➗ 数学工具Math （第 2 页）(page 2)
整数相乘Multiply Integers
把两个或多个整数相乘，支持任意大小。本地处理。可设置「千位逗号」。Multiply two or more integers of any size. Processed locally. Settings: Thousands commas.
去除整数中的千位逗号Remove Commas from an Integer
去掉整数写法里的千位逗号（1,234,567 → 1234567），可批量处理，保留负号。本地处理。Strip thousands commas from integer notation (1,234,567 -> 1234567), in bulk, keeping signs. Processed locally.
从整数列表中抽取Pick an Integer
从整数列表中随机抽取一个或多个（可设种子复现，可要求不重复）。适合抽签和抽样，本地处理。Pick one or more integers at random from a list (optional seed for reproducibility; optional unique picks). Handy for draws and sampling. Processed locally.
替换整数中的数字Replace Integer Digits
把每个整数里的指定数字替换成另一个（例如把所有 0 换成 9），保留符号。本地处理。Replace one digit with another in each integer (e.g. turn every 0 into 9), keeping the sign. Processed locally.
反转整数各位数字Reverse Integer Digits
把每个整数的数字左右颠倒（120 → 021），可选择去掉结果前导零。本地处理。Reverse the digits of each integer (120 -> 021); optionally drop leading zeros in the result. Processed locally.
把整数改写成若干数之和Rewrite Integer as Sum
把每个正整数改写成若干正整数之和：按固定加数（如全是 1）、按十进制位权展开，或随机拆分（可设种子）。本地处理。Rewrite each positive integer as a sum of positive integers: fixed addends (e.g. all 1s), decimal place-value expansion, or a random split (optional seed). Processed locally.
把整数改写成若干数之积Rewrite Integer as Product
把每个大于 1 的整数改写成因数乘积：质因数分解，或拆成若干大于 1 的因数（随机，可设种子）。本地处理。Rewrite each integer greater than 1 as a product of factors: prime factorization, or a random factor split (optional seed). Processed locally.
打乱整数各位数字Shuffle Integer Digits
随机打乱每个整数的各位数字（可设种子），保留符号。本地处理。可设置「种子（留空 = 真随机）」、「去掉前导零」。Randomly shuffle the digits of each integer (optional seed), keeping the sign. Processed locally. Settings: Seed (blank = truly random), Drop leading zeros.
打乱整数列表顺序Shuffle Integers
随机打乱整数列表的顺序（可设种子复现）。本地处理。可设置「种子（留空 = 真随机）」、「分隔」、「千位逗号」。Randomly shuffle the order of a list of integers (optional seed). Processed locally. Settings: Seed (blank = truly random), Separator, Thousands commas.
排序整数各位数字Sort Integer Digits
把每个整数的各位数字按升序或降序排列，保留符号。本地处理。可设置「顺序」、「去掉前导零」。Sort the digits of each integer ascending or descending, keeping the sign. Processed locally. Settings: Order, Drop leading zeros.
排序整数列表Sort Integers
把整数列表按数值升序或降序排序。本地处理。可设置「顺序」、「分隔」、「去重」、「千位逗号」。Sort a list of integers by numeric value, ascending or descending. Processed locally. Settings: Order, Separator, Remove duplicates, Thousands commas.
生成整数三元组（勾股等）Generate Integer Triples
生成整数三元组：勾股数（a^2+b^2=c^2）、连续三整数，或和为 S 的随机三元组（可设种子）。本地处理。Generate integer triples: Pythagorean triples, three consecutive integers, or random triples that sum to S (optional seed). Processed locally.
生成对称整数（回文数）Generate Symmetric Integers
生成回文整数：指定位数与个数，可设种子。本地处理。可设置「个数」、「位数」、「种子（留空 = 真随机）」。Generate palindromic integers of a chosen digit length and count; optional seed. Processed locally. Settings: How many, Digits, Seed (blank = truly random).
生成冯·诺依曼序数Generate von Neumann Ordinals
生成冯·诺依曼自然数（序数）的集合写法：0=∅, 1={∅}, 2={∅,{∅}}, …。可指定从 0 到 N。本地处理。Generate von Neumann naturals (ordinals) in set notation: 0 = empty set, 1 = set containing empty, 2 = set of those, and so on, from 0 through N. Processed locally.
生成策梅洛序数Generate Zermelo Ordinals
生成策梅洛自然数的集合写法：0=∅, 1={∅}, 2={{∅}}, 3={{{∅}}}, …。可指定从 0 到 N。本地处理。Generate Zermelo naturals in set notation: 0 = empty set, 1 = singleton of empty, 2 = singleton of that, and so on, from 0 through N. Processed locally.
拆分整数（按位数 / 块）Split an Integer
把整数按固定位数或固定块数切开（例如 123456 按 2 位 → 12 34 56），可从左或从右切，保留符号。适合分组显示和教学，本地处理。Split an integer into fixed-size digit chunks or a fixed number of parts (e.g. 123456 by pairs -> 12 34 56), from the left or right, keeping the sign. Handy for grouped display and teaching. Processed locally.
整数写成花体 / 数学字体Write Integers in Fancy Font
把整数的数字换成 Unicode 数学字母数字（粗体、无体、双线、无衬线粗体、等宽等），方便发帖和标题。本地处理。Rewrite integer digits in Unicode mathematical alphanumeric forms (bold, italic, double-struck, sans-bold, monospace, etc.) for posts and titles. Processed locally.
找出全部因数Find All Divisors
列出一个整数的全部正因数（含 1 和它本身），可按行或逗号输出，并显示因数个数与之和。本地处理。List every positive divisor of an integer (including 1 and itself), as lines or commas, with the count and sum. Processed locally.
