➗ 数学工具Math （第 5 页）(page 5)

终边相同的角计算器Coterminal Angle Calculator

求与给定角终边相同的角（θ ± 360°·k），给出 0°–360° 与 −180°–180° 内的标准角、所在象限和参考角，支持角度、弧度和 π 的倍数，也可判断两个角是否终边相同。Find angles coterminal with a given angle (θ ± 360°·k): its standard position in 0°–360° and −180°–180°, quadrant and reference angle, in degrees, radians or multiples of π — and check whether two angles are coterminal.

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抛物线计算器Parabola Calculator

由一般式 y = ax² + bx + c 或顶点式 y = a(x − h)² + k 求抛物线的顶点、焦点、准线、对称轴、开口方向、焦距、通径长度以及与坐标轴的交点。From y = ax² + bx + c or the vertex form y = a(x − h)² + k, find a parabola's vertex, focus, directrix, axis of symmetry, direction, focal length, latus rectum and axis intercepts.

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线性插值计算器Linear Interpolation Calculator

已知两点 (x₁, y₁)、(x₂, y₂)，用直线插值求任意 x 对应的 y（可一次输入多个 x），或由 y 反求 x；超出两点范围时会提示属于外推。Given two points (x₁, y₁) and (x₂, y₂), find y for any x on the straight line between them (several x values at once), or solve x from y — with a note when a value is outside the two points (extrapolation).

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区间表示法转换器Interval Notation Calculator

在不等式和区间表示法之间互相转换：输入如 -2 < x <= 5、x < 1 or x > 4 或 [3, inf)、(1, 3] U [2, 6)，得到区间表示、不等式和集合描述，并自动合并重叠区间。Convert between inequalities and interval notation: enter things like -2 < x <= 5, x < 1 or x > 4, [3, inf) or (1, 3] U [2, 6) to get the interval notation, inequality and set-builder form, with overlapping intervals merged.

笛卡尔符号法则计算器Descartes' Rule of Signs Calculator

输入多项式（如 x^3 - 3x^2 + 4）或系数列表，用笛卡尔符号法则统计 P(x) 和 P(−x) 的变号次数，列出正实根、负实根和虚根个数的所有可能组合。Enter a polynomial (e.g. x^3 - 3x^2 + 4) or its coefficients to count sign changes in P(x) and P(−x) with Descartes' rule of signs and list every possible mix of positive, negative and complex roots.

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分母有理化计算器Rationalize the Denominator Calculator

把 a ÷ (b√c) 或 a ÷ (b + d√c) 形式的分数化为分母不含根号的最简形式：单项式分母乘以 √c，二项式分母乘以共轭式，并显示每一步。Rationalize fractions of the form a ÷ (b√c) or a ÷ (b + d√c): multiply by √c for a single-term denominator or by the conjugate for a two-term one, simplified, with every step shown.

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最小公分母（LCD）计算器Least Common Denominator (LCD) Calculator

输入几个分数（可含整数、小数和带分数），求它们的最小公分母，并把每个分数改写成以 LCD 为分母的等值分数，同时给出它们的和。Enter several fractions (whole numbers, decimals and mixed numbers allowed) to find their least common denominator, rewrite each one over the LCD, and see their sum.

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小数转百分数计算器Decimal to Percent Calculator

把小数精确地转换为百分数（乘以 100，移动小数点两位，不产生浮点误差），也可反向把百分数转回小数；可一次转换多个数，并给出最简分数形式。Convert decimals to percentages exactly — multiply by 100 by moving the decimal point two places, with no floating-point error — or percentages back to decimals, several at once, with the simplest fraction too.

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乘法逆元计算器（倒数与模逆）Multiplicative Inverse Calculator

求一个数的乘法逆元：普通数的倒数（支持分数、带分数和小数，结果给出最简分数），或整数在模 m 下的模逆元 a⁻¹ mod m（扩展欧几里得算法，显示步骤，支持大整数）。Find a multiplicative inverse: the reciprocal of a number (fractions, mixed numbers and decimals, as a simplified fraction), or the modular inverse a⁻¹ mod m by the extended Euclidean algorithm, with steps and big integers.

