Math & Algebra Calculators

Calculate percent of a number, part-to-whole percentages, percent change, missing wholes, and reverse percent changes.

Find slope from two points, solve a missing coordinate from a known slope, or turn an angle or percentage grade into a slope. Includes equations, intercepts, perpendicular slope, graph, and step-by-step work.

Calculate sale price, stacked discounts, original price before a discount, or the discount percent between two prices.

Add, subtract, multiply, divide, simplify, compare, or convert decimals to fractions with steps and visual fraction bars.

Calculate percent error from experimental and accepted values with absolute error, signed error, and step-by-step substitution.

Find percentage increase or decrease between an original value and a new value with signed change, absolute change, and steps.

Simplify ratios, compare two quantities, and find unit rates with clear proportional reasoning steps.

Solve a missing value in a proportion using cross multiplication with clear substitution steps.

Convert regular numbers to scientific notation and show coefficient, exponent, and engineering notation.

Round a number to a chosen number of significant figures and see the rounding place and scientific notation.

Calculate square roots, simplify exact radicals for whole numbers, and identify perfect squares.

Calculate powers with positive, negative, and decimal exponents while showing the exponent rule used.

Calculate logarithms in any valid base with common log, natural log, and change-of-base steps.

Find the greatest common factor and least common multiple for a list of positive integers with factor steps.

Factor a positive integer into primes and show the factor tree style division steps.

Solve a 2x2 linear system with elimination/determinant steps for x and y.

Solve any triangle from SSS, SAS, ASA, AAS, or SSA. Get all sides and angles, area, perimeter, heights, medians, inradius, circumradius, classification, a scale drawing, and both solutions in the ambiguous case.

Solve a² + b² = c² for the hypotenuse or either leg with exact radical form (√50 = 5√2) and decimals, plus area, perimeter, both acute angles, a Pythagorean-triple check, steps, and a squares-on-sides diagram.

Solve a right triangle from any two values — two sides, a side and an angle, or area or perimeter with a side or angle. Get all sides, both acute angles, area, perimeter, altitude to the hypotenuse, inradius, circumradius, and steps.

Enter any one side of a 30-60-90 triangle and get the other two in exact radical form (x, x√3, 2x) and decimals, plus the area, perimeter, and height with the ratio steps written out.

Enter the leg or hypotenuse of a 45-45-90 triangle and get the other side in exact radical form (x, x, x√2) and decimals, plus area, perimeter, and the height to the hypotenuse with steps.

Find the area of a triangle from its three sides with every Heron's formula step: semiperimeter, the (s − a)(s − b)(s − c) product, the square root in exact form when possible, plus the three heights derived from the area.

Find the hypotenuse of a right triangle from two legs (c = √(a² + b²)) or from one leg and an acute angle, in exact radical form and decimal, with the other leg, both angles, and area included.

Find the height (altitude) of a triangle to any chosen base from three sides (via Heron's formula), from base and area, or from two sides and the included angle. Shows all three heights, the area, and where the altitude lands.

Solve an isosceles triangle from any two of legs, base, apex angle, base angle, height, area, or perimeter. Get every side, angle, height, area, perimeter, inradius, and circumradius with the steps.

Enter the side, height, area, perimeter, inradius, or circumradius of an equilateral triangle and get every other measurement with exact radicals (s√3/2, s²√3/4) and decimals, plus the steps.

Find the scale factor between two similar triangles from one pair of corresponding sides (or enter it directly), then get every missing side, both perimeters, both areas, and the perimeter and area ratios with steps.

Check whether three side lengths can form a triangle by testing all three inequalities, or enter two sides to get the exact range the third side must fall in — with the degenerate (flat) cases explained.

Solve a missing side or angle with the law of sines a/sin A = b/sin B. Flags the SSA ambiguous case and returns both possible angles, the third angle, the remaining side, and the classroom-style steps.

Generate every Pythagorean triple up to a hypotenuse limit (primitive or all), check whether three numbers form a triple and find its Euclid parameters (m, n), or scale a triple by a factor — with Euclid's formula shown.

Add, subtract, multiply, or divide binary numbers with the column method shown: carries, borrows, partial products, and a long-division tableau, plus decimal, hex, and octal equivalents.

Convert binary to decimal with the positional-weight table (bit × 2^n), fractional bits after the binary point, and an option to read the bits as a signed two's complement number.

Convert decimal to binary with the repeated-division steps, fractional parts by repeated multiplication (0.625 → 0.101), repeating-fraction detection, and negatives as a sign or two's complement.

Convert hexadecimal to decimal with the place-value table (digit × 16^n), the digit-to-value key for A-F, nibble grouping into binary, and signed two's complement reading.

Convert decimal to hexadecimal with the repeated-division-by-16 steps, remainders mapped to A-F, fractional parts, nibble grouping, and negatives in two's complement.

Convert hex to binary one nibble at a time: each hex digit becomes exactly four bits. Shows the nibble table, the decimal value, and octal, with fractional hex supported.

Convert binary to hex by grouping bits into nibbles from the right, padding the first group, and reading each group as one hex digit. Shows the grouping table plus decimal and octal.

Convert octal to binary, decimal, or hex and back, with the 3-bit grouping table (one octal digit = three bits), repeated division by 8, and fractional octal support.

Convert any number between any two bases from 2 to 36 with the steps shown: place-value expansion to decimal, repeated division into the target base, and fraction handling with repeat detection.

Add two binary numbers with every carry shown in an aligned column grid, plus the decimal check and the hex and octal equivalents of the sum.

Subtract binary numbers with each borrow shown in an aligned column grid, negative answers handled, and the difference checked in decimal, hex, and octal.

Multiply binary numbers with every shifted partial product listed in an aligned grid, then the final sum, plus the product in decimal, hex, and octal.