生成盈数（丰数）Generate Abundant Numbers
生成真因数之和大于自身的盈数（丰数）。可设个数与起点，本地处理。可设置「个数」、「起点」。Generate abundant numbers (proper-divisor sum greater than the number). Set how many and where to start. Processed locally. Settings: How many, Start.
生成几乎完全数Generate Almost Perfect Numbers
生成几乎完全数（真因数之和 = n−1）。已知的几乎完全数都是 2 的幂；本工具按此列出。本地处理。Generate almost perfect numbers (proper-divisor sum = n−1). All known examples are powers of two; this tool lists those. Processed locally.
生成亏数Generate Deficient Numbers
生成真因数之和小于自身的亏数。可设个数与起点，本地处理。可设置「个数」、「起点」。Generate deficient numbers (proper-divisor sum less than the number). Set how many and where to start. Processed locally. Settings: How many, Start.
生成 Baum–Sweet 序列Generate Baum–Sweet Sequence
生成 Baum–Sweet 序列：若 n 的二进制表示不含奇数长度的连续 0 块则为 1，否则为 0。本地处理。Generate the Baum–Sweet sequence: 1 if the binary of n has no odd-length run of zeros, else 0. Processed locally.
生成自然常数 e 的位数Generate e Digits
用级数 Σ 1/n! 和大整数精确生成 e 的小数点后 1–5000 位。本地处理。可设置「小数位数」。Generate 1–5000 decimal places of e = Σ 1/n! with exact big integers. Processed locally. Settings: Decimal places.
生成偶数Generate Even Numbers
生成一段偶数：可设个数、起点与步长（默认 2）。本地处理。可设置「个数」、「起点」、「步长」。Generate even numbers: set how many, the start value, and step (default 2). Processed locally. Settings: How many, Start, Step.
生成斐波那契质数Generate Fibonacci Primes
列出既是斐波那契数又是质数的项（按斐波那契下标）。可设要找多少个，本地用 Baillie–PSW 判断（大整数）。List Fibonacci numbers that are prime (by Fibonacci index). Set how many to find; primality uses the Baillie–PSW test locally.
生成斐波那契词Generate Fibonacci Words
生成斐波那契词：S0=0，S1=01（或 a/b 字母），S_n = S_{n-1}+S_{n-2}。可输出某一项或前几项，本地处理。Generate Fibonacci words: S0=0, S1=01 (or a/b letters), S_n = S_{n-1}+S_{n-2}. One term or the first few. Processed locally.
生成 Gijswijts 序列Generate Gijswijts Sequence
生成 Gijswijt 序列（OEIS A090822）：a(1)=1，之后每一项等于“目前序列末尾最多由同一段连续重复了几次”。1 1 2 1 1 2 2 2 3 1 1 2…，本地处理。Generate Gijswijt's sequence (OEIS A090822): a(1)=1, then each term is the largest k such that the sequence so far ends with some block repeated k times in a row. 1 1 2 1 1 2 2 2 3 1 1 2… Processed locally.
生成单位矩阵Generate Identity Matrix
生成 n×n 单位矩阵（对角线为 1，其余为 0），可调分隔符。本地处理。可设置「阶数 n」、「分隔」。Generate an n×n identity matrix (1 on the diagonal, 0 elsewhere); choose the separator. Processed locally. Settings: Order n, Separator.
生成 Kolakoski 序列Generate Kolakoski Sequence
生成 Kolakoski 序列（自描述的 1/2 游程序列）：1,2,2,1,1,2,1,2,2,1,…。本地处理。Generate the Kolakoski sequence (self-describing 1/2 run-length sequence): 1,2,2,1,1,2,1,2,2,1,… Processed locally.
生成外观数列（读数序列）Generate Look-and-Say Sequence
生成外观数列（look-and-say）：从种子出发，每次用「几个几」描述上一项。可设项数与种子，本地处理。Generate the look-and-say sequence: each term describes the previous as runs of digits. Set terms and seed. Processed locally.
生成卢卡斯数Generate Lucas Numbers
生成卢卡斯数列：L0=2，L1=1，之后每项为前两项之和。可设项数与起点下标，本地处理。Generate Lucas numbers: L0=2, L1=1, then each term is the sum of the previous two. Set count and start index. Processed locally.
生成卢卡斯质数Generate Lucas Primes
列出既是卢卡斯数又是质数的项。可设要找多少个，本地用 Baillie–PSW 判断（大整数）。可设置「个数」。List Lucas numbers that are prime. Set how many to find; primality uses the Baillie–PSW test locally. Settings: How many.
生成 Moser–de Bruijn 序列Generate Moser–de Bruijn Sequence
生成 Moser–de Bruijn 序列：把 n 的二进制位当作四进制数字得到的数（只含数字 0、1）。本地处理。Generate the Moser–de Bruijn sequence: interpret the binary digits of n as base-4 digits (only 0 and 1). Processed locally.
生成负斐波那契数Generate NegaFibonacci Numbers
生成负下标斐波那契数：F_{-n}=(-1)^{n+1} F_n。可设个数与起始负下标，本地处理。Generate negative-index Fibonacci numbers: F_{-n}=(-1)^{n+1} F_n. Set count and start index. Processed locally.
生成负卢卡斯数Generate NegaLucas Numbers
生成负下标卢卡斯数：L_{-n}=(-1)^n L_n。可设个数与起点，本地处理。可设置「项数」、「起点下标（负）」。Generate negative-index Lucas numbers: L_{-n}=(-1)^n L_n. Set count and start. Processed locally. Settings: Terms, Start index (negative).
生成奇数Generate Odd Numbers
生成一段奇数：可设个数、起点与步长（默认 2）。本地处理。可设置「个数」、「起点」、「步长」。Generate odd numbers: set how many, start, and step (default 2). Processed locally. Settings: How many, Start, Step.
