Math Visual Tools

📐

Every visual tool on the site

175 interactive visualizers across ten branches of mathematics. Each one animates a single idea step by step, with editable inputs and a written explanation beside the picture.

💡Tip: Pick a subject tab to see its tools grouped by topic.

175Tools

10Categories

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9 tools · 4 sub-groups

Arithmetic

Other2 tools

Base Conversion Visualizer

Enter a whole number and a target base from 2 to 36 and watch the conversion happen step by step: each division by the base, each remainder, and the digit it becomes, read off in the right order. Binary, octal and hexadecimal are the usual choices, but every base in the range works, with the letters A to Z standing in for digits above 9, and the same steps produce the same number in every base, which is the place-value idea the tool is built to show.

Fractions Visualizer - Interactive Learning Tool

Interactive fractions visualizer with circle diagrams. Learn fractions, decimals, percentages, and equivalent fractions through visual representation.

Diagrams1 tool

Arithmetic Diagrams

A gallery of every diagram already shown on the arithmetic pages: figures drawn for the lessons and frozen states of the visual tools. Browse by chapter or page, search captions, compare two figures side by side, and jump from any picture to the exact section where it appears or to the tool that drew it.

Divisibility & Remainders3 tools

Interactive Divisibility Table

Interactive divisibility table for numbers 0-100. Select divisors 1-12 to highlight patterns. Hover for rule explanations including digit sums, last digits, and more.

Divisibility Tiles Interactive Visualizer

Interactive divisibility visualization tool. Group tiles to see division, remainders, and check divisibility for numbers 1-100 with divisors 2-9.

Divisibility Decision Tree Interactive Tool

Interactive divisibility decision tree for testing numbers 1-9999. Visualize divisibility rules for 2-12 with step-by-step explanations and hover tooltips.

Primes, GCD & Modular Arithmetic3 tools

Sieve of Eratosthenes Visualization

Interactive Sieve of Eratosthenes visualization. Watch primes emerge as composites get crossed out with step-by-step explanations and color-coded divisibility.

Euclidean Algorithm Visualizer

Enter two positive integers and watch the Euclidean algorithm compute their greatest common divisor as a vertical chain of division equations. Amber remainder pills, dashed substitution arrows, and a purple GCD callout make every step of the algorithm's recursion visible. Random pair button and five curated presets cover coprime, divisor, and mid-complexity cases.

Modular Arithmetic Wheel Visualizer

Watch the integers 1 through N sort themselves into n remainder classes on a pie wheel — one slice per class, with the zero class (multiples of n) starred and centered at the top. Hover any slice for its formula and examples; switch the divisor from 2 to 9 to see how class structure changes.

11 tools · 6 sub-groups

Algebra

Binomials1 tool

Binomial Coefficient Visualizer

See where binomial coefficients come from through three connected views: a decision tree counting paths, an animated product expansion of (a+b)^n into like-term buckets, and Pascal's triangle as a path-counting grid. Adjust n from 1 to 5.

Equations · Quadratic1 tool

Completing the Square Calculator

Enter any quadratic ax^2 + bx + c and watch the geometric square build piece by piece — the x^2 block, two (b/2)·x strips, and the corner that completes the square. Click through the steps or hit play to see the vertex form derived visually.

Diagrams1 tool

Algebra Diagrams

A gallery of every diagram already shown on the algebra pages: figures drawn for the lessons and frozen states of the visual tools. Browse by chapter or page, search captions, compare two figures side by side, and jump from any picture to the exact section where it appears or to the tool that drew it.

Other3 tools

Equation Visual Explorer

Equation Visual Explorer

Drag a marble along the x-axis and watch where the curve crosses the horizontal target line — those crossings are the solutions of f(x) = n. Switch between linear, quadratic, and other equation families, play the solution back in steps, and read the sign chart showing where f(x) − n is positive, negative, or zero.

Inequality Visual Explorer

Inequality Visual Explorer

Drag a marble along the x-axis and watch which intervals satisfy the inequality. Switch between linear, quadratic, polynomial, and rational inequality families, toggle the direction between greater, less, and their non-strict variants, and read the sign chart that builds the solution set interval by interval.

Powers Table - Interactive Exponents Reference

Interactive powers table for bases 2 through 10 and exponents up to 16. Compare exponential growth, spot last-digit patterns, and reference exact values.

Algebraic Identities4 tools

Square of a Sum Visualization

Interactive visual proof of (a+b)² = a² + 2ab + b² through animated geometric dissection.

Square of a Difference Visualization

Interactive visual proof of (a-b)² = a² - 2ab + b² through animated geometric dissection.

Square of a Trinomial Visualization

Interactive visual proof of (a+b+c)² expansion through a 3x3 grid of pieces.

Difference of Squares Visualization

Interactive visual proof of a² - b² = (a+b)(a-b) through geometric slice and rearrange.

Polynomials1 tool

Polynomial Multiplication Visualizer

Multiply any two polynomials step by step with a visual grid: every term of P(x) times every term of Q(x) becomes a cell, color-coded by power of x. Like-term cells gather into buckets that combine into the final product. FOIL generalized to any size.

12 tools · 5 sub-groups

Functions

Function Properties5 tools

Function Asymptotes Visualizer

Find vertical, horizontal, and oblique asymptotes of common functions and watch them drawn as color-coded dashed lines over the curve. One-sided limit probes report whether each vertical asymptote tends to plus or minus infinity from each side, and four transformation sliders let you watch every asymptote track shifts and scales in real time.

Domain of a Function Visualizer

See the set of legal inputs of any function drawn directly on the x-axis as a colored band, with a horizontal number line below the plot and a draggable test point that checks domain membership and computes the function value for legal inputs. Pick from eleven base functions grouped by domain shape and watch the domain respond live to horizontal scale and shift — while vertical transformations leave it unchanged.

Range of a Function Visualizer

See the set of achievable outputs of any function drawn directly on the y-axis as a colored band, with a horizontal number line below the plot and a draggable test point that checks whether any specific y-value is reached. Pick from eleven base functions grouped by range shape and watch the range respond live to vertical scale and shift — while horizontal transformations leave it unchanged.

Function Symmetry Visualizer

See whether any function is even, odd, or neither by plotting f(x) alongside its y-axis reflection f(-x) and its origin rotation -f(-x) on one graph. Overlapping curves prove the symmetry visually, a color-coded verdict card states the result, and a step-by-step algebraic derivation walks through the test. Transformation sliders let you watch shifts break even or odd symmetry in real time.

Tangent Line Visualizer

Drag the x₀ slider and watch the tangent line follow the curve in real time. Pick a base function — linear, quadratic, cubic, reciprocal, exponential, logarithmic, square root, absolute, sine, cosine — apply affine transformations, and read both the point-slope and slope-intercept forms of the tangent equation alongside the slope m = f'(x₀). Critical points and undefined-tangent failures are flagged automatically.

