Interval Notation Calculator
Free Interval notation Calculator for algebra. Enter values to get step-by-step solutions with formulas and graphs. Get results you can export or share.
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Where square brackets [ ] indicate the endpoint is included (closed), parentheses ( ) indicate the endpoint is excluded (open), a is the lower bound, and b is the upper bound. The interval represents all real numbers x satisfying the given inequality conditions.
Last reviewed: December 2025
Worked Examples
Example 1: Converting Inequality to Interval Notation
Example 2: Testing Membership in an Interval
Background & Theory
The Interval Notation Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Interval Notation Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Formula
{x | a <= x < b} = [a, b)
Where square brackets [ ] indicate the endpoint is included (closed), parentheses ( ) indicate the endpoint is excluded (open), a is the lower bound, and b is the upper bound. The interval represents all real numbers x satisfying the given inequality conditions.
Worked Examples
Example 1: Converting Inequality to Interval Notation
Problem: Express the inequality -2 <= x < 7 in interval notation, set-builder notation, and determine the interval type.
Solution: The inequality -2 <= x < 7 means x is greater than or equal to -2 AND less than 7.\nLeft endpoint: -2 is included (<=) so we use a square bracket [\nRight endpoint: 7 is excluded (<) so we use a parenthesis )\nInterval notation: [-2, 7)\nSet-builder notation: {x | -2 <= x < 7}\nLength: 7 - (-2) = 9\nMidpoint: (-2 + 7) / 2 = 2.5\nType: Half-open (one endpoint included, one excluded)
Result: Interval: [-2, 7) | Length: 9 | Midpoint: 2.5 | Type: Half-Open
Example 2: Testing Membership in an Interval
Problem: Given the interval (1, 10], determine whether x = 1, x = 5.5, and x = 10 are members of the set.
Solution: The interval (1, 10] means 1 < x <= 10.\nTest x = 1: Is 1 > 1? No (not strictly greater). x = 1 is NOT in the interval.\nTest x = 5.5: Is 1 < 5.5 <= 10? Yes. x = 5.5 IS in the interval.\nTest x = 10: Is 1 < 10 <= 10? Yes. x = 10 IS in the interval.\nComplement: (-infinity, 1] U (10, infinity)
Result: x=1: Not in interval | x=5.5: In interval | x=10: In interval
Frequently Asked Questions
What is interval notation and why is it used in mathematics?
Interval notation is a compact mathematical shorthand used to describe a continuous range of real numbers between two endpoints. It uses brackets and parentheses to indicate whether endpoints are included or excluded from the set. Square brackets like [a, b] mean the endpoint is included (closed), while parentheses like (a, b) mean the endpoint is excluded (open). This notation is far more efficient than writing out full inequality statements, especially in calculus and analysis where intervals appear frequently. Mathematicians prefer interval notation because it clearly communicates both the range and boundary conditions in a single expression.
How do you convert between interval notation and inequality notation?
Converting between these notations is straightforward once you understand the bracket conventions. The interval [2, 7) translates to the inequality 2 <= x < 7, where the square bracket becomes a less-than-or-equal sign and the parenthesis becomes a strict less-than sign. Going the other way, if you have -3 < x <= 5, the strict inequality on the left means an open parenthesis and the inclusive inequality on the right means a closed bracket, giving you (-3, 5]. For unbounded intervals, use infinity symbols with always-open parentheses since infinity is not a number that can be reached or included.
What is set-builder notation and how does it relate to interval notation?
Set-builder notation describes a set by stating the properties its members must satisfy, typically written as {x | condition}. The vertical bar means such that, so {x | 2 <= x < 5} reads as the set of all x such that x is greater than or equal to 2 and less than 5. This is equivalent to the interval notation [2, 5). Set-builder notation is more flexible than interval notation because it can describe sets that are not simple intervals, such as {x | x is an even integer} or {x | x squared < 9}. However, for simple continuous ranges, interval notation is preferred because it is more concise and immediately conveys the boundary information.
Why do we always use parentheses with infinity in interval notation?
Infinity is not a real number but rather a concept representing unboundedness, so it can never be included as an endpoint in a set of real numbers. Since square brackets indicate inclusion, using them with infinity would incorrectly suggest that infinity is a member of the set. This is why we always write (negative infinity, a] or [b, infinity) with parentheses on the infinity side. This convention is universal across mathematics and prevents logical errors. The same rule applies to negative infinity. Even in extended real number systems where infinity is formally treated, standard interval notation conventions maintain parentheses for clarity and consistency.
How is interval notation used in calculus and real analysis?
Interval notation is fundamental in calculus for expressing domains, ranges, and regions of interest. When finding where a function is increasing, you might state f is increasing on (2, 7). The domain of the square root function is [0, infinity), and the range of the natural logarithm is (-infinity, infinity). In integration, definite integrals are computed over specific intervals. The Mean Value Theorem states that for a function continuous on [a, b] and differentiable on (a, b), there exists a point c in (a, b) where the instantaneous rate equals the average rate. Notice how the theorem carefully specifies closed versus open intervals for different conditions.
What is the complement of an interval and how do you find it?
The complement of an interval is the set of all real numbers that are NOT in the interval. For the interval [2, 5), the complement consists of all numbers less than 2 or greater than or equal to 5, written as (-infinity, 2) U [5, infinity). Notice that the bracket type reverses at each endpoint: the closed bracket at 2 in the original interval becomes an open parenthesis in the complement because 2 is included in the original and therefore excluded from the complement. Finding complements is important when solving inequalities by negation, computing probabilities of complementary events, and working with set theory problems in real analysis.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy