怎样证明素数有无穷多个？
How can one prove there are infinitely many primes?
欧几里得用反证法证明：假设素数只有有限个，把它们全部相乘再加 1，得到的新数要么本身是新的素数，要么含有列表中未出现的新素因子。无论哪种情况都出现了一个不在原列表中的素数，矛盾。因此素数永无穷尽。
Euclid proved it by contradiction: suppose there are only finitely many primes, multiply them all and add 1; the result is either itself a new prime or has a prime factor not in the list. Either way a prime outside the list appears, a contradiction. Therefore primes never run out.
难度 专业 · 适合 初中(12-15)
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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
#素数无穷 #欧几里得 #反证法
四元数是什么，为什么在三维旋转中有用？
What are quaternions and why are they useful for 3D rotation?
四元数是形如 a+bi+cj+dk 的超复数，其中 i、j、k 的平方都为 −1 且彼此相乘有特定规则。用单位四元数表示三维旋转可以避免欧拉角的“万向锁”问题，且插值平滑、计算稳定。它广泛用于计算机图形、机器人和航天姿态控制。
Quaternions are hypercomplex numbers of the form a+bi+cj+dk, where i, j, k square to −1 with specific multiplication rules. Using unit quaternions for 3D rotation avoids the gimbal lock of Euler angles and gives smooth interpolation and stable computation. They are widely used in computer graphics, robotics, and spacecraft attitude control.
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
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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
#四元数 #三维旋转 #复数推广
选择公理是什么？它和巴拿赫-塔斯基悖论有什么关系？
What is the Axiom of Choice, and how is it related to the Banach–Tarski paradox?
选择公理说：对任意一族非空集合，总能从每个集合中选出一个代表元组成新集合。对无限族而言这是非平凡的，因为无法写出选择规则。哥德尔证明了它与 ZF 相容，科恩 1963 年证明它独立于 ZF。巴拿赫-塔斯基悖论用它把单位球分成有限多块，经平移旋转重新拼成两个同样的球——关键在于这些块是不可测集合，没有可定义的体积。
The Axiom of Choice says that for any family of nonempty sets one can choose one element from each to form a new set. For infinite families this is nontrivial because no selection rule can be written down. Gödel proved it consistent with ZF; Cohen proved in 1963 that it is independent of ZF. The Banach–Tarski paradox uses it to split a unit ball into finitely many pieces which, by rigid motions, reassemble into two identical balls—the catch being that those pieces are non-measurable sets with no well-defined volume.
难度 专业 · 适合 高中(15-18)
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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
#选择公理 #巴拿赫-塔斯基悖论 #不可测集合
连续统假设说的是什么？
What does the continuum hypothesis state?
连续统假设由康托提出，猜测在可数无穷 ℵ₀ 与实数无穷之间不存在其他大小的无穷，即实数的基数就是下一个阿列夫 ℵ₁。20 世纪哥德尔与科恩证明：在标准集合论公理下，它既不能被证明也不能被否定。这意味着它是公理体系之外的独立命题。
The continuum hypothesis, proposed by Cantor, conjectures that no infinity lies strictly between the countable ℵ₀ and the infinity of the reals, i.e., the reals have cardinality the next aleph, ℵ₁. In the 20th century Gödel and Cohen showed it can be neither proved nor disproved from standard set-theory axioms. It is thus independent of the axiomatic system.
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
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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
#连续统假设 #阿列夫 #独立命题
什么是超限数（如 ℵ₀）？
What are transfinite numbers such as ℵ₀?
超限数是康托为衡量无穷集合“大小”而引入的数，记作 ℵ（阿列夫）。ℵ₀ 是可数无穷的基数，对应自然数集；更大的 ℵ₁ 等用于更大的无穷。它们不是普通意义上的“很大的数”，而是描述集合势的比较工具。
Transfinite numbers are quantities Cantor introduced to measure the size of infinite sets, denoted ℵ (aleph). ℵ₀ is the cardinality of countable infinity, matching the naturals; larger ones like ℵ₁ serve bigger infinities. They are not very large numbers in the ordinary sense but tools for comparing the cardinality of sets.