Other4 tools

Function Composition Visualizer

Pick an outer function f and an inner function g from ten base families to see both compositions plotted on the same graph — f composed with g in blue, g composed with f in amber. The visual contrast makes it immediate that composition is not commutative, while live symbolic expressions and special-case notes for identity and inverse pairs build the intuition behind f of g of x.

Inverse Function Visualizer

Plot any function alongside its inverse as a reflection across the line y = x. Pick a base function — linear, cubic, reciprocal, exponential, logarithmic, square root, quadratic, absolute, sine, or cosine — apply affine transformations, and watch the inverse re-derived symbolically in real time. For functions that need a domain restriction to be invertible, the discarded portion appears faded and the principal branch in bold.

Piecewise Function Builder

Build piecewise functions piece by piece — pick a shape from a catalog, set each piece's interval and open or closed endpoints, and watch the tool flag overlaps, gaps, jumps, and removable discontinuities in real time at every boundary.

Functions Families Gallery

Pick a function family from a sidebar of twelve standard families — linear, quadratic, cubic, power, rational, exponential, logarithmic, sine, cosine, tangent, absolute value, and square root — and adjust its parameters with live sliders. The plot, symbolic equation, and explanation panel update in real time, with the three trigonometric families grouped together for direct comparison.

Diagrams1 tool

Functions Diagrams

A gallery of every diagram already shown on the functions pages: figures drawn for the lessons and frozen states of the visual tools. Browse by chapter or page, search captions, compare two figures side by side, and jump from any picture to the exact section where it appears or to the tool that drew it.

Transformations · Reflections1 tool

Function Reflections Visualizer

Pick a base function — linear, quadratic, cubic, reciprocal, exponential, logarithmic, sine, cosine, absolute value, or square root — and switch between seven reflection types: across the x-axis, y-axis, y = x, the horizontal y = c, the vertical x = c, plus |f(x)| and f(|x|). Both curves draw together, with the axis of reflection highlighted in orange.

Transformations1 tool

Transformations of Functions Visualizer

Pick a base function — linear, quadratic, cubic, reciprocal, exponential, logarithmic, sine, cosine, absolute value, or square root — and apply the four affine transformations: g(x) = a · f(b(x − h)) + k. Isolate one transformation at a time with Auto-mode animation, or combine all four in the Custom tab. The dashed base curve stays visible alongside the transformed curve so every change reads at a glance.

18 tools · 9 sub-groups

Trigonometry

Angles2 tools

Angle Explorer

An interactive angle visualizer that lets you set any angle in degrees or radians and instantly see its type, quadrant, reference angle, and its complementary, supplementary, and coterminal partners. Toggle reference lines, jump to preset unit-circle angles, and read live values for sin, cos, tan, csc, sec, and cot, including exact special-angle forms.

Interactive Angle Types Explorer

An interactive hub for nine core angle concepts in trigonometry. Drag handles or use presets to explore basic classifications (acute, right, obtuse, straight, reflex), pairwise relationships (complementary, supplementary, vertical, adjacent), and trigonometric foundations (standard position, reference angles, coterminal angles, the 16 special unit-circle angles, and directed angles with even and odd verification).

Angles · Radian Measure1 tool

Arc and Sector Explorer

An interactive explorer for radian measure. Drag the central angle and the radius on one circle and watch the arc length s = r theta and the sector area one half r squared theta recompute live. Lay radius-length arcs around the circle to see what one radian is, show the sector as a fraction of the disc, and shrink to radius 1 to reach the unit circle.

Identities8 tools

Basic Trigonometric Identities Explorer

Drag the blue dot around the unit circle to watch sine, cosine, tangent, and their reciprocals emerge as signed legs and ratios. Switch between the six functions via tabs, step through a five-stage derivation for each, and use the multi-turn slider to explore periodicity and coterminal angles.

Double Angle Identities Explorer

Double-Angle identities explorer covering all six trigonometric functions. Animated geometric proofs derive sin(2θ) and cos(2θ) from a bisected isosceles triangle, while algebraic derivation cards handle tan(2θ), csc(2θ), sec(2θ), and cot(2θ) with one-click jumps back to their source proofs. A live formula table and verification cards confirm each identity numerically as you sweep the angle.

Half Angle Identities Explorer

Half-Angle identities explorer covering all six trigonometric functions. Animated geometric proofs derive sin(α/2) and cos(α/2) from a bisected isosceles triangle, while algebraic derivation cards handle tan(α/2), csc(α/2), sec(α/2), and cot(α/2) with one-click jumps back to their source proofs. A live formula table and verification cards confirm each identity numerically as you sweep α from 20° to 160°.

Negative Angle Identities Explorer

Negative-Angle identities explorer covering all six trigonometric functions. Animated reflection proofs derive sin(-θ) = -sin θ and cos(-θ) = cos θ from a point and its mirror image across the x-axis on the unit circle, while algebraic derivation cards handle tan, csc, sec, and cot. A live formula table and verification cards label each function's parity (even or odd) and confirm every identity numerically.

Pythagorean Identities Explorer

Pythagorean identities explorer covering all six trigonometric functions. The base identity sin²θ + cos²θ = 1 is proved geometrically inside the unit circle, then tan, csc, sec, and cot forms follow by dividing through and rearranging. Animated steps, algebraic derivation cards with jump-to-source buttons, and a live formula table verify every identity numerically as you sweep θ through the first quadrant.

Shift Identities Explorer

An interactive unit-circle proof of the six shift identities. Choose a shift of pi or pi/2, drag the angle through a full turn, and watch the point P rotate to P prime while the legs of both right triangles show which coordinate became which. Sine, cosine and tangent of the shifted angle are read off the rotated point and checked numerically, with all six identities in one table.

Supplementary Angle Identities Visualizer

Explore all six trig functions at π − θ across two complementary views. Sin and cos appear as reflections across the y-axis with live unit-circle geometry; tan, csc, sec, and cot derive algebraically from those two roots. Step through proofs, switch tabs, and verify each identity numerically at any angle.

Triple Angle Identities Explorer

An interactive explorer for the triple-angle identities. Play through the derivation of sin 3 theta and cos 3 theta one line at a time, from the angle sum through the double-angle and Pythagorean identities to a cubic in one function, with every line checked numerically at the chosen angle. Tangent, cosecant, secant and cotangent follow as derived cards, and a formula table holds all six.

Diagrams1 tool

Trigonometry Diagrams

A gallery of every diagram already shown on the trigonometry pages: figures drawn for the lessons and frozen states of the visual tools. Browse by chapter or page, search captions, compare two figures side by side, and jump from any picture to the exact section where it appears or to the tool that drew it.

Functions · Graph Transformations1 tool

Trigonometric Function Parameters Explorer

An interactive explorer for the general form y = A f(Bx - C) + D. Drag one parameter at a time and watch the feature it controls move on the curve: the amplitude bracket, the period bracket, the phase-shift bar and the midline, each drawn in the colour of its own slider. Works on sine, cosine, tangent and cotangent, and states plainly where amplitude stops having a meaning.