Divide binary numbers with the long-division tableau drawn out, the remainder shown, optional fraction bits past the binary point, and a decimal check.

Convert decimal to two's complement and back at 4, 8, 16, 32, or 64 bits with the invert-and-add-1 steps, the sign bit highlighted, overflow detection, and the unsigned reading of the same bits.

Find the one's complement of a binary pattern by flipping every bit, encode a signed decimal in one's complement at a chosen width, or decode a pattern, with the two's complement shown for comparison.

AND, OR, XOR, NOT, NAND, and NOR on binary, decimal, octal, or hex inputs at 4 to 64 bits, with an aligned bit grid, a bit-by-bit truth table, and the result in every base.

Left, logical right, and arithmetic right shifts at 4 to 64 bits with the before/after bit rows, bits shifted out, overflow detection, and the ×2^n or ÷2^n meaning of the shift.

Convert decimal fractions (or ratios like 1/3) to binary with repeating-block detection, or binary fractions back to exact decimals, with the multiply-by-2 table and place-value table shown.

Convert binary or decimal to Gray code and Gray code back to binary with the XOR step for every bit, the binary XOR (binary >> 1) shortcut, and a check that consecutive codes differ by one bit.

Convert text to binary, hex, and decimal bytes (UTF-8, one 8-bit group per byte) with a per-character table, or decode binary, hex, or decimal bytes back into text.

Solve |ax + b| = c and |ax + b| = |cx + d| with both cases shown, exact fraction answers, checks by substitution, and clear no-solution and all-real-numbers cases.

Evaluate sinh, cosh, tanh, coth, sech, and csch from their exponential definitions, or the inverse hyperbolic functions from their logarithmic forms with domain checks. Shows e^x and e^(−x), all six values at once, and the identity cosh² − sinh² = 1.

List every subset of a set with up to 6 elements, grouped by size, and count them with 2^n. Shows the C(n, k) count for each size, the number of proper subsets, and includes the empty set and the set itself.

Check whether one set is a subset, proper subset, or superset of another, or whether the two sets are equal. Lists the elements that break the relation, the common elements, and the number of subsets each set has.

Find the union A ∪ B, intersection A ∩ B, differences A − B and B − A, and symmetric difference of two sets, with Venn-diagram region counts and the inclusion–exclusion check |A ∪ B| = |A| + |B| − |A ∩ B|.

Compute C(n, k) = n choose k exactly for any size (big-integer arithmetic), with the cancelled factorial product, Pascal's identity, the symmetry C(n, k) = C(n, n - k), and the full row of Pascal's triangle.

Solve 2×2 and 3×3 systems of linear equations by elimination (addition method) with classroom steps: multiply equations to get opposite coefficients, add to eliminate a variable, back-substitute, and check. Exact fractions; detects no-solution and dependent systems.

Check whether ax^2 + bx + c is a perfect square trinomial and factor it as (px + q)^2, or find the missing a, b, or c that completes the square. Uses the b^2 = 4ac test with exact fractions and shows the expansion check.

Generate the complete truth table for a logical expression with up to 4 variables using ∧, ∨, ¬, →, ↔, ⊕ (or and, or, not, implies, iff, xor). Shows every intermediate column, flags tautologies and contradictions, and writes the disjunctive normal form.

Solve |ax + b| <, ≤, >, or ≥ c: rewrites it as a compound and/or inequality, solves exactly with fractions, gives interval and set-builder notation, draws the number line with open or closed dots, and catches no-solution and all-real-numbers cases.

Graph one inequality or a compound 'and'/'or' inequality on a number line with open and closed dots, and get the solution in inequality, interval, and set-builder notation. Shows each part of a compound inequality on its own line before combining.

Solve inverse variation y = k/x, joint variation z = kxy, and combined variation z = kx/y: find the constant k from known values, write the equation, and find the missing value exactly, with a hyperbola graph for the inverse case.

Solve a system of two linear equations by substitution, step by step: pick the easiest variable to isolate, substitute into the other equation, solve, back-substitute, and check. Exact fractions, with dependent and inconsistent systems detected.

Box method (area model) calculator that multiplies polynomials with an n × m box or factors a trinomial with the 2 × 2 box and ac method. Draws the box, labels every cell, and reads the factors off the edges.

Solve direct variation problems y = kx: find the constant of variation k from one point, write the equation, and find the missing x or y. Exact fractions, a ratio check, and a graph of the line through the origin.

Multiply any two polynomials with the distribution grid (box method) shown: every partial product, like terms grouped by degree, and the product in standard form with its degree and leading coefficient.

Decompose a rational function into partial fractions with exact fractions: distinct and repeated linear factors, irreducible quadratics, long division for improper fractions, the cover-up method, the coefficient system, and every step.

Add or subtract two polynomials by lining up like terms. Type each polynomial, see every column of like terms combined, the minus sign distributed, and the result in standard form with its degree and leading coefficient.

Find the conjugate of a complex number a + bi and see why z times its conjugate equals |z|^2. Also gives the modulus, the reciprocal 1/z, z + z̄, z - z̄, and the mirrored argument.

Add, subtract, multiply, divide, and raise complex numbers to powers exactly. Shows the modulus, argument in degrees and radians (with exact π fractions), conjugate, polar and exponential forms, De Moivre's theorem, and the point on the complex plane.

Find all n square, cube, or nth roots of a complex number with De Moivre's formula. Exact forms such as √3 + i when the angle is special, decimals otherwise, every root's angle, and a plot of the roots on a circle in the complex plane.

Complete the square on ax^2 + bx + c step by step: every algebraic line shown, exact fractions, vertex form a(x - h)^2 + k, the vertex, and the roots as simplified radicals.

Solve ax^3 + bx^2 + cx + d = 0 exactly when a rational root exists (rational root theorem + synthetic division + quadratic formula), otherwise with Cardano or the trigonometric method, with discriminant, factored form, complex roots, and a graph.