生成折纸序列Generate Paperfolding Sequence
生成规则折纸（龙曲线）序列 OEIS A014577：反复向同一方向对折纸条，展开后的折痕方向，1 表示谷折、0 表示峰折。本地处理。Generate the regular paperfolding (dragon curve) sequence, OEIS A014577: fold a strip in half repeatedly in the same direction and read the creases after unfolding, 1 for a valley fold and 0 for a mountain fold. Processed locally.
生成随机矩阵Generate Random Matrix
生成随机整数矩阵：可设行列、数值范围与种子。本地处理。可设置「行数」、「列数」、「下限」、「上限」、「种子（可选）」。Generate a random integer matrix: set rows, columns, value range and seed. Processed locally. Settings: Rows, Columns, Min, Max, Seed (optional).
生成 Rudin–Shapiro 序列Generate Rudin–Shapiro Sequence
生成 Rudin–Shapiro 序列：±1 值，与二进制中相邻 11 的个数奇偶相关。本地处理。可设置「项数」、「每行一个」。Generate the Rudin–Shapiro sequence of ±1 from the parity of adjacent 11-bits in n. Processed locally. Settings: Terms, One per line.
生成 Thue–Morse 序列Generate Thue–Morse Sequence
生成 Thue–Morse 序列：n 的二进制中 1 的个数的奇偶（0/1）。本地处理。可设置「项数」、「每行一个」。Generate the Thue–Morse sequence: parity of the number of 1-bits in n (0/1). Processed locally. Settings: Terms, One per line.
生成 Tribonacci 词Generate Tribonacci Words
生成 Tribonacci 词（OEIS A080843，字母 a、b、c 写作 0、1、2）。下标从 0 开始：S0 = 0，S1 = 01，S2 = 0102，之后 Sn = Sn−1 Sn−2 Sn−3（把前三项依次拼接）；Sn 也等于从 0 出发做 n 次替换 0→01、1→02、2→0。本地处理。Generate Tribonacci words (OEIS A080843, with the letters a, b, c written as 0, 1, 2). Indices start at 0: S0 = 0, S1 = 01, S2 = 0102, then Sn = Sn−1 Sn−2 Sn−3 (the previous three joined in order); Sn is also what n substitutions 0→01, 1→02, 2→0 make from 0. Processed locally.
矩阵求逆Invert Matrix
求方阵的逆：把输入的整数、小数或分数当作精确有理数，用高斯–约当消元精确求逆，再按需显示为小数或分数。本地处理。Invert a square matrix: integers, decimals and fractions are read as exact rationals and inverted exactly with Gauss–Jordan elimination, then shown as decimals or fractions. Processed locally.
矩阵乘法Matrix Multiply
计算两个矩阵的乘积 A×B，整数、小数和分数都精确计算。在上方输入框写 A，空一行再写 B；或者把 B 写进「矩阵 B」（用分号分行）。本地处理。Multiply two matrices, A×B, with integers, decimals and fractions computed exactly. Type A in the box above, then a blank line and B; or put B in the Matrix B field (rows separated by semicolons). Processed locally.
L 系统生成器L-System Generator
根据公理与改写规则迭代生成 L 系统字符串。本地处理。可设置「公理」、「规则（多条用分号分开，如 X → X+YF+; Y → -FX-Y）」、「迭代次数」。Iterate an L-system from an axiom and rewrite rules. Processed locally. Settings: Axiom, Rules (separate several with semicolons, e.g. X → X+YF+; Y → -FX-Y), Iterations.
计算数字积Calculate Digit Product
计算整数各位数字的乘积（可忽略 0）。本地处理。可设置「忽略 0」。Product of the decimal digits of an integer (optionally ignoring zeros). Processed locally. Settings: Skip zeros.
计算因数个数Calculate Number of Divisors
计算正整数的正因数个数 d(n)，可同时列出因数。本地处理。可设置「分隔」、「列出因数」。Count the positive divisors d(n) of a positive integer; optionally list them. Processed locally. Settings: Separator, List divisors.
计算数字列表乘积Calculate Number Product
计算一列整数的乘积。本地处理。Product of a list of integers. Processed locally.
计算数字列表之和Calculate Number Sum
计算一列整数的和。本地处理。Sum of a list of integers. Processed locally.
计算黄金比例 φ 的位数Calculate Phi Digits
用 φ=(1+√5)/2 和大整数开方精确计算 φ 的小数点后位数。本地处理。可设置「小数位数」。Compute decimal places of φ=(1+√5)/2 with an exact big-integer square root. Processed locally. Settings: Decimal places.
检查回文数Check Number Palindrome
判断整数是否为回文数（正反读相同），可忽略符号。本地处理。可设置「忽略负号」。Check whether an integer is a palindrome (same forwards and backwards); optional ignore sign. Processed locally. Settings: Ignore minus sign.
整数转负十进制Convert Number to Negadecimal
把整数转成负十进制（基为 −10）表示。本地处理。Convert an integer to negadecimal (base −10). Processed locally.
整数写成 10 的幂形式Convert Number to Power of Ten
把整数写成 a×10^k 的科学计数形式（或纯 10^k）。本地处理。Rewrite an integer as a×10^k scientific form (or pure 10^k). Processed locally.
整数写成 2 的幂形式Convert Number to Power of Two
判断并写成 2^k，或拆成若干 2 的幂之和（二进制展开）。本地处理。Write as 2^k when exact, or as a sum of powers of two (binary expansion). Processed locally.
生成 Unicode 分数Create Unicode Fractions
把 a/b 形式的分数转成 Unicode 横线分数（若有预组合字符则用之，否则用下标/上标）。本地处理。Turn a/b fractions into Unicode vulgar fractions when available, else superscript/subscript form. Processed locally.
