250 free mathematics calculators and tools. Solve real-world mathematics problems instantly with accurate, step-by-step results on ApexCalc.
Calculate greatest common divisor using Euclidean algorithm and least common multiple (LCM = a×b/GCD).
Decompose an integer into its prime factors using trial division. Every integer n>1 is uniquely the product of primes.
Compute a mod m, modular addition, subtraction, multiplication, and exponentiation (a^b mod m using fast exponentiation).
Calculate φ(n) = count of integers from 1 to n coprime to n. For prime p: φ(p)=p-1. For p^k: φ(p^k)=p^(k-1)(p-1).
Generate Fibonacci numbers F(n) = F(n-1) + F(n-2) with F(0)=0, F(1)=1. Ratio F(n+1)/F(n) converges to golden ratio φ≈1.6180.
Calculate nth Catalan number C(n) = C(2n,n)/(n+1) = (2n)!/(n!(n+1)!). First values: 1,1,2,5,14,42,132,429,1430,4862,16796.
Calculate C(n,k) = n!/(k!(n-k)!) — number of ways to choose k items from n without regard to order. Pascal triangle: C(n,k) = C(n-1,k-1) + C(n-1,k).
Calculate digital root by repeatedly summing digits until a single digit. Formula: if n=0 then 0; else 1 + (n-1) mod 9.
Calculate n! = n × (n-1) × ... × 2 × 1. Stirling approximation: n! ≈ √(2πn)(n/e)^n. 0!=1 by convention. 20!=2.43×10^18.
Calculate H(n) = 1 + 1/2 + 1/3 + ... + 1/n. Approximation: H(n) ≈ ln(n) + γ where γ ≈ 0.5772 (Euler-Mascheroni constant). H(n) diverges.
Test if a number is prime using trial division up to √n. A prime p has exactly two divisors: 1 and p. Sieve of Eratosthenes for ranges.
A perfect number equals the sum of its proper divisors. Known perfect numbers: 6, 28, 496, 8128, 33550336. Form: 2^(p-1)(2^p-1) where 2^p-1 is prime.
Find twin prime pairs (p, p+2) both prime. Example pairs: (3,5),(5,7),(11,13),(17,19),(29,31),(41,43),(59,61).
Verify Goldbach conjecture for even n: every even integer >2 is sum of two primes. Find all decompositions. Verified for all even numbers up to 4×10^18.
Calculate p(n) = number of ways to write n as sum of positive integers. p(0)=1,p(1)=1,p(2)=2,p(3)=3,p(4)=5,p(5)=7. Hardy-Ramanujan asymptotic formula.
Calculate μ(n): μ(1)=1; μ(n)=(-1)^k if n is product of k distinct primes; μ(n)=0 if n has a squared prime factor. Used in Möbius inversion.
Count divisors of n using prime factorization: if n = p1^a1 × p2^a2 × ..., then d(n) = (a1+1)(a2+1)... Example: d(12) = 3×2 = 6.
Calculate sum of all divisors of n using prime factorization: σ(p^k) = (p^(k+1)-1)/(p-1). Multiplicative function. Perfect number if σ(n)=2n.
Solve system of congruences x≡a1 mod m1, x≡a2 mod m2 when moduli are pairwise coprime. Unique solution modulo M=m1×m2×...×mk.
Generate Lucas numbers L(n): L(0)=2, L(1)=1, L(n)=L(n-1)+L(n-2). First values: 2,1,3,4,7,11,18,29,47,76. Relation: L(n) = F(n-1)+F(n+1).
Express a real number as a continued fraction [a0; a1, a2, ...]. Rational numbers terminate; quadratic irrationals are periodic. √2 = [1;2,2,2,...], π = [3;7,15,1,292,...].
Calculate Bernoulli numbers B(n). B(0)=1, B(1)=-1/2, all odd B(n)=0 for n>1. Used in Euler-Maclaurin formula and Riemann zeta function at even integers.
Evaluate ζ(s) = Σ 1/n^s for real s>1. Special: ζ(2)=π²/6, ζ(4)=π⁴/90, ζ(6)=π⁶/945. Euler product: ζ(s) = Π (1-p^(-s))^(-1) over primes.
Calculate signed Stirling numbers s(n,k) = number of permutations of n with exactly k cycles. Falling factorial x^(n) = Σ s(n,k)x^k.
Calculate S(n,k) = number of ways to partition n-element set into k nonempty subsets. Bell number B(n) = Σ S(n,k). Recurrence: S(n,k) = k×S(n-1,k) + S(n-1,k-1).
Solve ax²+bx+c=0: x = (-b ± √(b²-4ac))/(2a). Discriminant Δ=b²-4ac: Δ>0 two real roots, Δ=0 double root, Δ<0 two complex roots.
Solve depressed cubic t³+pt+q=0 using Cardano formula. Transform ax³+bx²+cx+d=0 by substitution x=t-b/3a. Discriminant Δ determines nature of roots.
Evaluate p(x) using Horner's method: p(x) = (...((a_n x + a_{n-1})x + a_{n-2})x + ...)x + a_0. Requires only n multiplications and n additions.
Divide polynomial p(x) by (x-c) using synthetic division to get quotient and remainder. Remainder theorem: p(c) = remainder. Factor theorem: if p(c)=0 then (x-c) is a factor.
Calculate det(A) = ad - bc for [[a,b],[c,d]]. Represents area of parallelogram spanned by row vectors. det(A)≠0 iff A is invertible.
Calculate det(A) using cofactor expansion along first row: det = a(ei-fh) - b(di-fg) + c(dh-eg). Multilinear, alternating function.
Calculate A^(-1) = (1/det(A)) × [[d,-b],[-c,a]] for A=[[a,b],[c,d]]. Exists iff det(A)≠0. Verifies A×A^(-1) = I.
Calculate C = A×B where C_ij = Σ A_ik × B_kj for 2×2 matrices. Matrix multiplication is associative but not commutative: AB ≠ BA in general.
