纳皮尔算筹（骨牌）是怎样帮人乘法的？
How did Napier's bones help with multiplication?
每根木条刻着一个数乘以1到9的结果，把对应木条排在一起，按对角线相加就能完成多位数乘法，是对数之前的手工计算神器。
Each rod shows a number times 1 through 9; lining up the rods and adding along diagonals performs multi-digit multiplication, a manual aid before logarithms.
难度 专业 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#数学工具 #纳皮尔算筹
对数计算尺为什么曾是工程师的标配？
Why was the slide rule once standard for engineers?
它把对数刻度刻在尺上，滑动就能做乘除和开方，速度快、便携。直到1970年代电子计算器普及，才渐渐退出舞台。
It engraved logarithmic scales so sliding performed multiplication, division and roots quickly and portably. It faded only when electronic calculators spread in the 1970s.
难度 专业 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#数学工具 #计算尺
对数表是怎么让大航海更安全的？
How did logarithm tables make ocean navigation safer?
航海要用球面三角算经纬度，手工乘除极繁且易错。纳皮尔发明对数后，人们编出对数表，把复杂运算简化为加减，显著减少了导航错误。
Navigation needed spherical trigonometry with tedious, error-prone multiplication. After Napier's logarithms, printed tables reduced it to addition and subtraction, sharply cutting navigational mistakes.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#数学工具 #对数航海
为什么计算机算0.1加0.2不等于0.3？
Why does a computer get 0.1 + 0.2 not equal to 0.3?
计算机用二进制浮点数（IEEE 754）近似小数，而0.1、0.2在二进制下是无限循环小数，只能存近似值，累加后产生微小误差。这不是bug，而是表示的局限。
Computers use binary floating point (IEEE 754); 0.1 and 0.2 are repeating binaries stored only approximately, so their sum drifts. It is a representation limit, not a bug.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
Goldberg,《What Every Computer Scientist Should Know About Floating-Point Arithmetic》,ACM Computing Surveys 第23卷,1991
D. Goldberg, What Every Computer Scientist Should Know About Floating-Point Arithmetic, ACM Computing Surveys 23, 1991
独立核验依据 · Independent verification: 核验依据 1 CROSSREF 标识 10.1145/103162.103163：What every computer sc
#数学工具 #浮点误差
对数是怎么把乘法变成加法的？
How does a logarithm turn multiplication into addition?
对数满足log(ab)=log a+log b，于是两个数相乘可先取对数相加、再查反对数还原。这让没有计算器的时代也能快速算大数乘法。
Logarithms satisfy log(ab)=log a+log b, so to multiply you add the logs and then exponentiate back. This let people compute big products quickly before calculators.
难度 专业 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#数学工具 #对数
算盘的进位思想和二进制有什么关系？
How is the abacus carry idea related to binary?
算盘每档满十进一，是十进制进位；二进制则是满二进一。两者都是位值制下的进位规则，只是基数不同，体现了进制的统一逻辑。
An abacus carries at ten per rod, decimal; binary carries at two. Both are place-value carry rules with different bases, showing the unified logic of numeral systems.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#数学工具 #进位
用尺规怎么把任意一个角平分？
How do you bisect any angle with compass and straightedge?
以顶点为圆心画弧交两边，再分别以两交点为圆心画等半径弧交于一点，连接顶点与该点即得角平分线。这是尺规作图的经典基本操作。
From the vertex draw an arc crossing both sides, then equal arcs from those crossings meet at a point; the line from vertex to that point bisects the angle. A classic straightedge-and-compass construction.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#数学工具 #尺规作图
计算器内部为什么用二进制而不是十进制？
Why do calculators use binary internally instead of decimal?
电路只有开和关两种稳定状态，正好对应0和1，用二进制做逻辑与运算最简单可靠。显示时再转换回我们熟悉的十进制。
Circuits have two stable states, on and off, mapping naturally to 0 and 1, making binary logic simplest and most reliable. The result is converted back to decimal for display.
难度 进阶 · 适合 大众
CC BY 4.0 · 来源待核验 · Source pending review
来源 · Source
出处备注：著录仅给书名，无版次与年份；同一书名常对应多个版次，不臆测具体版本。
Leibniz, G. W.《Explication de l'Arithmétique Binaire》(论只用0与1的二进制算术),1703;及与白晋(Joachim Bouvet)1701—1703年通信
G. W. Leibniz, 'Explication de l'Arithmétique Binaire' (1703); with the 1701–1703 correspondence with Joachim Bouvet
#数学工具 #二进制
算盘上怎么表示很大的数和小数？
How does an abacus represent very large numbers and decimals?
算盘每根杆代表一个数位，靠拨珠表示0到9，数位向左延伸就能表示任意大的整数。小数则约定某根杆为小数点位，规则相同。
Each rod is a place value with beads for 0–9, so moving left extends to arbitrarily large integers. Decimals just fix one rod as the decimal point with the same rules.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#数学工具 #算盘
世界上第一台机械计算器是谁造的？
Who built the first mechanical calculator?
1642年法国数学家帕斯卡为帮父亲算税，造出能加减的齿轮式计算器帕斯卡计算器。它开启了用机器代替手工计算的浪潮。
In 1642 the French mathematician Pascal built a geared adding-subtracting machine to help his tax-collector father. It launched the wave of mechanical calculation.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
《数学史》Boyer 与 Merzbach
A History of Mathematics by Boyer and Merzbach
独立核验依据 · Independent verification: 核验依据 1 ISBN 9787511704443（豆瓣书目条目: 数学史）
#数学工具 #计算史