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竖式加法计算器（列竖式）Long Addition Calculator

用竖式（列竖式）把两个或多个非负整数或小数相加，按小数点对齐，逐列显示进位和每一步，支持任意长度的大数，适合检查作业。Add two or more non-negative whole numbers or decimals by long addition, lined up on the decimal point, with the carry and working shown for each column and no limit on length — handy for checking homework.

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零空间（核）计算器Null Space Calculator

用分数精确计算矩阵的零空间（核）：化为行最简形（RREF），找出主元列和自由变量，给出零空间的一组基向量（可化为整数），以及秩和零度。Find the null space (kernel) of a matrix exactly with fractions: reduce it to row echelon form (RREF), identify pivot columns and free variables, and get a basis for the null space (optionally scaled to integers) with the rank and nullity.

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特征多项式计算器Characteristic Polynomial Calculator

用分数精确计算方阵的特征多项式 p(λ) = det(λI − A)（Faddeev–LeVerrier 算法），同时给出迹、行列式，以及所有有理数特征值。Compute the characteristic polynomial p(λ) = det(λI − A) of a square matrix exactly with fractions (Faddeev–LeVerrier algorithm), with the trace, determinant and any rational eigenvalues.

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格拉姆-施密特正交化计算器Gram–Schmidt Calculator

输入一组向量，用格拉姆-施密特过程精确求出正交基（分数）和标准正交基（小数，并给出每个向量的长度），线性相关的向量会被识别并跳过。Enter a set of vectors to get an orthogonal basis by the Gram–Schmidt process (exact fractions) and an orthonormal basis (decimals, with each vector's length); linearly dependent vectors are detected and skipped.

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四则运算顺序（逐步计算）Order of Operations Calculator

输入算式，按运算顺序（括号 → 乘方 → 乘除从左到右 → 加减从左到右，即 PEMDAS/BODMAS）一步一步算出结果，每步只做一次运算；用分数精确计算，同时给出小数。Enter an expression to evaluate it step by step in the order of operations — parentheses, exponents, multiplication and division left to right, then addition and subtraction left to right (PEMDAS/BODMAS) — one operation per step, exactly with fractions and with a decimal too.

圆周率 π 位数计算器（最多 10000 位）Pi Digits Calculator (up to 10,000 digits)

在浏览器里用梅钦公式（π = 16·arctan(1/5) − 4·arctan(1/239)）和大整数精确算出 π 小数点后 1–10000 位，可分组显示、复制，统计各数字出现次数，并查找生日等数字串第一次出现的位置。Compute π to 1–10,000 decimal places right in your browser with Machin's formula (π = 16·arctan(1/5) − 4·arctan(1/239)) and exact big-integer arithmetic. Group and copy the digits, count how often each digit appears, and find where a digit string such as a birthday first shows up.

黄金比例 φ 位数生成器（最多 10000 位）Golden Ratio Digits Generator (up to 10,000 digits)

用 φ = (1 + √5) / 2 和大整数平方根精确生成黄金比例小数点后 1–10000 位，也可以输出 1/φ（= φ − 1）或 φ²（= φ + 1），支持分组、复制和查找数字串。Generate the golden ratio φ = (1 + √5) / 2 to 1–10,000 decimal places with an exact big-integer square root — or its reciprocal 1/φ (= φ − 1) or square φ² (= φ + 1) — with grouping, copying and digit search.

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完全数生成器 / 判断器Perfect Number Generator & Checker

由已知的梅森素数 2ᵖ − 1 按欧几里得–欧拉公式 2ᵖ⁻¹(2ᵖ − 1) 精确生成前 1–25 个偶完全数（最大 13066 位），或判断一个数是完全数、盈数还是亏数并列出它的真因数。Generate the first 1–25 even perfect numbers exactly from the known Mersenne primes 2ᵖ − 1 with the Euclid–Euler formula 2ᵖ⁻¹(2ᵖ − 1) — the largest has 13,066 digits — or check whether a number is perfect, abundant or deficient and list its proper divisors.