Find eigenvalues λ by solving det(A-λI)=0. For 2×2: λ² - tr(A)λ + det(A) = 0. Eigenvectors from (A-λI)v=0.
Calculate a·b = Σ aᵢ × bᵢ = |a||b|cos(θ). Geometric: projection of a onto b times |b|. Result is zero if vectors are perpendicular.
Calculate a×b for 3D vectors: result is vector perpendicular to both. Magnitude: |a×b| = |a||b|sin(θ). Anti-commutative: a×b = -b×a.
Calculate |v| = √(Σvᵢ²). Normalize to unit vector: v̂ = v/|v|. Unit vector has magnitude 1 and preserves direction.
Project vector a onto vector b: proj_b(a) = (a·b/b·b) × b. Scalar projection: comp_b(a) = a·b/|b|. Used in Gram-Schmidt and least squares.
Solve 2×2 system by Cramer's rule: x = det(Ax)/det(A), y = det(Ay)/det(A). Only valid when det(A)≠0 (unique solution exists).
Solve 3-variable system by Gaussian elimination with partial pivoting. Augmented matrix [A|b], row reduce to row echelon form, then back substitution.
Rotate point (x,y) by angle θ: [x',y'] = [[cosθ,-sinθ],[sinθ,cosθ]][x,y]. Preserves distances and orientation. Inverse rotation: use -θ.
Find GCD of two polynomials using Euclidean algorithm for polynomials: GCD(a,b) = GCD(b, a mod b) where mod is polynomial remainder. Result is monic GCD.
Iterative root finding: x_{n+1} = x_n - f(x_n)/f'(x_n). Quadratic convergence near simple root. Diverges if f'(x_n)≈0 or starting point is far from root.
Construct degree-(n-1) polynomial through n points using: L(x) = Σ yᵢ × Π_{j≠i} (x-xⱼ)/(xᵢ-xⱼ). Unique for n distinct x-values.
Calculate det of n×n Vandermonde matrix V(x₁,...,xₙ) = Π_{i<j} (xⱼ - xᵢ). Zero iff any two xᵢ are equal. Used in polynomial interpolation and coding theory.
Calculate angle θ = arccos(a·b / (|a||b|)) between vectors a and b. Result in [0, π] radians. Works in any dimension.
Orthogonalize vectors: eₖ = vₖ - Σ_{i<k} (vₖ·eᵢ/eᵢ·eᵢ) eᵢ. Produces orthogonal basis. Normalize for QR decomposition.
Calculate trace tr(A) = Σ Aᵢᵢ (sum of diagonal elements). tr(A) equals sum of eigenvalues. Cyclic property: tr(AB) = tr(BA).
Calculate ||A||_F = √(Σᵢⱼ Aᵢⱼ²) = √(tr(AᵀA)). Equals square root of sum of squared singular values.
Form companion matrix C of polynomial p(x) = xⁿ + a_{n-1}xⁿ⁻¹+...+a₀: subdiagonal of 1s, last column [-a₀,-a₁,...,-a_{n-1}]ᵀ. Eigenvalues of C are roots of p(x).
Apply the power rule to differentiate polynomials: d/dx(ax^n) = nax^(n-1). Uses sum rule d/dx(f+g) = f’+g’ and constant rule d/dx(c) = 0 to differentiate any polynomial term by term.
Look up or compute standard trigonometric derivatives: d/dx(sin x)=cos x, d/dx(cos x)=-sin x, d/dx(tan x)=sec²x, d/dx(cot x)=-csc²x, d/dx(sec x)=sec x tan x, d/dx(csc x)=-csc x cot x.
Differentiate composite functions using the chain rule: d/dx[f(g(x))] = f’(g(x)) × g’(x). Extended to triple compositions: d/dx[f(g(h(x)))] = f’(g(h(x))) × g’(h(x)) × h’(x).
Differentiate products of two functions using the product rule: d/dx[f(x)g(x)] = f’(x)g(x) + f(x)g’(x). Mnemonic: first times derivative of second plus second times derivative of first.
Differentiate fractions using the quotient rule: d/dx[f/g] = (f’g - fg’)/g². Remember the mnemonic: low d-high minus high d-low over low-squared. Requires g(x) ≠ 0.
Resolve indeterminate forms 0/0 or ∞/∞ using L'Hôpital's Rule: lim[f(x)/g(x)] = lim[f'(x)/g'(x)]. Can be applied repeatedly. Handles 0×∞, ∞-∞, 0^0, 1^∞, ∞^0 via algebraic manipulation.
Approximate f(x) near x=a: T_n(x) = Σ f^(k)(a)/k! × (x-a)^k for k=0 to n. Maclaurin series sets a=0. Remainder is bounded by the (n+1)th derivative via the Lagrange remainder formula.
Numerically approximate ∫_a^b f(x)dx ≈ (b-a)/(2n) × [f(x_0) + 2f(x_1) + ... + 2f(x_{n-1}) + f(x_n)]. Error is O(h²) where h=(b-a)/n.
Numerically approximate ∫_a^b f(x)dx using Simpson's 1/3 rule: h/3 × [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + ... + f(x_n)] (n must be even). Error O(h^4) is much better than the trapezoidal rule.
Calculate the arc length of y=f(x) from a to b: L = ∫_a^b √(1 + (f’(x))²) dx. Parametric form: L = ∫ √((dx/dt)² + (dy/dt)²) dt.
Compute the volume of a solid formed by rotating y=f(x) around the x-axis: V = π∫_a^b [f(x)]² dx (disk method). Around y-axis use shell method: V = 2π∫_a^b x|f(x)| dx.
Compute the gradient ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z). The gradient points in the direction of steepest ascent. Its magnitude |∇f| is the rate of maximum increase. The gradient is perpendicular to level curves and surfaces.
Calculate the rate of change of f in direction û: D_û f = ∇f · û. Maximum value |∇f| in the gradient direction, minimum -|∇f| in the opposite direction, and zero perpendicular to the gradient.
