What Is Interval Notation? A Beginner’s Guide

Interval notation is a shorthand way to describe a range of numbers without writing out long inequalities. Instead of listing every number in a set, we use a compact format that shows the smallest and largest values in the range.

Interval notation looks like ordered pairs, but they don’t represent a single point. Instead, they describe a set of numbers between two endpoints.

In this post, you’ll learn what interval notation is, its benefits, how to write it for different cases, and how to represent intervals on a number line. You’ll also see examples of unions and intersections of intervals and how they relate to inequalities.

How to Write Interval Notation

Interval notation always has three parts:

1. A left endpoint (smallest number in the range).
2. A right endpoint (largest number in the range).
3. Brackets or parentheses that show whether the interval includes the endpoints.

The format looks like this:

(left endpoint, right endpoint)

The symbols tell us if the endpoints are included or excluded:

• Square brackets [ ] mean the endpoint is included (like ≤ or ≥).
• Parentheses ( ) mean the endpoint is not included (like < or >).

Closed Intervals: Includes Both Endpoints

If the interval includes both endpoints, use square brackets in the interval notation:

[a, b]

This notation means the interval includes every number between a and b, as well as a and b themselves.

Example

Write a closed interval from -3 to 5 as:

[−3, 5]

Here’s what that looks like on a number line:

Open Interval Notation: Excludes Both Endpoints

If the interval excludes both endpoints, use parentheses:

(a, b)

This notation means the interval includes every number between a and b, but not a or b themselves.

Example

Write an open interval from -3 to 5 as:

(−3, 5)

Here’s the number line for this interval:

Half-Open (Mixed) Interval Notation

Interval notation can include one endpoint but not the other. In these cases, we mix brackets and parentheses:

• Left closed, right open: [a, b)
• Left open, right closed: (a, b]

Example

The interval [−3, 5) includes -3 but not 5.

Here’s how that looks:

Intervals Extending to Infinity

Some intervals don’t have an upper or lower bound. In these cases, we use infinity symbols (∞ or −∞).

• −∞ indicates there’s no lower bound.
• ∞ indicates there’s no upper bound.
• Intervals never include infinity. Hence, we always use parentheses with ∞ or −∞.

Examples

1. The interval (−∞, 5] includes all numbers less than or equal to 5.
2. The interval [3, ∞) contains all numbers greater than or equal to 3.

Why Use Interval Notation?

Interval notation makes it easy to write and read ranges of numbers. Instead of using long inequalities like:

−3 ≤ x ≤ 5

We can simply write:

[−3, 5]

This notation is widely used in algebra, calculus, and data analysis. In statistics, we use it to represent Confidence Intervals.

Union and Intersection Interval Notation

Sometimes, we need to combine or find the overlap between two intervals. We use special symbols to represent these operations:

• Union ( ∪ ): The union of two intervals includes all numbers in either interval.
• Intersection ( ∩ ): The intersection of two intervals includes only the numbers that both intervals share.

Union Example ( ∪ )

If we take the intervals [-6, -2] and [-4, 2], their union includes all numbers covered by either interval:

[−6, −2] ∪ [−4, 2] = [−6, 2]

Here’s the number line representation:

Intersection Example ( ∩ )

The intersection includes only the numbers common to both intervals. In this case, the overlap is [-4, -2]:

[−6, −2] ∩ [−4, 2] = [−4, −2]

Here’s the number line representation:

Now that you know how to write interval notation, try practicing with a few examples! Learn about scientific notation, which is a shortcut way to represent numbers.