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杨辉三角（帕斯卡三角形）生成器Pascal's Triangle Generator

用大整数精确生成杨辉三角前 1–100 行（居中三角、左对齐或 CSV），或单独算出第 0–1000 行的全部二项式系数；还能显示奇偶图案，看到其中的谢尔宾斯基三角形。Generate the first 1–100 rows of Pascal's triangle exactly with big integers — centred, left-aligned or as CSV — or get every binomial coefficient in a single row from 0 to 1000. An odd/even view reveals the Sierpiński triangle inside it.

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中心极限定理计算器（样本均值的概率）Central Limit Theorem Calculator

输入总体均值 μ、标准差 σ 和样本量 n，按中心极限定理求样本均值 X̄ 的分布（均值 μ、标准误 σ/√n），并计算 X̄ 小于、大于或落在某个区间的概率。Enter the population mean μ, standard deviation σ and sample size n to get the sampling distribution of the mean X̄ by the central limit theorem (mean μ, standard error σ/√n), and the probability that X̄ is below, above or between given values.

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箱线图生成器（五数概括 + 离群值）Box Plot Maker (Five-Number Summary)

输入一组或多组数据，算出五数概括（最小值、Q1、中位数、Q3、最大值）、四分位距 IQR、1.5×IQR 围栏和离群值，并画出可下载的 SVG 箱线图；多组数据并排对比，四分位数可选 4 种常见算法。Enter one or more data sets to get the five-number summary (minimum, Q1, median, Q3, maximum), the IQR, the 1.5 × IQR fences and any outliers, drawn as a downloadable SVG box plot. Several groups are compared side by side, and the quartiles can use any of four common methods.

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坐标旋转计算器（绕原点或任意点）Rotation Calculator (Rotate Points)

把一个或多个点 (x, y) 绕原点或任意中心点旋转指定角度（逆时针或顺时针，角度或弧度），给出旋转后的坐标和旋转矩阵，适合几何作业、图形和 CAD 计算。Rotate one or more points (x, y) about the origin or any centre by a given angle — counter-clockwise or clockwise, in degrees or radians — and get the new coordinates and the rotation matrix. Handy for geometry homework, graphics and CAD.

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数轴生成器（点、区间、不等式）Number Line Generator

按起点、终点和刻度间隔画出数轴，可以标出实心点、空心点、区间（如 [−2, 3)）和不等式（如 x ≥ 1、−1 < x ≤ 4），同时给出区间记号，并下载为 SVG 图片。Draw a number line from a start, an end and a tick step, and mark closed or open points, intervals such as [−2, 3) and inequalities such as x ≥ 1 or −1 < x ≤ 4. Each mark is also given in interval notation, and the line downloads as an SVG image.

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极限计算器Limit Calculator

求函数的极限 lim f(x)：支持趋近于某个数、分数、pi 或 ±∞，可选双侧、左极限、右极限；符号计算给出精确值并用数值核对，能识别左右极限不相等、趋于无穷、振荡和无定义等“极限不存在”的情况。Find the limit of a function, lim f(x), as x approaches a number, a fraction, pi or ±∞ — two-sided, from the left or from the right. Exact values come from symbolic computation and are checked numerically, and it recognises limits that do not exist: unequal one-sided limits, oscillation or no real value nearby. Infinite limits are reported as ∞ or −∞.

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部分分式分解计算器Partial Fraction Decomposition Calculator

把有理函数（多项式之比）分解成部分分式之和：支持一次因式、重复因式和不可约二次因式，分子次数不低于分母时先分离出多项式部分；结果给出普通写法和 LaTeX，并在多个点代入核对。Split a rational function (a ratio of polynomials) into a sum of partial fractions — linear, repeated and irreducible quadratic factors are all handled, and when the numerator's degree is not lower than the denominator's the polynomial part is separated out first. Results come in plain notation and LaTeX, checked numerically at several points.

