Advanced Closed Interval Calculator

Enter interval data

Use the main interval fields to build a closed interval. The optional point and optional comparison interval add more analysis.

First endpoint One bound of the interval.

Second endpoint The other bound of the interval.

Test point (optional) Checks whether a point lies inside.

Comparison interval start (optional) Enter both comparison values to compare intervals.

Comparison interval end (optional) Useful for overlap, intersection, and union.

Formula used

Closed interval: If the entered endpoints are a and b, then the normalized closed interval is [min(a,b), max(a,b)].

Length: Length = upper bound - lower bound

Midpoint: Midpoint = (lower bound + upper bound) / 2

Radius: Radius = (upper bound - lower bound) / 2

Point membership: A point x belongs to the interval if lower ≤ x ≤ upper.

Intersection: For [a,b] and [c,d], the intersection exists when max(a,c) ≤ min(b,d).

Union: If intervals overlap or touch, the union is [min(a,c), max(b,d)]. Otherwise, the union remains two separate closed intervals.

How to use this calculator

Enter the two endpoints of your main interval.
Optionally add a test point to check membership.
Optionally enter a second interval for overlap, union, and intersection analysis.
Press the calculate button to view normalized bounds, interval properties, integer coverage, exports, and the graph.

Example data table

Main endpoints	Closed interval	Length	Midpoint	Test point	Membership
2 and 8	[2, 8]	6	5	6	Inside
9 and 3	[3, 9]	6	6	10	Outside
-4 and 1.5	[-4, 1.5]	5.5	-1.25	-2	Inside

Frequently asked questions

1. What is a closed interval?

A closed interval includes both endpoints. The notation uses square brackets, such as [a, b], meaning every value from a through b is included.

2. What happens if I enter the endpoints in reverse order?

The calculator automatically normalizes them. For example, entering 9 and 3 becomes the valid closed interval [3, 9].

3. How is the interval length found?

The length is the upper bound minus the lower bound. For [2, 8], the length is 8 - 2 = 6.

4. Why is the midpoint useful?

The midpoint shows the center of the interval. It helps when you want a balanced representative value between the bounds.

5. What does the radius mean here?

The radius is half of the interval length. It measures how far each endpoint is from the midpoint.

6. How does the point membership check work?

The calculator checks whether the optional test point satisfies lower ≤ x ≤ upper. If true, the point belongs to the closed interval.

7. Can I compare two closed intervals?

Yes. Enter the optional comparison interval to find the relation, overlap, intersection, and union between the two intervals.

8. Why does the integer list sometimes look shortened?

Large intervals may contain many integers. To keep results readable, the calculator shows a preview when the list becomes long.