What is Interval Notation?
Interval notation is a mathematical notation used to represent a set of real numbers that lie between two endpoints. It is a concise and elegant way to express intervals, commonly used in algebra, calculus, and set theory. Interval notation provides a standardized method to describe ranges of numbers, making it easier to communicate mathematical concepts and solve problems.
Unlike other mathematical notations, interval notation uses parentheses and brackets to clearly indicate whether the endpoints are included or excluded from the interval. This makes it particularly useful when working with inequalities, functions, and domain and range problems.
Understanding Interval Notation Symbols
Interval notation uses specific symbols to indicate whether endpoints are included:
- (Parentheses ( ) - Indicate an open interval, meaning the endpoint is not included in the set.
- [Square Brackets [ ] - Indicate a closed interval, meaning the endpoint is included in the set.
- ∞Infinity (∞) - Always uses parentheses, as infinity is never actually reached or included.
Interval Notation Converter
Enter an interval to see it in other formats and on a number line.
Types of Intervals in Interval Notation
1. Open Interval
An open interval excludes both endpoints. It is written using parentheses: (a, b)
Example:
(2, 5)
This represents all numbers greater than 2 and less than 5. The numbers 2 and 5 are not included.
2. Closed Interval
A closed interval includes both endpoints. It is written using square brackets: [a, b]
Example:
[2, 5]
This represents all numbers greater than or equal to 2 and less than or equal to 5. Both 2 and 5 are included.
3. Half-Open (or Half-Closed) Interval
A half-open interval includes one endpoint but excludes the other. It uses a combination of parentheses and brackets.
Examples:
[2, 5) - Includes 2, excludes 5
(2, 5] - Excludes 2, includes 5
These represent intervals where one endpoint is included and the other is not.
How to Write Interval Notation
Writing interval notation follows a simple format: [left endpoint, right endpoint] or (left endpoint, right endpoint)
Step-by-Step Process:
- Identify the smallest and largest values in your set
- Determine if each endpoint should be included (use brackets) or excluded (use parentheses)
- Write the smaller number first, followed by a comma, then the larger number
- Use the appropriate brackets or parentheses on each side
Interval Notation Examples
Example 1: Basic Open Interval
Interval Notation: (3, 7)
All numbers greater than 3 and less than 7. Neither 3 nor 7 is included.
Example 2: Closed Interval
Interval Notation: [-2, 4]
All numbers from -2 to 4, including both -2 and 4.
Example 3: Half-Open Interval
Interval Notation: [0, 10)
All numbers from 0 (included) to 10 (excluded). This is common when representing ages or counts.
Example 4: Interval with Infinity
Interval Notation: (-∞, 5]
All numbers less than or equal to 5. Extends infinitely to the left.
Example 5: All Real Numbers
Interval Notation: (-∞, ∞)
Represents the entire set of real numbers.
Interval Notation vs Set Notation
Interval notation is often compared to set-builder notation. While both represent sets of numbers, interval notation is more concise for continuous intervals. Learn more about the differences in our Interval Notation vs Set Builder Notation guide.
| Interval Notation | Set Notation | Description |
|---|---|---|
| (2, 5) | {x | 2 < x < 5} | Open interval |
| [2, 5] | {x | 2 ≤ x ≤ 5} | Closed interval |
| [2, 5) | {x | 2 ≤ x < 5} | Half-open interval |
Common Uses of Interval Notation
Domain and Range
Interval notation is extensively used to express the domain and range of functions in calculus and algebra. Learn more in our Domain and Range guide.
Inequality Solutions
When solving compound inequalities, the solution set is often expressed using interval notation.
Function Analysis
Mathematicians use interval notation to describe intervals where functions are increasing, decreasing, or constant. This is closely related to domain and range analysis.
Set Theory
In set theory, interval notation provides a clear way to represent subsets of real numbers. Explore union and intersection operations with intervals.
Practice Problems with Interval Notation
Problem 1
Write the interval notation for all numbers greater than -3 and less than or equal to 7.
Show Answer
Answer: (-3, 7]
Problem 2
Express the set of all real numbers less than 10 using interval notation.
Show Answer
Answer: (-∞, 10)
Tips for Mastering Interval Notation
- ✓Remember the symbols: Parentheses ( ) mean "not included" and brackets [ ] mean "included"
- ✓Always write smaller to larger: The left endpoint should always be smaller than the right endpoint
- ✓Infinity always uses parentheses: Since infinity is never actually reached, it always uses ( ) not [ ]
- ✓Practice with inequalities: Convert between inequality notation and interval notation regularly
- ✓Visualize on a number line: Drawing number lines can help you understand which endpoints are included
Conclusion
Interval notation is a fundamental mathematical tool that provides a clear, concise way to represent sets of real numbers. Whether you're working with functions, solving inequalities, or studying set theory, understanding interval notation is essential for success in mathematics.
By mastering the symbols, practicing with examples, and understanding when to use open, closed, or half-open intervals, you'll be well-equipped to use interval notation effectively in your mathematical studies. Remember that interval notation is not just a notation system—it's a powerful tool for communicating mathematical ideas clearly and precisely.
Ready to Master Interval Notation?
Practice makes perfect! Try converting inequalities to interval notation, work through examples, and visualize intervals on number lines to strengthen your understanding.
Related Topics
Explore these related topics to deepen your understanding of interval notation and its applications:
Compound Inequalities
Learn how to solve compound inequalities and express solutions using interval notation.
Domain and Range
Master finding and expressing domain and range of functions using interval notation.
Union and Intersection
Understand how to combine intervals using union and intersection operations.
Interval vs Set Builder
Compare interval notation with set builder notation and learn when to use each.