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三维空间两点距离计算器3D Distance Calculator

输入三维空间中两点的坐标 (x₁, y₁, z₁) 和 (x₂, y₂, z₂)，按 d = √(Δx² + Δy² + Δz²) 算出两点间的距离，同时给出各轴差值、距离平方和中点坐标。Enter two points (x₁, y₁, z₁) and (x₂, y₂, z₂) in 3D space to get the distance d = √(Δx² + Δy² + Δz²), along with the difference on each axis, the squared distance and the midpoint.

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菱形面积计算器Area of a Rhombus Calculator

按两条对角线、边长和高、或边长和内角三种方式计算菱形面积，同时给出边长、周长、两条对角线和内角。Find the area of a rhombus from its two diagonals, from side and height, or from side and an interior angle — with the side, perimeter, both diagonals and the interior angles.

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圆环体（环面）体积计算器Torus Volume Calculator

输入圆环的主半径 R（中心到管中心）和管半径 r，按 V = 2π²Rr² 计算体积、按 S = 4π²Rr 计算表面积；也可以改用内半径和外半径输入。Enter the major radius R (centre to tube centre) and tube radius r of a torus to get its volume V = 2π²Rr² and surface area S = 4π²Rr — or enter the inner and outer radii instead.

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椭球体积计算器Volume of an Ellipsoid Calculator

输入椭球的三个半轴 a、b、c，按 V = 4/3·πabc 计算体积，并用 Knud Thomsen 公式估算表面积（误差不超过约 1.06%）；三个半轴相等时就是球。Enter the three semi-axes a, b and c of an ellipsoid to get its volume V = 4/3·πabc and an estimate of its surface area from Knud Thomsen's formula (within about 1.06%). Equal semi-axes give a sphere.

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棱锥体积计算器Volume of a Pyramid Calculator

计算矩形底（含正方形）或正多边形底直棱锥的体积 V = 底面积 × 高 ÷ 3，并给出底面积、斜高、侧面积和总表面积。Work out the volume V = base area × height ÷ 3 of a right pyramid with a rectangular (or square) base or a regular-polygon base, plus the base area, slant height, lateral area and total surface area.

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反对数计算器Antilog Calculator

求以 10、e、2 或任意正数为底的反对数 y = bˣ（即已知对数 x 求原数），给出十进制结果和科学计数法；超过双精度范围的结果也能用科学计数法显示。Find the antilogarithm y = bˣ for base 10, e, 2 or any positive base — the number whose logarithm is x — as a decimal and in scientific notation, even when the result is beyond double-precision range.

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展开式（数位分解）计算器Expanded Form Calculator

把整数或小数按数位展开成加法式、乘法式（数字 × 位值）和 10 的幂形式，小数部分同时给出分数形式；支持负数和最多 40 位数字，按字符串精确处理不丢位。Write a whole number or decimal in expanded form — as a sum of place values, as digit × place value, and with powers of ten, plus fraction form for the decimal places. Negative numbers and up to 40 digits are handled exactly as text.

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埃及分数计算器Egyptian Fraction Calculator

用斐波那契–西尔维斯特贪心算法把分数 a/b 拆成互不相同的单位分数之和（如 4/13 = 1/4 + 1/18 + 1/468），大于 1 的分数先取出整数部分；用大整数精确计算并验算。Split a fraction a/b into a sum of distinct unit fractions with the Fibonacci–Sylvester greedy algorithm (e.g. 4/13 = 1/4 + 1/18 + 1/468), taking out the whole part first for fractions above 1. Big-integer arithmetic keeps it exact, and the sum is checked.

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考拉兹猜想（3n+1）计算器Collatz Conjecture Calculator

输入正整数 n，按「偶数除以 2、奇数乘 3 加 1」反复迭代，给出到达 1 的步数、过程中的最大值、奇偶步数和完整序列；用大整数计算，支持最多 100 位的起始数。Enter a positive whole number n and apply 'halve if even, 3n + 1 if odd' until it reaches 1 — with the number of steps, the highest value reached, the odd and even step counts and the full sequence. Big integers allow starting numbers up to 100 digits.