Functions2 tools

Interactive Trigonometric Functions Graphs

An interactive trigonometric functions visualizer that graphs sine, cosine, tangent, cosecant, secant, and cotangent in real time. Switch between degrees and radians, drag the angle slider through a full rotation, jump to special-angle presets, and watch live function values, periods, ranges, and asymptotes update on the plot and in the explanation panel.

Interactive Trigonometric Functions Signs by Quadrants

An interactive quadrant explorer that shows the sign of every trigonometric function in each of the four quadrants. Click a quadrant or press 1-4, pick a function, and watch the unit circle, coordinate signs, ASTC rule, and step-by-step reasoning update together. A clean way to internalize why sine, cosine, tangent, and their reciprocals flip sign as the angle moves around the circle.

Inverse Functions1 tool

Inverse Trigonometric Functions Explorer

An interactive explorer that builds the inverse trigonometric functions in four steps. Drag a horizontal line to watch the line test fail, drag two handles to cut the domain until the function is one-to-one, reflect the kept piece across y = x, and feed inputs through the composition to see where it folds. Works on sine, cosine and tangent, with the principal interval of each marked.

Triangle1 tool

Interactive Triangle Explorer

An interactive triangle visualizer with twelve preloaded scenarios covering classifications (equilateral, isosceles, acute, obtuse, right scalene), special right triangles (45-45-90, 30-60-90, 3-4-5, 5-12-13), and the laws of sines and cosines. Drag vertices, lock up to two angles, zoom freely, and read live stats and context-aware explanations including Pythagoras, SOH CAH TOA, and the trigonometric laws.

Other1 tool

What is Unit Circle?

Understand how numbers work in different bases (2-36) with interactive cube visualization. See binary, octal, decimal, and hexadecimal conversions in real-time. Ideal for computer science students and programmers learning number system fundamentals.

12 tools · 5 sub-groups

Calculus

Calculus · Derivatives4 tools

Average Rate of Change Visualizer

Drag two sample points along the cubic curve f(x) = ⅓x³ − x to watch the secant slope Δy/Δx update in real time. Four animated scenarios — ascending, descending, local max, and local min — walk through the geometry that turns the average rate of change into the derivative, with synchronized computation, meaning, and theory panels.

Inflection Points and Concavity Visualizer

Drag point c along the cubic f(x) = (1/3)x^3 - x to see how the sign of f''(c) decides whether the curve is concave up, concave down, or inflecting. Three pre-built scenarios animate the seven analysis steps from picking a region to evaluating f''(c) and shading the concavity gap between curve and tangent.

Newtons Method Visualizer

Drag a starting guess x₀ along the x-axis for f(x) = x³ − 2x − 5 and watch each Newton step draw its tangent, drop to the x-axis, and lift back to the curve. Three preset scenarios show a direct hit, a slow crossover, and a stalling failure when x₀ lands near a critical point where the tangent goes nearly horizontal.

Tangent Line Visualizer at a Point

Drag point c along the cubic f(x) = (1/3)x^3 - x to see the tangent line drawn through P = (c, f(c)) with slope f'(c). Four pre-built scenarios animate the six analysis steps for positive slopes, negative slopes, local maxima, and local minima, then write the tangent in point-slope form.

Limits and Continuity2 tools

Continuity Checker

Probe continuity at any point with a live three-condition checklist — f(c) defined, two-sided limit exists, f(c) equals the limit. Slide c through holes, jumps, asymptotes, and staircases and watch each row flip pass or fail in real time.

Limit Explorer

Watch left and right limits race toward each other as epsilon shrinks. Switch between seven canonical cases — continuous, hole, jump, infinite asymptote, oscillating, and one-sided — and read the verdict for both the limit and the continuity at c.

Derivatives3 tools

Derivative Visualizer

Move x0 along the graph and watch three pictures of one number lock together — the slope of the tangent on f, the height of f-prime at x0, and the numeric derivative value. Snap directly to roots, extrema, and inflection points to see why each one matters.

Mean Value Theorem Visualizer

Draw a secant between any two points on a smooth curve and the tool finds every interior c where the tangent has matching slope — the parallel-tangent guarantee at the heart of the Mean Value Theorem. Six function families illustrate the single-c, multi-c, and exact-midpoint cases.

Optimization Visualizer

Find every critical point of a smooth function on a chosen interval and watch the second-derivative test classify each as local max, local min, or inflection. Six families show single-extremum cases, W-shapes, and touch zeros where the test falls back to the first-derivative test.

Diagrams1 tool

Calculus Diagrams

A gallery of every diagram already shown on the calculus pages: figures drawn for the lessons and frozen states of the visual tools. Browse by chapter or page, search captions, compare two figures side by side, and jump from any picture to the exact section where it appears or to the tool that drew it.

Integrals2 tools

Fundamental Theorem of Calculus Visualizer

See both halves of the Fundamental Theorem of Calculus on one graph — slide x to grow the shaded area under f and watch the accumulator F match the area exactly, while the slope of F at every x equals f at the same point.

Riemann Sum Visualizer

Approximate a definite integral with rectangles or trapezoids, then watch the error shrink as you increase the partition count. Switch between left, right, midpoint, and trapezoid rules to see why some converge as one over n and others as one over n squared.

49 tools · 17 sub-groups

Linear algebra

Linear Algebra · Vector Spaces2 tools

Change of Basis 2D Visualizer

Drag basis vectors b1 and b2 to define a new coordinate system, then watch how vector v coordinates shift between the standard basis and your custom basis. Live computation of the change-of-basis matrix B, its inverse, and determinant, with ten preset scenarios covering orthonormal, oblique, singular, and orientation-reversed bases.

Span and Linear Independence 2D Visualizer

Drag two vectors in the plane and see their span directly: a filled parallelogram and a plane-covering lattice when they are independent, a collapsed segment and a single dashed line when one is a multiple of the other. Sweep one vector a full turn to watch the determinant trace a sine wave and cross zero at exactly the two dependent angles. Nine presets cover orthogonal, oblique, near-aligned, parallel, anti-parallel and zero-vector pairs, and a live card reports lengths, angle, determinant, area and rank.

Matrices · Decompositions4 tools

Cholesky Decomposition Visualizer

Watch A = L Lᵀ assemble for a symmetric positive definite matrix of your own numbers: each diagonal entry of L is a square root of what is left of the matching entry of A, each entry below it is a division, and the transpose fills in step on the right. Edits are mirrored to keep A symmetric, a shuffle produces guaranteed positive definite matrices, and two presets stop the run on purpose to show the unsymmetric and the indefinite cases.