Compute div F = ∂F_x/∂x + ∂F_y/∂y + ∂F_z/∂z = ∇·F. Positive divergence indicates a source (outflow), negative a sink (inflow), and zero means incompressible flow.
Compute curl F = ∇×F. In 3D: (∂F_z/∂y - ∂F_y/∂z, ∂F_x/∂z - ∂F_z/∂x, ∂F_y/∂x - ∂F_x/∂y). Curl measures the rotation or circulation of a vector field. Conservative fields have zero curl.
Optimize f(x,y) subject to constraint g(x,y)=c by solving ∇f = λ∇g. System: ∂f/∂x = λ∂g/∂x, ∂f/∂y = λ∂g/∂y, g(x,y)=c. The multiplier λ is the rate of change of the optimum with respect to the constraint constant.
Calculate the Hessian H = [[f_xx, f_xy],[f_yx, f_yy]] at a critical point. Second-order test: det(H)>0 and f_xx>0 → local min; det(H)>0 and f_xx<0 → local max; det(H)<0 → saddle point; det(H)=0 → inconclusive.
Compute ∬_R f(x,y) dA over rectangle [a,b]×[c,d] using Fubini’s theorem: ∫_c^d [∫_a^b f(x,y) dx] dy. Integration order is reversible for continuous f.
Calculate |J| = |∂(x,y)/∂(u,v)| = |∂x/∂u × ∂y/∂v - ∂x/∂v × ∂y/∂u|. Used in change-of-variables: ∬ f(x,y) dx dy = ∬ f(x(u,v), y(u,v)) |J| du dv.
Compute ∫_C f ds = ∫_a^b f(r(t)) |r’(t)| dt along curve C. Measures the weighted arc length of a curve through a scalar field.
Iteratively minimize f using x_{n+1} = x_n - α∇f(x_n). The learning rate α controls step size. Converges to a local minimum for convex functions when α is sufficiently small.
Compute Fourier coefficients of f on [-L,L]: a_n = (1/L)∫_{-L}^L f(x)cos(nπx/L)dx, b_n = (1/L)∫_{-L}^L f(x)sin(nπx/L)dx. Together these reconstruct periodic functions as sums of harmonics.
Convert a closed surface integral to a volume integral: ∯_S F·dA = ∭_V (∇·F) dV. Requires a closed surface enclosing volume V with smooth outward-pointing normal. Widely used in fluid flow and electrostatics.
Find extremals of a functional ∫_a^b L(x,y,y’)dx using the Euler-Lagrange equation: d/dx(∂L/∂y’) - ∂L/∂y = 0. This is the foundation of variational calculus and Lagrangian mechanics.
Relate rates of change using implicit differentiation: if F(x,y)=0, then dF/dt = ∂F/∂x(dx/dt) + ∂F/∂y(dy/dt) = 0. Classic problems: sliding ladder, draining tank, expanding circle, growing shadow.
Calculate mean μ = Σx/n, population variance σ² = Σ(x-μ)²/n, sample variance s² = Σ(x-x̄)²/(n-1), standard deviation, median, mode, and interquartile range IQR = Q3-Q1.
Calculate G = (x_1 × x_2 × ... × x_n)^(1/n) = exp(Σ ln(x_i)/n). Used for growth rates, financial returns, and ratios. Always ≤ the arithmetic mean (AM-GM inequality), with equality only when all values are equal.
Calculate z = (x - μ)/σ. The z-score measures how many standard deviations a value lies from the mean. Used for outlier detection (|z|>3), comparing values across scales, and finding probabilities from the standard normal table.
Known σ: CI = x̄ ± z_{α/2} × σ/√n. Unknown σ (use t-distribution): CI = x̄ ± t_{α/2, n-1} × s/√n. Width decreases proportionally to 1/√n as sample size grows.
Determine required sample size: n = (z_{α/2} × σ / E)² for estimating a mean with margin of error E. For proportion: n = z²_{α/2} × p̂(1-p̂)/E². Always round up to the nearest integer.
Test H0: μ = μ_0 using t = (x̄ - μ_0)/(s/√n). Compare to t_{α/2, n-1} critical value or use p-value. Assumes normality or n > 30 by the Central Limit Theorem.
Compare two independent means: t = (x̄_1 - x̄_2)/√(s_1²/n_1 + s_2²/n_2). Degrees of freedom estimated by the Welch-Satterthwaite formula. More robust than pooled t-test when group variances are unequal.
Test whether observed frequencies match an expected distribution: χ² = Σ(O_i - E_i)²/E_i with df = k - 1 - (estimated parameters). Requires E_i ≥ 5 in each cell for the approximation to be accurate.
Compute r = Σ(x_i-x̄)(y_i-ȳ) / √(Σ(x_i-x̄)² × Σ(y_i-ȳ)²). Range [-1,1]: ±1 is perfect linear correlation, 0 is no linear relationship. Significance test: t = r√(n-2)/√(1-r²).
Fit y = β_0 + β_1 x using OLS: β_1 = Σ(x_i-x̄)(y_i-ȳ)/Σ(x_i-x̄)², β_0 = ȳ - β_1 x̄. Coefficient of determination R² = 1 - SSE/SST measures proportion of variance explained.
R² = 1 - SS_res/SS_tot. Proportion of variance in y explained by the model. Adjusted R² = 1 - (1-R²)(n-1)/(n-p-1) penalizes for extra predictors. Allows fair comparison across models with different numbers of variables.
Calculate P(X=k) = C(n,k) p^k (1-p)^(n-k). Mean μ = np, variance σ² = np(1-p). Normal approximation is valid when np > 5 and n(1-p) > 5.
Calculate P(X=k) = e^(-λ)λ^k/k! for count k. Models rare events per unit interval. Mean = variance = λ. Normal approximation N(λ, λ) is valid when λ > 10.
Compute P(X ≤ x) = Φ((x-μ)/σ). The standard normal CDF Φ(z). Empirical rule: 68.27% of data within ±1σ, 95.45% within ±2σ, 99.73% within ±3σ.