找最大数Find Maximum Number
找出一列数字中的最大值。本地处理。Find the maximum in a list of numbers. Processed locally.
找最小数Find Minimum Number
找出一列数字中的最小值。本地处理。Find the minimum in a list of numbers. Processed locally.
生成类斐波那契数列Generate Fibonacci-like Series
由任意前两项按 F_n=F_{n-1}+F_{n-2} 生成数列。本地处理。可设置「项数」、「首项」、「第二项」。Generate a series from any two seeds with F_n=F_{n-1}+F_{n-2}. Processed locally. Settings: Terms, First, Second.
生成外观数列数值Generate Look-and-Say Numbers
生成外观数列各项（作为数字串），可设项数与种子。本地处理。可设置「项数」、「种子」。Generate look-and-say terms as digit strings; set count and seed. Processed locally. Settings: Terms, Seed.
生成 10 的幂Generate Powers of Ten
生成 10^0, 10^1, … 或指定指数范围。本地处理。可设置「个数」、「起始指数」。Generate 10^0, 10^1, … or a chosen exponent range. Processed locally. Settings: How many, Start exponent.
生成 2 的幂Generate Powers of Two
生成 2^0, 2^1, … 或指定指数范围。本地处理。可设置「个数」、「起始指数」。Generate 2^0, 2^1, … or a chosen exponent range. Processed locally. Settings: How many, Start exponent.
生成随机质数Generate Random Prime Numbers
在范围内随机生成质数（确定性 Miller-Rabin）。本地处理。可设置「个数」、「下限」、「上限」、「种子」、「不重复」。Random primes in a range (deterministic Miller-Rabin). Processed locally. Settings: How many, Min, Max, Seed, Unique.
生成随机向量Generate Random Vector
生成指定维数的随机实数/整数向量。本地处理。可设置「维数」、「下限」、「上限」、「种子」、「分隔」。Generate a random real/integer vector of a chosen dimension. Processed locally. Settings: Dimension, Min, Max, Seed, Separator.
生成立方数列Generate Sequence of Cubes
生成 n³ 数列：1³, 2³, … 或指定起点。本地处理。可设置「个数」、「起点 n」。Generate cubes n³: 1³, 2³, … or from a chosen start. Processed locally. Settings: How many, Start n.
生成平方数列Generate Sequence of Squares
生成 n² 数列：1², 2², … 或指定起点。本地处理。可设置「个数」、「起点 n」。Generate squares n²: 1², 2², … or from a chosen start. Processed locally. Settings: How many, Start n.
生成单位矩阵Generate Unit Matrix
生成 n×n 单位矩阵（与「生成单位矩阵 / identity」相同族）。本地处理。可设置「阶数 n」、「分隔」。Generate an n×n identity (unit) matrix. Processed locally. Settings: Order n, Separator.
线性递推计算器Linear Recurrence Calculator
计算常系数线性递推：给定初值与系数 a_n = c1 a_{n-1}+…。本地处理。可设置「项数」、「初值（空格分隔）」、「系数 c1 c2 …（对应 a_{n-1}, a_{n-2}, …）」。Compute a linear recurrence with constant coefficients from seeds and coeffs. Processed locally. Settings: Terms, Seeds (spaces), Coeffs c1 c2 … (for a_n-1, a_n-2, …).
质数判定Number Primality Test
判断一个整数是否为质数：大整数（BigInt）运算，3.3×10^24 以下用确定性的 Miller–Rabin 测试，更大的数给出概率性结论。本地处理。Test whether an integer is prime, using big-integer (BigInt) arithmetic: a deterministic Miller–Rabin test below 3.3×10^24 and a probable-prime test above that. Processed locally.
挑一个数Pick a Number
从一列数里按位置或随机挑出一个。本地处理。可设置「方式」、「序号（从 1 起）」、「随机种子」。Pick one number from a list by index or at random. Processed locally. Settings: Mode, Index (from 1), Random seed.
随机数字矩阵Random Number Matrix
生成随机整数矩阵（同「生成随机矩阵」）。本地处理。可设置「行数」、「列数」、「下限」、「上限」、「种子」。Generate a random integer matrix (same family as Generate Random Matrix). Processed locally. Settings: Rows, Columns, Min, Max, Seed.
打乱数字各位Shuffle Digits
随机打乱整数的各位数字（可保留符号，可设种子）。本地处理。可设置「种子」、「保留负号」。Randomly shuffle digits of an integer (optional keep sign; optional seed). Processed locally. Settings: Seed, Keep minus.
排序数字各位Sort Digits
把整数的各位数字按升序或降序排列。本地处理。可设置「顺序」。Sort digits of an integer ascending or descending. Processed locally. Settings: Order.
分数百分比计算器Marks Percentage Calculator
得分 ÷ 满分 × 100%；也可一次输入多门科目（每行“科目 得分/满分”），得出总百分比。本地处理。Score ÷ max marks × 100%; or enter several subjects (one per line, "subject score/max") for the overall percentage. Processed locally.
函数绘图计算器Graphing Calculator
绘制一个或多个 y = f(x) 函数（每行一个），带坐标刻度和网格，可缩放、拖动并导出 PNG。表达式由安全解析器求值，不执行代码。本地处理。Plot one or more y = f(x) functions (one per line) with axis ticks and a grid; zoom, drag to pan and export PNG. Expressions are evaluated by a safe parser, never run as code. Processed locally.