LU Decomposition Visualizer

Watch A = LU assemble in place on a matrix of your own numbers: Gaussian elimination turns a working copy of A into U while every multiplier is written into the matching position of L, and a zero pivot forces a row swap that turns the identity into PA = LU. Six presets cover the classic integer case, negative pivots, a forced swap, a Pascal matrix whose factors are Pascal triangles, and a singular matrix; the final scene reads the determinant off the diagonal of U.

QR Decomposition Visualizer

Watch A = QR assemble in place: Gram-Schmidt runs down the columns of A, each projection coefficient is written into R above the diagonal, each working column is shown as a growing expression inside Q, and each length goes onto the diagonal of R as the column is normalized. The zeros of R are drawn from the first scene, because the triangular shape follows from the order of the process. Shapes from 2×2 to 4×3, with the tall shapes that least squares uses.

Singular Value Decomposition Visualizer

Compute the SVD of a matrix of your own numbers, of any shape from 2×2 to 3×3, exactly as it is done by hand: form AᵀA, take its eigenvalues and eigenvectors, set σᵢ = √λᵢ and uᵢ = A vᵢ / σᵢ, complete U when the rank falls short, and assemble A = U Σ Vᵀ with a rectangular Σ. The factorization is multiplied out and rewritten as a sum of rank-one terms, and the final scene reports the rank, the spectral and Frobenius norms, the condition number and the four fundamental subspaces. Presets include a rank-one matrix, a rotation with repeated singular values, and wide and tall rectangular cases.

Linear Algebra · Eigenvalues2 tools

Complex Eigenvalues 2D Visualizer

Drag a start vector and watch a real 2×2 matrix with complex eigenvalues turn it step by step: the orbit spirals in, spirals out, or runs round an ellipse, and a continuous spiral traces A^k x₀ between the steps. The matrix card shows the rotation-scaling form A = P C P⁻¹ with the real and imaginary parts of a complex eigenvector as the columns of P, and the live card reads r from the determinant and θ from the trace. Seven presets from plain rotations to skewed spirals, editable entries, and a real-eigenvalue fallback.

Eigenvectors 2D Visualizer

Drag a test vector around the plane and watch when it aligns with its image Av to spot real eigendirections. Live computation of eigenvalues, characteristic polynomial, and structural classification across eleven scenarios spanning distinct real, isotropic, defective, and complex cases.

Linear Systems · Solving4 tools

Cramer's Rule Visualizer

Solve a 2×2 or 3×3 system of your own numbers by Cramer's rule, one unknown per scene: evaluate det A first, then swap b into each column of A in turn, evaluate that determinant, and divide. The right-hand side stays coloured so the swapped column is always visible, a singular preset shows the rule stopping at det A = 0, and the final scene multiplies the solution back to check it. Five presets and a shuffle that builds systems with small integer answers.

Gaussian Elimination Calculator

Row reduce an augmented matrix [A | b] one elementary operation at a time, stopping at row echelon form or carrying on to reduced row echelon form. Switch between the two targets on the same matrix to see exactly which extra operations Gauss-Jordan performs: both runs share the same pivots and the same below-pivot eliminations, and the reduced run adds the scalings and the above-pivot clearings. Every step names the operation in R notation, highlights the pivot row and the row it changes, and explains why the solution set is untouched. Results are exact fractions, the final step reports the rank and either the solution, the free variables, or the row that proves the system unsolvable. Edit the entries, pick a preset - swap-first, free variable, inconsistent, zero matrix - or shuffle a random system, up to 5 equations and 5 unknowns.

Least Squares Visualizer

Solve a least-squares problem with your own numbers exactly as it is done by hand: row reduce [A | b] to see that no exact solution exists, form the normal equations AᵀA x̂ = Aᵀb as matrix products, solve them, compute the projection p = A x̂, check that the residual b − p is perpendicular to every column, and build the projection matrix A(AᵀA)⁻¹Aᵀ. For design matrices the captions read off the fitted line or parabola and its errors. Presets include projection onto a line and a plane, a consistent system, and dependent columns.

Linear System Solutions Visualizer

Row reduce a system of your own numbers all the way to reduced row echelon form, one operation per scene, and read the solution set from the result: the solution vector when every column has a pivot, the row 0 = c that proves there is none, or a particular solution plus one direction per free variable when there are infinitely many. Six presets cover the three outcomes, a homogeneous system, an overdetermined but consistent one, and a plane of solutions with two free variables.

Diagrams1 tool

Linear Algebra Diagrams

A gallery of every diagram already shown on the linear algebra pages: figures drawn for the lessons and frozen states of the visual tools. Browse by chapter or page, search captions, compare two figures side by side, and jump from any picture to the exact section where it appears or to the tool that drew it.

Matrices · Eigenvalues4 tools

Eigenvalues and Eigenvectors Visualizer

Find the eigenvalues and eigenvectors of a matrix of your own numbers exactly as it is done by hand: subtract λ from the diagonal, expand the determinant into the characteristic polynomial, find its roots with the trace and determinant as checks, then row reduce A − λI for each eigenvalue and read an eigenvector from its free column. Presets cover distinct, repeated, defective, triangular, symmetric and complex cases, and the final scene says whether the matrix can be diagonalized.

Matrix Diagonalization Visualizer

Diagonalize a matrix of your own numbers as A = P D P⁻¹, exactly as it is done by hand: the eigenvectors become the columns of P, the eigenvalues fill the diagonal of D, P is inverted by row reduction, and the product is multiplied out and checked. Then pick a power k and watch A^k drop out of P D^k P⁻¹. Presets include a Markov chain settling to its steady state, the Fibonacci matrix, a symmetric matrix with orthogonal eigenvectors, and a defective shear that cannot be diagonalized.

Power Iteration Visualizer

Run the power method on a matrix of your own numbers: multiply a start vector by A, read the entry of largest magnitude as the eigenvalue estimate, rescale, and repeat for as many steps as you choose. Each step reports how much the estimate and the vector moved, and the final scene compares them with the exact dominant eigenvalue and eigenvector, gives the convergence ratio |λ₂/λ₁|, and explains ties, complex pairs and negative dominant eigenvalues. Presets range from a fast case to a crawling one and a swap matrix that never converges.

Spectral Decomposition Visualizer

Decompose a symmetric matrix of your own numbers as A = Q Λ Qᵀ, exactly as it is done by hand: eigenvalues, one integer eigenvector per eigenvalue, Gram–Schmidt inside any repeated eigenspace, normalization into an orthogonal Q, the check that Qᵀ Q = I, the factorization multiplied out, and A rewritten as a sum of rank-one projections λᵢ qᵢqᵢᵀ. The final scene classifies the quadratic form as definite, semidefinite or indefinite from the signs of the eigenvalues. Presets include a projection, a saddle, a repeated eigenvalue, and a non-symmetric shear that stops the run.