Compute posterior P(A|B) = P(B|A) × P(A) / P(B), where P(B) = P(B|A)P(A) + P(B|Aᶜ)P(Aᶜ). Updates prior belief P(A) with new evidence B to obtain posterior probability. Fundamental to Bayesian statistics and machine learning.
Compute the probability that at least two people share a birthday among n people: P = 1 - (365/365 × 364/365 × ... × (365-n+1)/365). Probability exceeds 50% at n=23 and 99.9% at n=70.
Calculate P(X=k) = C(K,k)C(N-K,n-k)/C(N,n). Models sampling k successes without replacement from population N containing K successes. Mean = nK/N. Used in quality control and ecology.
Calculate P(X=k) = (1-p)^(k-1) × p for k=1,2,3,... Waiting time for the first success in repeated Bernoulli trials. Mean = 1/p, variance = (1-p)/p². Exhibits the memoryless property: P(X>k+n|X>k) = P(X>n).
Calculate H(X) = -Σ p_i log_2(p_i) in bits. Maximum entropy = log_2(n) for a uniform distribution over n outcomes. Minimum = 0 for a degenerate distribution. Measures the average information or uncertainty in a random variable.
Calculate r_s = 1 - 6Σd_i²/(n(n²-1)) where d_i is the difference in ranks. A non-parametric alternative to Pearson that detects monotonic relationships and is robust to outliers and non-normal data.
For a 2-state Markov chain with transition matrix [[1-a, a],[b, 1-b]], compute the steady-state distribution π = [b/(a+b), a/(a+b)]. For larger chains, solve πP = π with Σπ_i = 1.
Compare means across k groups: F = (SS_between/df_between) / (SS_within/df_within). SS_between = Σn_j(x̄_j-x̄)², SS_within = ΣΣ(x_ij-x̄_j)². Reject H0: all means equal when F exceeds F-critical.
PI = ŷ ± t_{α/2,n-2} × s_e × √(1 + 1/n + (x*-x̄)²/Σ(x_i-x̄)²). Wider than the confidence interval for the mean because it includes the variability of individual future observations, not just the mean.
For 2 sets: |A∪B| = |A| + |B| - |A∩B|. For 3 sets: |A∪B∪C| = |A|+|B|+|C|-|A∩B|-|A∩C|-|B∩C|+|A∩B∩C|. Probability version: P(A∪B) = P(A)+P(B)-P(A∩B).
Calculate P(A|B) = P(A∩B)/P(B). Chain rule: P(A∩B) = P(A|B)P(B). Law of total probability: P(A) = Σ P(A|B_i)P(B_i). Independence: P(A|B) = P(A) when A and B are independent.
Calculate circle area (A = πr²), circumference (C = 2πr), arc length (rθ), sector area (r²θ/2), and annulus area (π(R²−r²)).
Compute triangle area using Heron's formula: s = (a+b+c)/2, Area = √(s(s−a)(s−b)(s−c)). Also supports (1/2)ab sin(C) and equilateral triangle cases.
Solve triangles with the law of sines: a/sin(A) = b/sin(B) = c/sin(C) = 2R. Handles ASA, AAS, and the ambiguous SSA case with circumradius output.
Apply the law of cosines c² = a² + b² − 2ab cos(C) to solve SAS and SSS triangles. Find any side or angle; generalises the Pythagorean theorem.
Compute regular n-gon area using Area = (n/4)s² cot(π/n) or Area = ½ × Perimeter × Apothem. Shows how the polygon approaches a circle as n → ∞.
Calculate ellipse area (πab, exact) and perimeter via Ramanujan approximation π(3(a+b) − √((3a+b)(a+3b))). Also computes eccentricity e = √(1 − b²/a²).
Compute sphere volume V = (4/3)πr³ and surface area SA = 4πr². Includes hemisphere calculations and the great-circle radius relationship.
Calculate cylinder volume V = πr²h, total SA = 2πr² + 2πrh, lateral SA = 2πrh, and open-top SA = πr² + 2πrh.
Find cone volume V = (1/3)πr²h, slant height l = √(r²+h²), and lateral SA = πrl. Also computes frustum (truncated cone) volume V = (πh/3)(R²+Rr+r²).
Compute torus (donut) volume V = 2π²Rr² and surface area SA = 4π²Rr, where R is the distance from the tube centre to the torus centre and r is the tube radius.
For edge length a: compute volume V = a³/(6√2), surface area SA = a²√3, height = a√(2/3), inradius = a/(2√6), and circumradius = a√6/4.
For edge a: volume V = (√2/3)a³, surface area SA = 2√3 a², height = a√2. The octahedron has 8 faces, 6 vertices, 12 edges, and is the dual of the cube.
Calculate circumradius R = abc/(4K) where K is triangle area, or equivalently R = a/(2 sin A). The circumcircle passes through all three vertices.
Compute inradius r = K/s (K = area, s = semiperimeter). Also r = 4R sin(A/2)sin(B/2)sin(C/2). The incircle is tangent to all three sides.
Find the centroid G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3). The centroid divides each median in a 2:1 ratio from vertex and is the centre of mass of a uniform triangle.
The nine-point circle has radius R/2 (half the circumradius) and passes through the midpoints of the sides, the feet of the altitudes, and the midpoints of the vertex-to-orthocenter segments.
Construct equilateral triangles on each side of any triangle; their outer centres form Napoleon's equilateral triangle. Compute its side length from the original triangle's sides and area.
Compute the power of a point P with respect to a circle: PO² − r². For an external point: power = PA × PB (secant). For an internal point: power = −PA × PB (chord).
Apply Menelaus' theorem: for a transversal cutting triangle ABC at D, E, F — (BD/DC)(CE/EA)(AF/FB) = −1 in signed ratios — to verify collinearity or find unknown segment lengths.