Count sign changes in f(x) and f(-x) to find the possible numbers of positive and negative real zeros of a polynomial, plus every consistent combination with nonreal zeros.

Find the discriminant b^2 - 4ac of a quadratic, classify the roots as rational, irrational, repeated, or complex, and preview the exact roots with steps.

Factor ax^2 + bx + c over the integers with the ac-method: the full factor-pair search table, splitting the middle term, grouping, and the roots when it does not factor.

Convert between inequalities, interval notation, set-builder notation, and a number-line description. Handles compound and/or inequalities, unions, infinity, and ≠.

Multiply binomials with FOIL or expand any product of polynomials. Type (2x + 3)(x - 4) and see the First, Outer, Inner, Last products, like terms combined, and the expanded polynomial.

Divide polynomials by long division or synthetic division. Type both polynomials and get the quotient, remainder, the full synthetic-division tableau or long-division rows, the check, and every step.

Solve ax^2 + bx + c = 0 with exact simplified radicals, reduced fractions, or complex roots, plus discriminant, vertex, factored and vertex forms, a graph, and every step.

Apply the rational root theorem: list every candidate ±p/q for a polynomial, test each one exactly, report the actual rational zeros with multiplicity, and show the deflated (remaining) factor.

Convert a quadratic between standard form ax^2 + bx + c and vertex form a(x - h)^2 + k in either direction, with the vertex, axis of symmetry, direction, intercepts, and steps.

Find the area of 11 shapes — rectangle, square, triangle (base-height, three sides, or SAS), circle, trapezoid, parallelogram, rhombus, ellipse, regular polygon, sector, and kite — with the formula, substitution, square-unit labels, and a diagram.

Find a chord's length from the radius and central angle, from the radius and its distance from the center, or find the radius from a chord and its sagitta. Also returns the arc length, sagitta, and segment area.

Enter any one circle measurement (radius, diameter, circumference, or area) to get all four, with exact π forms, unit-aware labels, arc length, sector area, and chord length for any central angle.

Find the circumference of a circle from its radius, diameter, or area. Shows the exact answer in terms of π (like 12π), the decimal, the radius and diameter, and every step.

Cone calculator: from radius and height (or slant height, diameter, or volume) get the volume, slant height, lateral and total surface area, apex angle, and base angle with exact π answers and steps.

Enter one cube measurement — edge, face diagonal, space diagonal, surface area, or volume — and get all five, with exact radicals (s√2, s√3), face area, inscribed and circumscribed sphere radii, and liters.

Cylinder calculator for volume, lateral area, total surface area, and diagonal from radius and height — or solve the height or radius from a known volume or lateral area. Exact π answers plus liters and gallons.

Write a circle's equation in standard form (x − h)² + (y − k)² = r² and general form x² + y² + Dx + Ey + F = 0 from the center and radius, convert between forms by completing the square, or find the circle through three points.

Hemisphere calculator for volume ((2/3)πr³), curved surface area (2πr²), and total surface area (3πr²) from radius or diameter — or find the radius from a known volume or area. Exact π answers and liters.

Regular hexagon calculator: enter the side, apothem, circumradius, area, or perimeter to get everything else, with the exact area (3√3/2)s² as a radical, interior angles, and the number of diagonals.

Regular octagon calculator: enter the side, apothem, circumradius, area, or perimeter to get the rest, using the exact area 2(1 + √2)s², with the 135° interior angle, 45° exterior angle, and 20 diagonals.

Solve a parallelogram from two sides and the angle between them, or from base and height (with an optional side). Returns area, perimeter, both diagonals, both heights, and both angles with steps.

Regular pentagon calculator: enter the side, apothem, circumradius, area, or perimeter and get all the others, plus the 108° interior angle, 72° exterior angle, and its five diagonals.

Find the perimeter of a rectangle, square, triangle, circle, trapezoid, parallelogram, regular polygon, or ellipse (Ramanujan approximation) with the formula, substitution, and unit-labeled answer.

Enter any two rectangle properties — length, width, diagonal, area, or perimeter — and get all five, plus the diagonal angle, with exact radicals for the diagonal and the steps used to solve.

Solve a rhombus from side and angle, from the two diagonals, or from side and height. Returns area, perimeter, both diagonals, height, inradius, and both angles, with exact radicals where they appear.

Find the area, arc length, and full perimeter of a semicircle from its radius or diameter — or work backward from a known area or perimeter. Exact answers in terms of π plus the centroid height.

Enter one sphere measurement — radius, diameter, circumference, surface area, or volume — and get all five with exact π forms (36π), the great-circle area, steps, and liters or gallons of capacity.

Enter one square measurement — side, diagonal, area, or perimeter — and get all of them with the diagonal in exact radical form (s√2), plus the inscribed and circumscribed circle radii.

Find the total surface area of a cube, box, cylinder, cone, sphere, rectangular pyramid, or triangular prism with the formula for each solid, the lateral and base breakdown, exact π forms, and square-unit labels.

Regular tetrahedron calculator: from the edge (or volume, surface area, or height) get the volume a³/(6√2), surface area √3·a², height a√(2/3), face area, inradius, midradius, and circumradius with exact radicals.

Torus (donut) calculator: enter the major radius (center to tube center) and minor radius (tube radius) to get the volume 2π²Rr², surface area 4π²Rr, outer and inner radii, and whether the torus is a ring, horn, or spindle.

Triangular prism calculator: enter the three sides of the triangular end (Heron's formula) or its base and height, plus the prism length, to get the volume, lateral area, and total surface area with steps.

Find the volume of 11 solids — cube, box, cylinder, cone, sphere, hemisphere, pyramid, triangular prism, ellipsoid, capsule, and frustum — with exact π forms, cubic-unit labels, surface area, and liters/gallons capacity.

