Last updated: July 31, 2026
Interval Notation Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
Use [ or ] when an endpoint is included, and ( or ) when an endpoint is excluded. The interval length is upperBound - lowerBound.
Where:
- l=Lower bound
- u=Upper bound
- [=Inclusive left endpoint marker
- (=Exclusive left endpoint marker
- ]=Inclusive right endpoint marker
- )=Exclusive right endpoint marker
Worked Examples
Half-open interval
Include the left endpoint but exclude the right endpoint.
- 1Choose [ because the lower bound is inclusive.
- 2Choose ) because the upper bound is exclusive.
- 3Compute the interval length as 5 - 2 = 3.
Open interval
Both endpoints are excluded.
- 1Use ( on the left and ) on the right.
- 2Write the bounds in order from lower to upper.
- 3Compute the length as 3 - (-3) = 6.
Closed interval
Both endpoints are included.
- 1Use square brackets on both sides.
- 2Compute the length as 10 - 0 = 10.
Single-point interval
Equal bounds are allowed when the interval is closed at both ends.
- 1Keep the same lower and upper value.
- 2Use brackets to indicate the point is included.
- 3Compute the length as 4 - 4 = 0.
Introduction
Interval notation is a compact way to describe sets of real numbers between two bounds. This calculator lets you choose the lower and upper endpoints, decide whether each is included, and instantly see the correct interval string, interval type, and length.
Brackets versus parentheses
A square bracket means the endpoint is included in the set, while a parenthesis means the endpoint is excluded. The choice changes the meaning even when the numeric bounds stay the same.
[2, 5] includes both 2 and 5.
(2, 5) excludes both 2 and 5.
[2, 5) includes 2 but excludes 5.
How interval types are classified
The calculator labels the interval based on the inclusivity of its endpoints.
- closed:
both endpoints included
- open:
both endpoints excluded
- half-open (left-closed):
only the lower endpoint included
- half-open (right-closed):
only the upper endpoint included
What interval length means
The interval length is the numerical distance between the upper and lower bounds. It depends only on the bounds, not on whether the endpoints are included.
[2, 5] and (2, 5) both have length 3.
A zero-length interval happens when both bounds are equal.
Inclusivity changes membership, not the distance between endpoints.
Valid bound ordering
The lower bound must be less than or equal to the upper bound. If the lower bound is larger, the notation would reverse the interval and the input is treated as invalid.
lower < upper creates a standard interval.
lower = upper creates a degenerate interval of length 0.
lower > upper is rejected.
A one-point interval such as [4, 4] is meaningful, but (4, 4) represents an empty set even though its numeric length is still reported as 0.
Where interval notation is used
Interval notation appears in algebra, calculus, inequality solving, domain and range descriptions, and real analysis.
Describe the solution set of inequalities.
State domains of functions.
Specify ranges of possible measurements.
Inclusive and exclusive endpoint language
Inclusive means the endpoint is part of the set; exclusive means it is not. Students often connect this to ≤ and ≥ versus < and >.
Inclusive left endpoint corresponds to x ≥ lowerBound.
Exclusive left endpoint corresponds to x > lowerBound.
Inclusive right endpoint corresponds to x ≤ upperBound.
Common mistakes to avoid
Most notation errors come from using the wrong bracket type or placing the bounds in reverse order.
Do not use brackets for excluded endpoints.
Keep the lower value on the left.
Remember that infinity is never written with a square bracket in standard interval notation.
FAQs
What is the difference between [ ] and ( )?
Square brackets include an endpoint, while parentheses exclude it.
Can the lower bound equal the upper bound?
Yes. That creates a zero-length interval. If both ends are inclusive, it represents a single point.
Why does a half-open interval matter?
It tells you that one endpoint belongs to the set while the other does not, which is important in inequalities and domains.
Does the interval length depend on inclusivity?
No. The length depends only on upperBound minus lowerBound.
What happens if the lower bound is greater than the upper bound?
That input is invalid for this calculator, so it returns an error asking for the lower bound to be less than or equal to the upper bound.
Can this calculator express infinity?
No. This version is designed for finite numeric bounds only.
How does interval notation relate to inequalities?
Inclusive endpoints match ≤ or ≥, while exclusive endpoints match < or >.
What does an open interval mean?
It contains every value strictly between the endpoints but not the endpoints themselves.
What does a closed interval mean?
It contains every value between the endpoints, including both endpoints.