Math

Enter the components of an n-dimensional vector to compute its L2 (Euclidean), L1 (Manhattan), and L∞ (maximum) norms plus the unit vector.

Compute the dot product, cross product (3D), angle, and norms of two vectors A and B.

Add or subtract two vectors of the same dimension component by component, and see the resulting vector, its norm, and dimension.

Multiply a vector by a scalar k and get the result vector plus the original and scaled norms. Any dimension.

Enter an m×n matrix and instantly get its transpose as a table, with the original size, transposed size, and whether the matrix is symmetric.

Compute the outer (tensor) product of two vectors A and B as an m×n matrix where C(i,j)=aᵢ×bⱼ, with rows, columns, and rank.

Compute the Frobenius norm, 1-norm (max column sum), infinity-norm (max row sum), and maximum absolute value of a matrix.

Enter a matrix to compute its rank via Gaussian elimination (row reduction). Shows the number of rows and columns, full-rank check, and the reduced row echelon form (RREF).

Compute the complex conjugate of a matrix by replacing each entry a+bi with a−bi. Detects Hermitian matrices too.

Compute the adjugate (classical adjoint) of a square matrix — the transpose of the cofactor matrix — plus its determinant and the inverse relation A⁻¹=adj(A)/det(A).

Add or subtract two same-size matrices element by element. Toggle between sum and difference and view the result as a table.

Multiply a matrix by a scalar k, scaling every entry by k to get the result matrix.

Multiply an m×n matrix by an n-dimensional vector to get the m-dimensional product vector, with dimension checking.

Multiply an m×n matrix by an n×p matrix to get the m×p product, shown as a table with size checks.

Compute the Hadamard product (element-wise product) of two same-size matrices. Enter each row and get the result matrix as a table.

Enter a square matrix A and an exponent n to compute Aⁿ and view the resulting matrix. n=0 gives the identity matrix; negative n uses the inverse matrix.

Enter the nine entries of a 3×3 matrix to compute its determinant with the rule of Sarrus. See each term and whether the matrix is invertible.

Compute the determinant of any n×n square matrix using Gaussian elimination with partial pivoting, and check whether the matrix is invertible.

Compute the inverse of a 3×3 matrix from its adjugate and determinant. If the determinant is 0, the matrix has no inverse.

Compute the inverse of an n×n square matrix via Gauss-Jordan elimination, with determinant and invertibility check.

Solve a system of three linear equations in three unknowns (ax+by+cz=d on each of 3 rows) with Cramer's rule to find x, y and z, plus the determinant and uniqueness of the solution.

Enter the augmented matrix [A|b] and this tool solves the system of linear equations Ax=b by Gauss-Jordan elimination. It detects a unique solution, no solution (inconsistent), or infinitely many solutions, and shows the solution vector and reduced row echelon form.

Solve a complex-coefficient system of linear equations Ax=b by entering the augmented matrix. Uses complex-number Gaussian elimination and reports the complex solution vector, number of variables, and uniqueness.

Decompose a square matrix into lower (L) and upper (U) triangular factors with partial pivoting (PA=LU), showing determinant and permutation.

Enter a matrix to compute its singular value decomposition A=UΣVᵀ: the singular values, numerical rank, and condition number, plus the U and V matrices for small inputs.

Enter a real number x to compute the reciprocal gamma function 1/Γ(x). It is an entire function defined for all real x, equal to exactly 0 at zero and the negative integers. Shows Γ(x) for reference plus a curve.

Compute the lower γ(s,x) and upper Γ(s,x) incomplete gamma plus regularized P(s,x) and Q(s,x) using series and continued fractions.

Compute the log gamma function lnΓ(x) for any real x using the Lanczos approximation. It also reports the value and sign of Γ(x) and handles negative non-integers via the reflection formula.

Enter an order n and argument x to evaluate the polygamma function ψ⁽ⁿ⁾(x), which becomes the digamma function when n=0, using recurrence and asymptotic expansion.

Enter x, a, b to compute the regularized incomplete beta function I_x(a,b) and B(x;a,b) via continued fractions.

Solve I_x(a,b)=p for x using Newton and bisection methods. Handy for beta distribution quantiles.

Evaluate the Riemann zeta function ζ(x) for a real argument. For x above 1 it uses a series plus the Euler–Maclaurin formula; for x below 1 it uses the functional equation (reflection formula), with notes on special values, the pole at x=1, and a graph.

