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Maths calculators

Percentages, fractions, algebra, number theory and trigonometry: the working shown alongside every answer.

PercentageThe three everyday percentage questions in one place.Percentage change & differenceIncrease, decrease and the symmetric difference.Fraction arithmeticAdd, subtract, multiply or divide two fractions and simplify.Decimal ↔ fractionClosest fraction to a decimal, at a chosen precision.GCD & LCMGreatest common divisor and lowest common multiple.Quadratic equationRoots, vertex and discriminant of ax² + bx + c.Ratio & proportionSolves a : b = c : d for the missing term.Exponents & rootsPowers, nth roots and scientific notation of the result.LogarithmLog in any base, plus the natural and common logs.Scientific notationConverts between plain numbers and a × 10ⁿ.Prime check & factorisationTests primality and lists the prime factors.Permutations & combinationsOrdered and unordered selections from a set.Number base converterBinary, octal, decimal, hex and any base to 36.Roman numeral converterArabic to Roman and back, 1 to 3999.Rounding & significant figuresRound to decimals, significant figures or a nearest step.Trigonometry: solve a triangleSine and cosine rules from two sides and the included angle.Trigonometric functionssin, cos, tan and their inverses for one angle.Modulo & integer divisionQuotient, remainder and the true mathematical modulo.Random number generatorDraws numbers from a range: press recalculate for a new set.Matrix determinant (2×2 or 3×3)Determinant of a 2×2 or 3×3 matrix from its entries.2×2 matrix inverseInverse of a 2×2 matrix using the adjugate formula.2×2 matrix operationsAdd, subtract or multiply two 2×2 matrices, and scale one by a constant.Vector dot productScalar dot product of two 3D vectors, and the angle between them.Vector cross productThe perpendicular vector from two 3D vectors, and the area of the parallelogram they span.Vector magnitude & unit vectorLength of a 3D vector and the unit vector pointing the same way.Arithmetic sequenceThe nth term and the sum of the first n terms of an arithmetic sequence.Geometric sequenceThe nth term and the sum of a geometric sequence, plus the infinite sum where it converges.Complex number arithmeticAdd, subtract, multiply or divide two complex numbers, plus modulus and argument.Synthetic divisionDivides a polynomial by (x − r) and returns the quotient and remainder.Equation of a lineSlope-intercept, point-slope and standard form of the line through two points.Distance & midpoint between two points (3D)Straight-line distance and midpoint between two points in 3D space (set z = 0 for 2D).Law of sines (solve a triangle)Finds the remaining sides and angles from two angles and one side.Arithmetic in standard formAdd, subtract, multiply or divide two numbers written as a × 10ⁿ.Binomial expansionExpands (x + y)ⁿ term by term using the binomial theorem.Distance from a point to a linePerpendicular distance from a point to a line given in general form.Geometric & harmonic meanThe geometric mean and the harmonic mean of a list of positive numbers.Golden ratioSplits a length into golden-ratio proportions, or scales a rectangle side by φ.Partial fraction decomposition (2 factors)Splits (ax+b)/((x−p)(x−q)) into simple fractions, including the repeated-root case.System of two linear equationsSolves a₁x+b₁y=c₁ and a₂x+b₂y=c₂ for x and y.Intersection of a circle and a lineWhere a line y = mx + c crosses a circle centred at (h, k).Matrix trace & transpose (2×2 or 3×3)Sum of the diagonal, and the matrix flipped over its diagonal.Hyperbolic functionssinh, cosh and tanh of a value, and their inverses.Circle equation: general to standard formConverts x²+y²+Dx+Ey+F=0 into centre-radius form.Absolute value equationSolves |ax + b| = c for x.Consecutive integers from a sumFinds n evenly-spaced integers (consecutive, even or odd) that add to a target sum.Divisibility testChecks which numbers from 2 to 12 divide a whole number evenly.Digital rootRepeatedly sums the digits of a number until one digit remains.Cubic equationReal roots of ax³ + bx² + cx + d = 0 via Cardano’s method.Eigenvalues of a 2×2 matrixThe two eigenvalues (real or complex) of [[a,b],[c,d]] from the characteristic polynomial.Solve a·bˣ = c for xUses logarithms to isolate x in an exponential equation.Rationalize a denominatorRemoves a square root from the denominator