Interval Notation Calculator - Symbolab

Interval notation is a concise way to represent sets of real numbers. This calculator helps you convert between interval notation and inequality notation, which is useful in algebra, calculus, and other mathematical fields.

What is Interval Notation?

Interval notation provides a shorthand method for describing ranges of real numbers. It's commonly used in mathematics to represent continuous sets of numbers between two endpoints.

There are four main types of intervals:

Closed interval: Includes both endpoints (e.g., [a, b])
Open interval: Excludes both endpoints (e.g., (a, b))
Half-open interval: Includes one endpoint but not the other (e.g., [a, b) or (a, b])
Infinite interval: Extends to infinity (e.g., [a, ∞) or (-∞, b])

Interval notation is particularly useful in calculus for describing domains and ranges of functions.

Converting Between Notations

The calculator helps you convert between interval notation and inequality notation. Here's how the conversion works:

Interval Notation	Inequality Notation	Description
[a, b]	a ≤ x ≤ b	Closed interval including both endpoints
(a, b)	a < x < b	Open interval excluding both endpoints
[a, b)	a ≤ x < b	Half-open interval including a but not b
(a, b]	a < x ≤ b	Half-open interval including b but not a
[a, ∞)	x ≥ a	All numbers greater than or equal to a
(-∞, b]	x ≤ b	All numbers less than or equal to b

Formula: The conversion between interval notation and inequality notation is a direct mapping of the bracket types to inequality symbols.

Common Interval Types

Here are some common interval types with examples:

Single point interval: [a, a] or (a, a) - represents just the number a

Note: (a, a) is an empty set since it contains no numbers between a and a.
All real numbers: (-∞, ∞) - represents every real number
Empty set: (a, a) - represents no numbers

Practical Applications

Interval notation is widely used in various mathematical fields:

Describing the domain and range of functions in calculus
Specifying solution sets to inequalities in algebra
Defining intervals for integration in calculus
Representing feasible regions in optimization problems

For example, if you're solving the inequality -3 ≤ 2x + 1 < 7, you can convert it to interval notation as [-2, 3).

FAQ

What is the difference between [a, b] and (a, b)?

The main difference is that [a, b] includes both endpoints a and b, while (a, b) excludes both endpoints. The square brackets indicate inclusion, and parentheses indicate exclusion.

How do I represent all real numbers in interval notation?

All real numbers are represented as (-∞, ∞) in interval notation. This indicates that there are no lower or upper bounds to the set of numbers.

What does an empty set look like in interval notation?

An empty set is represented as (a, a) where a is any real number. This is because there are no numbers between a and itself.

Can interval notation represent a single point?

Yes, a single point can be represented as [a, a] in interval notation. This indicates that the set contains only the number a.