Symbolab Interval Calculator

An interval represents a range of real numbers between two endpoints. This calculator helps you work with intervals, perform operations, and visualize them on a number line.

What is an Interval?

In mathematics, an interval is a set of real numbers that includes all numbers between two endpoints. Intervals are fundamental in calculus, analysis, and many other areas of mathematics.

Intervals can be open, closed, or half-open, depending on whether the endpoints are included or excluded. This calculator helps you understand and work with different types of intervals.

Interval Definition: An interval is a set of real numbers that includes all numbers between two endpoints, a and b, where a ≤ b.

Interval Notation

Interval notation provides a concise way to represent intervals. The most common notations are:

(a, b) - Open interval: includes all numbers between a and b, but not a and b themselves.
[a, b] - Closed interval: includes all numbers between a and b, including a and b.
(a, b] - Half-open interval: includes all numbers between a and b, but not a, and includes b.
[a, b) - Half-open interval: includes all numbers between a and b, but not b, and includes a.

Note: The parentheses and brackets in interval notation indicate whether the endpoints are included or excluded.

Interval Operations

You can perform various operations on intervals, such as addition, subtraction, multiplication, and division. This calculator helps you perform these operations and understand the results.

Addition of Intervals

To add two intervals [a, b] and [c, d], you add the lower bounds and the upper bounds:

[a, b] + [c, d] = [a + c, b + d]

Subtraction of Intervals

To subtract two intervals [a, b] and [c, d], you subtract the lower bounds and the upper bounds:

[a, b] - [c, d] = [a - d, b - c]

Multiplication of Intervals

To multiply two intervals [a, b] and [c, d], you consider all possible combinations of the endpoints:

[a, b] × [c, d] = [min(ac, ad, bc, bd), max(ac, ad, bc, bd)]

Interval Graphing

Graphing intervals on a number line helps visualize the range of numbers included in an interval. This calculator can generate a graph of your interval.

When graphing an interval, you represent the endpoints with open or closed circles, depending on whether the endpoints are included or excluded.

FAQ

What is the difference between an open and closed interval?: An open interval does not include its endpoints, while a closed interval includes both endpoints. For example, (2, 5) is an open interval, while [2, 5] is a closed interval.
How do I add two intervals?: To add two intervals [a, b] and [c, d], you add the lower bounds to get the new lower bound and the upper bounds to get the new upper bound: [a + c, b + d].
How do I multiply two intervals?: To multiply two intervals [a, b] and [c, d], you consider all possible combinations of the endpoints and take the minimum and maximum values: [min(ac, ad, bc, bd), max(ac, ad, bc, bd)].
Can I graph intervals with this calculator?: Yes, this calculator can generate a graph of your interval on a number line, showing the range of numbers included in the interval.
What is the difference between a half-open and a closed interval?: A half-open interval includes one endpoint but not the other. For example, [2, 5) includes 2 but not 5, while (2, 5] includes 5 but not 2.