Evaluate the alternating zeta series η(x)=Σ(-1)^(n-1)/n^x. Also shows the relation to the zeta function η(x)=(1-2^(1-x))ζ(x) and η(1)=ln2.

Compute the error function erf(x) and complementary error function erfc(x) for any real x with high precision, using series and continued fractions, with a curve graph.

Compute the inverse error function erf⁻¹(y) and inverse complementary erfc⁻¹(y) with Newton iteration. See the verification erf value and the convergence of each step.

Enter a degree n and x to evaluate the Legendre polynomial Pₙ(x) via recurrence. Shows neighboring degrees, the derivative, the coefficient list, and a graph.

Enter degree n, order m, and x to evaluate the associated Legendre function Pₙᵐ(x) via stable recurrence. A building block of spherical harmonics.

Enter an integer order n and x to compute the Bessel function of the first kind Jₙ(x) and the second kind Yₙ(x), with a table of adjacent orders and a graph.

Find the positive zeros of the Bessel functions Jn(x) and Yn(x) in ascending order. Choose the kind, order, and count to see the results in a graph and a table.

Enter an order n and x to evaluate the modified Bessel functions of the first kind Iₙ(x) and second kind Kₙ(x), with neighboring orders, a graph and an order-by-order table.

Compute the Hankel functions H^(1)ₙ(x)=Jₙ+iYₙ and H^(2)ₙ(x)=Jₙ−iYₙ as complex numbers from order n and x, with magnitude, argument and the Bessel values Jₙ, Yₙ.

Enter an order n and x to evaluate the spherical Bessel functions of the first kind jₙ(x) and second kind yₙ(x), with neighboring orders shown in a table.

Find the positive zeros of the spherical Bessel functions jₙ(x) or yₙ(x) in ascending order. Choose the kind, order, and count to list Newton-refined roots in a table and chart.

Enter an order n and x to compute the modified spherical Bessel functions of the first kind iₙ(x) and second kind kₙ(x), with a table of neighboring orders and a graph.

Enter the order n and argument x to compute the spherical Hankel functions h(1)n(x)=jn+i yn and h(2)n(x)=jn−i yn as complex numbers, along with the magnitude and the spherical Bessel components.

Enter x to compute the Airy functions Ai(x) and Bi(x). Evaluated with series and asymptotic expansions, with a curve chart.

Find the negative real zeros aₙ and bₙ of the Airy functions Ai(x) and Bi(x). Computed via asymptotic estimates refined with Newton's method, shown in a table.

Enter a real number x to compute the first derivatives of the Airy functions, Ai'(x) and Bi'(x), via power series. Reference values of Ai(x) and Bi(x) and a graph of Ai'(x) are also shown.

Find the negative real zeros of the Airy derivative Ai'(x) or Bi'(x). Enter how many you need and get them in ascending order, computed with asymptotic estimates refined by Newton's method, shown in a table and graph.

Enter x to evaluate the first-kind Kelvin functions ber(x) and bei(x) and the second-kind ker(x) and kei(x) by series, with a plot of ber and bei.

Enter the order n and argument x to evaluate the Struve function Hₙ(x) via its power series. The related Bessel function Jₙ(x) and a curve plot are shown too.

Enter the order n and x to compute the modified Struve function Lₙ(x) via its series expansion, with a graph.

Enter an order n and a positive x to compute the exponential integral Eₙ(x)=∫₁^∞ e^(-x t)/tⁿ dt with high accuracy, using series and continued fractions, plus neighboring orders and ratios.

Compute the exponential integral Ei(x) (principal value) with a convergent series and asymptotic expansion. Shows its relation to E1(x) and a curve graph.

Compute the logarithmic integral li(x) and the offset logarithmic integral Li(x). This special function approximates the prime-counting function and is evaluated via the exponential integral Ei.

Enter x to compute the sine integral Si(x) and cosine integral Ci(x) with high precision using series and asymptotic expansions, plus a Si(x) graph.

Enter x to compute the hyperbolic sine integral Shi(x) and hyperbolic cosine integral Chi(x) via power series, with a curve plot.

Compute the Fresnel integrals S(x) and C(x) and plot the Euler spiral (clothoid).

Enter a degree n to get the Bernoulli number Bₙ as an exact fraction and a decimal, plus a table of B0 through Bn.

Enter an index n to compute the Euler number Eₙ exactly via a recurrence, with a table of E0 through Eₙ.

Enter an integer n to compute the double factorial n!! (n·(n-2)·(n-4)…) exactly. See the parity, the relation to the ordinary factorial n!, the digit count, and scientific notation at a glance.

