Interval Notation Converter

About the Interval Notation Converter

Interval notation is a concise way to describe a range of numbers on the real number line. Instead of writing "all x such that 2 < x ≤ 5," mathematicians write (2, 5]. Parentheses indicate the endpoint is excluded (open), while brackets indicate it is included (closed). Mastering these conversions is essential for algebra, calculus, and beyond.

This calculator converts between four representations: interval notation (e.g., [−3, 7)), inequality notation (−3 ≤ x < 7), set-builder notation ({x ∈ ℝ | −3 ≤ x < 7}), and a visual number-line graph. It handles bounded, unbounded (half-line), and entire-real-line intervals, and computes useful properties like interval length, midpoint, and classification.

Whether you are learning algebra, checking homework, or need a quick conversion reference, it gives all four representations simultaneously with an interactive number-line visualization. When you review the result, pay close attention to whether each endpoint is open or closed and whether infinity is involved.

When This Page Helps

Interval notation is used throughout mathematics, from algebra through real analysis. It is the standard way to express domains of functions, ranges of solutions, confidence intervals in statistics, and constraint sets in optimization. Being fluent in converting between interval, inequality, set-builder, and graphical forms saves time and prevents errors.

This converter also helps students visualize abstract concepts by providing an instant number-line graph alongside the algebraic representations.

How to Use the Inputs

1. Select your input mode: interval notation or inequality.
2. For interval mode, enter the left and right endpoints and choose open or closed brackets.
3. Use "−Infinity" or "−∞" for unbounded endpoints.
4. For inequality mode, type an expression like "2 < x <= 5" or "x >= 3".
5. Click presets to load common interval examples.
6. Review all four notation forms in the output cards and conversion table.
7. Check the number-line graph for the visual representation.

Example Calculation

Tips & Best Practices

• Infinity endpoints always use parentheses — you can never "include" infinity.
• A single point can be written as the degenerate interval [a, a].
• On a number line, open dots (○) match parentheses and closed dots (●) match brackets.
• When converting from an inequality, identify which endpoints are strict (<, >) and which are non-strict (≤, ≥).
• The length of a bounded interval [a, b] is always b − a, regardless of bracket type.
• Union of disjoint intervals like (−∞, 2) ∪ [5, ∞) represents two separate regions.

Types of Intervals

Intervals are classified by their boundedness and bracket types. Bounded intervals have two finite endpoints and come in four flavors: open (a, b), closed [a, b], and two half-open variants [a, b) and (a, b]. Unbounded intervals extend to infinity in one or both directions: (−∞, b), [a, ∞), or (−∞, ∞). The last represents the entire real number line. Empty intervals (∅) arise when the endpoints are contradictory.

Interval Notation in Calculus

In calculus, interval notation is used extensively. The domain of f(x) = √x is [0, ∞). The range of sin(x) is [−1, 1]. Continuity, differentiability, and integrability are all defined on intervals. Open intervals are particularly important because many theorems (like the Mean Value Theorem) require the function to be continuous on a closed interval [a, b] and differentiable on the open interval (a, b).

Converting Between Representations

To convert from any representation to another, identify three pieces of information: (1) the left endpoint and whether it is included, (2) the right endpoint and whether it is included, and (3) whether either endpoint is infinite. From these three facts, you can write the interval, inequality, set-builder, and number-line forms. Practice all four conversions until they become automatic — this fluency pays dividends throughout mathematics.

Frequently Asked Questions

• What do parentheses and brackets mean in interval notation?

  Parentheses ( ) mean the endpoint is excluded (open). Brackets [ ] mean the endpoint is included (closed). For example, (3, 7] excludes 3 but includes 7.

• Why does infinity always get a parenthesis?

  Infinity is not a real number and cannot be "reached" or included. Therefore, endpoints at ±∞ always use parentheses: (−∞, 5] or [2, ∞).

• What is set-builder notation?

  Set-builder notation describes a set by stating the property its members satisfy. For example, {x ∈ ℝ | 2 < x ≤ 5} reads "the set of all real x such that x is greater than 2 and at most 5."

• What is a half-open interval?

  A half-open (or half-closed) interval includes one endpoint but not the other. Examples: [a, b) includes a but excludes b; (a, b] excludes a but includes b.

• Can an interval be empty?

  Yes. If the left endpoint exceeds the right, or if the endpoints are equal but at least one bracket is open, the interval is empty (∅).

• How do I convert an inequality to interval notation?

  Identify the endpoints and whether each is included (≤ or ≥ → bracket) or excluded (< or > → parenthesis). For example, −1 ≤ x < 4 becomes [−1, 4).