Put in Interval Notation Calculator

Interval notation is a concise way to represent sets of real numbers. This calculator helps you convert numbers and expressions into proper interval notation.

What is Interval Notation?

Interval notation is a mathematical shorthand used to describe a set of real numbers that lie between two endpoints. It's commonly used in calculus, algebra, and other branches of mathematics to represent ranges of values.

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 (or half-closed) interval: Includes one endpoint but not the other (e.g., [a, b) or (a, b])
Infinite interval: Represents numbers extending to infinity (e.g., [a, ∞) or (-∞, b])

Interval notation is particularly useful when working with inequalities, limits, and continuous functions.

How to Write Intervals

To write an interval in proper notation, follow these steps:

Identify the lower and upper bounds of your interval
Determine whether each endpoint is included or excluded
Use the appropriate bracket or parenthesis:
- Square bracket [ ] for included endpoints
- Parentheses ( ) for excluded endpoints
Write the lower bound first, followed by a comma, then the upper bound

Interval Notation Examples

All real numbers x such that 2 ≤ x ≤ 5: [2, 5]

All real numbers x such that -3 < x < 1: (-3, 1)

All real numbers x such that x ≥ 4: [4, ∞)

All real numbers x such that x < 0: (-∞, 0)

When working with inequalities, remember that:

≤ and ≥ symbols correspond to closed intervals
< and > symbols correspond to open intervals
∞ (infinity) is never included in an interval

Examples

Let's look at several examples of converting expressions to interval notation:

Expression	Interval Notation	Description
x > 3 and x < 7	(3, 7)	All numbers between 3 and 7, not including 3 and 7
x ≥ -2 and x ≤ 4	[-2, 4]	All numbers between -2 and 4, including both endpoints
x > -5 and x ≤ 0	(-5, 0]	All numbers greater than -5 and less than or equal to 0
x < 10	(-∞, 10)	All numbers less than 10
x ≥ 15	[15, ∞)	All numbers greater than or equal to 15

Notice how the type of inequality determines whether the endpoint is included or excluded in the interval notation.

FAQ

What is the difference between [ ] and ( ) in interval notation?

Square brackets [ ] indicate that an endpoint is included in the interval, while parentheses ( ) indicate that an endpoint is excluded. For example, [2, 5] includes 2 and 5, while (2, 5) does not include either 2 or 5.

How do I represent all real numbers in interval notation?

All real numbers can be represented as (-∞, ∞). This interval includes every possible real number, from negative infinity to positive infinity.

Can interval notation represent a single point?

Yes, a single point can be represented as [a, a] or (a, a), though the latter is not meaningful since it would represent an empty set. For example, the number 3 can be written as [3, 3].

What does an empty set look like in interval notation?

An empty set is represented by two parentheses with nothing between them, like ( ). This indicates there are no numbers that satisfy the given condition.