Enter n and k to compute the unsigned c(n,k) and signed s(n,k) Stirling numbers of the first kind via recurrence. Exact BigInt arithmetic for large values.

Enter a maximum order N to generate the full triangle of unsigned Stirling numbers of the first kind c(n, k) from n=0 to N. Each row sums to n factorial.

Compute the Stirling number of the second kind S(n,k), the number of ways to partition n elements into k non-empty subsets, with the related Bell number.

Enter a maximum order N to generate the triangular table of Stirling numbers of the second kind S(n,k) from n=0 to N, with the row sums (Bell numbers) shown alongside.

Enter an index n to get the Fibonacci number Fn, plus the neighboring terms, the golden-ratio approximation, and the recent sequence.

Compute the Pochhammer symbol, the rising factorial (x)ₙ=x(x+1)…(x+n-1). Also shows the falling factorial and term expansion, with exact integer results for integer inputs.

Compute the Brillouin function B_J(x) from total angular momentum J and argument x, with saturation and paramagnetism context.

Evaluate the Langevin function L(x)=coth(x)-1/x, with the small-x approximation x/3, an inverse-Langevin estimate, and a curve plot.

Enter x to compute the sigmoid σ(x)=1/(1+e^(-x)) with its first and second derivatives, plus a curve chart.

Compute the softplus function softplus(x)=ln(1+eˣ), its derivative (the sigmoid), and the gap from ReLU. Visualize this machine learning activation with a built-in graph.

Enter a numeric vector to compute the softmax probability of each component (summing to 1) and the component with the highest probability.

Compute the ReLU function relu(x)=max(0,x) and its derivative. Visualize this popular neural network activation function.

Compute the softsign function softsign(x)=x/(1+|x|) and its derivative, compared with tanh as an ML activation.

Compute the Leaky ReLU value f(x) and its derivative from an input x and slope alpha. Returns x when x is non-negative and alpha times x when negative, with a graph.

Compute the hyperbolic tangent tanh(x) with its first and second derivatives, the related sigmoid value, and a curve plot.

Enter the modulus k to compute the first kind complete elliptic integral K(k) and the second kind E(k) with high precision using the arithmetic geometric mean (AGM) method.

Enter the characteristic n and modulus k to compute the complete elliptic integral of the third kind Π(n,k) with high precision. Also shows K(k) and a curve over the modulus.

Compute the incomplete elliptic integral of the first kind F(φ,k) from amplitude φ and modulus k using the Carlson symmetric form, with the K(k) ratio.

Compute the incomplete elliptic integral of the second kind E(φ,k) from amplitude φ and modulus k using Carlson symmetric forms, with the complete integral E(k) and an integrand graph.

Enter the amplitude phi, characteristic n, and modulus k to compute the incomplete elliptic integral of the third kind Pi(phi,n,k) using Carlson symmetric forms.

Compute the Jacobi elliptic functions sn(u,k), cn(u,k), dn(u,k) from argument u and modulus k. Also shows derived functions sd, cd, nd, the parameter m=k^2, and an sn graph.

Enter u and the modulus k to compute the Jacobi amplitude am(u,k) in degrees and radians, along with the elliptic functions sn, cn, and dn.

Compute the inverse Jacobi elliptic functions arcsn(x,k), arccn(x,k), and arcdn(x,k). Enter the function type, argument x, and modulus k to get the u value expressed via the incomplete elliptic integral of the first kind.

Enter degree n, parameter lambda, and x to evaluate the ultraspherical polynomial C_n^lambda(x) via recurrence, with neighbor values and a curve.

Evaluate the physicists Hermite polynomial Hₙ(x) by a three term recurrence. See neighbouring degrees, the derivative, the probabilists Heₙ(x), a coefficient table and a graph.

Enter the degree n and x to evaluate the first-kind Chebyshev polynomial Tₙ(x) with the three-term recurrence. Shows neighboring degrees, the derivative, a coefficient table, and a graph.

Enter degree n and x to evaluate the Chebyshev polynomial of the second kind Uₙ(x) by recurrence. Shows neighboring degrees, derivative, coefficient table, and a plot.

Enter the degree n and x to compute the Laguerre polynomial Lₙ(x) with a three-term recurrence, plus neighboring degrees, the derivative and the coefficient list.

Enter degree n, parameter alpha, and x to evaluate the associated (generalized) Laguerre polynomial Lₙᵅ(x) via a stable three-term recurrence. Shows neighbors, derivative, coefficients, and a plot.

