➗ 数学工具Math （第 4 页）(page 4)
长方体Cuboid Calculator
体积 = lwh；表面积 = 2(lw+lh+wh)。本地处理。可设置「长」、「宽」、「高」。Volume = lwh; surface = 2(lw+lh+wh). Processed locally. Settings: Length, Width, Height.
贝叶斯定理Bayes Theorem
P(A|B) = P(B|A)P(A) / P(B)。本地处理。可设置「P(A)」、「P(B given A)」、「P(B)」。P(A|B) = P(B|A)P(A) / P(B). Processed locally. Settings: P(A), P(B given A), P(B).
Beta 分布（均值/方差）Beta Distribution (mean/var)
均值 = α/(α+β)；方差 = αβ/((α+β)²(α+β+1))。本地处理。可设置「α」、「β」。Mean = α/(α+β); variance = αβ/((α+β)²(α+β+1)). Processed locally. Settings: α, β.
指数分布Exponential Distribution
均值 = 1/λ；P(X≤x) = 1-e^(-λx)。本地处理。可设置「λ」、「x」。Mean = 1/λ; P(X≤x) = 1-e^(-λx). Processed locally. Settings: λ, x.
几何分布Geometric Distribution
首次成功均值 = 1/p（试验次数）。本地处理。可设置「成功概率 p」、「试验次数 k」。Mean trials until first success = 1/p. Processed locally. Settings: Success probability p, Trials k.
超几何分布Hypergeometric Distribution
P(X=k) = C(K,k)C(N-K,n-k)/C(N,n)。本地处理。可设置「总体 N」、「成功标签 K」、「抽取 n」、「抽中 k」。P(X=k) = C(K,k)C(N-K,n-k)/C(N,n). Processed locally. Settings: Population N, Success states K, Draws n, Observed k.
中位数绝对偏差 MADMedian Absolute Deviation
MAD = median(|x_i - median(x)|)。本地处理。可设置「数据」。MAD = median(|x_i - median(x)|). Processed locally. Settings: Data.
众数Mode Calculator
出现次数最多的数（可多个）。本地处理。可设置「数据」。Most frequent value(s). Processed locally. Settings: Data.
极差Range Calculator
极差 = max - min。本地处理。可设置「数据」。Range = max - min. Processed locally. Settings: Data.
均方根 RMSRoot Mean Square
RMS = √(Σx² / n)。本地处理。可设置「数据」。RMS = √(Σx² / n). Processed locally. Settings: Data.
威布尔分布（均值示意）Weibull Distribution (mean demo)
均值 ≈ λ Γ(1+1/k)（Γ 用数值近似）。本地处理。可设置「形状 k」、「尺度 λ」。Mean ≈ λ Γ(1+1/k) (Γ numerically approximated). Processed locally. Settings: Shape k, Scale λ.
等边三角形面积Equilateral Triangle Area
面积 = √3 / 4 × a²。本地处理。可设置「边长」。Area = √3 / 4 × a². Processed locally. Settings: Side.
Cholesky 分解（2×2）Cholesky Decomposition (2×2)
对 SPD 矩阵 [[a,b],[b,c]] 求 L。本地处理。可设置「a」、「b」、「c」。Factor SPD [[a,b],[b,c]] as L Lᵀ. Processed locally. Settings: a, b, c.
矩阵列空间（基与秩）Column Space Calculator
用行化简（RREF）找主元列，原矩阵的主元列构成列空间的一组基；同时给出秩。本地处理。Row-reduce (RREF) to find pivot columns; those columns of the original matrix form a basis of the column space. Rank is shown too. Processed locally.
数根（数字根）Digital Root Calculator
反复把各位数字相加直到只剩一位；也等于 1 + (n - 1) mod 9。支持很长的整数。本地处理。Repeatedly add the digits until one digit remains; equals 1 + (n - 1) mod 9. Works with very long integers. Processed locally.
整除判断（2-13）Divisibility Test Calculator
判断整数能否被 2-13 及自定义除数整除，并说明常用规则。本地处理。可设置「整数」、「自定义除数（可选）」。Check whether an integer is divisible by 2-13 and a custom divisor, with the usual rules. Processed locally. Settings: Integer, Custom divisor (optional).