Matrices · Vector Spaces2 tools

Four Fundamental Subspaces Visualizer

Find all four subspaces of a matrix of your own numbers, of any shape from 2×2 to 4×4, from one row reduction: the pivot columns of A for the column space, the non-zero rows of R for the row space, one special solution per free column for the null space, and the null space of Aᵀ for the left null space. Each null-space basis is checked by a matrix product, the two orthogonalities are shown as zero products, and the final scene counts the dimensions and says what they mean for solving Ax = b. Presets cover rank-deficient, full-rank, rectangular, invertible and zero matrices.

Span and Membership Visualizer

Enter spanning vectors and a test vector in R² to R⁴ and settle three questions exactly as they are settled by hand: row reduce V to find the dimension of the span and any dependencies among the vectors, row reduce [V | w] to decide whether w is a combination of them, and read off and verify the coordinates when it is. Dependent sets are diagnosed with their relations, non-unique coordinates are explained, and a vector outside the span is reported with the rank it adds. Presets cover a plane, a line, a dependent set, a basis of R³ and a plane in R⁴.

Vectors · Orthogonality1 tool

Gram-Schmidt Process Visualizer

Watch the Gram-Schmidt process run on two or three vectors, shown as the rows of one matrix: the first is kept, each later one loses its projections onto the vectors already made orthogonal, one subtraction per scene with the growing expression written in the cell, and a final pass divides each result by its length. The page explains why the projections go onto the new vectors rather than the originals, why order matters, and how the discarded coefficients become the R of a QR decomposition.

Matrices · Matrix Operations10 tools

Hadamard Product Visualizer

Watch the element-wise product A circle-dot B build one cell at a time, with animated arrows linking each pair of operand cells to its destination. Adjust the shared shape of A and B from 1x1 to 5x5 and see how the Hadamard product differs from the standard matrix product.

Matrix Addition and Subtraction Visualizer

Watch A + B = C or A - B = C build one cell at a time, with animated arrows linking each pair of operand cells to its destination in the result matrix. Toggle between addition and subtraction, adjust the shared shape of A and B from 1x1 to 5x5, and step through the element-wise rule at your own pace.

Matrix Inverse Visualizer

Watch the inverse of a 2×2 or 3×3 matrix build by the adjugate formula — strike a row and a column for every entry to read off its cofactor, transpose the cofactor matrix into the adjugate, expand the determinant along the first row from cofactors already in hand, then divide every entry by it. Every cofactor is written out symbolically, and the page explains why the transpose is necessary and why a zero determinant stops the construction.

Linear Combination of Matrices Visualizer

Watch αA + βB build in three phases — scale A by α, scale B by β, then add the scaled matrices cell by cell into C. Adjust the shared shape of A and B from 1x1 up to 5x5 and see how a linear combination decomposes into scalar multiplication followed by matrix addition.

Matrix Multiplication by Columns

Build a matrix times a vector the way the definition says, then watch the same products regroup into whole columns. Each entry of the vector becomes a weight on one column of the matrix, and each scaled column is drawn in the plane as an arrow laid tail to head on the one before it. A second tab repeats the argument with a matrix on the right, assembling the product one column at a time. The reading leads straight to span, column space, rank and the question of when a system can be solved.

Matrix Multiplication by Rows

Put the vector on the left and the matrix hands over its rows. The tool fills the product one term at a time by the definition, then regroups the same terms across instead of down, so each entry of the vector becomes a weight on a whole row. Every scaled row is drawn in the plane, laid tail to head on the one before it. A second tab repeats the argument with a matrix on the left, building the product one row at a time. This is the reading behind elementary matrices, row equivalence and elimination.

Matrix Multiplication by Rows and Columns

One product, two decompositions, and the relationship between them. The tool builds AB a column at a time out of the columns of A, then a row at a time out of the rows of B, then puts a column and a row of the finished product in the same plane so you can see the single entry they share. Colour marks the role each factor plays rather than its name, so the swap between readings is visible as it happens. This page explains how the two readings fit together rather than teaching either one.

Matrix Power Visualizer

Watch n copies of a square matrix collapse pairwise into Aⁿ, one bracket-and-collapse per stage, with the exponent growing by one each time. Cells of A² are written out in full as sums of products; from A³ upward each cell shows a Σ summary, and hovering it lists every term with the count in the header. Sizes 2×2 to 4×4 and exponents up to 5, which is enough to see the term count double or triple at every collapse.

Matrix Scalar Multiplication Visualizer

Watch C = kA build one cell at a time, with every entry of A multiplied by the same scalar k. Adjust the shape of A from 1x1 up to 5x5 and see how scalar multiplication preserves shape while uniformly scaling every entry.

Matrix Transpose Visualizer

Watch A become A^T through four equivalent methods — cell-by-cell, row-as-column, column-as-row, or a single diagonal reflection. Set the shape of A from 1x1 up to 5x5, switch methods with a tabbed control panel, and see how each mental model breaks the same operation into a different number of steps.

Linear Algebra · Linear Transformations5 tools

Kernel and Image 2D Visualizer

Drag a vector on the domain canvas and watch its image Av appear on the codomain canvas next to it. The kernel shows as a red dashed line on the left, the image as a green line on the right. Edit matrix A directly, sweep v around a full circle to trace its image, and explore ten scenarios spanning full rank, rank one projections, nilpotent, and the zero map.

Linear Transformation 2D Visualizer

Drag the t slider to morph the identity matrix continuously into any 2x2 matrix A and watch the plane reshape. The unit square fills with color to show orientation, eigenvectors highlight as magenta lines, kernel and image lines appear when A is singular, and a live readout tracks determinant, trace, eigenvalues, and rank across twelve preset scenarios.

Matrix Composition 2D Visualizer

Watch a vector travel through two matrices in the plane along a two-segment trail, first B then A, while the grid and unit square morph in two phases. Flip the order and see the dashed alternative path land somewhere else, or on the same point when the pair commutes. Nine presets cover commuting pairs, non-commuting pairs, two reflections composing to a rotation, and a shear cancelled by its inverse, in a single-canvas trail view or a three-canvas stages view.

Projection 2D Visualizer

Drag a vector in the plane and watch its orthogonal projection slide along a line as the foot of the perpendicular, with the kernel line, the dropline and a right-angle mark drawn in. Morph the grid from the identity to the projection matrix to see the plane flatten onto the line, read the 2×2 matrix and its invariants live, and set the line by preset or by an angle slider. Seven presets cover the coordinate axes, the diagonals and custom angles.

Reflection 2D Visualizer

Drag a vector in the plane and watch its mirror image across a line through the origin, joined by a perpendicular whose midpoint is the projection onto the mirror. Morph the grid from the identity to the reflection and see it pass through the projection halfway, with the unit square flattening and then flipping colour as the determinant crosses from +1 to −1. Read the 2×2 matrix and its invariants live, and set the mirror by preset or angle slider.

