Free Online Mathematics Tools
Evaluate number-theory functions, distributions, special polynomials and coordinate geometry with online mathematics tools.
Choose the right tool
Choose number theory for exact integer relationships, distribution tools for a declared random model, and geometry tools for coordinates. A density is not a probability and a statistical model does not establish a real-world outcome.
Available tools
- Extended Euclidean Algorithm Calculator
Online extended euclidean algorithm calculator. Returns Bézout coefficients for nonnegative integers with positive second input.
- Modular Multiplicative Inverse Calculator
Online modular multiplicative inverse calculator. Requires coprime inputs and modulus at least two.
- Two Congruence Chinese Remainder Solver
Online two congruence chinese remainder solver. Positive coprime moduli at most 10000.
- Linear Modular Congruence Solver
Online linear modular congruence solver. Solves ax congruent to b modulo n; reports a base and step.
- Euler Totient Calculator
Online euler totient calculator. Positive integer up to one million in this interface.
- Carmichael Lambda Calculator
Online carmichael lambda calculator. Exponent of the multiplicative group of integers modulo n.
- Mobius Function Calculator
Online mobius function calculator. Square factors produce zero; input must be positive.
- Liouville Number Theory Function
Online liouville number theory function. Counts prime factors with multiplicity to determine parity.
- Integer Radical Calculator
Online integer radical calculator. Product of distinct prime factors, not a square root.
- Generalized Divisor Sum Calculator
Online generalized divisor sum calculator. Sums the supplied power over positive divisors.
- Distinct Prime Factor Count Calculator
Online distinct prime factor count calculator. Counts distinct factors rather than their multiplicities.
- Multiplicative Order Modulo N
Online multiplicative order modulo n. Coprime base and modulus; modulus at most 10000.
- Jacobi Symbol Calculator
Online jacobi symbol calculator. Denominator must be positive and odd.
- Legendre Symbol via Euler Criterion
Online legendre symbol via euler criterion. Euler criterion for an odd prime modulus; composite moduli are rejected.
- Prime Counting Function Calculator
Online prime counting function calculator. Sieve capped at one million.
- Nth Prime Number Calculator
Online nth prime number calculator. Index from one to 50000.
- Prime Numbers in an Interval
Online prime numbers in an interval. Inclusive integer range ending at most one million.
- Neighboring Prime Gap Calculator
Online neighboring prime gap calculator. Lower prime is at or below the input; upper prime is strictly above it.
- Twin Prime Pair Counter
Online twin prime pair counter. Counts pairs differing by two with both primes at or below limit.
- Goldbach Pair Finder
Online goldbach pair finder. Lists unordered prime pairs summing to an even integer up to one million.
- Coprime Integer List Generator
Online coprime integer list generator. Lists positive integers up to n coprime to n; n at most 10000.
- Aliquot Sum Calculator
Online aliquot sum calculator. Sum of positive divisors excluding the number itself.
- Collatz Steps and Peak Calculator
Online collatz steps and peak calculator. Positive integer with bounded iterations and exact-integer checks.
- Four Digit Kaprekar Routine
Online four digit kaprekar routine. Pads with leading zeros and reports convergence or a cycle.
- Catalan Number Calculator
Online catalan number calculator. Floating arithmetic; very large values lose integer precision.
- Derangement Count Calculator
Online derangement count calculator. Permutations with no fixed positions; n at most 170.
- Stirling Number of the Second Kind
Online stirling number of the second kind. Unlabeled nonempty groups; n at most 100.
- Bell Number Calculator
Online bell number calculator. Counts all partitions of a set; n at most 25.
- Integer Partition Count Calculator
Online integer partition count calculator. Unordered sums of positive integers; n at most 200.
- Repeated Item Permutation Calculator
Online repeated item permutation calculator. Three labeled item types; identical items within a type.
- Birthday Collision Probability Calculator
Online birthday collision probability calculator. Independent equally likely birthdays with no leap-day model.
- Coupon Collector Expected Draws
Online coupon collector expected draws. Independent uniform sampling with replacement.
- Hypergeometric Probability Calculator
Online hypergeometric probability calculator. Sampling without replacement; counts must be feasible.
