Definition and properties of absolute value

Absolute value of a number represents its distance from zero on a number line. Since it represents a distance it is always non-negative. The smallest possible absolute value is that of zero, it equals zero and the absolute value of every other number, even the negative ones, is a positive number. It ignores in which direction from zero the number lies, it only matters how far it is. This distance is also called a module of a number.

For example, looking at the points: $\ A (4)$, $\ B(2)$, $C(-2)$ marked on the number line one can observe their distance from the origin. More generally, the absolute value of the difference of two real numbers is the distance between them.

Absolute value of a number 2023

We can see that the distance between points $A$ and $O$ is 4, so the absolute value of the number 4 equals 4. The distance both from the points $B$ and $C$ to $O$ is 2, so their absolute value equals 2. Notice how the absolute value of positive numbers is exactly that number, and the absolute value of negative numbers is the positive version of that number.

The same applies to all real numbers, rational, irrational, integers and natural numbers.

Whether $x$ is positive or negative, its absolute value will always be positive.

$|x| = 0$ if and only if $ x = 0$.

$|x| \ge 0$ for every $ x \in \mathbb{R}$.

Absolute value has these properties:

  1. $|a \cdot b| = |a| \cdot |b|$ for every $ a, b \in \mathbb{R}$
  2. $\displaystyle{\left| \frac{a}{b}\right| = \frac{|a|}{|b|}}$, $ b \not= 0$ for every $ a \in \mathbb{R}$, $ b \in \mathbb{R} \setminus \{ 0 \}$
  3. $|a + b| \le |a| + |b|$, for every $ a, b \in \mathbb{R}$; called the triangle inequality.


Since the square root symbol represents the unique positive square root (when applied to a positive number), it follows that ${\displaystyle |x|={\sqrt {x^{2}}}}$ and $|x| = \sqrt{x^2}$ are equivalent to the definition above, and may be used as an alternative definition of the absolute value of real numbers.