Rewrite any trig function as its cofunction of the complementary angle (sin θ = cos(90° − θ)) in degrees or radians, verify the values match, and see all six cofunction pairs at once.

Find the complement (adds to 90°), supplement (adds to 180°), and explement (adds to 360°) of any angle in degrees or radians, with a clear note when a pair does not exist.

Find cos θ for any angle in degrees or radians with exact values like √3/2 and -1/2, solve θ from a cosine value with all solutions, or use cosine to find a right-triangle side or angle.

Find the smallest positive and largest negative coterminal angles of any angle in degrees or radians, plus the general form θ + 360°k, the quadrant, and the reference angle.

Convert degrees to radians as an exact π fraction (150° = 5π/6) and a decimal, radians back to degrees, and degrees-minutes-seconds to decimal degrees, with gradians and turns as a bonus.

Compute sin 2θ, cos 2θ (all three forms), and tan 2θ from the angle itself or from a known sin θ, cos θ, or tan θ and its quadrant, with exact fractions and radicals and every step written out.

Compute sin(θ/2), cos(θ/2), and tan(θ/2) from the angle or from a known cos θ or sin θ with its quadrant, choosing the correct sign for θ/2 and showing exact radicals and every step.

Compute arcsin, arccos, arctan, arccsc, arcsec, and arccot of a value in degrees and radians with exact π fractions, domain errors like arcsin 2, and every solution from 0° to 360°.

Find the reference angle of any angle in degrees or radians, including negative angles and angles past 360°, with the quadrant, coterminal angles, the rule used, and the trig values it unlocks.

Find sec θ, csc θ, and cot θ for any angle in degrees or radians with exact rationalized forms like 2√3/3, undefined detection where sin or cos is zero, and the sin/cos/tan values they come from.

Find sin θ for any angle in degrees or radians with exact values like 1/2 and √3/2, solve θ from a sine value with every solution, or use sine to find a right-triangle side or angle.

Expand sin(α ± β), cos(α ± β), and tan(α ± β) for any two angles with exact radicals when both are special, or decompose an angle like 75° into 45° + 30° to find its exact trig values.

Find tan θ for any angle in degrees or radians with exact values like √3 and √3/3, detect undefined tangents at 90°, solve θ from a tangent value with all solutions, or find a right-triangle side or angle.

Enter any angle to numerically verify the Pythagorean, quotient, reciprocal, cofunction, even-odd, and double-angle identities, with the complete identity reference table.

Find sin, cos, tan, csc, sec, and cot of any angle in degrees, radians, π-fractions, gradians, or turns, with exact radical values, quadrant, reference angle, coterminal angles, and a unit-circle diagram.

Pick a special angle, type any angle, or go backward from a trig value to the angles that produce it. Get exact unit-circle coordinates (cos θ, sin θ), radians, all six trig functions, both solutions in one turn, and the complete special-angle table from 0° to 360°.

Find the distance between two points in 3D space with the exact radical form and decimal, plus the midpoint, the direction vector from P to Q, its unit vector, and direction angles.

Find the angle between two 2D or 3D vectors in degrees and radians using the dot product, with the exact cosine, magnitudes, and an acute/obtuse/perpendicular classification.

Find the average rate of change of a function between two x-values from a polynomial or from two points, with the exact fraction, the secant line equation, and every step.

Interpolate a value inside a 2×2 grid of known points: two linear interpolations in x, then one in y, with exact fractions, the corner weights, and an extrapolation warning for points outside the cell.

Find the catenary y = a cosh(x/a) of a hanging cable from its span and sag (or from a and the span): the parameter a, cable length, tension ratio, support angle, optional forces from the weight per unit length, and a graph against the parabola approximation.

Compute the cosine similarity of two vectors of any length with the dot product, both magnitudes, the exact rationalized value when possible, the angle between the vectors, and the cosine distance.

Compute the cross product of two 3D vectors with the determinant expansion shown, the magnitude (parallelogram area), the unit normal vector, and a perpendicularity check.

Convert between rectangular (x, y, z) and cylindrical (r, θ, z) coordinates, and from cylindrical to spherical, with exact radicals, quadrant-correct angles in degrees and radians, and step-by-step work.

Find the perpendicular distance from a point to a line in 2D (any line form) or from a point to a plane in 3D, with the exact radical form, the foot of the perpendicular, and formula steps.

Find the perpendicular distance from a point to a plane ax + by + cz = d with the formula substituted step by step, the exact radical, the foot of the perpendicular, and a mode for the distance between two parallel planes.

Compute the dot product of two 2D or 3D vectors with the angle between them, both magnitudes, the scalar projection, and an orthogonality check, with every multiplication shown.

Find the equation of a plane from three points, a point and a normal vector, or a point and two direction vectors: standard form with integer coefficients, normal and unit normal, intercepts, parametric and vector forms, with cross-product steps and a 3D sketch.

Find the equation of a sphere from its center and radius, from the general form by completing the square in x, y, and z, from the endpoints of a diameter, or from four points; get standard and general forms, center, radius, surface area, and volume.

Compute the gradient vector ∇f = ⟨∂f/∂x, ∂f/∂y⟩ of a polynomial in x and y, evaluated at a point, with both partial derivatives shown, the magnitude (rate of steepest ascent), the direction as an angle and unit vector, and the steepest-descent direction.

Find where two lines intersect from any form of their equations. Exact-fraction intersection point, parallel and coincident detection, the angle between the lines, a graph, and Cramer's-rule steps.

Find the area of any polygon from its vertex coordinates with the shoelace formula, every cross term shown, plus the perimeter, centroid, vertex orientation, a self-intersection check, and a plot of the shape.

Find the length and endpoints of the latus rectum of a parabola, ellipse, or hyperbola from its equation, with the focus, the line containing the chord, exact fractions and radicals, and every step shown.