Enter the degree n, parameters α and β, and a point x to evaluate the Jacobi polynomial P_n^(α,β)(x) with a three-term recurrence, plus its derivative, a per-degree value table, and a plot.

Compute the real part, imaginary part, magnitude, and normalization of the spherical harmonic Y(l,m) from degree l, order m, and the angles theta and phi.

Compute the Gauss hypergeometric function 2F1(a,b;c;z) by power series. Enter a, b, c, and z to see the value, convergence status, term count, and how the partial sums settle.

Enter the parameters a and c with the argument z to compute the confluent hypergeometric function of the first kind (Kummer function) M(a;c;z)=₁F₁(a;c;z) from its power series, with term count and convergence shown.

Compute the Tricomi function U(a;c;z) from the connection formula using Kummer's function M. Enter a, c, and z (z>0) to get U and its related M values.

Enter parameters kappa, mu and positive z to compute the first kind M and second kind W Whittaker functions via confluent hypergeometric functions.

Enter one value, the leg, hypotenuse, or area, to find the area, hypotenuse, perimeter, height, inradius, and circumradius of a right isosceles triangle.

Enter any two values, two sides or one side and one acute angle, to solve a right triangle for its sides, area, perimeter, angles, height, and inradius and circumradius.

From the equal leg and either the base or the apex angle, find the area, perimeter, height, base and apex angles, and the inradius and circumradius.

Enter three sides, or two sides with the included angle, to compute area (Heron formula), perimeter, all three angles, every height, the inradius and circumradius, and the centroid. The triangle inequality is checked automatically.

Find the area of a triangle from its base and height. The formula is area = base times height divided by 2, shown with a clear diagram of the triangle.

Find the area of a triangle from two side lengths and their included angle using (1/2)ab sin C. Also shows the opposite side and perimeter.

Enter the three sides a, b and c to compute the area of a triangle with Heron's formula. It also shows the perimeter and semi perimeter s, and draws the triangle to scale.

Find the area, the remaining two sides, and the third angle of a triangle from one side and its two adjacent angles (ASA) using the law of sines.

Find the area, perimeter, and diagonal of a square from its side, area, or diagonal.

Enter the height and width to instantly find a rectangle area, perimeter, and diagonal.

Find the area and midline of a trapezoid from its two bases and height.

Find the area of a rhombus from its two diagonals (p and q), or from a side and height. Side length, perimeter and diagonals are computed too.

Find the area of a parallelogram from its base and height. Enter both values and see the base times height result with a clear diagram.

Find the area of a parallelogram from two sides and the included angle using ab sin θ. Also shows the perimeter and both diagonals with a scaled diagram.

Compute the area of a regular polygon from the number of sides and the side length, plus perimeter, apothem, circumradius and interior angle.

Enter the number of sides and the area of a regular polygon to get the side length, perimeter, inradius, and circumradius.

Calculate the area of a circle from its radius or diameter. Also shows circumference and diameter.

Enter a circle area to instantly find its radius, diameter, and circumference, shown with a clear diagram.

Find the area, arc length, and perimeter of a circular sector from its radius and central angle. Works in degrees and radians.

Find the area of a circular segment from its radius and central angle. Chord length, arc length, and height (sagitta) are calculated too.

Calculate the area of a circular segment from the circle radius and the height (sagitta). It also finds the central angle, chord length, and arc length, with a diagram.

Enter the chord length and height (sagitta) to instantly find a circular segment area, plus its radius, central angle, and arc length.

Find the area of an ellipse from its semi major axis a and semi minor axis b, plus its perimeter, eccentricity and focal distance.

Enter the four sides a, b, c, d to find the area of a cyclic quadrilateral using Brahmagupta formula. It also returns the perimeter, semiperimeter, and circumradius with a scaled diagram.

Compute the area of any quadrilateral from its four side lengths and the sum of two opposite angles using the Bretschneider formula. Perimeter and semiperimeter are shown too.

Enter the semi-major and semi-minor axes plus the start and end angles from the center to find the area of an elliptical sector, with the full ellipse area and swept fraction.

Find the area of an elliptical segment cut off by a vertical chord from the semi-major and semi-minor axes and cut position.

Find the area of a parabolic segment bounded by a chord and a parabola from the chord width (base) and height. The area equals two thirds of base times height, with ratio and arc length too.

Find the area of a hyperbolic segment bounded by x²/a²−y²/b²=1 and a vertical chord, computed by numerical integration from a, b and the cut position x.