of a/(b + c√n) using the conjugate.Pythagorean triple generatorBuilds a right-triangle integer triple from two generator numbers (Euclid’s formula).Clock hand angleThe angle between the hour and minute hands at a given time.Reference & coterminal angleReduces any angle to its 0–360° coterminal angle, quadrant and reference angle.Round to the nearest fractionRounds a decimal to the nearest 1/2, 1/4, 1/8, 1/16 or any denominator you choose.Percent errorHow far a measured or estimated value is from the accepted true value.Set operationsUnion, intersection and differences of two sets of numbers.Polygon area from vertices (shoelace formula)Area, perimeter and centroid of a simple polygon from its ordered vertices.Triangle incenter & inradiusIncenter coordinates and inscribed-circle radius from a triangle’s three vertices.Perfect, abundant & deficient number checkClassifies a whole number by comparing it to the sum of its own proper divisors.Amicable numbers checkTests whether two numbers form an amicable pair: each is the sum of the other’s divisors.Continued fraction expansionExpands a fraction p/q into continued-fraction terms and lists its convergents.Linear Diophantine equation (ax + by = c)Finds all integer solutions to ax + by = c using the extended Euclidean algorithm.System of three linear equations (Cramer’s rule)Solves a 3×3 system of linear equations using determinants.Vector projectionProjects vector a onto vector b in 3D (set z = 0 for 2D vectors).Distance from a point to a planePerpendicular distance from a point to a plane given as ax + by + cz + d = 0.Chinese remainder theorem (two congruences)Finds x satisfying x≡a₁ (mod n₁) and x≡a₂ (mod n₂), including non-coprime moduli.Modular multiplicative inverseFinds x such that a·x ≡ 1 (mod n), using the extended Euclidean algorithm.nth roots of a complex numberAll n complex roots of a+bi using De Moivre’s theorem, in polar and rectangular form.Fibonacci sequenceThe nth Fibonacci number, the sequence around it and the sum so far.Egyptian fraction decompositionSplits a fraction into a sum of distinct unit fractions using the greedy algorithm.Linear interpolationEstimates a missing y (or x) on the straight line through two known points.Modular exponentiationComputes baseᵉˣᵖ mod m efficiently using square-and-multiply.Luhn algorithm (checksum)Validates a number sequence or computes the check digit that makes it valid.Manhattan (taxicab) distanceGrid-style distance between two points, moving only along axes.Collatz conjecture (3n+1)Steps to reach 1 by halving even numbers and applying 3n + 1 to odd ones.Cosine similarityAngular similarity between two vectors, from −1 (opposite) to 1 (same direction).Descartes' rule of signsMaximum possible positive and negative real roots of a polynomial, from its sign changes.Sum of digitsAdds up the individual digits of a whole number in one pass (not repeated to a single digit).Order of magnitudeThe power of ten closest to a number, used to compare sizes at a glance.Unit rateSimplifies a ratio of two quantities to a rate per single unit of the second quantity.Golden rectangleFinds the sides, diagonal and area of a rectangle whose side ratio equals the golden ratio.Magic squareFinds the magic constant of an n×n magic square and builds it for odd n via the Siamese method.Discrete convolutionConvolves two finite number sequences to produce their discrete convolution sequence.Two's complementConverts a signed decimal integer to its binary two’s-complement form at a chosen bit width.Harmonic numberSums the reciprocals of the first n positive integers to give the nth harmonic number.Gamma functionEvaluates the gamma function Γ(x) for any real x, extending the factorial to non-integers.Error functionComputes the Gauss error function erf(x) and its complement erfc(x) for a given x.Involute functionFinds the involute of a pressure angle, used in gear-tooth geometry: φ = tan(α) − α.Absolute value inequalitiesSolves |ax + b| compared to c (<, ≤, >, ≥) and shows the solution in interval notation.Euler's formula for polyhedraFinds the missing vertex, edge or face count of a convex polyhedron from V − E + F = 2.Cyclomatic complexityComplexity of a control-flow graph from its edge, node and connected-component counts.Bilinear interpolationEstimates a value at a point inside a rectangular grid from the four surrounding known values.

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