2×2 矩阵特征值与特征向量Eigenvalue & Eigenvector Calculator (2×2)
解 det(A - λI) = 0 得特征值（含复数），并给出对应的特征向量。本地处理。可设置「a₁₁」、「a₁₂」、「a₂₁」、「a₂₂」。Solve det(A - λI) = 0 for the eigenvalues (complex included) and give eigenvectors. Processed locally. Settings: a₁₁, a₁₂, a₂₁, a₂₂.
误差函数 erf / erfcError Function Calculator
erf(x) = 2/√π ∫₀ˣ e^(-t²) dt，用高精度级数/连分式计算，并给出 erfc = 1 - erf。本地处理。erf(x) = 2/√π ∫₀ˣ e^(-t²) dt via high-precision series / continued fraction, plus erfc = 1 - erf. Processed locally.
伽马函数 Γ(x)Gamma Function Calculator
用 Lanczos 近似和反射公式计算 Γ(x)（x 可为负的非整数），同时给出 ln|Γ(x)|。正整数时 Γ(n) = (n - 1)!。本地处理。Γ(x) via the Lanczos approximation and reflection formula (negative non-integers allowed), plus ln|Γ(x)|. For positive integers Γ(n) = (n - 1)!. Processed locally.
黄金矩形Golden Rectangle Calculator
黄金比 φ = (1 + √5)/2 ≈ 1.618。已知长边或短边，求另一边、面积和周长。本地处理。Golden ratio φ = (1 + √5)/2 ≈ 1.618. From the long or short side, get the other side, area and perimeter. Processed locally.
双曲函数 sinh / cosh / tanhHyperbolic Functions Calculator
计算 sinh、cosh、tanh、coth、sech、csch 以及反双曲函数 asinh、acosh、atanh。本地处理。可设置「x」。sinh, cosh, tanh, coth, sech, csch and the inverses asinh, acosh, atanh. Processed locally. Settings: x.
不规则多边形面积（坐标法）Irregular Polygon Area Calculator
按顺序输入各顶点坐标（每行一个"x y"），用鞋带公式计算面积和周长。本地处理。Enter the vertices in order (one "x y" per line); the shoelace formula gives the area and perimeter. Processed locally.
幻方生成器Magic Square Generator
生成 n 阶幻方（n = 3-20）：奇数阶用暹罗法，4 的倍数用对角线互补法，其余偶数用 LUX 法；每行、每列、两条对角线之和都相等。本地处理。Generate an n×n magic square (n = 3-20): Siamese method for odd n, diagonal complement for multiples of 4 and the LUX method for other even n. Every row, column and both diagonals have the same sum. Processed locally.
矩阵幂 AⁿMatrix Power Calculator
用快速幂计算方阵的 n 次幂（n = 0-1000），每行一行、空格分隔输入矩阵。本地处理。Raise a square matrix to the power n (0-1000) by repeated squaring; enter one row per line, space separated. Processed locally.
矩阵的秩（含行最简形）Matrix Rank Calculator
用高斯-若尔当消元求矩阵的秩，并显示行最简形（RREF）和零度（列数 - 秩）。每行一行、空格分隔。本地处理。Find the rank by Gauss-Jordan elimination, with the reduced row echelon form (RREF) and nullity (columns - rank). One row per line, space separated. Processed locally.
矩阵的迹Matrix Trace Calculator
方阵的迹 = 主对角线元素之和，等于全部特征值之和；同时给出主对角线元素。本地处理。The trace of a square matrix is the sum of its main-diagonal entries, which equals the sum of its eigenvalues. Processed locally.
负二项分布Negative Binomial Distribution Calculator
每次成功概率为 p，第 r 次成功恰好出现在第 k 次试验的概率 P(X = k) = C(k-1, r-1) pʳ (1-p)ᵏ⁻ʳ，以及累计概率、均值和方差。本地处理。With success probability p, the chance that the r-th success happens exactly on trial k is P(X = k) = C(k-1, r-1) pʳ (1-p)ᵏ⁻ʳ; plus cumulative probabilities, mean and variance. Processed locally.