分数与算式化简计算器Simplify Calculator (fractions & arithmetic)
用精确分数（BigInt 有理数）计算并化简算式：12/18 → 2/3，(1/3 + 1/6) × 4 → 2，同时给出带分数和小数（含循环节）。支持 + − × ÷、^ 或 ** 整数次幂、括号、abs/min/max/floor/ceil；表达式由安全解析器计算，不执行代码。本地处理。Evaluate and simplify arithmetic exactly with fractions (BigInt rationals): 12/18 → 2/3, (1/3 + 1/6) × 4 → 2, with the mixed number and decimal (including repeating digits). Supports + − × ÷, integer powers with ^ or **, brackets and abs/min/max/floor/ceil; a safe parser evaluates it, never running it as code. Processed locally.
百分比转分数计算器（精确）Percent to Fraction Calculator (exact)
把百分比按输入的小数原样精确化为最简分数（BigInt 运算，0.0001% → 1/1000000），同时给出带分数、小数和“最接近的简单分数”。本地处理。Convert a percent to a fraction in lowest terms exactly as typed (BigInt arithmetic, 0.0001% → 1/1000000), with the mixed number, decimal and the closest simple fraction. Processed locally.
有效数字计算器（修约 / 计数）Significant Figures Calculator (round & count)
按指定有效数字位数修约（按输入的十进制数字四舍五入，9.995 → 10.0 / 1.00×10¹），或数一个数有几位有效数字（含科学计数法，并指出整数末尾 0 的歧义）。本地处理。Round to N significant figures (half-up on the decimal digits you type: 9.995 → 10.0 / 1.00×10¹), or count the significant figures in a number (scientific notation supported; ambiguous trailing zeros in whole numbers are flagged). Processed locally.
投手防御率 ERA（支持 6.1 局写法）ERA Calculator (with 6.1 innings notation)
ERA = 自责分 × 9 ÷ 投球局数。局数可按棒球记分写法输入（6.1 = 6⅓ 局，6.2 = 6⅔ 局），也可选十进制局数；同时填写被安打和四坏球时还会算出 WHIP。本地处理。ERA = earned runs × 9 ÷ innings pitched. Innings can be entered in baseball notation (6.1 = 6⅓, 6.2 = 6⅔) or as decimal innings; fill in hits and walks to also get WHIP. Processed locally.
临界值计算器（z、t、卡方、F）Critical Value Calculator (z, t, chi-square, F)
求标准正态 z、t（自由度 df）、卡方（df）和 F（df₁, df₂）分布在任意显著性水平 α 下的左尾、右尾或双侧临界值；用不完全 Beta / Gamma 函数精确求逆，结果与统计软件一致到小数点后 6 位。本地处理。Critical values of the standard normal z, t (df), chi-square (df) and F (df1, df2) distributions for any significance level α, left-tailed, right-tailed or two-tailed; computed by exactly inverting the incomplete beta / gamma functions, matching statistical software to 6 decimals. Processed locally.
协方差计算器（样本 / 总体，含相关系数）Covariance Calculator (sample or population, with correlation)
计算两组数据的协方差（样本除以 n−1 或总体除以 n），同时给出均值、标准差和 Pearson 相关系数 r。本地处理。Covariance of two data sets (sample, divided by n−1, or population, divided by n), with the means, standard deviations and the Pearson correlation r. Processed locally.
重心计算器（三角形 / 多边形）Centroid Calculator (triangle or polygon)
求三角形（三个顶点）或任意简单多边形（按顺序列出顶点，用鞋带公式）的面积重心和面积，三点共线 / 面积为 0 时给出提示。本地处理。Area centroid and area of a triangle (three vertices) or any simple polygon (vertices in order, shoelace formula), with a warning when the points are collinear / the area is 0. Processed locally.
相对频率与频率分布表Relative Frequency Calculator & Frequency Table
两种模式：单个事件的相对频率 = 次数 ÷ 总数；或粘贴原始数据（空格、逗号或换行分隔）或“值:次数”表，生成频率分布表（频数、相对频率、百分比、累计频率）。本地处理。Two modes: relative frequency of one event = count ÷ total, or paste raw data (separated by spaces, commas or line breaks) or a value:count table to get a frequency table with counts, relative frequency, percentage and cumulative frequency. Processed locally.
判别式与一元二次方程求根Discriminant & Quadratic Roots Calculator
计算 ax² + bx + c 的判别式 Δ = b² − 4ac，并给出根（两个实根、重根或共轭复根 a ± bi）和顶点。系数按十进制精确计算，所以 0.1、0.2 这类小数不会因为浮点误差判错 Δ 的正负。本地处理。Discriminant Δ = b² − 4ac of ax² + bx + c, with the roots (two real roots, a double root or complex roots p ± qi) and the vertex. Coefficients are handled as exact decimals, so values such as 0.1 and 0.2 never get the sign of Δ wrong through floating-point error. Processed locally.
线段端点与中点计算器（二维、三维）Endpoint & Midpoint Calculator (2D, 3D)
已知中点和一个端点，求另一个端点：B = 2M − A；或反过来由两个端点求中点 M = (A + B) ÷ 2。支持二维和三维坐标。本地处理。Find the other endpoint from the midpoint and one endpoint, B = 2M − A, or the midpoint of two endpoints, M = (A + B) ÷ 2. Works in 2D and 3D. Processed locally.
正弦计算器Sine Calculator
sin(θ)，可选度/弧度。本地处理。可设置「单位」、「θ」。sin(θ) in degrees or radians. Processed locally. Settings: Unit, θ.
余弦计算器（含 sin、tan、arccos 与余弦定理）Cosine Calculator (with sin, tan, arccos & Law of Cosines)
计算 cos θ（也可选 sin、tan），或由余弦值求角度（arccos），或用余弦定理求三角形的边或角。角度可用度或弧度，弧度可直接写 pi/2 这样的表达式；90°、270°、π/2 等特殊角给出精确的 0。本地处理。Compute cos θ (or sin and tan), the angle for a cosine value (arccos), or a side or angle of a triangle with the law of cosines. Angles in degrees or radians, and radians can be written as expressions such as pi/2; special angles such as 90°, 270° and π/2 give an exact 0. Processed locally.