难度 专业 · 适合 高中(15-18)
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Courant & Robbins,《What Is Mathematics?》,Oxford University Press,1996,第三版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 3rd ed., 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
#超限数 #阿列夫 #基数
实数比整数“多”吗？康托尔怎么用一个“对角线”证明多出来？
Are there more real numbers than integers? How did Cantor prove it with a “diagonal” argument?
假设 (0,1) 里的实数能与正整数一一列出成无限表，每行是某个数的小数展开。构造新数：第 n 位取与表中第 n 行第 n 位不同的数字（比如相差一个的循环），它就与每一行都不相等，却理应也在表里，矛盾。所以 (0,1) 不可数，实数比整数“真正更多”。康托尔由此建立基数理论：整数是可数无穷 ℵ₀，实数是不可数无穷，而且 2^ℵ₀>ℵ₀。
Suppose the reals in (0,1) are listed in a countable table, each row being one number's decimal expansion. Construct a new number whose n-th digit differs from the n-th entry of the n-th row (say by cycling the digit). It differs from every row, yet should be in the list — a contradiction. Hence (0,1) is uncountable, and the reals are genuinely more numerous than the integers. Cantor's cardinal theory grew from this: integers have cardinality ℵ₀, the continuum is uncountable, and 2^ℵ₀>ℵ₀.
难度 专业 · 适合 高中(15-18)
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《什么是数学》(柯朗)（第8章：无穷集合）
What Is Mathematics? (Courant), ch. 8 on infinite sets
#对角线论证 #不可数 #连续统
复数乘法为什么等价于平面旋转与伸缩？
Why is complex multiplication equivalent to rotation and scaling in the plane?
把复数写成模长与幅角的形式后，两数相乘会让模长相乘、幅角相加。幅角相加正是平面上的旋转，模长相乘正是缩放。因此乘以一个复数就同时完成了一次旋转加伸缩，这正是它在二维几何和信号相位中如此有用的原因。
Writing a complex number by its magnitude and angle, multiplying two complexes multiplies their magnitudes and adds their angles. Adding angles is exactly rotation in the plane, and multiplying magnitudes is scaling. So multiplying by a complex number performs a rotation plus scaling at once, which is why it is so useful in 2D geometry and signal phase.
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
#复数 #旋转 #幅角
连续统假设为什么既不能被证明也不能被否证？
Why can the continuum hypothesis be neither proved nor disproved?
连续统假设（CH）断言：不存在基数严格大于自然数集、又严格小于实数集的集合，康托尔猜测它为真。1940 年哥德尔证明 CH 与 ZFC 公理系统相容；1963 年科恩用他发明的力迫法证明 CH 的否定也与 ZFC 相容。因此 CH 在 ZFC 中独立——公理系统本身不足以回答这个问题，这是不完备现象在集合论中的深刻体现。
The continuum hypothesis (CH) asserts there is no set whose cardinality lies strictly between that of the natural numbers and the reals; Cantor believed it true. In 1940 Gödel proved CH is consistent with the ZFC axioms; in 1963 Cohen used forcing, which he invented, to show the negation of CH is also consistent with ZFC. Thus CH is independent of ZFC—the axioms themselves cannot settle it, a profound manifestation of incompleteness in set theory.
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
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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
#连续统假设 #力迫法 #独立于ZFC
什么是素数？
What is a prime number?
素数（质数）是大于 1 且只能被 1 和自身整除的自然数，如 2、3、5、7。1 不是素数，2 是唯一的偶素数。算术基本定理说每个大于 1 的整数都能唯一分解为素数的乘积，因此素数被称为“数字的原子”。
A prime is a natural number greater than 1 divisible only by 1 and itself, such as 2, 3, 5, 7. One is not prime, and 2 is the only even prime. The fundamental theorem of arithmetic says every integer above 1 factors uniquely into primes, so primes are called the atoms of numbers.
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
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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
#素数 #质因数 #数论
实数和有理数有什么不同？
How do real numbers differ from rational numbers?