Verify or apply Ceva's theorem: cevians AD, BE, CF in triangle ABC are concurrent iff (AF/FB)(BD/DC)(CE/EA) = 1. Identifies centroid, incenter, circumcenter, and orthocenter as special cases.
Apply Ptolemy's theorem for cyclic quadrilateral ABCD: AC × BD = AB × CD + AD × BC. Verify whether a quadrilateral is cyclic and derive classical trigonometric identities.
Verify V − E + F = 2 (Euler characteristic χ = 2) for any convex polyhedron. Check Platonic solids, explore the torus (χ = 0), and validate graphs of polyhedra.
Compute the cross-ratio (A,B;C,D) = (AC/AD)/(BC/BD). This projective invariant is preserved under all Möbius transformations; a cross-ratio of −1 defines a harmonic range.
Invert point P in circle (centre O, radius r): image P' on ray OP with OP × OP' = r². Lines through O map to lines; other lines/circles map to circles. Used in Apollonius and Peaucellier problems.
Calculate solid angle in steradians: for a cone with half-angle θ, Ω = 2π(1 − cos θ). For a spherical triangle: Ω = A + B + C − π (spherical excess). Full sphere = 4π sr.
Compute ordered arrangements P(n,r) = n!/(n−r)! = n×(n−1)×…×(n−r+1). P(n,n) = n! for all n items. Covers counting problems where order matters.
Count permutations with no element in its original position: D(n) = n! Σ_{k=0}^{n} (−1)^k/k! ≈ n!/e. Probability of no fixed point ≈ 1/e ≈ 36.8%.
Arrange n objects in a circle: (n−1)! ways (divide by n for rotational equivalence). For necklaces (reflection also equivalent): (n−1)!/2. Accounts for rotational and reflective symmetry.
Check whether a connected undirected graph has an Eulerian circuit (all even degrees) or Eulerian path (exactly 2 odd-degree vertices). Based on the solution to the Königsberg bridge problem.
For a connected planar graph: V − E + F = 2. Check planarity: simple planar graphs satisfy E ≤ 3V − 6; triangle-free graphs E ≤ 2V − 4. K₅ and K₃,₃ are minimal non-planar graphs (Kuratowski).
Compute chromatic number bounds: χ(G) ≤ Δ+1 (greedy). Bipartite: χ = 2. K_n: χ = n. Four Colour Theorem: χ ≤ 4 for planar graphs. Identifies clique number as lower bound.
Compute A^k: entry [i][j] gives the number of walks of length k from vertex i to j. A²[i][i] = degree of vertex i. Count triangles via tr(A³)/6.
For n nodes: minimum height ⌊log₂(n)⌋ (complete tree), maximum height n−1 (degenerate). Perfect binary tree of height h has 2^(h+1)−1 nodes. AVL trees maintain O(log n) height.
Load factor α = n/m. Separate chaining expected search time O(1+α). Open addressing: expected probes = 1/(1−α) successful, 1/(1−α)² unsuccessful. Optimal α ≤ 0.75.
Solve T(n) = aT(n/b) + f(n) using the Master theorem: compare f(n) to n^(log_b a). Three cases give Θ(n^(log_b a)), Θ(n^(log_b a) log n), or Θ(f(n)).
Work with ordinary generating functions F(x) = Σ a_n x^n. Common: 1/(1−x) = Σ x^n, 1/(1−x)² = Σ(n+1)x^n. Extract [x^n] coefficients using partial fractions or differentiation.
Count distinct colourings under group symmetry: |X/G| = (1/|G|) Σ |X^g|. For n-bead necklaces with k colours: (1/n) Σ_{d|n} φ(d) k^(n/d).
Compute the length of a De Bruijn sequence: k^n characters for alphabet size k and subsequence length n, with every n-character string appearing exactly once as a window. Used in keyless entry and LFSR design.
Relax all edges |V|−1 times: d[v] = min(d[v], d[u]+w(u,v)). Detects negative cycles on the |V|-th iteration. O(VE) complexity. Handles negative-weight edges, unlike Dijkstra.
Greedy shortest path: select unvisited vertex with minimum tentative distance, relax its neighbours. Requires non-negative edge weights. O((V+E) log V) with a min-heap priority queue.
Determine a topological ordering of a DAG using Kahn's algorithm (remove 0 in-degree vertices iteratively) or DFS-based post-order. Detects cycles via DFS vertex colouring.
Apply Kruskal's algorithm: sort edges by weight, add each edge that does not form a cycle (use union-find), yielding an MST of n−1 edges with minimum total weight.
For cache capacity k over n uniformly accessed items, expected hit rate ≈ k/n. Working set model: if working set fits in cache, hit rate approaches 1. Compares LRU with Bélády's optimal policy.
For n elements with coin-flip promotion (p=0.5), expected height ≈ log₂(n) + constant. Expected space O(n log n). Expected search time O(log n) with high probability.
Count spanning trees of graph G using any cofactor of the Laplacian L = D − A (Kirchhoff's theorem). For complete graph K_n: n^(n−2) spanning trees (Cayley's formula).
Expected comparisons for n elements: 2n ln n ≈ 1.386 n log₂ n. Worst case Θ(n²) with sorted input and first-element pivot. Randomised pivot guarantees O(n log n) expected.
Solve T(n) = 2T(n/2) + n via the Master theorem: T(n) = Θ(n log n). Merge sort is stable, uses O(n) auxiliary space, and extends naturally to external sorting.
Check max-heap: parent ≥ children; 1-indexed array with parent at i and children at 2i and 2i+1. Build heap O(n). Heap sort: O(n) build + n × O(log n) extractions = O(n log n).
Compute the upper bound R(s,t) ≤ C(s+t−2, s−1). Known values: R(3,3)=6, R(4,4)=18. R(s,t) is the minimum n such that any red-blue colouring of K_n contains a red K_s or blue K_t.
Count monotone lattice paths from (0,0) to (m,n): C(m+n, m). Paths not crossing the diagonal (ballot problem): C(m+n,m) − C(m+n,m−1). Generalises the Catalan number formula.
