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Interval Calculator Mathway

Reviewed by Calculator Editorial Team

An interval represents the range between two numbers on the number line. This calculator helps you determine the size of an interval, visualize it, and understand its mathematical properties.

What is an Interval?

In mathematics, an interval is a set of real numbers with every number between two endpoints. Intervals are commonly used in calculus, statistics, and engineering to describe ranges of values.

There are four types of intervals:

  1. Closed interval [a, b] - includes both endpoints
  2. Open interval (a, b) - excludes both endpoints
  3. Half-open intervals [a, b) and (a, b] - include one endpoint and exclude the other
  4. Infinite intervals [a, ∞) and (-∞, b] - extend to infinity

Note: Intervals can be continuous (like real numbers) or discrete (like integers). This calculator works with continuous intervals.

How to Use the Interval Calculator

  1. Enter the lower bound of your interval in the first input field
  2. Enter the upper bound of your interval in the second input field
  3. Select the interval type (closed, open, or half-open)
  4. Click "Calculate" to see the interval size and visualization
  5. Use "Reset" to clear all inputs

The calculator will display the size of the interval and show a visual representation on the number line.

Interval Formula

The size of an interval is calculated as:

Interval Size = Upper Bound - Lower Bound

For example, the interval [3, 7] has a size of 7 - 3 = 4.

For open intervals, the endpoints are not included in the count, but the size calculation remains the same.

Worked Examples

Example 1: Closed Interval

If you have the interval [5, 12]:

  1. Lower bound = 5
  2. Upper bound = 12
  3. Interval size = 12 - 5 = 7

The interval includes all numbers from 5 to 12, inclusive.

Example 2: Open Interval

For the interval (10, 20):

  1. Lower bound = 10
  2. Upper bound = 20
  3. Interval size = 20 - 10 = 10

The interval includes all numbers between 10 and 20, not including 10 and 20 themselves.

Example 3: Half-Open Interval

For the interval [15, 25):

  1. Lower bound = 15
  2. Upper bound = 25
  3. Interval size = 25 - 15 = 10

The interval includes 15 but not 25.

Frequently Asked Questions

What is the difference between an interval and a range?
An interval refers specifically to a set of real numbers between two endpoints, while range can refer more generally to the difference between maximum and minimum values in a dataset.
Can intervals be negative?
Yes, intervals can be negative. For example, [-5, -1] is a valid interval containing all numbers from -5 to -1.
How do I represent an interval with a single number?
A single number can be represented as a closed interval with identical endpoints, like [3, 3].
What is the difference between an interval and a segment?
In geometry, a segment is a part of a line with two endpoints, while in mathematics, an interval refers to the set of numbers between two endpoints.
Can intervals be used in probability?
Yes, intervals are commonly used in probability to describe ranges of possible outcomes, such as confidence intervals.