Absolute Value¶

Definition¶

Let $x$ be a real number. The absolute value $|x|$ of $x$ is defined as follows:

Let $x$ and $y$ be real numbers. The quantity $|x - y|$ is called the distance between $x$ and $y$ and is thus denoted by $d(x,y)\text{,}$ i.e., $d( x, y) = | x - y|\text{.}$

Interpretation¶

The absolute value measures how far a real number lies from zero on the real line, whichever side it lies on. The case definition only removes the sign: $|x|$ keeps $x$ when $x\ge0$ and switches to $-x$ when $x<0$$0$.

The graph is a V with its corner at the origin: the left branch is the line $y=-x$ and the right branch is $y=x$$x$. The marked point shows $|-2|=2$$2$: the number $-2$ lies at distance $2$ from $0$, just as $2$ does.

Applied to a difference, $|x-y|$ is the distance between $x$ and $y$, and $|x-y|=|y-x|$$x|$, so the order does not matter. This is how inequalities with absolute values should be read: $|x-a|<\delta$ describes the points within distance $\delta$ of $a$, that is, the interval $(a-\delta,a+\delta)$$\delta )$. This reading comes back later in the course, in the definitions of limits and continuity, and the same idea of distance is later extended to vectors and to more general spaces.

A common slip is to read $-x$ as "a negative number". For $x<0$ the number $-x$ is positive, which is exactly why $|x|=-x$ there. For the same reason $\sqrt{x^2}=|x|$$|x|$, not $x$.

Key Results¶

• $|x| \ge 0$ for all $x \in \mathbb{R}\text{.}$
• $|x| = 0$ if and only if $x = 0\text{.}$
• $|x - y|$ defines a distance function on $\mathbb{R}\text{.}$

Examples¶

• $|3| = 3\text{,}$ $|-5| = 5$
• $|x| = |-x|$

Prerequisites¶

• Real Number

Related Concepts¶

• Inequality
• Complex Absolute Value
• Trichotomy of Real Numbers
• Negation
• Euclidean Norm
• Chebyshev Inequality

Sources¶

• Math I: Calculus · Lecture 2

Aliases: modulus Keywords: distance on the real line Symbols: delta, element of, real numbers, square root

Last updated: 20 October 2026 03:02