平行与垂直线Parallel and Perpendicular Lines
由斜率得平行/垂直斜率。本地处理。可设置「已知斜率」。Parallel/perpendicular slopes from a given slope. Processed locally. Settings: Slope.
质心计算器（一维、二维、三维）Center of Mass Calculator (1D, 2D, 3D)
按 x̄ = Σmᵢxᵢ ÷ Σmᵢ 计算一组质点的质心，支持一维、二维和三维坐标，位置可以为负。每行一个质点：质量后跟坐标。本地处理。Center of mass of point masses, x̄ = Σmᵢxᵢ ÷ Σmᵢ, in 1D, 2D or 3D; positions may be negative. One point per line: the mass followed by its coordinates. Processed locally.
十分位数计算器（D1–D9，含位置与插值）Decile Calculator (D1–D9 with Positions)
粘贴一组数据，一次算出全部九个十分位数 D1–D9，可选两种常用定义：(n+1)p（Excel PERCENTILE.EXC、Minitab）或 (n−1)p+1（Excel PERCENTILE.INC、NumPy 默认），并列出每个十分位的位置和插值。本地处理。Paste a data set to get all nine deciles D1–D9 at once, using either of two common definitions: (n+1)p (Excel PERCENTILE.EXC, Minitab) or (n−1)p+1 (Excel PERCENTILE.INC, NumPy default), with the position and interpolation for each decile. Processed locally.
组距计算器（含斯特奇斯组数与频数表）Class Width Calculator (with Sturges Classes & Frequency Table)
由最小值、最大值和组数算组距 =（最大 − 最小）÷ 组数并向上取整；也可粘贴原始数据，自动取最值，未填组数时按斯特奇斯公式 k = ⌈1 + log₂n⌉ 定组数，并列出各组区间和频数。本地处理。Class width = (max − min) ÷ number of classes, rounded up. Or paste the raw data: the min and max are taken from it, the number of classes defaults to the Sturges rule k = ⌈1 + log₂n⌉, and every class interval is listed with its frequency. Processed locally.
样本比例 p̂ 计算器（标准误与置信区间）P-Hat Calculator (Sample Proportion, Standard Error & Confidence Interval)
由成功次数 x 和样本量 n 算样本比例 p̂ = x/n、q̂ = 1 − p̂、标准误 √(p̂q̂/n)，并给出 90%、95% 或 99% 的 Wald 与 Wilson 置信区间，检查正态近似条件 np̂ ≥ 10、nq̂ ≥ 10。本地处理。From the successes x and the sample size n, get the sample proportion p̂ = x/n, q̂ = 1 − p̂, the standard error √(p̂q̂/n) and a 90%, 95% or 99% Wald and Wilson confidence interval, with a check of the normal-approximation condition np̂ ≥ 10 and nq̂ ≥ 10. Processed locally.
百分等级计算器（由数据或人数）Percentile Rank Calculator (from Data or Counts)
算某个分数的百分等级 PR =（低于该分的人数 + 0.5 × 等于该分的人数）÷ 总人数 × 100；可以直接粘贴全部分数，也可以只填人数，同时给出“不高于该分”的百分比。本地处理。Percentile rank of a score: PR = (number below + 0.5 × number equal) ÷ total × 100. Paste all the scores or enter just the counts; the percentage at or below the score is shown too. Processed locally.
准确率计算器（混淆矩阵与测量误差）Accuracy Calculator (Confusion Matrix & Measurement Error)
两种准确率：分类模型或诊断检测，由 TP、TN、FP、FN 算准确率、精确率、召回率（灵敏度）、特异度、F1 和平衡准确率；测量值，由测量值和真实值算百分误差和准确度。本地处理。Two kinds of accuracy: for a classifier or diagnostic test, accuracy, precision, recall (sensitivity), specificity, F1 and balanced accuracy from TP, TN, FP and FN; for a measurement, the percent error and accuracy from the measured and true values. Processed locally.
指数函数计算器（求值、反解 x、两点拟合）Exponential Function Calculator (Evaluate, Solve for x, Fit Two Points)
处理 y = a·b^(kx)：给定 x 求 y，给定 y 反解 x，或由两个点求出 a 和 b；同时给出增长/衰减率、翻倍时间或半衰期，以及等价的 a·e^(rx) 形式。本地处理。Work with y = a·b^(kx): find y from x, solve for x from y, or find a and b from two points, with the growth or decay rate, the doubling time or half-life, and the equivalent a·e^(rx) form. Processed locally.
二项分布计算器（单点、累积概率与分布表）Binomial Distribution Calculator (Exact, Cumulative & Table)
由试验次数 n、成功概率 p 和成功次数 k 算 P(X = k)、P(X ≤ k)、P(X < k)、P(X ≥ k)、P(X > k)，以及均值 np、方差和标准差；n 不大时列出整张分布表。n 最多 10,000。本地处理。From the number of trials n, the success probability p and the number of successes k, get P(X = k), P(X ≤ k), P(X < k), P(X ≥ k) and P(X > k), plus the mean np, variance and standard deviation, and the full distribution table for small n. n up to 10,000. Processed locally.
平均百分比计算器（简单平均与加权平均）Average Percentage Calculator (Simple & Weighted)
算一组百分比的简单平均；如果填写每个百分比对应的样本量（人数、件数等），还会算加权平均，也就是合并后的总百分比，并说明两者为什么不同。本地处理。Average a list of percentages. If you add the sample size behind each percentage (people, items and so on) you also get the weighted average, which is the combined overall percentage, and why the two differ. Processed locally.
