Interval notation

Interval notation is a notation used to denote all of the numbers between a given set of numbers (an interval). For example, "all of the integers between 12 and 16 including 12 and 16" would include the numbers 12, 13, 14, 15, and 16. Even with such a small range of numbers, it is already cumbersome to list them. Interval notation, as well as a couple other methods, allow us to more efficiently denote intervals.

To use interval notation we need to first understand some of the commonly used symbols:

Open and closed intervals

A closed interval is an interval that includes the values on the end. The example above would be denoted as

[12, 16]

since both 12 and 16 are included. An open interval is one in which the values on the end are not included, and would be denoted as:

(12, 16)

It is also possible to have a combination of the two. If 12 were included, but 16 were not, we can denote it in interval notation as follows:

[12, 16)

The above are examples of finite intervals. It is also possible to have infinite intervals. Both negative infinity and positive infinity are considered open since it is not really possible to quantify infinity. All real numbers greater than or equal to 12 can be denoted in interval notation as:

[12, ∞)

Union and intersection

Unions and intersections are used when dealing with two or more intervals. For example, the set of all real numbers excluding 1 can be denoted using a union of two sets:

(-∞, 1) ∪ (1, ∞)

Intersection is used to denote the interval over which two sets overlap.

(-∞, 4] ∩ [2, 22]

The above reads as "the intersection between the sets (-∞, 4] and [2, 22]," which is [2, 4].

Number lines and inequalities

Intervals can also be denoted using number lines and inequalities.

Open and closed intervals

Closed intervals on number lines are denoted using filled-in circles at the endpoints; open intervals use circles that are not filled in for the endpoints:


closed interval


open interval


To express the same intervals above using inequalities:

closed interval

0 ≤ x ≤ 4

open interval

0 < x < 4