4.2 The absolute value

For a real number $x$ we set

\begin{equation}\label{eq-absh}\tag{4.4} |x| \href{This means: for \(x\) greater or equal to \(0\), then \(|x|\) is equal to \(x\), otherwise for \(x\) smaller thant, then \(|x|\) is equal to \(-x\).}{\class{mathpopup bg-info-subtle}{:=}} \begin{cases} x, & x \geq 0, \\ -x, & x < 0. \end{cases} \end{equation}

The function $|x|$ is called absolute value. The notation used in equation (4.4) should be interpreted as follows: For given $x$ grater or equal to $0$, then $|x|$ is defined as $x$ and in the case $x$ is negative, $|x|$ is defined as $-x$. The graph of the function absolute value is plotted in picture 4.1.

Figure 4.1: Graph of the absolute value.

Now let us summarize a few basic properties of the absolute value. Its The domain of the absolute values is the whole set of real numbers, i.e. $D_{|x|} = \mathbb{R}$. The image set of the absolute value is given by the set of all non-negative real umbers, thus $H_{|x|} = [0, +\infty)$. Indeed, from definition (4.4) we obtain the inequality $|x| \geq 0$ for every $x$ and on the other hand for any $y \geq 0$ it holds $|y| = y$. Furthermore, directly from definition (4.4) it clearly follows that for every real $x$ and $y$ it holds

\begin{equation}\label{eq-absminus}\tag{4.5} |-x| = |x|, \quad x \leq |x|, \quad -x \leq |x| \end{equation}

and (think about it!)

\begin{equation*} |x\cdot y| = |x| \cdot |y|, \quad \left| \frac{x}{y} \right| = \frac{|x|}{|y|} \quad \text{for} \ y\neq 0. \end{equation*}

An important property of absolute value is the so-called triangular inequality.

Theorem 4.1 (triangular inequality)

For every real $x$ and $y$ the following inequality holds

\begin{equation*} |x + y| \href{The value on the left is smaller or than the value on the right.}{\class{mathpopup bg-info-subtle}{\leq}} |x| + |y|. \end{equation*}

Show proof

Consider any real $x$ and $y$. We have

  • if $x+y \geq 0$, then $|x+y| = x+y \leq |x| + |y|$,

  • if $x+y < 0$, then $|x+y| = -(x+y) = -x -y \leq |x| + |y|$.

Obviously, for every real $z$ it applies $z \leq |z|$.


Question 4.1

Prove or refute the claim: for every $x\in\mathbb{R}$ it holds $\sqrt{x^2} = x$.

Show answer

The statement is not true, consider any negative number $x$. For every real number $x$ it holds $\sqrt{x^2} = |x|$.