Find the line where two planes meet: direction vector from the cross product of the normals, a point on the line, and the parametric, symmetric, and vector equations, with parallel and coincident planes detected.

Interpolate a value between two known points with the formula y = y1 + (x - x1)(y2 - y1)/(x2 - x1), exact fractions, the slope and line equation, inverse interpolation (find x for a given y), an extrapolation warning, and a graph.

Compute the Manhattan (taxicab, L1) distance between two points in 2D, 3D, or any number of dimensions, alongside the Euclidean and Chebyshev distances for comparison, with every coordinate difference shown.

Find the missing endpoint of a segment from one endpoint and the midpoint (or the midpoint from two endpoints), with the segment length, the line through the points, and steps.

Find the vertex, focus, directrix, axis of symmetry, focal length, latus rectum, and intercepts of a parabola from standard form, vertex form, or focus and directrix, with a graph.

Find the equation of the line parallel to a given line through a point. Accepts y = mx + b, two points, point-slope, or standard form, and graphs both lines with the distance between them.

Find the equation of the line perpendicular to a given line through a point using the negative reciprocal slope. Accepts any line form, graphs both lines, and shows where they meet at a right angle.

Write the equation of a line in point-slope form y - y1 = m(x - x1) from a point and a slope or from two points, then convert it to slope-intercept and standard form with a graph.

Convert rectangular (x, y) coordinates to polar (r, θ) and back, with exact radicals and π fractions at special angles, all-quadrant atan2 steps, degrees and radians, and a polar grid plot.

Find the point that divides a segment AB in a given ratio m:n, internally or externally, using the section formula with exact fractions, distance checks, the parameter t, and a graph of A, B, and P.

Rotate a point about the origin or any center by any angle, clockwise or counterclockwise, with exact coordinates for special angles, the rotation matrix, the 90°/180°/270° shortcut rules, and a before-and-after graph.

Write the equation of a line in slope-intercept form y = mx + b from two points, a point and a slope, or standard form. Exact fractions, intercepts, a graph, and every other form of the line.

Convert between rectangular (x, y, z), spherical (ρ, θ, φ), and cylindrical coordinates with a math-vs-physics convention toggle, exact radicals at special angles, degrees and radians, and full steps.

Convert a line between standard form Ax + By = C and slope-intercept form y = mx + b in either direction, with integer-cleared coefficients, exact fractions, intercepts, and steps.

Locate an unknown point from two known positions and the bearing or direction angle to it from each, using the law of sines with every step shown, the distances from both observers, the triangle's angles, and a plotted fix.

Normalize a 2D or 3D vector to a unit vector with exact rationalized radical components and decimals, plus the magnitude, direction angle, and the opposite unit vector.

Add or subtract two or more 2D or 3D vectors component by component. Get the resultant vector, its magnitude and direction, and a tip-to-tail parallelogram diagram with steps.

Find the magnitude (length) and direction angle of a 2D or 3D vector from its components or from two points. Exact radical magnitude, unit vector, direction cosines, and an arrow diagram.

Project vector a onto vector b in 2D or 3D. Get the vector projection, the scalar projection (component of a along b), the rejection (perpendicular part), and the angle between them, with steps.

Find the volume of a parallelepiped from three edge vectors or four vertices using the scalar triple product a · (b × c), with the 3×3 determinant expanded step by step, the tetrahedron volume, base area, height, and a 3D sketch.

Find the x-intercept and y-intercept of a line given in any form, or the intercepts of a quadratic y = ax^2 + bx + c, with exact fractions, radicals, and the substitution steps.

Compute the cofactor matrix and adjugate (adjoint) of any square matrix up to 8 x 8 with exact fractions. Shows every signed minor, the transpose step, the identity A·adj(A) = det(A)·I, and the inverse via adj(A)/det(A).

Find the characteristic polynomial det(λI - A) of any square matrix up to 8 x 8 with exact coefficients. Shows the Faddeev-LeVerrier steps, trace and determinant checks, the factored form, the roots (eigenvalues), and a Cayley-Hamilton verification.

Solve 2 x 2, 3 x 3, and up to 8 x 8 linear systems with Cramer's rule. Shows D and every D_x, D_y, D_z determinant with the replaced column, exact fraction answers, and explains what happens when D = 0.

Find the eigenvalues and eigenvectors of a square matrix up to 8 x 8. Shows the characteristic polynomial, exact eigenvalues (including fractions, square roots, and complex pairs), eigenvector bases, multiplicities, and whether the matrix is diagonalizable.

Factor a square matrix up to 8 x 8 as PA = LU with partial pivoting and exact fractions. Shows P, L, U, every elimination multiplier, the verification PA = LU, and the determinant from the pivots.

Add or subtract two matrices of the same size up to 8 x 8 with exact fractions. Shows the entry-by-entry work, checks that the dimensions match, and displays the other operation for comparison.

Find the determinant of any square matrix up to 8 x 8 with exact fractions. Shows cofactor expansion along any row or column for 3 x 3 (with a rule-of-Sarrus check), row-reduction steps for larger matrices, and tells you whether the matrix is invertible.

Invert any square matrix up to 8 x 8 with exact fractions. Gauss-Jordan elimination on [A | I] with every row operation shown, the adjugate route for small matrices, a singular-matrix explanation, and an A·A⁻¹ = I check.

Multiply two matrices up to 8 x 8 with exact fractions. Checks that the dimensions line up, shows the dot product for every entry of the product, compares AB with BA, and explains why order matters.

Find a basis for the null space (kernel) of any matrix up to 8 x 8 with exact integer-scaled vectors, plus the column space and row space bases, rank, nullity, free variables, and every row operation.

Raise a square matrix up to 8 x 8 to any integer power from -30 to 30 with exact fractions. Shows each intermediate power, handles A^0 = I and negative powers through the inverse, and checks det(A^k) = det(A)^k.

