Interval Numbers Calculator

An interval number represents a range of values between two endpoints. This calculator helps you perform operations on interval numbers including finding overlaps, unions, and differences between intervals.

What is an Interval Number?

An interval number is a mathematical concept that represents a range of values between two endpoints. It's commonly written in the form [a, b], where a is the lower bound and b is the upper bound. The square brackets indicate whether the endpoints are included in the interval.

There are four types of intervals:

Closed interval: [a, b] - includes both endpoints
Open interval: (a, b) - excludes both endpoints
Half-open intervals: [a, b) or (a, b] - includes one endpoint but not the other

Interval numbers are commonly used in mathematics, engineering, and computer science for representing ranges of values, constraints, and uncertainties.

Interval Operations

This calculator can perform several operations on interval numbers:

Union: Combines two intervals into one that covers all values from both intervals
Intersection: Finds the overlapping values between two intervals
Difference: Subtracts one interval from another
Complement: Finds all numbers not in the given interval

For two intervals A = [a₁, a₂] and B = [b₁, b₂]:

Union: A ∪ B = [min(a₁, b₁), max(a₂, b₂)]

Intersection: A ∩ B = [max(a₁, b₁), min(a₂, b₂)] (if they overlap)

Difference: A - B = [a₁, min(a₂, b₁)] ∪ [max(a₁, b₂), a₂]

How to Use This Calculator

Enter the first interval in the format [a, b] or (a, b)
Enter the second interval in the same format
Select the operation you want to perform
Click "Calculate" to see the result
View the visualized result in the chart below

The calculator will display the result in interval notation and show a visual representation of the intervals and their operation.

Worked Examples

Example 1: Union of Two Intervals

Interval A: [2, 5]

Interval B: [4, 8]

Operation: Union

Result: [2, 8]

Explanation: The union combines both intervals, creating a new interval that covers all values from 2 to 8.

Example 2: Intersection of Two Intervals

Interval A: [3, 7]

Interval B: [5, 9]

Operation: Intersection

Result: [5, 7]

Explanation: The intersection finds the overlapping values between the two intervals, which is from 5 to 7.

Frequently Asked Questions

What is the difference between closed and open intervals?: A closed interval includes both endpoints (e.g., [a, b]), while an open interval excludes both endpoints (e.g., (a, b)). Half-open intervals include one endpoint but not the other.
Can I perform operations on more than two intervals?: This calculator currently supports operations on two intervals at a time. For more complex operations, you may need to perform them step by step.
What happens if the intervals don't overlap?: For intersection operations, if the intervals don't overlap, the result will be an empty set (∅). For union operations, the result will be the combination of both intervals.
How accurate are the calculations?: The calculator uses standard interval arithmetic formulas and provides exact results based on the input values. The visual chart helps verify the results.
Can I use negative numbers in the intervals?: Yes, the calculator accepts both positive and negative numbers in interval notation.