What Is Interval Notation? A Complete Definition

Interval notation is a compact way to describe a set of real numbers that lie between two endpoints—or extend forever in one direction. Instead of writing long inequalities or listing numbers, mathematicians use brackets and parentheses to show exactly which numbers are included. For example, the set of all numbers greater than 3 but less than 7 is written as (3, 7) in interval notation. This notation is a cornerstone of algebra, calculus, and many other areas of mathematics.

Origins and Why Interval Notation Matters

Interval notation was developed in the 19th and 20th centuries as mathematicians sought clearer ways to express the continuous nature of real numbers. It replaced clunky descriptions like “all real numbers between a and b, including a but not b” with a simple symbol: [a, b).

Why does it matter? Interval notation makes it easy to:

• Write solutions to inequalities quickly.
• Talk about domains and ranges of functions in calculus.
• Combine intervals using set operations like union and intersection.
• Visualize ranges on a number line.

If you need a step-by-step guide on writing intervals, see our How to Write Interval Notation: Step-by-Step Guide 2026.

How Interval Notation Is Used

Interval notation has four basic forms, depending on whether the endpoints are included (closed) or excluded (open):

• Open interval: (a, b) means all numbers where a < x < b. Both endpoints are excluded.
• Closed interval: [a, b] means a ≤ x ≤ b. Both endpoints are included.
• Half‑open (or half‑closed) interval: [a, b) includes a but not b; (a, b] includes b but not a.
• Unbounded intervals: Use infinity (∞ or −∞) with a parenthesis because infinity is a concept, not a number. For example, (−∞, 5] means all numbers less than or equal to 5.

To see what different interval patterns mean and how to read them, check out our page on Interval Notation Ranges: What Different Intervals Mean (2026).

Worked Example: Solving an Inequality

Let’s solve 2x – 3 < 7 and write the answer in interval notation.

1. Add 3 to both sides: 2x < 10
2. Divide by 2: x < 5

The solution is all numbers less than 5. In interval notation, that’s (−∞, 5). The parenthesis at 5 means 5 is not included, and the parenthesis at −∞ indicates that the interval extends forever leftward. On a number line, we draw an open circle at 5 and shade everything to the left with an arrow.

Now try a compound inequality: −1 ≤ x ≤ 3. Both endpoints are included, so the interval is [−1, 3].

Common Misconceptions

Because interval notation is so concise, beginners often make these mistakes:

• Confusing parentheses and brackets. Remember: a parenthesis means “not including” the endpoint; a bracket means “including” it. For x > 2, write (2, ∞), not [2, ∞).
• Thinking infinity can be included. Infinity is not a real number, so it always gets a parenthesis. [−∞, 5] is never correct.
• Writing intervals in the wrong order. The smaller number always comes first: (5, 2) is meaningless. Always list left endpoint then right endpoint.
• Forgetting that interval notation only works for continuous ranges of real numbers. If you want to describe a set like {1, 2, 3}, you need set‑builder or roster notation.
• Mixing operations incorrectly. When combining intervals, use ∪ (union) for “or” and ∩ (intersection) for “and”. For union, it’s often helpful to draw a number line.

If you have more questions, our 10 Frequently Asked Questions About Interval Notation (2026) page provides clear answers to common doubts.

Final Thoughts

Interval notation is a simple yet powerful tool for describing sets of real numbers. Once you learn the meaning of parentheses and brackets, you can quickly write and interpret inequalities, domains, and ranges. Practice with a few examples, and soon it will become second nature. And if you ever need to convert interval notation to another format or perform operations like union or intersection, try our Interval Notation Calculator for quick, accurate results.