1.3. Absolute values

The absolute value is denoted by vertical bars and indicate the positive number corresponding to the numeric value of an expression.

Let \(a\) be a positive number, \(a>0\)

Then we have:

(1.12)\[\begin{split}\begin{aligned} |a|=a\\ |-a|=a \end{aligned}\end{split}\]

For inequalities, we have two notations that mean the same:

(1.13)\[\begin{split}\begin{aligned} |a|\leq1\\ -1\leq a\leq 1 \end{aligned}\end{split}\]

Inequalities of sums and differences:

(1.14)\[\begin{split}\begin{aligned} |a+b|&=\sqrt{a^2+b^2}\\ |a-b|&=\sqrt{a^2+(-b)^2}\\ \end{aligned}\end{split}\]

(1.15)\[\left|\frac{a}{b}\right|=\frac{|a|}{|b|}\]

Combining the equations above gives the following:

(1.16)\[\begin{split}\begin{aligned} \left|\frac{a+b}{a-b}\right|&=\frac{|a+b|}{|a-b|}\\ &=\frac{\sqrt{a^2+b^2}}{\sqrt{a^2+(-b)^2}}\\ &=1 \end{aligned}\end{split}\]

The triangle inequality can be useful in some stabilty analyses:

(1.17)\[|x+y|\leq|x|+|y|\]