Interval Notation Generator

Define a set of numbers on the number line using interval notation

Specify the endpoints and whether they are inclusive or exclusive to generate the corresponding interval notation, set-builder notation, and a description.

Examples

Click on any example to load it into the calculator

Closed Interval

closed-interval

An interval where both endpoints are included.

Interval: [-2, 5]

Open Interval

open-interval

An interval where neither endpoint is included.

Interval: (0, 10)

Half-Open Interval (Left-Closed)

half-open-left

An interval where the left endpoint is included and the right is not.

Interval: [3, 8)

Half-Open Interval (Right-Closed)

half-open-right

An interval where the right endpoint is included and the left is not.

Interval: (-100, 100]

Other Titles
Understanding Interval Notation: A Comprehensive Guide
Master interval notation to describe sets of numbers, a fundamental skill in algebra, calculus, and beyond for defining domains, ranges, and solution sets.

What is Interval Notation? Core Concepts

  • A concise way to represent a continuous range of numbers.
  • Uses parentheses `()` for exclusive endpoints and brackets `[]` for inclusive endpoints.
  • Essential for defining the domain and range of functions and expressing solutions to inequalities.
Interval notation is a streamlined method for describing a set of numbers. Instead of using words or complex inequalities, it uses a pair of delimiters—parentheses or brackets—to signify a range on the number line. This notation is fundamental in mathematics for its clarity and efficiency.
The Four Types of Bounded Intervals
1. Open Interval (a, b): Represents all numbers between 'a' and 'b', but does not include 'a' or 'b'. It corresponds to the inequality a < x < b.
2. Closed Interval [a, b]: Represents all numbers between 'a' and 'b', including both 'a' and 'b'. It corresponds to the inequality a ≤ x ≤ b.
3. Half-Open Interval [a, b): Includes 'a' but excludes 'b'. It corresponds to a ≤ x < b.
4. Half-Open Interval (a, b]: Excludes 'a' but includes 'b'. It corresponds to a < x ≤ b.

Basic Interval Examples

  • The set of numbers from -1 to 5, not including the endpoints, is written as `(-1, 5)`.
  • The temperatures from 0°C to 100°C, including both, is `[0, 100]`.
  • A passing grade from 60 (inclusive) to 100 (exclusive) could be `[60, 100)`.

Step-by-Step Guide to Using the Interval Notation Generator

  • Enter the interval endpoints.
  • Select the appropriate brackets for inclusion or exclusion.
  • Instantly generate all corresponding notations.
Our calculator simplifies the process of creating and understanding interval notation. Follow these steps to generate your results:
Input Fields:
  • Endpoints (a and b): Enter the numerical start and end values for your interval. The left endpoint 'a' must be smaller than the right endpoint 'b'.
  • Brackets: For each endpoint, choose a bracket. Use [ or ] if the number should be included in the interval (inclusive). Use ( or ) if the number should be excluded (exclusive).
  • Variable: Optionally, change the default variable 'x' for the set-builder notation output.
Generating and Interpreting Results:
Click 'Generate Notation' to see the output. The calculator provides three forms of the result for a complete understanding: the standard interval notation, the formal set-builder notation, and a plain-language description.

Practical Usage Examples

  • Input: `[` `-5` `10` `)` → Interval: `[-5, 10)`, Set-Builder: `{ x | -5 ≤ x < 10 }`
  • Input: `(` `0` `1` `)` → Interval: `(0, 1)`, Set-Builder: `{ x | 0 < x < 1 }`
  • Input: `[` `100` `200` `]` → Interval: `[100, 200]`, Set-Builder: `{ x | 100 ≤ x ≤ 200 }`

Real-World Applications of Interval Notation

  • Defining tolerance in manufacturing and engineering.
  • Specifying valid input ranges in computer programming.
  • Expressing confidence intervals in statistics.
Interval notation is a practical tool used across various professional fields to define precise ranges and constraints.
Engineering and Manufacturing
Engineers specify acceptable measurement tolerances using intervals. A machine part might need to have a length within the interval [10.01, 10.03] cm to pass quality control.
Computer Science
Programmers use intervals to validate user inputs. For example, a password length might be restricted to the interval [8, 32] characters.
Statistics and Data Analysis
Statisticians use confidence intervals to express the uncertainty of an estimate. A 95% confidence interval for a poll result might be [48.5%, 53.5%], indicating where the true population preference likely lies.

Industry Applications

  • Safe operating temperature for a CPU: T ∈ [0, 95] degrees Celsius.
  • A movie rated for ages 17 and up: Age ∈ [17, ∞). Note: our calculator focuses on bounded intervals.
  • Acceptable pH level for a swimming pool: pH ∈ (7.2, 7.8).

Common Misconceptions and Correct Methods

  • Always write the smaller number first.
  • Distinguish correctly between `()` and `[]`.
  • Infinity `∞` is a concept, not a number, and always uses a parenthesis.
Misconception 1: Incorrect Order of Endpoints
  • Wrong: (10, 2). Interval notation must always go from the smaller value to the larger value as read from left to right on a number line.
  • Correct: (2, 10).
Misconception 2: Confusing Parentheses and Brackets
  • Wrong: Using (5, 9] when the description is 'greater than 5 and less than or equal to 9' but you intend to include 5. The notation (5, 9] explicitly excludes 5.
  • Correct: [5, 9] means '5 ≤ x ≤ 9', while (5, 9) means '5 < x < 9'. Match the bracket to the inequality symbol.
Misconception 3: Using Brackets with Infinity
  • Wrong: [0, ∞]. Infinity is not a number that can be 'included' in a set.
  • Correct: Always use a parenthesis with ∞ and -∞. The correct form is [0, ∞). (Note: this calculator focuses on bounded intervals between two numbers).

Correction Examples

  • For 'x is greater than or equal to 3': Use `[` for the 3, as in `[3, ...)`.
  • For 'x is strictly between -1 and 1': Use `(-1, 1)`.
  • Never write `[10, 5]`. It must be `[5, 10]`.

Mathematical Derivation and Connections

  • Intervals are connected subsets of real numbers (ℝ).
  • Interval notation is a shorthand for set-builder notation.
  • The union `∪` symbol is used to combine disjoint intervals.
In formal mathematics, an interval is defined as a subset of the real numbers (ℝ) with the property that any number lying between two elements of the set is also in the set. This 'no gaps' property is called connectedness.
From Set-Builder to Interval Notation
Interval notation is a direct, more legible translation of set-builder notation for these specific sets:
  • Set-Builder: { x ∈ ℝ | a ≤ x ≤ b } becomes Interval: [a, b]
  • Set-Builder: { x ∈ ℝ | a < x < b } becomes Interval: (a, b)
Union of Disjoint Intervals
When a solution set consists of two or more separate ranges, the union symbol ∪ is used. For example, the solution to x² > 4 is x < -2 or x > 2. This is written in interval notation as (-∞, -2) ∪ (2, ∞). Our calculator focuses on creating single, connected intervals.

Advanced Context

  • The domain of the function `f(x) = sqrt(x)` is `[0, ∞)`.
  • The range of the sine function `sin(x)` is `[-1, 1]`.
  • The solution to `|x| ≥ 3` is `(-∞, -3] ∪ [3, ∞)`.