模逆元计算器（扩展欧几里得，支持大整数）Modular Inverse Calculator (Extended Euclid, Big Integers)
求 a 在模 m 下的乘法逆元 x，使 a·x ≡ 1 (mod m)，用扩展欧几里得算法，支持负数和任意大的整数，列出每一步并验证结果；不互质时给出最大公约数。本地处理。Find the multiplicative inverse x of a modulo m, so that a·x ≡ 1 (mod m), with the extended Euclidean algorithm. Negative and arbitrarily large integers work; every step is listed and the answer is checked, and if a and m are not coprime the gcd is shown. Processed locally.
单位圆计算器（六个三角函数、精确值、象限）Unit Circle Calculator (Six Trig Functions, Exact Values, Quadrant)
输入角度（度、弧度或 π 的倍数，如 3π/4）得到单位圆上的点 (cos θ, sin θ)，以及 tan、sec、csc、cot、所在象限、参考角和 0°–360° 内的同终边角；特殊角给出精确值（如 √2/2）。本地处理。Enter an angle (degrees, radians or a multiple of π such as 3π/4) to get the point (cos θ, sin θ) on the unit circle, plus tan, sec, csc and cot, the quadrant, the reference angle and the coterminal angle in 0°–360°. Special angles get exact values such as √2/2. Processed locally.
圆的方程计算器（标准式、一般式互化）Equation of a Circle Calculator (Standard & General Form)
由圆心和半径、圆心和圆上一点、直径的两个端点或三个点写出圆的标准方程 (x − h)² + (y − k)² = r² 和一般式 x² + y² + Dx + Ey + F = 0；也可以把一般式化回圆心和半径。同时给出面积和周长。本地处理。Write the standard equation (x − h)² + (y − k)² = r² and the general form x² + y² + Dx + Ey + F = 0 of a circle from its centre and radius, its centre and a point, the two ends of a diameter or three points, or turn a general form back into the centre and radius. The area and circumference are shown too. Processed locally.
绝对变化计算器（含相对变化与倍数）Absolute Change Calculator (with Relative Change & Ratio)
算从旧值到新值的绝对变化 = 新值 − 旧值，以及变化的大小、相对（百分比）变化和新旧之比；两个值本身是百分比时，提示用“百分点”表示绝对变化。本地处理。Absolute change from the old to the new value = new − old, with its size, the relative (percentage) change and the new-to-old ratio. When both values are percentages, it reminds you that the absolute change is in percentage points. Processed locally.
两向量夹角计算器（二维/三维）Angle Between Two Vectors Calculator (2D/3D)
输入两个二维或三维向量（如 1, 2, 3），用 cos θ = a·b /(|a||b|) 求夹角，给出角度、弧度、点积、两个向量的模，并判断是否垂直或平行。本地处理。Enter two 2D or 3D vectors (such as 1, 2, 3) to get the angle from cos θ = a·b / (|a||b|), in degrees and radians, with the dot product, both magnitudes and whether the vectors are perpendicular or parallel. Processed locally.
决定系数 R² 计算器（数据拟合或 SS）Coefficient of Determination (R²) Calculator (from Data or SS)
两种方式：输入成对的 x、y 数据，求最小二乘直线 y = a + bx、相关系数 r、R² 和调整 R²；或直接由残差平方和 SS_res 与总平方和 SS_tot 求 R²。本地处理。Two ways: enter paired x and y data to get the least-squares line y = a + bx, the correlation r, R² and adjusted R², or get R² straight from the residual and total sums of squares. Processed locally.
余函数计算器（sin/cos、tan/cot、sec/csc）Cofunction Calculator (sin/cos, tan/cot, sec/csc)
输入角度（度或弧度，弧度可写 π），把六种三角函数写成余函数形式，例如 sin θ = cos(90° − θ)，并给出两边的数值以便核对。本地处理。Enter an angle in degrees or radians (π allowed) to rewrite any of the six trig functions as its cofunction, such as sin θ = cos(90° − θ), with the value of both sides so you can check them. Processed locally.
比例常数计算器（正比例/反比例，表格检验）Constant of Proportionality Calculator (Direct or Inverse, Table Check)
由一对 x、y 求比例常数 k，或输入一组 x、y 数据，检验它们是否成正比例（y = kx）或反比例（y = k/x），并给出每一对的比值。也可以用 k 求另一个未知值。本地处理。Find the constant k from one x, y pair, or enter a table of x and y values to check whether they are directly (y = kx) or inversely (y = k/x) proportional, with the ratio for each pair. You can also use k to find a missing value. Processed locally.
自由度计算器（t 检验、卡方、方差分析、回归）Degrees of Freedom Calculator (t-tests, Chi-square, ANOVA, Regression)
按检验类型计算自由度：单样本或配对 t 检验 (n − 1)、两独立样本 t 检验（合并方差 n₁ + n₂ − 2 或 Welch 近似）、卡方拟合优度 (k − 1)、列联表 (r − 1)(c − 1)、单因素方差分析（组间 k − 1、组内 N − k）和线性回归 (n − p − 1)。本地处理。Get the degrees of freedom for your test: one-sample or paired t (n − 1), two independent samples (pooled n₁ + n₂ − 2 or the Welch approximation), chi-square goodness of fit (k − 1), contingency table (r − 1)(c − 1), one-way ANOVA (between k − 1, within N − k) and linear regression (n − p − 1). Processed locally.
差商计算器（任意多项式，化简并求极限）Difference Quotient Calculator (Any Polynomial, Simplified with the Limit)
输入多项式 f(x) 的系数（从最高次到常数项，如 1, 0, -3, 2 表示 x³ − 3x + 2），把差商 [f(x + h) − f(x)] / h 展开化简成关于 x 和 h 的式子，给出 h → 0 时的极限（导数 f′(x)），并可代入具体的 x 和 h 求值。本地处理。Enter the coefficients of a polynomial f(x) from the highest power to the constant (for example 1, 0, -3, 2 for x³ − 3x + 2) to expand and simplify the difference quotient [f(x + h) − f(x)] / h in x and h, get its limit as h → 0 (the derivative f′(x)), and evaluate it at a chosen x and h. Processed locally.