- Negative Binomial Failure Count Probability
Online negative binomial failure count probability. Failures before the requested success count, with independent constant probability.
- Geometric First Success Probability
Online geometric first success probability. Trial index starts at one.
- Beta Distribution Density Calculator
Online beta distribution density calculator. Density is not a probability; endpoint singularities are rejected.
- Gamma Distribution Density Calculator
Online gamma distribution density calculator. Shape-scale parameterization with positive x.
- Lognormal Distribution Density Calculator
Online lognormal distribution density calculator. Mean and deviation describe the natural logarithm of the variable.
- Laplace Distribution Density Calculator
Online laplace distribution density calculator. Two-sided exponential density with positive scale.
- Cauchy Distribution Density Calculator
Online cauchy distribution density calculator. Location-scale parameterization; mean and variance are undefined.
- Student T Probability Density Calculator
Online student t probability density calculator. Reports density, not a tail p-value.
- Chi Square Density Calculator
Online chi square density calculator. Positive x and degrees of freedom; no significance decision.
- F Distribution Density Calculator
Online f distribution density calculator. Density under the specified degrees of freedom.
- Continuous Uniform Interval Probability
Online continuous uniform interval probability. Interval must be ordered; probability outside support is clipped.
- Triangular Distribution Density Calculator
Online triangular distribution density calculator. Requires lower less than mode less than upper.
- Binary Shannon Entropy Calculator
Online binary shannon entropy calculator. Entropy of a two-outcome distribution in bits.
- Binary Kullback Leibler Divergence
Online binary kullback leibler divergence. Both probabilities strictly between zero and one; direction matters.
- Binary Jensen Shannon Divergence
Online binary jensen shannon divergence. Equal mixture and probabilities strictly inside the unit interval.
- Binary Gini Impurity Calculator
Online binary gini impurity calculator. Classification impurity for two exhaustive classes.
- Probability Logit Calculator
Online probability logit calculator. Probability strictly between zero and one.
- Logistic Sigmoid Calculator
Online logistic sigmoid calculator. Maps real log odds to a probability.
- Softplus Activation Calculator
Online softplus activation calculator. Numerically stable natural-log softplus function.
- Three Class Softmax Calculator
Online three class softmax calculator. Subtracts the maximum logit for numerical stability.
- Poisson Exceedance Probability Calculator
Online poisson exceedance probability calculator. Probability strictly above an integer threshold; threshold at most 170.
- Benford Leading Digit Probability
Online benford leading digit probability. Theoretical first-digit distribution; does not prove fraud or data quality.
- Binary Hamming Ball Size Calculator
Online binary hamming ball size calculator. Counts binary words within the supplied integer Hamming distance.
- Pentagonal Number Calculator
Online pentagonal number calculator. Positive figurate-number index; zero returns zero.
- Hexagonal Number Calculator
Online hexagonal number calculator. Ordinary hexagonal figurate numbers, not centered hexagonal numbers.
- Tetrahedral Number Calculator
Online tetrahedral number calculator. Sum of the first n triangular numbers.
- Centered Hexagonal Number Calculator
Online centered hexagonal number calculator. Index starts at one.
- Lucas Number Calculator
Online lucas number calculator. Binet expression; floating accuracy decreases for large indices.
- Harmonic Number Calculator
Online harmonic number calculator. Finite reciprocal sum, n up to 100000.
- Alternating Harmonic Partial Sum
Online alternating harmonic partial sum. Starts with positive one; n up to 100000.
- Sum of Integer Powers Calculator
Online sum of integer powers calculator. Sums k to the supplied power for k from one to n.
- Consecutive Fibonacci Ratio Calculator
Online consecutive fibonacci ratio calculator. Ratio F(n+1)/F(n), with index at least one.
- Bounded Ackermann Function Calculator
Online bounded ackermann function calculator. Closed forms for m up to three and n up to ten.
- Positive Gamma Function Calculator
Online positive gamma function calculator. Lanczos approximation for positive real arguments.
- Euler Beta Function Calculator
Online euler beta function calculator. Positive real arguments; finite floating results only.
