Maths calculators

Algebra, trigonometry, geometry and statistics: the working shown alongside every answer.

Maths

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.

Statistics

Averages, spread, distributions, confidence and correlation for data sets you paste straight in.

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Descriptive statisticsMean, median, mode, spread and quartiles for a data set.Weighted averageAverage where each value carries its own weight.Z-score & percentileHow far a value sits from the mean, in standard deviations.Normal distribution probabilityArea under the bell curve between two points.Confidence interval for a meanMargin of error around a sample average.Required sample sizeHow many responses a survey needs.Correlation & linear regressionPearson r and the line of best fit for paired data.Binomial probabilityChance of exactly k successes in n independent trials.Probability of eventsCombines two event probabilities: and, or, neither.Odds ↔ probabilityConverts between fractional odds, decimal odds and percent.GPA / weighted gradeGrade point average across units with credit weights.Grade needed on the finalWhat the last assessment must score to hit a target.Coefficient of variationRelative variability of a data set: standard deviation as a percentage of the mean.SkewnessSample skewness: whether the data leans left or right of the mean.Poisson probabilityChance of a given number of independent events in a fixed interval, given the average rate.Geometric probabilityChance the first success happens on a given trial, for repeated independent trials.Hypergeometric probabilityChance of drawing an exact number of successes from a finite population, without replacement.Chi-square goodness of fitTests whether observed category counts match expected counts.Cohen's d (effect size)Standardised difference between two group means, in pooled standard deviations.Margin of error (proportion)Margin of error for a survey proportion at a chosen confidence level.Relative riskCompares event risk between an exposed and an unexposed (or treated and control) group.T-statistic (one or two sample)How many standard errors a sample mean is from the reference value, or from another mean.Z-test for a proportionTests a sample proportion against a hypothesised population proportion.Expected value of a discrete distributionMean and standard deviation of a discrete random variable from its outcomes and probabilities.Outlier fences (Tukey)The IQR-based cutoffs beyond which a data point counts as an outlier.CovarianceHow two variables move together, in sample or population form.Standard error of the meanHow much a sample mean is expected to vary from the true population mean.F-test for equal variancesCompares the spread of two samples using the ratio of their variances.Negative binomial probabilityChance that the rth success happens on exactly the kth independent trial.Percentile rankWhat percentage of a data set falls at or below a given value.Value at a percentileFinds the data value at any percentile using linear interpolation.Mean absolute deviationThe average distance of each data point from the mean (or median).Bayes’ theoremUpdates the probability of an event given new evidence.Class width for a histogramHow wide each bin should be, from either a target class count or Sturges’ rule.Spearman’s rank correlationMeasures a monotonic relationship between two variables using their ranks, ties included.Chebyshev’s theoremMinimum share of any data set within k standard deviations of the mean, whatever its shape.Two-proportion z-testTests whether two independent samples have significantly different proportions.Pooled standard deviationCombines two sample standard deviations into one pooled estimate for equal-variance tests.Z-test for a single meanTests a sample mean against a hypothesised mean when the population standard deviation is known Quartile deviationHalf the interquartile range: a robust spread measure, less sensitive to outliers than the SD.Range rule of thumb for standard deviationA quick estimate of standard deviation from just the minimum and maximum of a data set.Mean & standard deviation from grouped dataEstimates the mean and standard deviation of a frequency table using class midpoints.Empirical rule (68-95-99.7)The share of normal data expected within 1, 2 and 3 standard deviations of the mean.KurtosisSample excess kurtosis: how heavy-tailed a data set is versus a normal distribution.Diagnostic test accuracy metricsSensitivity, specificity, predictive values and accuracy from a 2×2 test result table.Birthday paradox probabilityChance that at least two people in a group share the same birthday.Bonferroni correctionAdjusts the significance threshold when running many statistical tests at once.Benford's law leading-digit checkCompares a data set's leading digits against the distribution natural data follows.One-way ANOVA (F-statistic)Tests whether three or more group means differ, from the raw data in each group.McNemar's testTests whether paired binary outcomes changed, using the two discordant-pair counts.Mann-Whitney U test (Wilcoxon rank-sum)Compares two independent samples for a shift in distribution, using ranks instead of means.MidrangeThe midpoint between the smallest and largest value in a data set, a quick measure of centre.Sum of squaresTotal squared deviation of each data point from the mean, the building block of variance.Confusion matrixTurns the four counts of a binary confusion matrix into accuracy, precision, recall and F1.Youden's indexCombines a test’s sensitivity and specificity into one score of diagnostic performance.Weibull distributionComputes the PDF, CDF, mean and variance of a Weibull distribution from its shape and scale.Rayleigh distributionComputes the PDF, CDF, mean and variance of a Rayleigh distribution from its scale parameter.Fisher's exact testRuns a two-tailed Fisher exact test on a 2×2 contingency table of counts.Process capability indexCalculates Cp and Cpk from specification limits, the process mean and standard deviation.Degrees of freedomWorks out degrees of freedom for a one-sample, two-sample or chi-square test.Index of qualitative variationMeasures diversity among categorical frequencies on a 0 to 1 scale using the IQV formula.Beta distributionPDF, mean and variance of a beta distribution from its two shape parameters.Monty Hall problemWin probability from staying versus switching after the host reveals losing doors.Statistical power analysisAchieved power for a given sample size, and the sample size needed for a target power.Error propagationPropagates the uncertainty of two measurements through a sum, difference, product or quotient.

Geometry

Area, perimeter, volume and surface area for flat shapes and solids, plus coordinate geometry.

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CircleArea, circumference and diameter from any one measure.Circle sector & arcA wedge of a circle: arc length, area and chord.Triangle from three sidesArea, angles and centres via Heron’s formula.Right triangle / PythagorasThird side, angles and area of a right-angled triangle.Rectangle & squareArea, perimeter and diagonal.TrapezoidArea and perimeter of a four-sided figure with parallel sides.Parallelogram & rhombusArea from base and height, or from sides and angle.EllipseArea, perimeter (Ramanujan) and eccentricity.Regular polygonArea, angles and radii of an n-sided regular shape.SphereVolume and surface area from the radius.CylinderVolume, surface area and capacity of a tube or tank.ConeVolume, slant height and surface area.Rectangular prism & cubeVolume, surface area and space diagonal of a box.PyramidVolume and surface area of a rectangular-based pyramid.Torus (ring)Volume and surface of a doughnut shape.Distance, midpoint & slopeEverything about the line between two points.Area & volume unit scalingHow area and volume change when a shape is scaled.