) denotes where we are on the real (or horizontal) axis, and $b$ (or $y$) denotes where we are on the imaginary (or vertical) axis.

That means the distance that a number $a + b i$ is from the origin (i.e. its absolute value) is calculated as

$\left\mid \left(a + b i\right) \right\mid = \sqrt{{a}^{2} + {b}^{2}}$

For the complex number $10 - 7 i$, this is calculated to be

$\left\mid \left(10 - 7 i\right) \right\mid = \sqrt{{10}^{2} + {\left(\text{-7}\right)}^{2}}$
$\textcolor{w h i t e}{\left\mid \left(10 - 7 i\right) \right\mid} = \sqrt{100 + 49}$
$\textcolor{w h i t e}{\left\mid \left(10 - 7 i\right) \right\mid} = \sqrt{149} \text{ } \approx 12.21$.