Interval Notaion Calculator

Interval notation is a concise way to represent sets of real numbers. This calculator helps you understand, convert, and work with interval notation in mathematics.

What is Interval Notation?

Interval notation is a method of representing a set of real numbers using parentheses and brackets. It's commonly used in calculus, algebra, and other branches of mathematics to describe ranges of values.

Key Symbols in Interval Notation

( ) - Parentheses indicate that the endpoint is not included in the interval
[ ] - Brackets indicate that the endpoint is included in the interval
(∞, a) - All numbers less than a
(a, ∞) - All numbers greater than a
(-∞, ∞) - All real numbers

Interval notation provides a compact way to represent continuous ranges of numbers, which is particularly useful when dealing with functions, inequalities, and limits.

How to Use This Calculator

Our interval notation calculator allows you to:

Enter interval notation expressions
Convert between interval notation and inequality notation
Visualize intervals on a number line
Solve interval problems involving unions and intersections

For best results, enter intervals in standard notation format. For example: [1, 5) or (-∞, 0].

Interval Notation Examples

Here are some common interval notation examples and their meanings:

Interval Notation	Description	Inequality Notation
[a, b]	All numbers from a to b, including a and b	a ≤ x ≤ b
(a, b)	All numbers from a to b, not including a and b	a < x < b
[a, b)	All numbers from a to b, including a but not b	a ≤ x < b
(a, b]	All numbers from a to b, not including a but including b	a < x ≤ b
(-∞, a)	All numbers less than a	x < a
(a, ∞)	All numbers greater than a	x > a

Converting Between Notations

Converting between interval notation and inequality notation is straightforward:

Conversion Rules

Brackets [ ] in interval notation correspond to ≤ in inequality notation
Parentheses ( ) in interval notation correspond to < or > in inequality notation
Infinite intervals (-∞, a) and (a, ∞) correspond to x < a and x > a respectively

For example, the interval [3, 7] in interval notation is equivalent to 3 ≤ x ≤ 7 in inequality notation.

Common Interval Problems

Here are some common interval problems you might encounter:

Finding the union of two intervals
Finding the intersection of two intervals
Determining if a number is in an interval
Solving inequalities involving intervals

When working with multiple intervals, remember that the union (∪) combines all numbers from both intervals, while the intersection (∩) only includes numbers common to both intervals.

FAQ

What is the difference between [ ] and ( ) in interval notation?: Brackets [ ] indicate that the endpoint is included in the interval, while parentheses ( ) indicate that the endpoint is not included.
How do I represent all real numbers in interval notation?: All real numbers are represented as (-∞, ∞) in interval notation.
Can I use interval notation for discrete sets of numbers?: Interval notation is specifically for continuous ranges of real numbers. For discrete sets, you would typically list the numbers or use set notation.
How do I combine two intervals using union or intersection?: For union, you combine all numbers from both intervals. For intersection, you only include numbers that are in both intervals. The calculator can help visualize these operations.