Calculate partial derivatives of multivariable functions. For f(x,y): df/dx treats y as constant, df/dy treats x as constant. Supports second order and mixed partials via Clairaut theorem.
Perform implicit differentiation on equations F(x,y)=0 using dy/dx = -(dF/dx)/(dF/dy). Perfect for circles, ellipses, and curves not expressible as explicit functions.
Find derivatives and arc length for parametric curves x=x(t), y=y(t). Computes dy/dx = (dy/dt)/(dx/dt), second derivative, and arc length integral of sqrt((dx/dt)^2+(dy/dt)^2) dt.
Calculate derivatives, area, and arc length for polar curves r=f(theta). Computes dy/dx via polar formula, area A=(1/2) integral r^2 d-theta, and arc length from the polar arc length formula.
Compute line integrals of vector fields along a curve C: integral of F dot dr = integral F(r(t)) dot r-prime(t) dt. Calculates work done by force F. Identifies path-independent conservative fields.
Evaluate surface integrals over parametric and explicit surfaces. The area element dS = |r_u cross r_v| dA for parametric surfaces. Handles both scalar integrals and flux integrals of vector fields.
Apply Stokes' theorem: the surface integral of curl F over S equals the line integral of F around the boundary curve. Converts surface integrals of curl to boundary line integrals. Generalizes Green's theorem to 3D.
Apply Green's theorem: the line integral of (P dx + Q dy) around a closed curve equals the double integral of (dQ/dx - dP/dy) over the enclosed region. Converts between line and area integrals.
Evaluate double integrals using polar coordinates. The Jacobian factor r must be included: dA = r dr d-theta. Ideal for circular and rotationally symmetric regions where Cartesian integration is difficult.
Set up and evaluate triple integrals in rectangular, cylindrical, and spherical coordinates. Jacobians: cylindrical = r, spherical = rho-squared times sin(phi). Computes volume, mass, and moments of 3D regions.
Test convergence of improper integrals with infinite limits or discontinuities. Applies p-integral test: integral from 1 to infinity of x^(-p) dx converges iff p>1. Comparison and limit comparison tests included.
Apply the Mean Value Theorem for derivatives and integrals. For differentiable f on [a,b]: finds c where f-prime(c) = (f(b)-f(a))/(b-a). Integral MVT: finds c where f(c) equals the average value of f.
Find local and global extrema using first and second derivative tests. Locates critical points where f-prime(x)=0 or undefined. First derivative sign changes identify maxima/minima. Second derivative test: f-double-prime > 0 is local min, < 0 is local max.
Find the radius and interval of convergence for power series. Uses ratio test R = lim|a_n/a_{n+1}| and Cauchy-Hadamard formula. Endpoint behavior checked separately after radius is found.
Test infinite series for convergence using ratio, root, integral, comparison, and alternating series tests. Ratio test: converges if lim|a_{n+1}/a_n| < 1. Integral test links series and integral convergence.
Calculate the centroid and center of mass of 2D regions using double integrals. x-bar = (1/A) double-integral x dA, y-bar = (1/A) double-integral y dA. For non-uniform lamina with density rho(x,y), divides by total mass M.
Calculate moments of inertia for 2D regions: I_x = double-integral y^2 rho dA, I_y = double-integral x^2 rho dA, polar moment I_0 = I_x + I_y. Applies parallel axis theorem I = I_cm + md^2.
Compute surface area for parametric surfaces r(u,v): SA = double-integral |r_u cross r_v| dA. For explicit z=f(x,y): SA = double-integral sqrt(1 + (df/dx)^2 + (df/dy)^2) dA.
Apply Fubini's theorem to evaluate double integrals as iterated integrals in either order. For continuous f on a rectangle, the order of integration can be reversed without changing the result.
Classify critical points of f(x,y) using the Hessian determinant D = f_xx times f_yy minus f_xy squared. D>0 and f_xx>0: local min; D>0 and f_xx<0: local max; D<0: saddle point; D=0: inconclusive.
Test whether a vector field F=(P,Q,R) is conservative by checking curl F = 0. Finds potential function phi where grad phi = F on simply-connected domains. Establishes path independence of line integrals.
Apply the Gauss Divergence Theorem: the flux of F through a closed surface equals the volume integral of the divergence of F. Converts surface flux integrals to volume integrals. Fundamental to Gauss's law.
Find the normal vector to a surface at a given point. For implicit F(x,y,z)=c: normal = grad F. For explicit z=f(x,y): normal = (-f_x, -f_y, 1). Used to write tangent plane equations and orient surface integrals.
Find the tangent plane to z=f(x,y) at point (x_0, y_0, z_0). The plane equation uses partial derivatives f_x and f_y at the point. Provides the best linear approximation to the surface near that point.
Calculate volumes of 3D solids using triple integrals. In cylindrical coordinates: include Jacobian factor r. Sphere of radius R has volume (4/3) pi R^3 confirmed by spherical coordinate integration. Handles complex bounded regions.
Apply the Rank-Nullity theorem: dim(V) = rank(T) + nullity(T) for linear transformation T from V to W. Rank = dim(image), Nullity = dim(kernel). For matrix A: rank(A) + dim(null space) = number of columns.
Explore inner product spaces with properties: positive definiteness, linearity, and conjugate symmetry. Computes Cauchy-Schwarz inequality and induced norms. Foundation for Hilbert spaces and orthogonal projections.
Compute singular values for 2x2 matrices as square roots of eigenvalues of A-transpose times A. Decomposes A = U Sigma V-transpose. Calculates condition number = sigma_max / sigma_min. Determines rank from nonzero singular values.
Decompose matrix A = QR where Q is orthogonal and R is upper triangular. Solve Ax=b via Rx = Q-transpose b. Used for least squares problems, eigenvalue algorithms, and numerically stable linear system solutions.
Factor matrix A = LU (L lower triangular, U upper triangular). Solve Ax=b in O(n^3) via forward substitution Ly=b then back substitution Ux=y. With partial pivoting PA = LU for numerical stability.