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伴随矩阵计算器Adjoint (Adjugate) Matrix Calculator

输入 2×2 到 6×6 的方阵，精确计算伴随矩阵 adj(A)（余子式矩阵的转置）、余子式矩阵和行列式，并在可逆时给出分数形式的逆矩阵 A⁻¹ = adj(A) / det(A)。整数和最多 6 位小数都按精确有理数计算。Enter a square matrix from 2×2 to 6×6 to get its adjugate adj(A) — the transpose of the cofactor matrix — exactly, together with the cofactor matrix, the determinant and, when it exists, the inverse A⁻¹ = adj(A) / det(A) as fractions. Whole numbers and decimals with up to 6 places are handled as exact rationals.

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QR 分解计算器QR Decomposition Calculator

用 Householder 反射把 m×n 矩阵（m ≥ n，最大 8×8）分解为 A = QR：Q 的列为标准正交向量，R 为上三角矩阵（对角线取非负）。给出精简形式或完整形式，并验算 QR 与 A 的最大误差和 QᵀQ 的正交性。Factor an m×n matrix (m ≥ n, up to 8×8) as A = QR with Householder reflections: Q has orthonormal columns and R is upper triangular with a non-negative diagonal. Choose the reduced or full form, and see the largest error of QR against A and how close QᵀQ is to I.

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茎叶图生成器Stem and Leaf Plot Calculator

把一组数据整理成茎叶图：自动或手动选择叶单位，可拆分茎（每个茎分两行），支持负数和小数，并给出个数、最小值、最大值、中位数和众数。Turn a list of numbers into a stem-and-leaf plot, with an automatic or chosen leaf unit, optional split stems (two lines per stem), support for negatives and decimals, and the count, minimum, maximum, median and mode.

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直方图生成器Histogram Maker

粘贴一组数据，按 Sturges、平方根或 Freedman–Diaconis 规则自动分组（组距取整到 1、2、2.5、5 × 10ⁿ），或自定义组数、组距和起点，生成直方图和频数、频率、累计频率表，可下载 SVG 或复制 CSV。Paste your data to build a histogram, with bins from Sturges', the square-root or the Freedman–Diaconis rule (widths rounded to 1, 2, 2.5 or 5 × 10ⁿ) or your own bin count, width and start — plus a table of counts, relative and cumulative frequencies, an SVG download and CSV to copy.

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韦恩图计算器Venn Diagram Calculator

按集合大小和交集数量（两个或三个集合），用容斥原理算出韦恩图每个区域的数量、并集和「都不属于」的个数；也可以直接列出各集合的元素，自动求交集、并集和差集。Work out every region of a two- or three-set Venn diagram from the set sizes and overlaps with inclusion–exclusion, including the union and the number in none of the sets — or list each set's items and get the intersections, union and differences automatically.

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二进制加法计算器（多个二进制数求和）Add Binary Numbers

把两个或多个二进制数相加：每行一个，或用空格、逗号、加号分隔，支持很长的数（任意位数）、负数和 0b 前缀；可显示带进位的竖式，并同时给出十进制和十六进制结果，还能设置最少位数补 0、每 4 位分组。适合学习二进制加法、检查作业和数字电路练习，数据在本地处理。Add two or more binary numbers. Put one per line or separate them with spaces, commas or plus signs. Numbers of any length, negative numbers and the 0b prefix all work. Show the column addition with carries, see the result in decimal and hex too, pad to a minimum width and group every 4 bits. Handy for learning binary addition, checking homework and digital-logic exercises. Processed locally.

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二进制乘法计算器（多个二进制数相乘）Multiply Binary Numbers

把两个或多个二进制数相乘：每行一个，或用空格、逗号、* 或 × 分隔，支持任意位数、负数和 0b 前缀；两个数相乘时可显示移位相加的竖式（部分积），同时给出十进制和十六进制结果，还能设置最少位数补 0、每 4 位分组。适合学习二进制乘法、计算机组成原理和数字电路练习，数据在本地处理。Multiply two or more binary numbers. Put one per line or separate them with spaces, commas, * or ×. Any length, negative numbers and the 0b prefix work. For two numbers you can show the shift-and-add working with each partial product, and see the result in decimal and hex too, padded to a minimum width or grouped every 4 bits. Handy for learning binary multiplication, computer architecture and digital-logic exercises. Processed locally.

