Interval Notation
Quickly convert number ranges with open or closed endpoints into standard bracket notation [a,b], (a,b), and half-open forms. Free interval converter for algebra and precalculus.
About This Calculator
The Interval Notation Calculator converts number ranges with specified endpoint types into proper interval notation. Enter a lower bound, upper bound, and choose whether each endpoint is open (exclusive, using parentheses) or closed (inclusive, using brackets). The calculator immediately generates the correct notation such as [a, b], (a, b], [a, b), or (a, b).
Interval notation is a compact way to describe continuous sets of real numbers. It is widely used in algebra for inequality solutions, in calculus for domains and ranges, in statistics for confidence intervals, and in physics for measurement ranges. Understanding the difference between open and closed intervals is crucial for correctly interpreting mathematical statements and solving inequality problems.
This tool is designed for students learning interval notation for the first time, teachers demonstrating concepts in class, and professionals who need quick conversions between range descriptions and formal notation. Every generated result is accompanied by the exact notation string that can be copied for use in homework, exams, or mathematical writing.
Frequently Asked Questions
What is interval notation?
Interval notation is a mathematical notation used to represent a set of real numbers between two endpoints. A square bracket [ or ] indicates the endpoint is included (closed interval), while a parenthesis ( or ) indicates the endpoint is excluded (open interval). For example, [2, 5] includes 2 and 5, while (2, 5) excludes both.
How do I use the interval notation calculator?
Enter the lower and upper bounds of your interval in the input fields. Then choose whether each endpoint is open (exclusive) or closed (inclusive) using the dropdown menus. The calculator generates the proper interval notation such as [a, b], (a, b], [a, b), or (a, b) depending on your selections.
What is the difference between open and closed intervals?
A closed interval [a, b] includes both endpoints a and b. An open interval (a, b) excludes both endpoints. A half-open interval [a, b) includes a but excludes b, while (a, b] excludes a but includes b. In set-builder notation, [a, b] corresponds to { x ∈ ℝ | a ≤ x ≤ b } and (a, b) corresponds to { x ∈ ℝ | a < x < b }.
When would I use interval notation?
Interval notation is commonly used in algebra, precalculus, and calculus to express domains, ranges, solution sets of inequalities, and continuous intervals. It is also used in statistics for confidence intervals, in physics for ranges of values, and in computer science for specifying valid input ranges.
Is this interval notation tool free?
Yes, all calculators on Calculy are completely free to use with no registration or hidden charges. You can bookmark or share your interval notation results via the URL.
How do I convert an inequality to interval notation?
For an inequality like 2 < x <= 5, the lower bound 2 is exclusive (open) and the upper bound 5 is inclusive (closed), so the interval notation is (2, 5]. For x >= 3 with no upper bound, use [3, ∞). For x < 2 with no lower bound, use (-∞, 2). Always use parentheses for infinity since infinity is not a specific value.
How do I write all real numbers in interval notation?
The entire set of real numbers in interval notation is written as (-∞, ∞). This represents all numbers from negative infinity to positive infinity. Parentheses are used for both infinities because infinity is not a specific, reachable endpoint.
What is the connection between interval notation and set-builder notation?
Interval notation is a compact shorthand for set-builder notation. For example, the set { x ∈ ℝ | a <= x <= b } is written as [a, b] in interval notation. Similarly, { x ∈ ℝ | a < x < b } becomes (a, b). Interval notation is faster to write and easier to read for continuous ranges.