Find the area of a hyperbolic sector of x²/a²−y²/b²=1 from the semi-axes a and b and the parameter t or a point on the curve. Area = (a·b/2)·t.

Find the volume of a cube from its edge length, plus surface area, space diagonal, and total edge length.

Find a cube edge length from its volume, plus surface area and diagonals.

Enter length, width, and height to find the volume of a rectangular box, plus its surface area and space diagonal.

Find the volume of a tetrahedron (triangular pyramid) from base area and height, or from the three base sides and height, with a shape diagram.

Find the volume of a regular tetrahedron from its edge length, plus surface area, height, circumradius and inradius.

Find the edge length of a regular tetrahedron from its volume, plus surface area, height, and circumscribed and inscribed sphere radii.

Calculate the volume of a regular triangular prism from its base edge and height. Base area, lateral area, and surface area are computed too.

Find the volume of a square prism from its base edge and height. Base area, surface area, and the space diagonal are computed too.

Find the volume of a regular hexagonal prism from its base edge and height. Base area, surface area and lateral area are computed too.

Find the volume of a right square pyramid from its base edge and height, plus base area, slant height, lateral area and surface area.

Find the volume, height, slant height, and surface area of a right square pyramid from its base edge and lateral edge.

Find the volume of a square frustum from the bottom edge, top edge, and height. Surface area and slant height are computed too.

Enter the base area and height to find the volume of a pyramid using V equals one third times base area times height. Works for any pyramid.

Find the volume of a pyramid frustum from its lower base area, upper base area, and height using V=(h/3)(S1+S2+square root of S1 S2).

Find the volume of a wedge with a rectangular base from its sides, ridge, and height.

Find the volume of a right circular cylinder from its radius and height (πr²h). Surface area, lateral area, base area and diameter are computed too.

Find the volume, wall thickness, and surface area of a hollow cylinder (pipe) from its outer radius, inner radius, and height.

Compute the volume of a right circular cone from its radius and height, plus slant height and surface area.

Find the volume of a conical frustum from its bottom radius, top radius, and height. Slant height, lateral area, and total surface area are computed too.

Find the volume of an elliptic cylinder from its semi major axis, semi minor axis, and height. Base area, lateral area, and surface area are included.

Find the volume and lateral area of a cylinder with a slanted top from its radius and minimum and maximum heights.

From the radius and the two heights of the short and tall sides, work out the volume, lateral area, slanted elliptical top area and surface area of an obliquely cut cylindrical solid.

Find the volume of an elliptic cone from its semi-major axis, semi-minor axis, and height. Base area included.

Find the volume of an elliptic cone frustum from the semi-axes of its bottom and top elliptical faces and its height. Shows both base areas too.

Find the volume of a sphere with V = 4/3 pi r cubed, plus surface area, diameter and great-circle circumference from the radius or diameter.

Compute the volume, base circle radius, curved surface area, and total surface area of a spherical cap from the sphere radius and cap height.

Find the volume and lateral surface area of a spherical zone, the middle slice of a sphere cut by two parallel planes, from its two base radii and height.

Enter the center radius R and tube radius r to compute a torus (donut) volume with V = 2 pi squared R r squared, plus surface area, outer diameter, and inner diameter.

Find the volume of an ellipsoid from its three semi-axes a, b, and c, with an approximate surface area too.

Find the volume of the five Platonic solids from one edge length. Tetrahedron, cube, octahedron, dodecahedron and icosahedron, with surface area, circumscribed and inscribed sphere radii, and face vertex edge counts.

Find the height of an equilateral triangular prism from its side length and volume. Also shows base area, lateral area, and total surface area.

Find the height of a square prism from its base side length and volume. Base area and surface area are computed as well.

Find the height of a regular hexagonal prism from its side length and volume, plus base area and surface area.

Compute the volume and surface area of an n-dimensional sphere (hypersphere) from the dimension n and radius r using the gamma function.

Find the volume of a cylindrical wedge (ungula) cut from a right circular cylinder by a slanted plane through a base diameter, from the radius and height. Base, curved, and cut areas are shown too.

Find the volume of a cylinder cut by a slanted plane from its radius and minimum and maximum heights using the mean height.

Find the volume of the solid formed by revolving a circular segment about an axis, using the Pappus theorem from radius, central angle, and axis distance.

Find the inscribed circle radius, incenter coordinates, area, and semiperimeter of a triangle from its three sides a, b, c.

Enter the three sides a, b, c of a triangle to find the circumscribed circle radius, diameter, area, plus the triangle area and angles, with a diagram.

Find the inscribed circle radius (apothem) of a regular polygon from its number of sides and side length. Also shows the incircle area, polygon area, and their ratio with a diagram.

Find the circumscribed circle radius and area of a regular polygon from its side count and side length, plus polygon area and fill ratio.

Find the circumscribed circle radius of a rectangle from its height and width, plus the diagonal, circle area, and diameter.

Find the equation of the line through two points: slope, y-intercept, general form ax+by+c=0, distance and midpoint, plotted on a coordinate plane.

Enter the x-intercept a and y-intercept b to find the slope, equation, general form, distance from the origin, and intercept triangle area of a line, with a coordinate diagram.

Solve a1x+b1y=c1 and a2x+b2y=c2 with Cramer's rule to find the intersection point. Detects parallel and coincident lines, and shows each slope, the angle between the lines, and a coordinate graph.

Enter the coordinates of three vertices to find the triangle area with the shoelace formula. Also shows the perimeter, centroid, and each side length with a coordinate diagram.

Enter vertex coordinates in order to compute a polygon area with the shoelace formula. Also returns vertex count, perimeter, and centroid, handles concave shapes, and draws a coordinate diagram.

Rotate a coordinate point around any center by a given angle and find the new coordinates. Uses the rotation matrix and shows the point before and after on a coordinate diagram.

Find the new coordinates of a point when the coordinate axes are rotated by an angle. Enter the original point and rotation angle to get the new coordinates with a diagram.

Convert Cartesian coordinates (x, y) into polar coordinates (r, θ). See the radius r and angle θ in both degrees and radians, with a coordinate diagram.

Convert polar coordinates (r, θ) into Cartesian coordinates (x, y). Enter the angle in degrees or radians and see the point on a coordinate plane.

Find the area, side length, perimeter, and circle area ratio of a regular polygon inscribed in a circle from its circumradius and number of sides.

From a circle radius and side count, compute the area, side length, perimeter, and area ratio of the regular polygon circumscribed about the circle.

Divide a length by the golden ratio phi=(1+sqrt5)/2 about 1.618. Enter the whole, the longer part, or the shorter part to get the others and a golden rectangle diagram.

Find the shortest distance between two lines in 3D space from each line point and direction vector. Detects skew, parallel and intersecting cases and shows closest points.

Find the equation of the plane ax+by+cz+d=0 that passes through three points in 3D space, with the normal vector and distance from the origin.

Find the volume of a tetrahedron from the coordinates of its four vertices using the scalar triple product, plus each face area, centroid, surface area and total edge length.

Compute the distance from a point (x0,y0,z0) to the plane ax+by+cz+d=0, with the foot of the perpendicular and the unit normal.

Convert Cartesian coordinates (x, y, z) to spherical coordinates (r, θ, φ). Shows the radius r, polar angle (colatitude) θ, and azimuthal angle φ in both degrees and radians.

Convert Cartesian coordinates (x, y, z) into cylindrical coordinates (ρ, φ, z). Get the radial distance, azimuth in degrees and radians, and height at once.

Convert spherical coordinates (r, θ, φ) into Cartesian coordinates (x, y, z). Polar and azimuthal angles support both degrees and radians.

Convert spherical coordinates (r, θ, φ) into cylindrical coordinates (ρ, φ, z). Computes ρ = r sinθ and z = r cosθ instantly, with angles shown in degrees and radians.

Convert cylindrical coordinates (rho, phi, z) into Cartesian coordinates (x, y, z). Supports degrees and radians and shows an xy-plane projection.

Convert cylindrical coordinates (ρ, φ, z) to spherical coordinates (r, θ, φ). Get the radius r, polar angle θ in degrees and radians, and azimuth φ with a cross-section diagram.

Rotate a point (x, y, z) about the x, y, and z axes by given angles and get the rotated coordinates.

Enter a term index n to compute the n-th tetrahedral number n(n+1)(n+2)/6. Shows the matching triangular number, the first term, and a table of the sequence.

Enter an angle in degrees to instantly compute all six trigonometric ratios sin, cos, tan, csc, sec, and cot, plus the radian conversion, with a unit circle diagram.

Find arcsin, arccos, and arctan of a value x in degrees and radians, with a unit circle.

Compute sin, cos, tan, csc, sec and cot from an angle in radians, with a degree conversion and a unit circle diagram.

Compute arcsin, arccos, and arctan of a value x in radians and degrees, with domain and range for each function.

Enter one acute angle and the base to instantly get the hypotenuse, height, area, and trig ratios.

Enter an acute angle and the height (opposite side) to find a right triangle base, hypotenuse, area, and trig ratios.

Enter an acute angle and the hypotenuse to get the base, height, area, and trig ratios of a right triangle.

Enter the base and height of a right triangle to find the hypotenuse, angle, area, and trig ratios.

Enter the base and hypotenuse of a right triangle to find its height, angle, area, and trig ratios.

Enter the height and hypotenuse to find the base, angle, and area. Solves a right triangle with the Pythagorean theorem and arcsine.

Enter the three sides a, b and c of a triangle to find all three interior angles using the law of cosines, plus area and a diagram.

Enter x to compute sinh, cosh and tanh along with csch, sech and coth at once. Includes a graph and a check of the identity cosh squared minus sinh squared equals 1.

Enter a number and a base to compute log_b(x). It also shows the natural log ln, common log log10, and binary log log2, with a curve graph.

Enter a base a and an exponent x to compute the exponential function a^x. It also returns e^x, 2^x, and 10^x with a curve graph.

Find the square root, cube root, and nth root of any number at once. Supports real roots of negative numbers with odd indices, with a graph.

Enter a base a and an exponent b to compute a to the power of b. Supports integers, decimals and negative exponents, and also shows the square, cube and reciprocal.

Enter a non-negative integer n to compute the factorial n! exactly, along with its digit count, Stirling approximation, and the previous factorial. Large values are shown in scientific notation.

Find the conjugate, modulus, and argument of a complex number z = a + bi, shown on the complex plane.

Add, subtract, multiply, or divide two complex numbers z1=a+bi and z2=c+di, then see the modulus, argument, and the result plotted on the complex plane.

Find the greatest common divisor (GCD) and least common multiple (LCM) of several integers using the Euclidean algorithm, with a step table and coprime check.

Enter a positive integer to get its prime factorization, plus the divisor count, divisor sum, full list of divisors, and a primality check.

Enter coefficients a and b to solve the linear equation ax+b=0. Get x=−b/a instantly, including the special cases when a is zero.

Solve a x²+b x+c=0 with the quadratic formula. The discriminant detects real, double, and complex roots, and the vertex, axis, and parabola graph are shown.

Enter the coefficients of ax³+bx²+cx+d=0 to find all three roots (real and complex) with Cardano's method, plus the discriminant and the number of real roots.

Solve ax⁴+bx³+cx²+dx+e=0 for all four roots. Real and complex roots are shown in x+yi form, with the count of real roots.

Enter coefficients from the highest degree and get every root of an n-degree polynomial, including complex roots, via the Durand-Kerner method.

Enter a linear inequality of the form ax+b compared with 0 to find the solution range, boundary value, and direction, shown on a number line.

Enter a x squared plus b x plus c and an inequality sign to find the solution range. It uses the discriminant and the parabola direction, shown with a graph.

Solve a cubic inequality ax³+bx²+cx+d against 0 and see the solution as intervals on a number line.

Solve a quartic inequality ax to the 4th plus bx cubed plus cx squared plus dx plus e. Get the real roots and the solution intervals on a number line.

Solve a system of two linear equations a1x+b1y=c1 and a2x+b2y=c2 with Cramer's rule. Get the solution (x, y), classify it as unique, none, or infinite from the determinant, and see the two lines plotted.

Find the sum of an arithmetic series from the first term, common difference, and number of terms.

Find the sum and last term of a geometric series from the first term, ratio, and count.

Add up natural numbers, squares, cubes, or odd numbers from the closed-form formula by entering the number of terms.

Enter the first term a and common ratio r to find the sum of the infinite geometric series a + a r + a r² + …. It computes S = a / (1 − r) when the absolute value of r is below 1 and tells you whether the series converges or diverges.

Divide or scale a length by the silver ratio (1+sqrt2 ≈ 2.414) or the paper ratio 1:sqrt2, with a silver rectangle diagram.

Compute the number of ways to choose r items from n without regard to order, with the related permutation and symmetry.

Enter a maximum n and see every nCr value for n from 0 up to that maximum, laid out as a lower-triangular table. It is Pascal's triangle at a glance.

Count how many ways to choose r items from n types when repetition is allowed and order does not matter.

Enter a maximum n and a maximum r to build a reference table of nHr, the number of ways to choose r items from n types with repetition allowed. Values are computed exactly with BigInt.

Calculate the permutation nPr, the number of ordered arrangements of r items chosen from n. Also shows nCr and n factorial.