可重复排列Permutations with Repetition Calculator
从 n 种元素中可重复地取 r 个排成一列：nʳ 种；也可计算多重集排列 n! / (n₁! n₂! …)（例如 MISSISSIPPI 的字母排列）。结果用大整数精确计算。本地处理。Arrange r items chosen from n kinds with repetition: nʳ ways; or permutations of a multiset n! / (n₁! n₂! …), e.g. letters of MISSISSIPPI. Exact big-integer results. Processed locally.
相对标准偏差 RSD（变异系数）Relative Standard Deviation Calculator
RSD = 标准差 ÷ |平均值| × 100%，也叫变异系数（CV）。可选样本标准差（n - 1）或总体标准差（n）。本地处理。RSD = standard deviation ÷ |mean| × 100%, also called the coefficient of variation (CV). Choose sample (n - 1) or population (n) standard deviation. Processed locally.
球面方程Sphere Equation Calculator
由球心 (a, b, c) 和半径 r 写出标准方程 (x - a)² + (y - b)² + (z - c)² = r² 与一般方程，并给出体积和表面积。本地处理。From centre (a, b, c) and radius r, write the standard equation (x - a)² + (y - b)² + (z - c)² = r² and the general form, plus volume and surface area. Processed locally.
长方体表面积Surface Area of a Rectangular Prism
表面积 = 2(lw + lh + wh)，同时给出侧面积 2h(l + w)、体积和体对角线长度。本地处理。Surface area = 2(lw + lh + wh), plus lateral area 2h(l + w), volume and space diagonal. Processed locally.
矩阵张量积（克罗内克积）Tensor Product Calculator (Kronecker Product)
A ⊗ B：把 A 的每个元素 aᵢⱼ 换成 aᵢⱼB，得到 (m·p) × (n·q) 的矩阵。也适用于向量（单行或单列）。本地处理。A ⊗ B replaces each entry aᵢⱼ of A with aᵢⱼB, giving an (m·p) × (n·q) matrix. Works for vectors too (a single row or column). Processed locally.
三角形重心Triangle Centroid Calculator
重心 G = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3)，是三条中线的交点；同时给出面积。本地处理。Centroid G = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3), where the three medians meet; the area is shown too. Processed locally.
长方体体积Volume of a Rectangular Prism
体积 V = 长 × 宽 × 高，并换算成升和立方英尺（按米输入时）；同时给出表面积。本地处理。Volume V = length × width × height, also in litres and cubic feet (for metre inputs), plus the surface area. Processed locally.
同底数幂相乘Multiplying Exponents Calculator
同底数幂相乘指数相加：aᵐ × aⁿ = aᵐ⁺ⁿ；幂的乘方指数相乘：(aᵐ)ⁿ = aᵐⁿ；同指数不同底：aⁿ × bⁿ = (ab)ⁿ。给出化简式和数值。本地处理。Same base: add exponents, aᵐ × aⁿ = aᵐ⁺ⁿ; power of a power: (aᵐ)ⁿ = aᵐⁿ; same exponent: aⁿ × bⁿ = (ab)ⁿ. Shows the simplified form and the value. Processed locally.
分数乘法（自动约分）Multiplying Fractions Calculator
输入两个或多个分数（如 2/3 × 3/4，也支持整数和带分数"1 1/2"），分子乘分子、分母乘分母并约分，给出最简分数、带分数和小数。本地处理。Enter two or more fractions (e.g. 2/3 × 3/4; whole numbers and mixed numbers like "1 1/2" work too). Numerators and denominators are multiplied and reduced to the simplest fraction, mixed number and decimal. Processed locally.
减法计算器（精确小数）Subtraction Calculator
计算 a - b - c …，用十进制精确运算，避免 0.3 - 0.1 = 0.19999… 这类浮点误差；支持很长的数字。本地处理。Compute a - b - c … with exact decimal arithmetic, avoiding floating-point artefacts such as 0.3 - 0.1 = 0.19999…; very long numbers work. Processed locally.