Numerically integrate using Gaussian quadrature, exact for polynomials of degree up to 2n-1 with n quadrature points. Two-point rule: integral from -1 to 1 of f(x) dx is approximately f(-1/sqrt(3)) + f(1/sqrt(3)).
Solve boundary value problems y-double-prime = f(x,y,y-prime) with y(a)=alpha, y(b)=beta using the shooting method. Guesses initial slope, integrates the IVP, adjusts via bisection or Newton until the boundary condition is met.
Solve ODEs numerically with the classical 4th-order Runge-Kutta method. Computes four slope estimates k1, k2, k3, k4 per step and combines them as a weighted average. Local and global error are O(h^4).
Solve initial value problems using Euler's method: y_{n+1} = y_n + h f(t_n, y_n). Simple first-order method with O(h) global error. Also demonstrates Heun's improved predictor-corrector method for second-order accuracy.
Compute and apply Fourier transforms. Key properties: convolution theorem (transform of convolution = product of transforms), and Parseval theorem (energy conserved in both domains). Essential for signal processing and PDE solutions.
Calculate Laplace transforms by integrating f(t) times e^(-st) from 0 to infinity. Key pairs: L{1}=1/s, L{e^(at)}=1/(s-a), L{sin(wt)}=w/(s^2+w^2), L{t^n}=n!/s^(n+1). Converts ODEs to algebraic equations in s-domain.
Compute Z-transforms as the sum of x[n] times z^(-n). Key pairs: Z{delta[n]}=1, Z{u[n]}=z/(z-1), Z{a^n u[n]}=z/(z-a). Analyzes discrete-time systems and difference equations. ROC determines system stability.
Compute the matrix exponential e^A as the power series sum. For diagonalizable A = P D P-inverse: e^A = P e^D P-inverse, where e^D is diagonal. Solves linear ODE systems y-prime = Ay with solution y(t) = e^(At) y(0).
Compute the characteristic polynomial p(lambda) = det(lambda I - A) of a matrix. For 2x2: p(lambda) = lambda^2 - tr(A) lambda + det(A). Roots are eigenvalues. Cayley-Hamilton theorem: A satisfies p(A) = 0.
Diagonalize a matrix A = P D P-inverse where D is diagonal with eigenvalues and P has eigenvectors as columns. Computes matrix powers A^n = P D^n P-inverse efficiently. Identifies non-diagonalizable (defective) matrices.
Compute orthogonal projection matrices onto subspaces. Projection onto column space of A: P = A (A-transpose A)-inverse A-transpose. Properties: P squared = P (idempotent), P-transpose = P (symmetric).
Orthonormalize three linearly independent vectors using Gram-Schmidt. Successively removes previously computed direction components and normalizes, producing an orthonormal basis. Foundation for QR decomposition.
Calculate the spectral radius rho(A) = maximum absolute eigenvalue. Power iteration converges to the dominant eigenvector. The spectral radius is bounded above by any induced matrix norm and determines iterative method convergence.
Apply the Cauchy-Schwarz inequality: |inner product of u and v| <= ||u|| times ||v||. Discrete form: (sum of a_i b_i)^2 <= (sum of a_i^2)(sum of b_i^2). Equality holds if and only if u and v are proportional.
Apply the AM-GM inequality: the arithmetic mean of non-negative numbers is always >= their geometric mean. Equality holds if and only if all values are equal. Weighted form and applications in optimization proofs.
Apply the Intermediate Value Theorem: if f is continuous on [a,b] and k is strictly between f(a) and f(b), then there exists c in (a,b) with f(c)=k. Guarantees root existence and underpins the bisection method.
Evaluate limits using the Squeeze (Sandwich) Theorem: if g(x) <= f(x) <= h(x) near a and both g and h approach L, then f also approaches L. Classic application: the limit of x sin(1/x) as x approaches 0 is 0.
Analyze convex functions using the chord-above-graph definition or second derivative test. Jensen's inequality: E[f(X)] >= f(E[X]) for convex f and random variable X. Essential for optimization and probability bounds.
Apply the Integral Mean Value Theorem: if f is continuous on [a,b], then there exists c in (a,b) where f(c) equals the average value of f. Computes average value = (1/(b-a)) times the definite integral of f.
Convert numbers between decimal (base 10), binary (base 2), octal (base 8), and hexadecimal (base 16). Uses repeated division for integer parts and multiplication for fractional parts. Covers two's complement for signed binary integers.
Calculate Fourier series coefficients (aₙ, bₙ) for periodic functions. Decomposes a signal into its harmonic components for spectral analysis and signal processing.
Compute the Discrete Fourier Transform of a sampled sequence. Converts a finite time-domain signal into its frequency-domain representation for digital signal processing.
Calculate the exact frequency represented by each FFT bin given sample rate and FFT size. Essential for interpreting spectral analysis results correctly.
Apply the Laplace transform to convert time-domain differential equations into algebraic equations in the s-domain. Widely used in control systems and circuit analysis.
Compute the Z-transform of discrete-time sequences. The discrete-time counterpart of the Laplace transform, used in digital filter design and difference equation analysis.
Compute the discrete convolution of two finite sequences. Fundamental operation in digital signal processing, used for FIR filtering, correlation, and polynomial multiplication.
Calculate the Pearson r correlation coefficient to measure the linear relationship between two continuous variables. Includes significance test and confidence interval.
Compute Spearman's rank correlation coefficient (ρ) to measure monotonic associations between two variables. A non-parametric alternative to Pearson r.
Calculate Kendall's τ (tau) rank correlation coefficient by counting concordant and discordant pairs. Robust to ties and preferred for small samples.
Compute the autocorrelation function (ACF) at lag k for a time series. Essential for detecting seasonality, periodicity, and appropriate ARIMA model orders.
Determine the optimal ARIMA(p,d,q) order for a time series using AIC/BIC criteria and autocorrelation diagnostics. Essential for accurate time series forecasting.
