Interval notation is one of the most efficient tools in mathematics for describing continuous sets of numbers. Whether you are working on algebra, precalculus, or calculus, knowing how to write interval notation allows you to express domains, ranges, and solution sets with clarity and precision. Instead of writing out long inequalities or listing every possible value, interval notation uses brackets and parentheses to communicate exactly where a set begins and ends. This guide walks you through everything you need to know about interval notation, from basic symbols to advanced combinations, so you can confidently apply it in your coursework and beyond.
What Is Interval Notation?
Interval notation is a shorthand method used to describe a set of real numbers that fall between two endpoints. That's why it is commonly used in algebra and calculus to define the domain and range of functions, the solution sets of inequalities, and intervals on the number line. While inequality notation uses symbols like <, >, ≤, and ≥, and set-builder notation uses braces and variables, interval notation relies on a compact system of brackets and parentheses The details matter here..
The core idea is simple: an interval represents all the numbers between a starting point and an ending point. Whether those endpoints are included in the set determines which symbols you use.
The Building Blocks: Parentheses vs. Brackets
To write interval notation correctly, you must understand the difference between parentheses and brackets.
- Parentheses ( ) indicate that an endpoint is excluded from the interval. This corresponds to strict inequalities such as < or >. Take this: the interval (2, 5) includes every number greater than 2 and less than 5, but it does not include 2 or 5 themselves.
- Brackets [ ] indicate that an endpoint is included in the interval. This corresponds to non-strict inequalities such as ≤ or ≥. To give you an idea, [2, 5] includes every number from 2 to 5, including 2 and 5.
When infinity is involved, you always use a parenthesis because infinity is not a specific number that can be reached or included. Take this: all numbers greater than 3 are written as (3, ∞), never [3, ∞].