Matrices · Properties3 tools

Determinant Visualizer

Watch a determinant assemble term by term for any square matrix from 2×2 to 5×5. Four strategies show the same number four ways: the diagonal product at 2×2, Sarrus's rule lighting six diagonals on an augmented layout at 3×3, and cofactor expansion along any row or any column you choose, each term striking a row and a column, drawing the minor alongside the original and attaching the checkerboard sign. The sign pattern gets a scene of its own, a running formula grows in the step log, and the page explains why the shortcut has no 4×4 version.

Matrix Rank Visualizer

Row reduce a matrix of your own numbers to echelon form one operation at a time — find a pivot, swap rows when needed, clear the entries below, skip the columns that have no pivot — and count the pivots to read off the rank. Six presets cover rank-deficient, full rank, rank 1, wide, tall and zero matrices, a shuffle produces random ones, and the final scene highlights the pivot columns in the original matrix and reports the nullity.

Matrix Trace Visualizer

Watch tr(A) build one diagonal entry at a time, with off-diagonal cells fading into the background. Set the square dimension of A from 2x2 up to 10x10, follow the running sum, and see why the trace is the cleanest scalar invariant of a square matrix.

Special Matrices1 tool

Square Matrix Types Generator

Generate a square matrix of any of the classic special types — identity, zero, scalar, diagonal, upper and lower triangular, symmetric, skew-symmetric, or random — at the size you choose, and see which entries are forced and which are free. Each type comes with a short explanation, its defining pattern in colour, and the properties that follow from it: what its determinant and eigenvalues look like, what it does under multiplication, and where it turns up in decompositions and physics.

Matrices · Orthogonality1 tool

Orthogonal Matrices Visualizer

Test a matrix of your own numbers for orthogonality exactly as it is done by hand: multiply QᵀQ and compare with the identity, read the determinant to separate rotations from reflections, watch lengths and dot products of a test vector survive, verify that the transpose is the inverse, and identify the transformation, as a rotation angle or mirror line in the plane or an axis and angle in space. Near misses are diagnosed: perpendicular columns of the wrong length are normalized on the spot. Presets include a permutation, a Householder reflection and a scaled rotation.

Vectors · Operations3 tools

Vector Addition and Subtraction Visualizer

Watch u + v = w or u - v = w build one component at a time, with animated arrows linking each pair of operand components to its destination in the result vector. Toggle between addition and subtraction, adjust the shared length of u and v from 1 to 5, and step through the component-wise rule at your own pace.

Linear Combination of Vectors Visualizer

Watch αu + βv build in three phases — scale u by α, scale v by β, then add the scaled vectors component by component into w. Adjust the shared length of u and v from 1 up to 5 components and see how a linear combination decomposes into scalar multiplication followed by vector addition.

Vector Scalar Multiplication Visualizer

Watch w = kv build one component at a time, with every component of v multiplied by the same scalar k. Adjust the length of v from 1 up to 5 components and see how scalar multiplication preserves dimension while uniformly scaling every component.

Vectors · Products4 tools

Cross Product of Vectors Visualizer

Watch u × v build one component at a time for two vectors in three dimensions. Choose the component formula — skip a row, cross-multiply the other two in cyclic order — or the i, j, k determinant expansion that strikes a row and a column, takes the minor and applies the cofactor sign. Both methods produce the same perpendicular vector, and the tool shows exactly where the sign of the middle component comes from.

Outer Product of Vectors Visualizer

Watch the outer product u vᵀ fill as a matrix — every entry of u against every entry of v — cell by cell, row by row, or column by column. Set the two lengths independently, from 1 to 5 each, and see why no matching rule applies, why every row is a copy of vᵀ and every column a copy of u, and why that makes the result a rank-1 matrix, the building block of matrix multiplication and the singular value decomposition.

Vector Projection Visualizer

Watch the projection of u onto v build in six phases — pair the components into a dot product, form the squared length of v, divide to get the coefficient, scale v by it, and subtract to expose the perpendicular remainder. Choose projection only or the full orthogonal decomposition, set the shared length from 1 to 5, and see why the divisor is the squared length and why the remainder is at right angles to v.

Inner Product Visualizer

Pair, multiply, sum — watch the inner product build into a single scalar one term at a time. Switch between the classical vector dot product and the Frobenius matrix inner product, set the shape of the operands, and follow a running expanded sum with per-term color coding.

Vectors · Properties1 tool

Vector Magnitude and Unit Vector Visualizer

Watch the magnitude of a vector build in three phases — square each component, sum the squares and take the root, then divide every component by that one number to get the unit vector. Choose magnitude only or magnitude followed by normalization, set the length from 1 to 6 components, and see why the direction survives normalization while the length becomes exactly 1.

Other1 tool

Interactive Matrix Multiplication Visualizer

Visualize matrix multiplication step-by-step with our interactive tool. Watch animated calculations, understand dot products, customize matrix dimensions. Free online matrix calculator for learning linear algebra.

30 tools · 9 sub-groups

Probability

Probability Functions & CDF5 tools

Cumulative Distribution Function Visualizers

Understand cumulative distribution functions through interactive charts for discrete (Binomial, Poisson, Geometric) and continuous distributions (Normal, Exponential, Uniform). Adjust parameters to see how probability accumulates across different distributions with real-time CDF visualizations.

Continuous Distributions CDF Visualizer

Visualize smooth cumulative distribution functions for continuous probability distributions. Adjust parameters, explore curves, calculate probabilities.

Discrete Distributions CDF Visualizer

Visualize cumulative distribution functions for discrete probability distributions. Adjust parameters, explore step functions, calculate probabilities interactively.

Discrete Probability Distributions

Interactive discrete probability distribution visualizer with Binomial, Poisson, Geometric and more. Explore probability mass functions with real-time parameter controls and calculations.

Continuous Probability Distributions

Interactive continuous probability distribution visualizer with Normal, Exponential, Uniform distributions. Explore probability density functions with real-time PDF/CDF toggle and parameter controls.

Simulators2 tools

Coin Toss Probability Simulator and Calculator

Flip a fair or biased coin thousands of times and watch the Law of Large Numbers pull results toward theory, or map the complete sample space for up to six flips and calculate the probability of any pattern you highlight.

Dice Roll Probability Simulator and Calculator

Roll up to six dice and watch frequency distributions converge to theoretical patterns, or explore the complete sample space and filter by conditions like sum equals 7 to compute exact probabilities.

Conditional Probability6 tools

Probability Tree Diagram Maker

Free probability tree diagram maker. Set P(A), P(B|A) and P(B|Aᶜ) and get joint, total and Bayes' theorem probabilities step by step.

Venn Diagram - Conditional Probability Visualization

Interactive Venn diagram showing conditional probability through partitioned sample space. Click compartments to see P(A|Bᵢ) calculations with area-based explanations.

Waffle Chart - Conditional Probability Visualization

Interactive waffle chart showing conditional probability across four regions. Adjust event distribution to see how P(Event|Region) differs from total probability P(Event).

Contingency Table - Conditional Probability Visualization

Interactive 2×2 contingency table with adjustable P(A), P(B|A), P(B|Aᶜ). See joint, marginal, and conditional probabilities with clickable highlighting.

