2D Rotation Matrix Calculator
Compute the entries of a 2-dimensional rotation matrix R(θ) = [[cos θ, −sin θ], [sin θ, cos θ]] and the result of rotati...
Everyday arithmetic, algebra, and geometry — worked out clearly.
Percentages, fractions, ratios, geometry, and the little calculations that come up all the time — with formulas you can reuse anywhere.
Compute the entries of a 2-dimensional rotation matrix R(θ) = [[cos θ, −sin θ], [sin θ, cos θ]] and the result of rotati...
Compute the n-th triangular number T_n = n(n + 1) / 2 — the number of dots in a triangle with n dots on each side.
Compute the n-th tetrahedral number Te_n = n(n + 1)(n + 2) / 6 — the number of stacked spheres in a tetrahedron with n s...
Perform one Newton-Raphson iteration to refine an estimate of a root: x_{new} = x − f(x) / f'(x), given the current esti...
Compute Chebyshev's inequality bound on the tail probability of any distribution with finite variance: P(|X − μ| ≥ k·σ)...
Apply a single iteration of Newton's method (Newton-Raphson) given the current estimate x_n, the function value f(x_n),...
Compute the Wilson-score confidence interval for a binomial proportion p̂ = x/n. Unlike the standard Wald CI, the Wilson...
Compute the one-tailed and two-tailed p-values from a standard-normal z-score using the Polya approximation Φ(z) ≈ ½·(1...
Compute the apothem (perpendicular distance from the centre to the midpoint of a side) of a regular n-sided polygon with...
Compute the number of combinations C(n, r) — the number of ways to choose r items from n items without regard to order:...
Compute the mean, variance, mode and standard deviation of a Beta(α, β) distribution from its two shape parameters.
Compute the interior angle of a regular n-gon and the sum of all interior angles: each = (n − 2) · 180 / n; sum = (n − 2...
Apply Fisher's r-to-z transformation to a Pearson correlation coefficient so the resulting z is approximately normal wit...
Perform one bisection-method step: compute the midpoint m of the bracket [a, b], evaluate the new bracket based on the s...
Compute the OLS slope coefficient of a simple linear regression y = a + b·x directly from the five paired-sample summary...
Compute a Wilson score confidence interval for a single binomial proportion (95 % default). Outperforms the Wald interva...
Compute the cosine similarity between two vectors when only the dot product and the two L2-norms are pre-computed: cos θ...
Test whether two independent sample proportions differ significantly. Pooled estimate p̄ = (x₁ + x₂)/(n₁ + n₂); z = (p̂₁...
Compute the volume of a parallelepiped formed by three edge vectors a, b, c using the scalar triple product: V = |a · (b...
Compute the confidence interval for a population mean using the t-distribution (sample standard deviation, df = n − 1):...
Compute the adjusted coefficient of determination for a linear regression with n observations and p predictors (excludin...
Compute the L¹ (Manhattan / taxicab) distance between two points in 3-D space: d = Σ |coord₂ − coord₁|. The distance you...
Compute the minimum sample size needed for a desired margin of error E around a proportion p̂ at a given confidence leve...
Compute the number of permutations P(n, r) — the number of ways to arrange r items from n distinct items where order mat...