商和余数Quotient and Remainder Calculator
整数除法：商 = trunc(被除数 ÷ 除数)，余数 = 被除数 - 商 × 除数。可设置「被除数」、「除数」。Integer division: quotient = trunc(dividend ÷ divisor), remainder = dividend - quotient × divisor. Settings: Dividend, Divisor.
五数概括5-Number Summary Calculator
Min / Q1 / 中位数 / Q3 / Max。Min / Q1 / median / Q3 / Max.
45-45-90 三角形45-45-90 Triangle Calculator
斜边 = 腿 × √2。可设置「已知」、「长度」。hyp = leg × √2. Settings: Given, Length.
周长转直径Circumference to Diameter Calculator
圆：d = C / π，也可由直径求周长。可设置「已知」、「数值」。Circle: d = C / π; also C from diameter. Settings: Given, Value.
30-60-90 三角形30-60-90 Triangle Calculator
边比 1 : √3 : 2（短直角边 : 长直角边 : 斜边）。可设置「已知」、「长度」。Side ratio 1 : √3 : 2 (short leg : long leg : hypotenuse). Settings: Given, Length.
以 2 为底的对数Log Base 2 Calculator
log₂(x) = ln(x)/ln(2)。可设置「x」。log₂(x) = ln(x)/ln(2). Settings: x.
10 的幂Power of 10 Calculator
计算 10ⁿ，也可由 y 求 n = log₁₀(y)。可设置「模式」、「数值」。Compute 10ⁿ, or n = log₁₀(y). Settings: Mode, Value.
单位面积速查Unit Area Reference
查看 1 个面积单位等于多少其他单位。可设置「单位」。See what 1 of an area unit equals in others. Settings: Unit.
变异系数计算器Coefficient of Variation Calculator
输入一组数据，计算变异系数 CV = 标准差 ÷ 平均值，同时给出平均值和标准差；可选样本或总体标准差，用来比较量纲或平均水平不同的数据的离散程度。Enter a data set to get the coefficient of variation, CV = standard deviation ÷ mean, along with the mean and standard deviation. Choose sample or population SD; CV lets you compare spread between data with different units or levels.
经验法则（68-95-99.7）计算器Empirical Rule Calculator (68-95-99.7)
输入正态分布的平均值和标准差，给出平均值 ±1、±2、±3 个标准差的区间，以及各区间内数据所占的比例（约 68%、95%、99.7%），还可以查某个数值的 z 分数。Enter the mean and standard deviation of a normal distribution to get the ranges within ±1, ±2 and ±3 standard deviations and the share of data in each (about 68%, 95% and 99.7%), plus the z-score of any value.
几何平均数计算器Geometric Mean Calculator
计算一组正数的几何平均数（n 个数乘积的 n 次方根），也可以输入每期增长率（%）求平均增长率；同时给出算术平均数作对比。Calculate the geometric mean of positive numbers (the n-th root of their product), or enter growth rates in percent to get the average growth rate, with the arithmetic mean for comparison.
调和平均数计算器Harmonic Mean Calculator
计算一组正数的调和平均数 n ÷ Σ(1/x)，适合求平均速度、平均单价等「比率」的平均；可选填权重求加权调和平均，并给出算术、几何平均作对比。Calculate the harmonic mean n ÷ Σ(1/x) of positive numbers — the right average for rates such as speeds or unit prices — with optional weights, and the arithmetic and geometric means for comparison.
泊松分布计算器Poisson Distribution Calculator
输入平均发生次数 λ 和次数 k，计算泊松分布的 P(X = k)、P(X ≤ k)、P(X < k)、P(X ≥ k) 和 P(X > k)，并给出均值、方差和标准差。Enter the average rate λ and a count k to get the Poisson probabilities P(X = k), P(X ≤ k), P(X < k), P(X ≥ k) and P(X > k), with the mean, variance and standard deviation.