Find the rank of any matrix up to 8 x 8 by row reduction with every step shown. Reports pivot columns, nullity from the rank-nullity theorem, full-rank status, and the determinant for square matrices.

Find the trace of a square matrix up to 8 x 8 (the sum of its diagonal entries) with exact fractions. Shows the running sum, the eigenvalue check trace = λ1 + … + λn, the determinant, tr(A²), and the trace properties.

Transpose any matrix up to 8 x 8 and check whether it is symmetric, skew-symmetric, or orthogonal. Shows AᵀA and AAᵀ, the row-to-column steps, and the transpose rules for products, determinants, and traces.

Reduce any matrix up to 8 x 8 to reduced row echelon form with every row operation shown, or enter A and b to solve Ax = b with unique, infinite (parametric), and no-solution reporting. Exact fractions throughout.

Multiply a matrix up to 8 x 8 by a scalar (integer, decimal, or fraction) with every entry shown. Explains why det(kA) = kⁿ·det(A) and how scalar multiplication differs from matrix multiplication.

Find |x| for any number with the number-line meaning, the distance |a - b| between two numbers, or solve absolute value equations |ax + b| = c by splitting into two cases, all with steps.

Solve systems of simultaneous congruences with the Chinese remainder theorem: checks the moduli are pairwise coprime, builds the N_i, the modular inverses, and the sum step by step, returns the smallest solution and the general solution mod N, and handles non-coprime moduli when the system is consistent.

Compare numbers of any form (integers, decimals, fractions, mixed numbers, percents, scientific notation) exactly: shows which is greater with the correct symbol, orders up to six numbers from least to greatest, and reveals the common denominator or decimal form used to decide.

Find k consecutive integers (or consecutive even or odd integers) whose sum equals a given number by setting up and solving x + (x + 1) + … = S, or whose product equals a given number; shows every step and explains when no integer solution exists.

Find the cube root of any number with an exact answer for perfect cubes, simplified radical form (∛54 = 3∛2), the decimal value, nearest perfect cubes, and full support for negative and decimal inputs.

Find the sum of the digits of any whole number, the digital root (repeated digit sums down to one digit) with every iteration, the mod-9 shortcut, the divisibility-by-3 and -9 verdicts it implies, and the digit product.

Apply the distributive property to expand a product of two sums, numeric or algebraic: a(b + c), 3(2x + 4), (x + 3)(x - 5), (2a + b)(a - 3b); shows each partial product (FOIL order), combines like terms, and checks numeric cases by evaluating both sides.

Test any whole number against the divisibility rules for 2 through 13 with the actual rule work shown (digit sums, last digits, the 7 and 13 chop tricks), plus a custom divisor with quotient and remainder.

Expand a fraction p/q into an Egyptian fraction, a sum of distinct unit fractions, using the greedy algorithm: each step takes the largest unit fraction that fits, shows the leftover, and stops when it reaches zero, with the full sum verified at the end.

Write any number in expanded form (4,825.3 = 4,000 + 800 + 20 + 5 + 0.3), expanded exponential form with powers of ten, expanded fraction form for decimals, and word form; or convert an expanded expression back to standard form.

Compute n! exactly for large n with every digit shown, trailing zeros, scientific notation, Stirling's approximation, and step-by-step expansion; also double factorial n!! and subfactorial !n.

Find all factors of a whole number, the factor pairs, how many divisors it has and their sum, the prime factorization, and whether the number is prime, perfect, abundant, or deficient.

Compute the floor ⌊x⌋ and ceiling ⌈x⌉ of any real number together with rounding, truncation, and the fractional part, all exact for decimals and negatives; a second mode gives the floor division ⌊a/b⌋, ceiling division, and remainder of two integers.

Apply the fundamental counting principle (multiplication rule) to count outcomes: multiply the number of choices at each stage, or count ways to fill k positions from n options with repetition allowed (n^k) or not (n × (n-1) × …), with the full product shown and the link to permutations explained.

Find the greatest common factor (GCF, also called GCD or HCF) of any list of whole numbers with three worked methods: listing factors, the prime-factorization table, and Euclid's division algorithm.

Compute the harmonic mean n / (1/x₁ + 1/x₂ + … + 1/xₙ) of a list of positive numbers with every reciprocal shown, compare it with the arithmetic and geometric means (AM ≥ GM ≥ HM), and solve the classic average-speed-over-equal-distances problem.

Find the least common multiple (LCM) of any list of whole numbers with three worked methods: listing multiples, the prime-factorization exponent table, and lcm(a, b) = a × b ÷ gcf(a, b) using Euclid's algorithm.

Add or subtract two numbers with the column method drawn like a worksheet, showing every carry or borrow, lined-up decimal points, the sign rules for negatives, and a check using the inverse operation.

Multiply two numbers with the classic column method laid out like a worksheet: each partial product, the place-value shift, the final addition, decimal-point placement, negative signs, and a division check.

Build normal magic squares of order 3, 4, 5, 7, 8, 9, 11, or 12 with the construction explained and the magic constant n(n² + 1)/2, or paste your own square to verify every row, column, and diagonal sum and spot the lines that fail.

Compute a^b mod n exactly for big numbers using square-and-multiply with every squaring and reduction tabulated, find a modular inverse a^-1 mod n with extended Euclid steps, or the ordinary multiplicative inverse (reciprocal) of a number.

Find a mod n for integers, negative numbers, and decimals with the quotient, remainder, division equation, congruence class, clock visual, and a clear explanation of floored vs truncated remainders.

Find the nth root of a number for any index: exact simplified radical form, decimal value, the rational-exponent form x^(m/n), and fractional powers such as 81^(3/4) = 27, with steps.

