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 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¶

• Endpoint and Bounded Interval
• Singleton Set
• Empty Set

Related Concepts¶

• Riemann Sum
• Interval

Sources¶

• Math I: Calculus · Lecture 10

Aliases: interval length

Last updated: 20 October 2026 03:02