What is Interval Notation?

Interval notation is how we write out a set of numbers in a short and organized way. It is the “notation” for an “interval.” The notation is given as:

• Closed and closed: [a, b]
• Closed and open: [a, b)
• Open and closed: (a, b]
• Open and open: (a, b)

Where a is the lower limit of the interval and b is the upper limit of the interval. A square bracket means that limit is included in the interval. A parenthesis means that limit is not included in the interval.

An interval is a set of real numbers (meaning there are no imaginary numbers) that includes all numbers between the two limit numbers of the set and sometimes also includes the limit numbers. For example, the interval (2, 5) is a set of every single real number between 2 and 5. This interval can also be written in inequality notation as 2 < x < 5.

How to Write Interval Notation

Let’s write out the notation for values of x on the interval between 2 and 5, such that 2 < x < 5. Since the limit numbers are not included in the interval, the limits are considered open and open. The interval will be notated as (2, 5).

If a limit is included in the interval, it is called closed, and we use a square bracket for that endpoint in our notation. The notation for the interval 2 ≤ x ≤ 5 is [2, 5]. If one of the end points is included but the other isn’t, we use a combination of a parenthesis and a square bracket. The interval 2 < x ≤ 5 is called open and closed and notated as (2, 5]. Likewise, our notation for the interval 2 ≤ x <5 will be called closed and open and notated as [2, 5).

Interval Notation Example Problems

Let's go through a couple of example problems together to reinforce our understanding of interval notation.

Example Problem 1

What is the interval notation for the set of numbers that is between 7 and 19 including, 7?

Solution:

1. This interval is closed and open because the lower limit is included and the upper limit is not included.
2. The interval notation for the given set of numbers is [7, 19).

Example Problem 2

What is the interval notation for the set of numbers that is between 5 and -4?

Solution:

1. This interval is open and open since both limits are not included.
2. The interval notation for the given set of numbers is (-4, 5).