散度计算器（输入向量场，二维/三维）Divergence Calculator (Enter a Vector Field, 2D or 3D)
输入向量场 F = (P, Q) 或 (P, Q, R) 的分量表达式（如 x^2*y、sin(z)），在指定点用数值微分求 ∂P/∂x、∂Q/∂y、∂R/∂z 和散度 ∇·F，并说明该点是“源”还是“汇”。也保留由网格上相邻函数值做中心差分估算的方法。本地处理。Enter the components of a vector field F = (P, Q) or (P, Q, R) as expressions (such as x^2*y or sin(z)) to get ∂P/∂x, ∂Q/∂y, ∂R/∂z and the divergence ∇·F at a point by numerical differentiation, and see whether the point is a source or a sink. The central-difference estimate from neighbouring grid values is still available. Processed locally.
定义域和值域计算器（常见函数类型）Domain and Range Calculator (Common Function Types)
选择函数类型并输入参数，求定义域、值域、截距和关键点：一次函数、二次函数（顶点）、有理函数 (ax + b)/(cx + d)（渐近线）、平方根 a√(bx + c) + d、对数 a·ln(bx + c) + d、指数 a·bˣ + c 和绝对值 a|x − h| + k，结果用区间表示。本地处理。Pick a function type and enter its parameters to get the domain, range, intercepts and key points: linear, quadratic (vertex), rational (ax + b)/(cx + d) (asymptotes), square root a√(bx + c) + d, logarithm a·ln(bx + c) + d, exponential a·bˣ + c and absolute value a|x − h| + k, in interval notation. Processed locally.
二倍角公式计算器（角度或已知函数值）Double-Angle Formula Calculator (from an Angle or a Known Value)
由角度 θ（度或弧度），或由已知的 sin θ、cos θ 值和 θ 所在象限，计算 sin 2θ、cos 2θ、tan 2θ，并列出所用的二倍角公式及代入过程。本地处理。From an angle θ in degrees or radians, or from a known sin θ or cos θ and the quadrant of θ, get sin 2θ, cos 2θ and tan 2θ with the double-angle formulas used and the substituted values. Processed locally.
二重积分计算器（矩形或 y 介于两曲线之间的区域）Double Integral Calculator (Rectangle or Region Between Two Curves)
输入被积函数 f(x, y)（如 x*y、x^2 + y^2、2），在矩形区域 [x₀, x₁] × [y₀, y₁] 上，或在 x 从 a 到 b、y 从 g₁(x) 到 g₂(x) 的区域上，用二维辛普森法数值计算二重积分，同时给出区域面积和平均值。本地处理。Enter an integrand f(x, y) (such as x*y, x^2 + y^2 or 2) and integrate it numerically with 2D Simpson rule over a rectangle [x₀, x₁] × [y₀, y₁] or over the region where x runs from a to b and y from g₁(x) to g₂(x), with the region area and the average value. Processed locally.
误差传递计算器（任意公式、和/差/积/商/幂）Error Propagation Calculator (Any Formula, Sum, Difference, Product, Quotient, Power)
输入公式（如 a*b/c、a^2*sin(b)）和各变量的测量值与不确定度，按一阶泰勒展开（偏导数平方和开方）计算结果的不确定度、相对不确定度和各变量的贡献；也可以用和、差、积、商、幂的快速公式。假设各变量误差独立。本地处理。Enter a formula (such as a*b/c or a^2*sin(b)) with each variable value and uncertainty to get the uncertainty of the result by first-order Taylor expansion (root sum of squared partial terms), the relative uncertainty and each variable contribution, or use the quick rules for sums, differences, products, quotients and powers. Errors are assumed independent. Processed locally.
FOIL 展开计算器（两个二项式相乘）FOIL Calculator (Multiply Two Binomials)
把 (ax + b)(cx + d) 按 FOIL 法展开：首项 (First)、外项 (Outer)、内项 (Inner)、末项 (Last) 分别相乘，再合并同类项得到 acx² + (ad + bc)x + bd。可以改变量名，也可以代入 x 的值求结果。本地处理。Expand (ax + b)(cx + d) by FOIL: multiply the First, Outer, Inner and Last terms, then combine like terms to get acx² + (ad + bc)x + bd. You can rename the variable and substitute a value of x. Processed locally.
乘法原理计算器（基本计数原理）Fundamental Counting Principle Calculator
输入每一步的选择数（如 3 5 2），按乘法原理求总方案数并列出乘法过程；也可以按“每步选择数相同、共 k 步”计算 nᵏ（例如密码或车牌组合）。使用大整数，结果很大时也能给出精确值和位数。本地处理。Enter the number of choices at each stage (such as 3 5 2) to get the total number of outcomes by the multiplication principle with the working shown, or use the same n choices over k stages for nᵏ (such as PINs or licence plates). Big integers give the exact value and number of digits even for huge results. Processed locally.
半角公式计算器（角度或已知 cos θ）Half-Angle Formula Calculator (from an Angle or a Known cos θ)
由角度 θ（度或弧度），或由已知 cos θ 的值和 θ/2 所在象限，用半角公式求 sin(θ/2)、cos(θ/2)、tan(θ/2)，正负号按象限确定，并列出公式和代入过程。本地处理。From an angle θ in degrees or radians, or from a known cos θ and the quadrant of θ/2, use the half-angle formulas to get sin(θ/2), cos(θ/2) and tan(θ/2), with signs set by the quadrant and the formulas and values shown. Processed locally.