- Error Function Calculator
Online error function calculator. Polynomial approximation with typical absolute error below 2e-7.
- Normalized Sinc Function Calculator
Online normalized sinc function calculator. sin(pi x)/(pi x), defined as one at zero.
- Chebyshev Polynomial Value Calculator
Online chebyshev polynomial value calculator. First-kind recurrence; degree up to 1000.
- Legendre Polynomial Value Calculator
Online legendre polynomial value calculator. Ordinary Legendre recurrence, not associated polynomials.
- Laguerre Polynomial Value Calculator
Online laguerre polynomial value calculator. Ordinary Laguerre polynomials with alpha zero.
- Physicists Hermite Polynomial Calculator
Online physicists hermite polynomial calculator. Physicists convention H1(x)=2x, not the probabilists convention.
- Arithmetic Geometric Mean Calculator
Online arithmetic geometric mean calculator. Thirty positive-input iterations.
- Cubic Smoothstep Interpolation Calculator
Online cubic smoothstep interpolation calculator. Unit interval easing with zero endpoint slopes.
- Quintic Smootherstep Calculator
Online quintic smootherstep calculator. Unit interval easing with zero first and second endpoint derivatives.
- Logistic Growth Model Calculator
Online logistic growth model calculator. Constant carrying capacity; model parameters are supplied scenarios.
- Gompertz Growth Model Calculator
Online gompertz growth model calculator. Parametric growth model with supplied constants.
- Richardson Extrapolation Calculator
Online richardson extrapolation calculator. Requires refinement ratio above one and assumed asymptotic error order.
- Newton Square Root Iteration Calculator
Online newton square root iteration calculator. Demonstrates bounded iteration rather than arbitrary expression solving.
- Simpson Integration of Sine Calculator
Online simpson integration of sine calculator. Even subdivision count; integrates sine with radian bounds.
- Point to Infinite Line Distance Calculator
Online point to infinite line distance calculator. Line ax+by+c=0; coordinate units must match.
- Point Reflection Across a Line Calculator
Online point reflection across a line calculator. Infinite line ax+by+c=0; a and b cannot both be zero.
- Two Implicit Lines Intersection
Online two implicit lines intersection. Parallel or coincident lines have zero determinant and are rejected.
- Angle Between Two Planar Vectors
Online angle between two planar vectors. Nonzero two-dimensional vectors; angle from zero to 180 degrees.
- Planar Vector Projection Calculator
Online planar vector projection calculator. Projects the first vector onto the nonzero second vector.
- Rotate a Point Around a Pivot
Online rotate a point around a pivot. Counterclockwise rotation in Cartesian coordinates.
- Triangle Inradius from Three Sides
Online triangle inradius from three sides. Requires strict triangle inequalities.
- Triangle Circumradius from Three Sides
Online triangle circumradius from three sides. Nondegenerate triangle only.
- Triangle Coordinate Centroid Calculator
Online triangle coordinate centroid calculator. Uniform planar triangular lamina.
- Ellipse Eccentricity and Focal Distance
Online ellipse eccentricity and focal distance. Semimajor must be at least semiminor.
- Hyperbola Focal Distance Calculator
Online hyperbola focal distance calculator. Standard nondegenerate hyperbola with transverse semiaxis a.
- Parabola Focus and Directrix Calculator
Online parabola focus and directrix calculator. Vertex at origin for y=ax squared; a cannot be zero.
- Spherical Cap Surface Calculator
Online spherical cap surface calculator. Cap height must not exceed twice the sphere radius.
- Circular Cone Solid Angle Calculator
Online circular cone solid angle calculator. Cone half angle must lie between zero and 180 degrees.
- Catenary Height and Arc Length
Online catenary height and arc length. Symmetric ideal catenary with minimum height set to zero.
- Cycloid Point Calculator
Online cycloid point calculator. Circle rolls without slipping along the x axis.
- Circular Helix Length Calculator
Online circular helix length calculator. Pitch is axial advance per full revolution.
- Archimedean Spiral Radius Calculator
Online archimedean spiral radius calculator. Signed radial rate in distance per radian.