⅟ 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.
使用说明与常见问题How-to & FAQ
「乘法逆元计算器（倒数与模逆）」是什么？What is the Multiplicative Inverse Calculator tool?
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.
如何使用「乘法逆元计算器（倒数与模逆）」？How do I use the Multiplicative Inverse Calculator tool?
在「乘法逆元计算器（倒数与模逆）」中，填写「类型」、「数 x（如 3/4、2 1/3、0.2）」、「整数 a」、「模数 m」等参数，结果会随输入实时计算并显示在下方。In the Multiplicative Inverse Calculator: Fill in "Type", "Number x (e.g. 3/4, 2 1/3, 0.2)", "Integer a", "Modulus m"; results are recalculated instantly as you type.
什么是乘法逆元？What is a multiplicative inverse?
与 x 相乘等于 1 的数。对普通数就是倒数 1/x，例如 3/4 的倒数是 4/3；0 没有倒数。A number that multiplies with x to give 1. For ordinary numbers it is the reciprocal 1/x — 3/4 has inverse 4/3 — and 0 has none.
模逆元怎么求？How is a modular inverse found?
求整数 b 使 a × b ≡ 1 (mod m)。用扩展欧几里得算法求 gcd(a, m) 并同时得到系数；只有 gcd(a, m) = 1 时逆元存在，例如 3 × 4 = 12 ≡ 1 (mod 11)。Find b with a × b ≡ 1 (mod m). The extended Euclidean algorithm finds gcd(a, m) along with the coefficients; an inverse exists only when gcd(a, m) = 1, e.g. 3 × 4 = 12 ≡ 1 (mod 11).
计算精度如何？How precise are the results?
「乘法逆元计算器（倒数与模逆）」使用 JavaScript 双精度浮点数（约 15–16 位有效数字），部分功能使用 BigInt 处理大整数。The Multiplicative Inverse Calculator uses double-precision floating point (about 15–16 significant digits); some features use BigInt for large integers.
「乘法逆元计算器（倒数与模逆）」免费吗？需要注册吗？Is the Multiplicative Inverse Calculator free? Do I need an account?
「乘法逆元计算器（倒数与模逆）」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Multiplicative Inverse Calculator is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.