二次回归计算器Quadratic Regression Calculator
输入若干对 (x, y) 数据，用最小二乘法拟合二次函数 y = ax² + bx + c，给出系数、决定系数 R²、残差平方和，并可代入 x 求预测值。Fit a quadratic y = ax² + bx + c to (x, y) data by least squares, with the coefficients, R², residual sum of squares and a prediction for any x.
二项式系数（组合数）计算器Binomial Coefficient Calculator
计算组合数 C(n, k) = n! ÷ (k!(n − k)!)，即从 n 个不同元素中取 k 个的方法数，用大整数精确计算并给出位数和科学记数法；n 最大 100,000。Calculate the binomial coefficient C(n, k) = n! ÷ (k!(n − k)!), the number of ways to choose k items from n, exactly with big integers, with the digit count and scientific notation; n up to 100,000.
效应量计算器（Cohen's d）Effect Size Calculator (Cohen's d)
输入两组的平均值、标准差和样本量，计算 Cohen's d（合并标准差）、小样本校正的 Hedges' g 和 Glass's Δ，并按 Cohen（1988）的常用基准给出大小参考。Enter the mean, standard deviation and sample size of two groups to get Cohen's d (pooled SD), the small-sample-corrected Hedges' g and Glass's Δ, with Cohen's (1988) conventional benchmarks.
误差范围计算器（调查抽样）Margin of Error Calculator (Surveys)
输入样本量、置信水平和样本比例，计算调查结果的误差范围 ± z·√(p(1 − p)/n)，可填总体规模做有限总体校正，并给出置信区间。Calculate a survey's margin of error, ± z·√(p(1 − p)/n), from the sample size, confidence level and sample proportion, with an optional finite-population correction and the confidence interval.
点估计计算器（比例）Point Estimate Calculator (Proportion)
输入成功次数和试验次数，计算比例的四种点估计：最大似然（x/n）、拉普拉斯、Jeffreys 和 Wilson，并按常用规则给出推荐值。Enter the number of successes and trials to get four point estimates of a proportion — maximum likelihood (x/n), Laplace, Jeffreys and Wilson — with the one a common rule recommends.
相关系数计算器Correlation Coefficient Calculator
计算 Pearson r、Spearman ρ、r² 和协方差，含显著性检验（P 值）与置信区间。Pearson r, Spearman ρ, r² and covariance — with significance test (p-value) and confidence interval.
残差计算器（线性回归）Residual Calculator (Linear Regression)
输入成对的 (x, y) 数据，拟合最小二乘直线 ŷ = a + bx，列出每个点的预测值和残差 y − ŷ，并给出残差平方和、残差标准误和最大残差。Enter paired (x, y) data to fit the least-squares line ŷ = a + bx and list each point's predicted value and residual y − ŷ, with the residual sum of squares, residual standard error and the largest residual.
单位向量计算器Unit Vector Calculator
输入 2–10 维向量，求它的模长和同方向的单位向量 û = v ÷ |v|；整数分量时给出带根号的精确形式，二维和三维还给出方向角。Enter a vector with 2 to 10 components to get its magnitude and the unit vector in the same direction, û = v ÷ |v| — exact with square roots for whole-number components, plus direction angles in 2D and 3D.
向量投影计算器Vector Projection Calculator
求向量 a 在向量 b 上的投影向量 proj_b a = (a·b ÷ |b|²) b、标量投影（a·b ÷ |b|）、与 b 垂直的分量（a − 投影）以及两向量夹角，支持 2–10 维。Project vector a onto vector b: the vector projection proj_b a = (a·b ÷ |b|²) b, the scalar projection a·b ÷ |b|, the component of a perpendicular to b (a − projection) and the angle between them, in 2 to 10 dimensions.
半圆面积计算器Area of a Semicircle Calculator
已知半径、直径、面积或周长中的任意一个，求半圆的面积 πr²/2、弧长 πr、周长（弧长 + 直径）以及形心到直径的距离 4r/(3π)。Given a semicircle's radius, diameter, area or perimeter, find its area πr²/2, arc length πr, perimeter (arc + diameter) and the distance from the diameter to its centroid, 4r/(3π).