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Math Calculator Interval Notation

Reviewed by Calculator Editorial Team

Interval notation is a concise way to represent sets of real numbers. It's commonly used in mathematics, particularly in calculus and algebra, to describe ranges of values. This guide explains how to read and write interval notation, provides examples, and includes a calculator to help you work with intervals.

What is Interval Notation?

Interval notation is a method for representing a set of real numbers that lie between two endpoints. It's often used to describe the domain and range of functions, the solution set of inequalities, and other mathematical concepts involving ranges of numbers.

The two main types of intervals are open and closed. A closed interval includes its endpoints, while an open interval does not. There are also half-open intervals that include one endpoint but not the other.

Interval notation is particularly useful in calculus for describing the domain of functions and the intervals over which integrals are evaluated.

How to Use Interval Notation

To write an interval in interval notation, you use square brackets [ ] for closed intervals and parentheses ( ) for open intervals. The endpoints are written in order from smallest to largest, separated by a comma.

Basic Interval Notation Rules

  • [a, b] - Closed interval from a to b, including both endpoints
  • (a, b) - Open interval from a to b, excluding both endpoints
  • [a, b) - Half-open interval from a to b, including a but excluding b
  • (a, b] - Half-open interval from a to b, excluding a but including b

For infinite intervals, you can use infinity symbols:

  • (a, ∞) - All numbers greater than a
  • (-∞, b) - All numbers less than b
  • (-∞, ∞) - All real numbers
Example: The interval from 2 to 5, including 2 but not 5, is written as [2, 5).

Common Interval Notation Examples

Here are some common examples of interval notation and their meanings:

Interval Notation Description Number Line Representation
[3, 7] All real numbers from 3 to 7, including 3 and 7 ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••