Find the order of magnitude of a number as the exponent in its scientific notation, the nearest power of ten, and the log-scale position between powers; or compare two numbers to see how many orders of magnitude larger one is than the other.

Test whether a number is a perfect cube with the exact cube root, the prime-factor reason, the nearest perfect cubes below and above (negatives included), and a list of all perfect cubes in any range, computed exactly for numbers up to 60 digits.

Test whether a number is a perfect square with the exact root, the prime-factor reason, the nearest perfect squares below and above, and a list of all perfect squares in any range, computed exactly for numbers up to 60 digits.

See every digit of a whole number or decimal in a place value chart (ones through quintillions, tenths through billionths) with the value of each digit, expanded form, powers-of-ten form, fraction form, and the word name.

Simplify i^n for any integer exponent (huge or negative) by reducing n mod 4 and reading the cycle i, -1, -i, 1, with every step shown; a second mode simplifies (c·i)^n for a rational coefficient c, giving exact answers like (2i)^5 = 32i.

Test whether a number is prime (up to 24 digits, exactly), see the trial-division or Miller–Rabin reasoning, get the prime factorization for composites, the nearest primes above and below, or list all primes in a range.

Classify decimals, fractions, repeating decimals, square and cube roots, and π/e expressions as rational or irrational with the reason, the exact fraction and decimal expansion for rational inputs, and a second mode that tests whether a fraction p/q terminates using the 2^a × 5^b rule.

Rationalize any denominator: a/√b, a/∛b, a/(√b ± c), and a/(√b ± √c). Shows the multiplier or conjugate, the difference-of-squares step, the simplified radical, the reduced answer, and a decimal check.

Find the reciprocal (multiplicative inverse) of an integer, decimal, fraction, or mixed number as an exact fraction, a mixed number, and a decimal, with the conversion steps and the check x × 1/x = 1.

Test whether numbers are relatively prime (coprime, gcd = 1) with every Euclidean division shown and the shared prime factors when they are not; a second mode computes Euler's totient φ(n), how many numbers from 1 to n are coprime to n, from the prime-factor product formula.

Divide two whole numbers and get the quotient and remainder, the check equation a = b × q + r, the mixed-number, fraction, and decimal forms of the same division, and both sign conventions for negatives.

Simplify √, ∛, ∜, and fifth roots to simplest radical form using a prime-factor table, with coefficients (5√72 = 30√2), negative radicands, the decimal value, and the rational-exponent form.

Write an ordinary number in standard form A × 10^n with 1 ≤ A < 10 (UK and GCSE usage; the same thing as scientific notation), or expand A × 10^n to an ordinary number, exactly for very large and very small values, with the decimal-point shift and the sign of n explained.

Compute the nth triangular number T_n = n(n + 1)/2, test whether a number is triangular with the 8N + 1 perfect-square test and recover its index, list the triangular numbers up to a limit, and sum the first n triangular numbers.

Divide whole numbers with every classic long-division step shown — divide, multiply, subtract, bring down — plus quotient, remainder, decimal answer, and a check line.

Convert between improper fractions and mixed numbers or add, subtract, multiply, and divide mixed numbers with exact reduced fractions, decimal equivalents, and step-by-step work.

Round numbers to any place value or to significant figures with five rounding methods, exact tie handling, both neighbor candidates, and the rounding error shown.

Add up to five fractions or mixed numbers. Finds the least common denominator with prime factors, rewrites each fraction, adds, simplifies, and shows the mixed-number and decimal forms with a fraction-strip picture.

Average a list of percentages correctly: a simple mean when the groups are equal, or a weighted average by sample size when they are not, with both shown side by side and the difference explained.

Compare and order two to six fractions, mixed numbers, or decimals from least to greatest and greatest to least, using the common-denominator method, decimals, and cross products.

Solve a proportion a/b = c/d for any missing term by cross multiplying, with the products, the one-step equation, the exact fraction answer, the decimal, and a check that both cross products match.

Convert any terminating or repeating decimal to an exact fraction with steps: powers of 10 for terminating decimals, the subtraction trick for repeating ones like 0.1(6), plus simplified, mixed-number, and percent forms.

Divide fractions, mixed numbers, and whole numbers using keep-change-flip. Shows the reciprocal, cross-cancelling, and the quotient as a simplified fraction, mixed number, decimal, and percent.

Generate equivalent fractions, rewrite a fraction with a chosen denominator, or check whether two fractions are equivalent using cross products, with simplest form and decimal shown.

Evaluate a base raised to a fractional (rational) exponent such as 8^(2/3) exactly when possible, with the root-then-power steps, simplest radical form, decimal value, and negative-exponent handling.

Convert any fraction or mixed number to a decimal with long-division steps, automatic detection of repeating digits with overline notation, rounding to your chosen places, and the percent form.

Convert an improper fraction to a mixed number by dividing numerator by denominator, with the quotient and remainder shown, the fraction part simplified, and decimal and percent forms.

Find the least common denominator of two to six fractions using prime factorization, then see every fraction rewritten over the LCD with its multiplier, ready for adding, subtracting, or comparing.

Convert a mixed number to an improper fraction with the multiply-and-add steps shown, the simplified form, the decimal, and the percent. Handles negative mixed numbers.

Multiply two or three fractions, mixed numbers, or whole numbers with cross-cancelling shown before multiplying. Get the simplified product, mixed number, decimal, and percent.

Reduce a number by a percent, find the percent decrease between two values, or reverse a decrease to recover the original amount, with the multiplier, amount removed, and the percent remaining.

Find the symmetric percentage difference between two values (difference divided by their average), and see how it compares with the percent change in each direction, with steps and a plain explanation of which to use.

Increase a number by a percent, find the percent increase between two values, or work backwards from the final value to the original, with the multiplier, increase amount, and step-by-step formulas.

Compare two percentages the right way: get the change in percentage points and the relative percent change side by side, or add points to a rate, with a clear explanation of why the two numbers differ.