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二进制转负二进制（base −2）Convert Binary to Negabinary

把普通二进制数（可以带负号）转换成负二进制（以 −2 为底的进位制）：每行一个或用空格分隔，支持任意位数；可同时显示十进制值，或显示“输入 → 结果”对照。负二进制不需要负号就能表示负数，常见于编程题和数制练习，数据在本地处理。Convert ordinary binary numbers (a minus sign is allowed) into negabinary, the base −2 number system. One value per line or separated by spaces, any length. Optionally show the decimal value or an input → result table. Negabinary writes negative numbers without a sign and turns up in programming puzzles and number-system exercises. Processed locally.

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十进制转负二进制（负二进制编码）Encode Negative Binary

把十进制整数（正数、负数或 0）编码成负二进制（以 −2 为底的进位制），每行一个或用空格分隔，支持任意大的整数；可显示“输入 → 结果”对照、各位权重展开式（如 7 = 16 − 8 − 2 + 1），也能设置最少位数补 0。适合学习数制、做编程题和验证算法，数据在本地处理。Encode decimal integers (positive, negative or zero) as negabinary, the base −2 number system. One value per line or separated by spaces, integers of any size. Show an input → result table or the place-value expansion (e.g. 7 = 16 − 8 − 2 + 1), and pad to a minimum width. Handy for learning number systems, programming puzzles and checking algorithms. Processed locally.

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负二进制解码（base −2 转十进制）Decode Negative Binary

把负二进制数（以 −2 为底的进位制）解码成十进制整数，也可以同时给出普通二进制（带负号）：每行一个或用空格分隔，支持任意位数；可显示“输入 → 结果”对照和各位权重展开式（如 11011 = 16 − 8 − 2 + 1 = 7）。适合学习数制、做编程题和检查编码结果，数据在本地处理。Decode negabinary numbers (base −2) into decimal integers, optionally with the ordinary signed binary too. One value per line or separated by spaces, any length. Show an input → result table or the place-value expansion (e.g. 11011 = 16 − 8 − 2 + 1 = 7). Handy for learning number systems, programming puzzles and checking encoded results. Processed locally.

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二进制减法计算器（带借位竖式）Subtract Binary Numbers

用第一个二进制数依次减去后面的数：每行一个，或用空格、逗号、加号或减号分隔，支持任意位数、负数和 0b 前缀。两个非负数相减时可显示带借位的竖式；结果为负数时默认用负号表示，也可以按指定位数显示成补码。同时给出十进制和十六进制结果，可设置最少位数补 0、每 4 位分组。适合学习二进制减法、补码和数字电路练习，数据在本地处理。Subtract binary numbers: the first value minus each of the others. Put one per line or separate them with spaces, commas, plus signs or minus signs. Any length, negative numbers and the 0b prefix work. For two non-negative values you can show the column subtraction with borrows. A negative result is written with a minus sign, or as two's complement in a width you choose. See the result in decimal and hex too, pad to a minimum width or group every 4 bits. Handy for learning binary subtraction, two's complement and digital-logic exercises. Processed locally.

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十六进制加法计算器Add Hex Numbers

把两个或多个十六进制数相加：每行一个或用空格、逗号、加号分隔，支持任意位数、负数和 0x 前缀；可同时显示十进制和二进制结果，设置最少位数补 0。适合学习十六进制运算和检查作业，数据在本地处理。Add two or more hexadecimal numbers. One per line or separated by spaces, commas or plus signs; any length, negatives and 0x prefixes work. Also show decimal and binary, pad to a minimum width. Handy for learning hex arithmetic and checking homework. Processed locally.

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十六进制乘法计算器Multiply Hex Numbers

把两个或多个十六进制数相乘：每行一个或用空格、逗号、*、× 分隔，支持任意位数、负数和 0x 前缀；可同时显示十进制结果。适合学习十六进制运算，数据在本地处理。Multiply two or more hexadecimal numbers. One per line or separated by spaces, commas, * or x; any length, negatives and 0x. Optionally show decimal. Handy for hex arithmetic. Processed locally.

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