Apply simple, double, or triple exponential smoothing (Holt-Winters) to a time series. Produces smoothed estimates and one-step-ahead forecasts with optimized alpha.
Forecast time series with trend and seasonality using the Holt-Winters triple exponential smoothing method. Supports additive and multiplicative seasonal models.
Determine the minimum sampling rate required to faithfully reconstruct a band-limited signal. Calculates Nyquist rate, aliasing thresholds, and anti-aliasing filter cutoffs.
Calculate Shannon entropy H(X) = −Σ p(x) log₂ p(x) for a discrete probability distribution. Measures information content, uncertainty, and compressibility of a source.
Compute the KL divergence D_KL(P || Q) between two probability distributions. Quantifies how much information is lost when Q is used to approximate P.
Compute the Jensen-Shannon divergence, a symmetric and smoothed version of KL divergence. Values range from 0 (identical distributions) to 1 (maximally different, log base 2).
Estimate the mutual information I(X;Y) between two discrete random variables. Measures the reduction in uncertainty about one variable given the other.
Calculate the Hamming distance between two strings or binary vectors — the number of positions at which corresponding symbols differ. Used in error-correcting codes and genetics.
Compute the minimum edit distance (insertions, deletions, substitutions) between two strings using dynamic programming. Fundamental to spell checking and bioinformatics alignment.
Calculate the Jaccard similarity index J(A,B) = |A∩B| / |A∪B| between two sets or binary vectors. Widely used in text mining, recommendation systems, and genomics.
Compute the cosine similarity between two vectors as the cosine of the angle between them. Standard metric in NLP, information retrieval, and recommendation systems.
Calculate the Minkowski distance of order p between two vectors. Generalizes Manhattan (p=1) and Euclidean (p=2) distances. Used in k-NN classifiers and clustering.
Compute the Mahalanobis distance between a point and a distribution, accounting for covariance. Detects multivariate outliers and is used in discriminant analysis.
Calculate the proportion of variance explained by each principal component. Helps determine the optimal number of components to retain in dimensionality reduction.
Calculate the condition number κ(A) = ‖A‖ · ‖A⁻¹‖ of a matrix to assess numerical stability of linear systems. High condition numbers indicate ill-conditioned problems.
Determine the rank of a matrix using Gaussian elimination or SVD. Identifies linear dependence among rows/columns and whether a linear system has a unique solution.
Select the optimal regularization parameter λ for ridge regression (L2 penalty) using cross-validation. Reduces overfitting by shrinking large coefficients toward zero.
Calculate LASSO regression (L1 penalty) coefficients using coordinate descent. Performs automatic feature selection by shrinking some coefficients exactly to zero.
Optimize the elastic net mixing parameter α (balance between L1 and L2 penalties) alongside the regularization strength λ. Combines LASSO feature selection with ridge stability.
Analyze the effect of learning rate on gradient descent convergence. Computes the maximum stable learning rate from the Lipschitz constant and visualizes loss trajectories.
Analyze Newton's method iterations for root-finding. Computes successive approximations xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ) and confirms quadratic convergence near the root.
Calculate the number of bisection iterations required to achieve a desired root precision. The bisection method guarantees convergence for continuous functions on a bracketing interval.
Determine the optimal step size h for the classical 4th-order Runge-Kutta ODE solver. Balances local truncation error O(h⁵) against computational cost.
Calculate crystallization fraction using the Avrami equation X(t) = 1 − exp(−kt^n). Determine Avrami exponent n and rate constant k from experimental data.
Calculate activation energy Ea from the Arrhenius equation k = A·exp(−Ea/RT). Determine Ea from two-point or multi-point rate constant measurements at different temperatures.
Estimate the standard error of a Monte Carlo simulation as σ/√N. Calculate required sample size N to achieve target precision and analyze convergence behavior.
Calculate the confidence interval for a population mean using z or t distribution depending on known or unknown variance. Supports 90%, 95%, and 99% confidence levels.
Determine the required sample size to achieve a target statistical power (1−β) for a given effect size, significance level α, and test type (one-tailed or two-tailed).
Calculate Cohen's d effect size for the difference between two means, standardized by the pooled standard deviation. Interprets magnitude as small (0.2), medium (0.5), or large (0.8).
Apply the Bonferroni multiple comparisons correction. Adjusts the significance threshold to α/m for m simultaneous tests to control the family-wise error rate (FWER).
Apply the Benjamini-Hochberg procedure to control the false discovery rate at level q. More powerful than Bonferroni correction for large-scale simultaneous testing in genomics and neuroimaging.
Calculate the Area Under the ROC Curve (AUC) from binary classifier outputs. AUC measures discrimination ability: 0.5 = random, 1.0 = perfect classifier.
Calculate Youden's J index = sensitivity + specificity − 1 to identify the optimal classification threshold on a ROC curve. Maximizing J maximizes diagnostic performance.
Perform the likelihood ratio test (LRT) statistic Λ = −2·ln(L₀/L₁). Under H₀, Λ follows a chi-squared distribution for comparing nested statistical models.
Calculate the Bayes factor BF₁₀ = P(data|H₁) / P(data|H₀) to compare two hypotheses. Provides continuous evidence scale without requiring a binary reject/fail-to-reject decision.
Calculate posterior probabilities using Bayes' theorem: P(H|D) = P(D|H)·P(H) / P(D). Update prior beliefs with observed evidence across any number of hypotheses.
Calculate information gain IG(S,A) = H(S) − Σ |Sv|/|S| · H(Sv) for decision tree attribute selection. Measures the reduction in entropy from splitting on attribute A.
Calculate the Gini impurity G = 1 − Σ pᵢ² for a node in a decision tree. Measures the probability of misclassifying a randomly chosen element if labeled according to the class distribution.
Calculate the cross-entropy loss H(p,q) = −Σ p(x)·log q(x) between true labels and model predictions. Standard loss function for classification neural networks and logistic regression.