Write Interval Notation and Graph The Interval Calculator

Interval notation is a concise way to represent sets of real numbers. This guide explains how to write interval notation and graph intervals using our interactive calculator.

What is Interval Notation?

Interval notation is a mathematical shorthand used to describe ranges of real numbers. It's commonly used in calculus, algebra, and other branches of mathematics to represent continuous sets of numbers between two endpoints.

Interval notation provides a compact way to represent intervals on the real number line without having to write out every number in the set. This makes it easier to work with intervals in equations and inequalities.

Interval notation is different from set notation, which uses curly braces to list individual elements. For example, {1, 2, 3} is a set with three elements, while [1, 3] represents all real numbers between 1 and 3, including 1 and 3.

How to Write Interval Notation

Writing interval notation involves understanding the different types of intervals and their corresponding symbols. Here's a basic guide to writing interval notation:

Closed Interval: Uses square brackets [ ] to indicate that both endpoints are included in the interval. For example, [a, b] includes a and b.
Open Interval: Uses parentheses ( ) to indicate that the endpoints are not included in the interval. For example, (a, b) does not include a or b.
Half-Open/Half-Closed Interval: Uses a combination of brackets and parentheses to indicate that one endpoint is included while the other is not. For example, [a, b) includes a but not b, and (a, b] includes b but not a.
Infinite Intervals: Uses the symbols ∞ (infinity) and -∞ (negative infinity) to represent intervals that extend infinitely in one or both directions. For example, [a, ∞) includes all numbers greater than or equal to a, and (-∞, b] includes all numbers less than or equal to b.

Interval Notation Formula

[a, b] = {x | a ≤ x ≤ b}

(a, b) = {x | a < x < b}

[a, b) = {x | a ≤ x < b}

(a, b] = {x | a < x ≤ b}

(-∞, b] = {x | x ≤ b}

[a, ∞) = {x | x ≥ a}

(-∞, ∞) = All real numbers

Graphing Intervals

Graphing intervals on the real number line helps visualize the range of numbers represented by the interval notation. Here's how to graph different types of intervals:

Closed Interval [a, b]: Draw a solid dot at both endpoints a and b, then draw a solid line connecting them.
Open Interval (a, b): Draw an open circle at both endpoints a and b, then draw a solid line connecting them.
Half-Open/Half-Closed Intervals: Use a combination of solid dots and open circles to indicate which endpoints are included. For example, [a, b) uses a solid dot at a and an open circle at b.
Infinite Intervals: Use arrows to indicate that the interval extends infinitely in one or both directions. For example, [a, ∞) uses a solid dot at a and an arrow extending to the right, while (-∞, b] uses an arrow extending to the left and a solid dot at b.

When graphing intervals, it's important to clearly label the endpoints and use the correct symbols to indicate whether the endpoints are included or excluded from the interval.

Common Interval Types

There are several common types of intervals that you'll encounter in mathematics. Here are some examples:

Single Point Interval: Represents a single number on the real number line. For example, [5, 5] or {5} represents just the number 5.
Empty Interval: Represents no numbers at all. This is often written as ∅ or (a, a) where a is any real number.
Union of Intervals: Represents the combination of two or more intervals. For example, [1, 3) ∪ (4, 6] represents all numbers from 1 to 3 (not including 3) and from 4 to 6 (including 6).
Complement of an Interval: Represents all numbers not included in the interval. For example, the complement of [2, 5] is (-∞, 2) ∪ (5, ∞).

Interval Notation Examples

Here are some examples of interval notation and their corresponding graphs:

Interval Notation	Description	Graph
[2, 5]	All real numbers from 2 to 5, including 2 and 5	Solid dot at 2, solid line to solid dot at 5
(-3, 1)	All real numbers from -3 to 1, not including -3 or 1	Open circle at -3, solid line to open circle at 1
[0, ∞)	All real numbers greater than or equal to 0	Solid dot at 0, solid line extending to the right with an arrow
(-∞, 4]	All real numbers less than or equal to 4	Arrow extending to the left, solid line to solid dot at 4
(1, 3) ∪ (5, 7)	All real numbers from 1 to 3 (not including 1 and 3) and from 5 to 7 (not including 5 and 7)	Open circle at 1, solid line to open circle at 3, open circle at 5, solid line to open circle at 7

FAQ

What is the difference between interval notation and set notation?

Interval notation uses brackets and parentheses to represent ranges of numbers, while set notation uses curly braces to list individual elements. Interval notation is more concise and is typically used to represent continuous ranges of numbers.

How do you graph an open interval?

To graph an open interval, draw an open circle at each endpoint and connect them with a solid line. Open circles indicate that the endpoints are not included in the interval.

What does the symbol ∪ mean in interval notation?

The ∪ symbol represents the union of two intervals, which means combining the numbers from both intervals into a single set.