Enter a maximum n to generate a table of permutation values nPr for every n and r. Computed exactly with BigInt.

Compute permutations with repetition n^r for choosing and arranging r items from n types. Also finds the number of possible PIN codes.

Build a table of permutations with repetition, n to the power r, across a range of n and r values. Enter the maximum n and maximum r to see every value at once.

Find the number of circular permutations of n distinct items, which is (n minus 1) factorial. Also shows the linear permutation n factorial and the ratio.

Enter a maximum number of items n to list the count of circular permutations (n−1)! for n = 1 up to that value. Counts arrangements around a circle where rotations are treated as identical, computed exactly with BigInt.

Count necklace permutations (n−1)!/2 for n items arranged in a circle where flips are treated as identical, compared with circular and linear permutations.

Enter a maximum n to list the necklace permutation count (n−1)!/2 for every size from n=3 up to n. Computed exactly with BigInt.

Compute the nth square pyramidal number n(n+1)(2n+1)/6, the running sum of squares, shown with a sequence table.

Find the quotient and remainder of a divided by b (a mod b). Shows the truncated, floored, and Euclidean definitions, plus how negative numbers behave.

Calculate the catenary curve of a chain or cable hanging between two supports. Find the sag, arc length, and tension ratio from the parameter or arc length, with a drawn curve.

Enter y=ax²+bx+c to find the vertex, focus, directrix, axis, latus rectum, and intercepts, with a parabola plot.

Compute the arithmetic-geometric mean (AGM) of two positive numbers a and b by iterating the arithmetic and geometric means until they converge, with the full iteration table.

Find the modular inverse x of an integer a modulo m (a times x is congruent to 1 mod m) using the extended Euclidean algorithm, with an existence check via gcd and full working.

Approximate pi by bounding it between inscribed and circumscribed regular polygons of a unit circle. Watch Archimedes method converge as the side count doubles, step by step.

Approximate pi using Viete infinite product formula. Enter the number of terms to see each partial product and the error against the true value of pi.

Approximate pi using the accelerated series inspired by the Japanese wasan mathematician Takebe Katahiro. Enter a term count to see the approximation and its error.

Approximate pi with Machin's formula pi/4 = 4 arctan(1/5) − arctan(1/239). Choose the number of Taylor-series terms to see the approximation, the error against true pi, the matching digits, and the convergence table.

Approximate pi using the three-term Machin-like formula pi/4 = arctan(1/2) + arctan(1/5) + arctan(1/8). Choose the number of series terms and see each arctan value, the error against true pi, the correct digits, and the convergence table.

Approximate pi quickly with a Machin-like formula built from four arctangent terms. Set the number of series terms to see the approximation, the error against true pi, the matching digits, and each term broken down.

Approximate pi with Ramanujan's rapidly converging series. Choose how many terms to add and see the approximation, the error against true pi, the correct digit count, and a term-by-term convergence table.

Compute pi using the Gauss-Legendre algorithm based on the arithmetic-geometric mean (AGM). Pick the iteration count to see each approximation, its error, correct digits, and the convergence table.

Square integers ending in 5 such as 35 or 65 using the Indian (Vedic) mental math trick. It breaks down each step: multiply the tens part by the next number, then append 25.

Subtract numbers using the Indian complement method. See each step broken down so you can do borrow-free subtraction in your head.

Multiply two numbers from 11 to 19 using the Indian mental-math trick. See each step: add one number to the other's ones digit, times ten, plus the product of the ones digits.

Multiply two numbers close to 100 using their differences from the base 100, with a clear step-by-step mental-math breakdown.

Multiply two two-digit numbers with different tens digits using the Indian rounding method: round one factor to a clean ten, then adjust by the difference. Shows every step.

Multiply by 99, 999 and other strings of nines using the complement trick N times 100 minus N, with each step shown.

Multiply two numbers close to a base of 200 to 900 using the Indian mental-math method, with each step of the cross-sum and correction shown.

Multiply two numbers the Indian box method way: split each number by place value, fill a grid with partial products, then add them up. Shows the full grid and steps.

Multiply by drawing lines for each digit and counting where they cross. This calculator recreates the line method (crossing technique) with a diagram, per place intersection counts, and the product.

Multiply two numbers digit by digit using the Indian vertically-and-crosswise method. Each place value is built from crosswise digit products with carries, shown step by step in a table.

Find the quotient and remainder when dividing by 9 using the Indian mental-math running-sum method. Shows the steps and verifies with ordinary division.