Length of an Interval¶

Definition¶

For a bounded interval $I$ represented by $[a, b]\text{,}$ $(a, b)\text{,}$ $[a, b)\text{,}$ or $(a, b]$ where $a < b$ are real numbers, we define the length of $I\text{,}$ denoted by $|I|\text{,}$ as $|I|=b-a\text{.}$ If $I$ is a singleton set or the empty set, then $|I|=0\text{.}$

Interpretation¶

The length of an interval measures how much of the real line it covers. For a bounded interval it is the distance $b-a$ between the endpoints, so it depends only on where the interval starts and ends.

The half-open interval from 1 to 4 on a number line with an arrow marking its length 4 - 1 = 3, the same as for the closed and open intervals with these endpoints.

The figure shows the half-open interval $[1,4)$ : the filled dot at $1$ belongs to it, the open dot at $4$ does not, and the arrow above marks its length $4-1=3$ $3$ . The intervals $[1,4]$ and $(1,4)$ have the same length: $[1,4]$ is $[1,4)$ together with the single point $4$ , and a singleton has length $0$ .

The notation $|I|$ echoes the absolute value: since $a<b$ $b$ , $|I|=b-a=|b-a|$ $a ∣$ , the distance from $a$ to $b$ . Lengths are the widths of the rectangles in a Riemann sum, and over a partition of $[a,b]$ the lengths of the subintervals add up to $b-a$ $a$ .

A common slip is to count points instead of measuring: $[1,4]$ contains the four integers $1,2,3,4$ , but its length is $3$ , and it contains infinitely many real numbers.

Key Results¶

The length depends only on the endpoints.
Singletons and empty sets have length zero.

Examples¶

The interval $(2,5)$ has length $3\text{.}$
The interval $[a,b]$ has length $b-a\text{.}$

Prerequisites¶

Sources¶

Math I: Calculus · Lecture 10

Aliases: interval length

Last updated: 20 October 2026 03:02