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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.Arc LengthArc length of a circle from its radius and central angle, in degrees or radians.CuboidVolume, surface area, individual face areas and space diagonal of a cuboid from its three edge lengths.Power of 22 raised to any exponent, with a reference table of powers of 2 from 2⁰ to 2²⁰.Inverse Trigonometry (Theta)Solves backwards from a known sine, cosine or tangent ratio to the angle theta.Lowest Common Denominator CalculatorFinds the lowest common denominator of two or more fractions and rewrites each one over it.Triangular Numbers CalculatorThe nth triangular number, plus a reference table of the first several terms of the sequence.Perfect Square CalculatorChecks whether a number is a perfect square, and shows the nearest perfect squares below and above it.Fraction to Percentage CalculatorConverts a fraction to a percentage, or a percentage back to a simplified fraction.Angle of depressionThe angle below the horizontal from an elevated point down to a target, from height and horizontal distance.Percentage DecreaseNew value after a percentage decrease, or the percentage decrease between two values.Fraction to DecimalConverts a fraction to its decimal expansion, marking any repeating block clearly.Exponential FunctionEvaluates y = a × bˣ at a chosen x, with the underlying growth or decay rate.XOR CalculatorBitwise XOR of two whole numbers, with AND, OR and NOT shown alongside for comparison.Square Root CalculatorPrincipal square root of a number, whether it is a perfect square, and the nearest ones either side.Exponential GrowthProjects a value forward under a fixed percentage growth rate per period, from a starting value and number of periods.Arctan (Inverse Tangent)The angle, in degrees and radians, whose tangent equals the value you enter.Pascal's TriangleGenerates any row of Pascal's triangle as binomial coefficients, with the surrounding rows shown for context.Percentage Difference Between Two NumbersThe symmetric percentage difference between two numbers, using their average as the base, distinct from a directional percentage increase or decrease.Polar Form of a Complex NumberConverts a complex number between rectangular (a + bi) form and polar (r∠θ) form, in either direction.Similar Triangles CalculatorScale factor and missing side lengths between two similar triangles from one matching pair of sides.Factoring CalculatorFactors a quadratic trinomial ax² + bx + c into (x − r1)(x − r2) form using its roots.Simplifying Fractions CalculatorReduces a fraction to its simplest form and shows the decimal and mixed-number equivalents.Divisor CalculatorLists every divisor of a whole number, with the count and sum of divisors.Long Division CalculatorShows the full step-by-step working for long division: bring down each digit, divide, subtract, repeat.Greater Than or Less Than CalculatorCompares two numbers and shows the correct inequality symbol: greater than, less than or equal to.Million in Numbers CalculatorExpands a figure written in thousands, millions, billions or trillions into its full digit form.Power of 1010 raised to any exponent, with a reference table of powers of 10 from 10⁰ to 10¹⁰.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.Point-Slope Form of a LineWrites the equation of a line in point-slope form from one point and the slope, and converts it to slope-intercept form.Trapezoidal RuleApproximates the area under a curve (a definite integral) from a set of x,y data points using the trapezoidal rule.Arcsin (Inverse Sine)The angle, in degrees and radians, whose sine equals the value you enter.Cubic Metres to Tonnes ConverterConvert cubic metres to tonnes and back, using an editable density for the material involved.Logarithm Laws CalculatorDemonstrates the product, quotient and power laws of logarithms on two numbers you choose.Arccos (Inverse Cosine)The angle, in degrees and radians, whose cosine equals the value you enter.Sum of Series CalculatorThe sum of the first n terms of an arithmetic or geometric series, with the nth term shown too.Decimal to Percentage CalculatorConverts a decimal to a percentage, or a percentage back to a decimal, by multiplying or dividing by 100.Vector AdditionAdds two 2D or 3D vectors component by component and gives the resultant vector and its magnitude.Point of IntersectionWhere two straight lines cross, from their slope-intercept equations, with parallel and identical lines handled.Sine, Cosine & TangentSine, cosine and tangent of an angle you enter, in degrees or radians.Engineering NotationConverts a number to engineering notation, with a multiple-of-3 exponent and SI prefix.Constant of Proportionality CalculatorFinds k in y = kx from one x-y pair, with an optional second pair to check the relationship is genuinely proportional.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.Percentage IncreaseNew value after a percentage increase, or the percentage increase between two values.Volume of Common ShapesVolume and surface area of a cube, box, sphere, cylinder or cone from its dimensions.AverageMean, sum and count for a list of numbers.TrigonometrySolves a right-angled triangle: the missing sides and angle from one angle and one known side.Fraction CalculatorAdd, subtract, multiply or divide two fractions or mixed numbers, simplified automatically.Gradient (Slope) of a LineGradient between two coordinate points, plus the equation of the line through them.SUVAT equations solverSolve any of the five SUVAT equations of motion for the two unknowns, given the other three.Midpoint between two pointsThe point exactly halfway between two coordinates.Hypotenuse from two legsThe hypotenuse of a right triangle from its two shorter sides.X out of Y as a percentageTurns any "X out of Y" ratio, such as 32 out of 40, into a percentage.

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