Simplify ratios with two to six parts like 12:18:30 to 2:3:5, scale a ratio so one part hits a target, and see each part as a fraction and percent of the whole, plus the n:1 and 1:n forms.

Reduce any fraction, mixed number, or decimal to lowest terms with prime-factor GCF steps, a list of equivalent fractions, mixed-number form, decimal, and percent, plus a strip picture showing both forms are equal.

Subtract fractions and mixed numbers with the least common denominator, borrowing shown for mixed numbers, and the answer as a simplified fraction, mixed number, decimal, and percent with a strip visual.

Turn any rate like 120 miles in 3 hours into both unit rates (per 1 hour and per 1 mile), scaled rates per 10 and per 100, and compare two options by unit price to find the better deal.

Calculate the antilogarithm b^y for base 10, 2, e, or any custom base, with exact answers for whole-number exponents, scientific notation, and the inverse log check.

Find the sum of an arithmetic series from the first term, common difference, and number of terms, or from the first and last terms, with a partial-sum table and Gauss's pairing explained.

Evaluate log_b(x) in any base with the change of base formula, showing both the ln and log₁₀ routes, exact answers when x is a power of b, and every step.

Run the 3n + 1 (Collatz) process from any starting number: the full hailstone sequence, the number of steps to reach 1, the peak value, odd and even step counts, and a log-scale trajectory chart.

Condense a sum or difference of logarithms into a single logarithm, applying the power, product, and quotient rules in order, combining like terms, and evaluating exact numeric results.

Divide powers with the same base (subtract exponents) or the same exponent (divide bases), with the quotient rule named, negative results rewritten as reciprocals, and exact values.

Get Euler's number e to any number of decimal places, compute e^x or the x-th root of e, and see the limit definition (1 + 1/n)^n and the series 1 + 1/1! + 1/2! + ... converge.

Expand a logarithm of a product, quotient, or power into a sum of simpler logs, naming the product, quotient, and power rule at each step and evaluating exact numeric logs.

Solve exponential decay for the remaining amount, initial amount, decay rate, or time using y = a(1 - r)^t, y = a·e^(-kt), or the half-life form, with half-life, a table of values, and a graph.

Evaluate y = a·b^x, classify it as growth or decay, build a table of values, find the base b and coefficient a from two points, or solve for x with logarithms, all with steps.

Solve exponential growth for the final amount, starting amount, rate, or time using y = a(1 + r)^t or y = a·e^(kt), with doubling time, a table of values, and a graph.

Find the nth Fibonacci number exactly up to n = 10,000, list the first n terms, sum them, test whether a number is Fibonacci, and watch the ratios converge to the golden ratio with Binet's formula.

Sum the first n terms of a geometric series or the whole infinite series when |r| < 1, with a convergence check, partial-sum table, the ratio-1 special case, and step-by-step work.

Compute the harmonic number H(n) = 1 + 1/2 + ... + 1/n as an exact fraction for n up to 500, as a decimal for any n, and compare it with the asymptotic estimate ln(n) + γ.

Calculate log base 2 of any positive number with an exact answer for powers of 2, the decimal value otherwise, and the bits and doublings interpretation.

Multiply powers with the same base (add exponents) or the same exponent (multiply bases), with the rule named, symbolic or numeric bases, and the exact evaluated value.

Calculate the natural logarithm ln(x) or its inverse e^x, with exact answers for powers of e, the matching log₁₀ and log₂ values, series terms, and step-by-step work.

Rewrite a negative exponent as a reciprocal and evaluate it exactly: 2^-3 = 1/2^3 = 1/8, (2/3)^-2 = (3/2)^2 = 9/4, with numeric or variable bases and step-by-step work.

Generate Pascal's triangle up to 40 rows exactly, look up any binomial coefficient C(n, k) with the row highlighted, and see row sums, Pascal's rule, and the factorial formula.

Raise any number to any power with every digit shown exactly for huge integer results (up to 20,000 digits), exact fractions for rational answers, and decimals plus scientific notation otherwise.

Simplify (a^m)^n by multiplying the exponents, with numeric or variable bases, negative and fractional exponents, and the exact evaluated value shown with steps.

Sum arithmetic and geometric series with the closed-form formulas, partial sums, and the infinite geometric series when |r| < 1, with a convergence check and step-by-step work.

Find the distance between two coordinate points with dx, dy, squared distance, and square-root steps.

Find the midpoint between two coordinate points with coordinate-average steps.

Calculate triangle area from base and height with unit-friendly steps.

Calculate trapezoid area from two bases and height with average-base steps.

Calculate ellipse area from semi-major and semi-minor axes with the pi a b formula and equal-area circle check.

Calculate circle sector area from radius and central angle in degrees.

Calculate arc length from radius and central angle in degrees.

Calculate regular polygon area from side length and number of sides.

Calculate interior angle sum, each regular interior angle, and exterior angle for a polygon.

Calculate cylinder volume, base area, and lateral surface area from radius and height.

Calculate cone volume, base area, and slant height from radius and height.

Calculate sphere volume, surface area, and diameter from radius.

Calculate rectangular prism volume, surface area, and space diagonal from length, width, and height.

Calculate rectangular prism surface area, face areas, and volume from length, width, and height.

Calculate rectangular pyramid volume from base length, base width, and perpendicular height.

Calculate ellipsoid volume from three semi-axis lengths, plus a sphere-equivalent radius check.

Calculate circular segment area from radius and central angle, with sector, triangle, and arc-length checks.

Calculate sine, cosine, tangent, reciprocal trig ratios, and acute angle from right-triangle sides.

Solve the third side of a triangle from two sides and included angle using the law of cosines.

Calculate the nth term and partial sum of an arithmetic sequence.

Calculate the nth term and partial sum of a geometric sequence.