Interval Notation Calculator
Express a range of numbers using interval notation. Choose your bounds and whether they're included (brackets) or excluded (parentheses), and see both interval and inequality forms.
Understanding Interval Notation
Interval notation provides a compact way to describe sets of real numbers. Instead of writing "all x such that 2 ≤ x < 5," you write [2, 5). The brackets and parentheses tell you instantly whether endpoints are included.
Square brackets [ ] mean "closed" or "inclusive." The endpoint is part of the interval. Parentheses ( ) mean "open" or "exclusive." The endpoint is not included. So [2, 5) includes 2, excludes 5, and contains every number in between.
For unbounded intervals, use infinity symbols. (−∞, 3] means all numbers less than or equal to 3. [4, ∞) means all numbers greater than or equal to 4. Since infinity isn't a number you can reach, always use parentheses with ∞. The notation (−∞, ∞) represents all real numbers.
Converting Between Notations
Interval notation and inequality notation express the same idea differently. [2, 5] as an inequality is 2 ≤ x ≤ 5. (2, 5) becomes 2 < x < 5. [2, 5) is 2 ≤ x < 5, and (2, 5] is 2 < x ≤ 5. Each pair of symbols (bracket vs. parenthesis and ≤ vs. <) corresponds directly.
To convert from inequality to interval, identify the bounds and check whether they use ≤ or <. Use [ or ] for ≤, ( or ) for <. For example, −1 < x ≤ 4 becomes (−1, 4]. Practice reading both notations fluently—they're used interchangeably in algebra, calculus, and beyond.
Unions of intervals use ∪. If x ≤ 2 or x > 5, write (−∞, 2] ∪ (5, ∞). This represents two separate pieces of the number line. Interval notation cleanly handles these disjoint sets.
Uses of Interval Notation in Mathematics
Interval notation appears throughout math. In calculus, domains and ranges are often intervals. The domain of √x is [0, ∞) because you can't take the square root of a negative number. The range of x² is [0, ∞) because squares are never negative.
When solving inequalities, the solution set is naturally expressed in interval notation. Solve 2x − 3 < 7, get x < 5, write (−∞, 5). When graphing, intervals show where a function is positive, negative, increasing, or decreasing.
In statistics, confidence intervals are written in interval notation. A 95% confidence interval [3.2, 4.8] means the true parameter likely lies between 3.2 and 4.8. Interval notation is concise, universal, and essential for communicating ranges clearly and precisely.
Frequently Asked Questions
What is interval notation?
Interval notation is a shorthand for describing a range of numbers. Square brackets [ ] mean the endpoint is included, parentheses ( ) mean it's excluded. For example, [2, 5) includes 2 but not 5.
What's the difference between [ ] and ( )?
Square brackets [ ] indicate a closed endpoint (the number is part of the interval). Parentheses ( ) indicate an open endpoint (the number is not included). [a, b] includes both a and b, (a, b) includes neither.
How do you write an interval that goes to infinity?
Use (−∞, b] or [a, ∞). Infinity is never included, so always use parentheses with ∞ or −∞. For example, (−∞, 3] means all numbers less than or equal to 3.
Can an interval have one endpoint closed and the other open?
Yes. [a, b) includes a but not b. (a, b] includes b but not a. These are called half-open or half-closed intervals.
How do you convert interval notation to an inequality?
[a, b] becomes a ≤ x ≤ b. (a, b) becomes a < x < b. [a, b) becomes a ≤ x < b. (a, b] becomes a < x ≤ b. The inequality shows the same range with comparison symbols.