Interactive Contingency Tables Visualizer

Interactive contingency tables for visualizing joint, marginal, and conditional probabilities. Click cells and conditional rows to highlight relationships and see how Bayes' theorem connects different probability perspectives. Available in 2×2, 2×3, 2×4, and 3×3 configurations with detailed explanations for each state.

Total Probability Visualizer

Calculate marginal probabilities using the law of total probability with an interactive tree diagram. The tool shows how probabilities flow through partitioned sample spaces, displaying joint probabilities, conditional probabilities, and Bayes' theorem calculations in real-time.

Diagrams1 tool

Probability Diagrams

A gallery of every diagram already shown on the probability pages: figures drawn for the lessons and frozen states of the visual tools. Browse by chapter or page, search captions, compare two figures side by side, and jump from any picture to the exact section where it appears or to the tool that drew it.

Distributions · Discrete6 tools

Binomial Distribution Calculator

Calculate binomial probabilities for n trials with success probability p. Interactive tool with PMF/CDF visualization, parameter controls, and step-by-step probability calculations.

Geometric Distribution Calculator

Calculate geometric probabilities for trials until first success. Interactive calculator with memoryless property visualization, PMF/CDF charts, and comprehensive probability tools.

Negative Binomial Distribution Calculator

Calculate negative binomial probabilities for trials needed to achieve r successes. Interactive tool with PMF/CDF visualization and comprehensive probability calculations.

Poisson Distribution Calculator

Calculate Poisson probabilities for events in fixed intervals. Interactive rate parameter λ control with PMF/CDF visualization and comprehensive probability tools.

Hypergeometric Distribution Calculator

Calculate hypergeometric probabilities for sampling without replacement from finite populations. Interactive N, K, n parameter controls with comprehensive probability calculations.

Discrete Uniform Distribution Calculator

Calculate discrete uniform probabilities for equally likely outcomes. Interactive tool for finite uniform distributions with parameter controls and probability calculations.

Distributions · Continuous3 tools

Normal Distribution Calculator

Calculate normal distribution probabilities with interactive μ and σ controls. Includes PDF/CDF visualization, Z-score calculator, and 68-95-99.7 empirical rule display.

Exponential Distribution Calculator

Calculate exponential distribution probabilities for time between events. Interactive λ parameter control with PDF/CDF visualization and memoryless property demonstration.

Continuous Uniform Distribution Calculator

Calculate continuous uniform probabilities over intervals [a,b]. Interactive bounds control with constant PDF visualization and linear CDF display.

Expected Value & Variance3 tools

Discrete Expected Value Visualization

Interactive discrete expected value visualization with adjustable probability sliders. See E[X] formula, bar chart, and contribution breakdown for PMF.

Weighted Expected Value Visualizer

Interactive visualization showing expected value as probability-weighted average. See how probability weights pull E[X] with preset distributions and animation.

Interactive Variance Calculator and Visualizer

Drag data points and watch variance update in real time as deviations from the mean become visually tangible through interactive bars and step-by-step calculations.

Inequalities2 tools

Markov Inequality Visualizer

Interactive Markov inequality tool showing P(X ≥ a) ≤ E[X]/a across 9 distributions. Compare bounds to actual tail probabilities in real time.

Chebyshev Inequality Visualizer

Interactive Chebyshev inequality tool showing P(|X-μ| ≥ a) ≤ σ²/a² across 9 distributions. Visualize two-tailed bounds with mean and variance controls.

Venn Diagrams2 tools

2-Set Venn Diagram Tool

Interactive 2-set Venn diagram for probability calculations. Click segments, explore intersections, unions, and complements with step-by-step solutions.

3-Set Venn Diagram Tool

Interactive 3-set Venn diagram for complex probability problems. Visualize 8 regions, solve systems of equations, and explore triple intersections with calculations.

12 tools · 3 sub-groups

Combinatorics

Permutations5 tools

Circular Permutation Visualizer

Watch (n-1)! circular permutations build one by one — one item anchored at the top breaks the rotational symmetry while the remaining items cycle through every distinct seating around the circle. Adjust n and step through the build animation.

Full Permutation Visualizer

Build every n! permutation of n distinct items one ball at a time. Each arrangement starts with a different first item, and rows group all permutations sharing that starter — so you literally see why n! splits as n × (n−1)!

Partial Permutation Visualizer

Pick r items from n and arrange them in order, one ball at a time. Each partial permutation is grouped by its first item — making the n × (n−1) × … × (n−r+1) structure of P(n,r) visible at a glance.

Permutation with Identical Items Visualizer

Cycle through six multisets — AAB, AAAB, AABB, AABC, AAABB, AABBC — and watch every distinct permutation build one position at a time. Only the specific copy in use dims in the source row, exposing why n! divides by k! for each repeated group.

Permutation with Repetition Visualizer

Build every n^r arrangement of r positions where each spot can be any of n items — source balls never dim because copies are unlimited. Cycle n and r to compare n^r against n!/(n−r)! and see how reuse multiplies the count.

Combinations6 tools

Simple Combination Visualizer

Pick r items from n where order doesn't matter — watch each unordered subset build canonically and group by its smallest element. The varying row sizes expose why C(n,r) = P(n,r)/r! and how Pascal's identity sums to the binomial coefficient.

Distribution into Cells Visualizer

Send n distinct items into k labeled cells where each item picks its own cell — watch the assignment tuple fill in below the cells and the k^n outcomes group by item 1's destination. Same math as permutations with repetition, framed from the items' point of view.

Partition into Groups Visualizer

Split n distinct items into k labeled boxes of fixed sizes — watch each partition build position by position and group by which box receives item 1. Cycle the partition shape to see how the multinomial formula n!/(n1!·n2!·...) reshapes the count.

Pascal's Triangle Visualizer

Click any cell to focus a binomial coefficient C(n, r) and switch between four identity modes — Pascal's rule, the hockey-stick sum, the row sum of 2^n, and C(n, r) = C(n, n − r). A polynomial banner shows the focused row's expansion of (a + b)^n live.

Strong Composition Visualizer

Drop k−1 bars into the n−1 gaps between identical items to split them into k nonempty bins — every bin must hold at least one. The gap-selection encoding makes C(n−1, k−1) visible directly, with live bin brackets and a chosen-gaps readout updating as the bars land.

Weak Composition Visualizer

Drop k−1 bars into a strip of n+k−1 cells to split n identical items into k bins — empty bins allowed. The stars-and-bars encoding makes C(n+k−1, k−1) visible directly, with live bin brackets, a composition tuple readout, and bar-position readout updating as the bars land.

Diagrams1 tool

Combinatorics Diagrams

A gallery of every diagram already shown on the combinatorics pages: figures drawn for the lessons and frozen states of the visual tools. Browse by chapter or page, search captions, compare two figures side by side, and jump from any picture to the exact section where it appears or to the tool that drew it.

11 tools · 9 sub-groups

Set theory

Diagrams1 tool

Set Theory Diagrams

A gallery of every diagram already shown on the set theory pages: figures drawn for the lessons and frozen states of the visual tools. Browse by chapter or page, search captions, compare two figures side by side, and jump from any picture to the exact section where it appears or to the tool that drew it.

Counting and Cardinality · Inclusion-Exclusion1 tool

Inclusion-Exclusion Principle Explorer

Enter the sizes of two or three sets and their overlaps, then apply the inclusion-exclusion formula one term at a time. Each region of the diagram shows how many times it has been counted so far, so the over-counting appears, the subtractions overshoot on the triple overlap, and the final term repairs it. The walk-through ends when every region reads exactly one.

Sets and Subsets · Power Sets1 tool

Power Set Explorer

Draw the power set of a set of up to five elements as a Hasse diagram of the subset lattice. Click any subset and its subsets light up below it while its supersets light up above it, so containment reads as a direction on the page. A level table counts each size as a binomial coefficient and totals them to 2 to the n, and a roster panel lists every subset in set notation.

Set Notation · Set-Builder1 tool

Set-Builder Notation Explorer

Assemble a set-builder expression one piece at a time: choose a domain, add conditions such as even, prime or less than, and join them with and or or. Every candidate in the domain appears as a chip that turns green when it passes, clicking one shows it tested condition by condition, and a roster panel prints the finished set — or says why it cannot be listed.

Venn Diagrams · Three Sets2 tools

Three-Set Venn Diagram Basic Identities Explorer

Shade the three-circle Venn diagram for any three-set identity — triple union and intersection, complements, the various differences, counting identities like exactly two of A, B, C, and De Morgan's laws for three sets. Switch between six category tabs, pick an identity from the formula buttons or the Jump-to dropdown, and customize the shading color and opacity. Each highlighted region updates with an explanation panel showing the set-builder notation.

Three-Set Venn Diagram Laws and Complex Identities Explorer

Verify three-set algebraic laws by shading both sides of each identity on a pair of side-by-side three-circle Venn diagrams. Browse 12 laws across four categories — associative laws for union, intersection, and symmetric difference; distributive laws; three-set De Morgan's laws; and five difference identities — and watch the match indicator confirm each visual proof across all eight regions.

Venn Diagrams · Two Sets2 tools

Two-Set Venn Diagram Basic Identities Explorer

Shade the Venn diagram for any two-set identity — union, intersection, complement, the three differences, De Morgan's laws, and set relations like subset and disjoint. Switch between categories using the tabs, pick an identity from the formula buttons or Jump-to dropdown, and customize the shading color and opacity. Each highlighted region updates with an explanation panel showing the set-builder notation.

Two-Set Venn Diagram Laws and Identities Explorer

Verify two-set algebraic laws by shading both sides of each identity on a pair of side-by-side Venn diagrams. Browse 27 laws across eight categories — idempotent, commutative, identity, complement, De Morgan's, absorption, difference, and compound complements — and watch the match indicator confirm each visual proof.

Set Operations · Indexed Families1 tool

Indexed Union and Intersection Explorer

Pick a family of sets indexed by 1, 2, 3 and upward, then raise the index and watch the big union and the big intersection accumulate in opposite directions. Interval families are drawn on a shared number line with endpoint marks that matter, discrete families as a membership grid, and the finite families on a Venn diagram. Each family is built to make one limit surprising.

Venn Diagrams · Multiple Sets1 tool

Venn Diagram Generator

Shade any set expression on a 2, 3, 4, or 5 set Venn diagram. Type it with keyboard shortcuts, build it from symbol buttons, or load a preset. A region strip lists every region and whether it is shaded, a compare box tests two expressions for equivalence, and finished diagrams export to SVG or PNG.

Logic and Sets · Truth Tables1 tool

Venn Diagram and Truth Table Explorer

One set expression shown two ways at once — as shaded regions of a two or three set Venn diagram, and as a truth table with a result column. Clicking a row outlines the region it describes, and clicking a region selects its row. The tool also rewrites every expression into logical notation and flags tautologies, contradictions, and expressions that ignore one of their sets.

11 tools · 5 sub-groups

Complex numbers

Arithmetic3 tools

Complex Addition & Subtraction Visualizer

Add and subtract complex numbers geometrically. Drag two points and watch the parallelogram rule for addition and the difference vector for subtraction update instantly, with step-by-step component breakdowns and the triangle inequality verified in real time.

Complex Division Visualizer

Visualize complex division geometrically — moduli divide, angles subtract. See the conjugate multiplication method step by step alongside the polar interpretation, with a zoom inset for small quotients.

Complex Multiplication Visualizer

See complex multiplication as rotation and scaling. Drag two points and watch the product vector update — moduli multiply, angles add. Follow both the FOIL algebraic method and the geometric polar method side by side.

Modulus & Geometry2 tools

Complex Conjugate and Modulus Visualizer

Drag a point on the Argand plane to see z and its conjugate z̄ mirrored across the real axis. Watch the modulus circle, right triangle decomposition, and the proof that z · z̄ = |z|² update in real time.

Complex Distance & Midpoint Calculator

Compute and visualize the distance |z₁ − z₂| and midpoint between two complex numbers. Drag two points to see the right triangle, step-by-step calculations, and a circle of radius equal to the distance centered at z₁.

Plane & Representation2 tools

Complex Number Explorer

Visualize complex numbers on an interactive Argand plane. Drag points or enter coordinates to see real parts, imaginary parts, modulus, and conjugates. Watch the right triangle form as you explore different quadrants with real-time calculations and explanations.

Polar-Rectangular Complex Number Converter

Convert between rectangular form a + bi and polar form r·eⁱᶿ interactively. Drag a point on the Argand plane to see modulus, argument, and live conversion formulas update in real time, with the right triangle bridging both representations.

Powers & Exponential3 tools

De Moivre's Theorem Visual Calculator

Raise any complex number to any integer power and watch the result animate on the plane. See modulus scale and angle multiply, follow the step-by-step polar calculation, and trace intermediate powers as a spiral trail.

Euler's Formula Explorer

Explore eⁱᶿ = cos θ + i sin θ on an interactive unit circle. Drag the angle or use the slider, see the right triangle projections, hit landmark angles like Euler's identity eⁱᵖ = −1, and scale with r for the full polar form.

Powers of i Calculator

Calculate any power of i instantly. See the 4-step cycle in a dynamic diagram, follow the division-by-4 method step by step, and look up results from i⁰ to i¹⁰⁰ in the reference table.

Diagrams1 tool

Complex Numbers Diagrams

A gallery of every diagram already shown on the complex numbers pages: figures drawn for the lessons and frozen states of the visual tools. Browse by chapter or page, search captions, compare two figures side by side, and jump from any picture to the exact section where it appears or to the tool that drew it.