有理数可以写成两个整数的比（分数），而实数还包括无法这样表示的无理数，如 √2 和 π。从几何上看，实数填满了整条数轴，没有任何空隙。实数系是连续、完备的，是微积分得以成立的基础。
Rational numbers can be written as a ratio of two integers (a fraction), while real numbers also include irrationals that cannot, such as √2 and π. Geometrically, the reals fill the entire number line with no gaps. The real number system is continuous and complete, the foundation on which calculus rests.
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
#实数 #有理数 #无理数
什么是有理数？
What are rational numbers?
有理数是能写成两个整数之比 a/b（b≠0）的数，包括整数、有限小数和无限循环小数。它们在数轴上处处稠密，任意两个有理数之间还有别的有理数。有理数加上无理数才构成完整的实数系。
Rational numbers are those writable as a ratio a/b of two integers (b≠0), including integers, terminating decimals, and repeating decimals. They are dense on the number line, with more rationals between any two. The reals are completed only by adding irrationals to the rationals.
难度 进阶 · 适合 大众
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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
#有理数 #分数 #数系
无理数是什么，能举例子吗？
What are irrational numbers, with examples?
无理数是不能写成两个整数之比的实数，其小数展开既不终止也不循环。经典例子有 √2（正方形对角线长）和圆周率 π。古希腊人发现 √2 无理，曾颠覆了“一切数皆可比”的信念。
Irrational numbers are real numbers that cannot be written as a ratio of two integers; their decimal expansions neither terminate nor repeat. Classic examples are √2 (the diagonal of a unit square) and π. The Greeks' discovery that √2 is irrational overturned the belief that all numbers were commensurable.
难度 进阶 · 适合 大众
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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
#无理数 #根号2 #圆周率
为什么 √2 不能写成分数？这个发现为什么曾经引发“危机”？
Why can √2 not be written as a fraction, and why did that discovery once cause a crisis?
假设 √2=p/q 是最简分数，两边平方得 p²=2q²，于是 p² 是偶数，p 也是偶数；设 p=2k，又推出 q 也是偶数，和“最简”矛盾，所以假设不成立。据传毕达哥拉斯学派的希帕索斯最先证明了这一点，而学派坚持“万物皆数（有理数）”，无法接受这个结论。
Suppose √2=p/q in lowest terms. Squaring gives p²=2q², so p² and hence p are even; writing p=2k then forces q to be even too, contradicting the fraction being reduced. So no such fraction exists. Legend says Hippasus of the Pythagorean school first proved this, shocking a community that held that everything is ratio (rational) — a genuine intellectual crisis.
难度 进阶 · 适合 初中(12-15)
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《什么是数学》(柯朗)（数的扩张：无理数）
What Is Mathematics? (Courant), the expansion of numbers (irrationals)
独立核验依据 · Independent verification: 核验依据 2 DOUBAN ISBN 标识 9787309128109（按标识回读: 什么是数学（原作名 What Is Ma
#无理数 #反证法 #毕达哥拉斯
数学里的“无穷大”符号 ∞ 表示什么？
What does the infinity symbol ∞ mean in mathematics?
∞ 表示“没有尽头、可以任意大”的概念，常用于极限、区间和无尽过程的描述，而不是一个普通的数。在扩展实数系中它可以参与某些运算，但要遵守特殊规则（如 ∞+1 仍记 ∞）。它帮助我们简洁地谈论趋向无穷的趋势。
∞ represents the idea of endless, arbitrarily large, used in limits, intervals, and unending processes rather than as an ordinary number. In the extended real line it can enter some operations under special rules (e.g., ∞+1 is still ∞). It lets us speak concisely about tendencies toward infinity.
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
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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
#无穷大 #极限 #无穷
整数和分数（有理数）哪个更多？康托尔为什么说它们“一样多”？
Are there more integers or more fractions (rationals)? Why did Cantor say they are equally many?
直觉上分数应该多得多，但康托尔证明了有理数能和整数一一配对。把分数按分子分母之和从小到大排成一张可以“走对角线”的表，就能给每个分数一个唯一的编号，一个也不会漏。能与正整数配对的集合叫可数集，整数、偶数、有理数都是可数集，它们都有“同样多”的元素。
Intuitively fractions seem far more numerous, yet Cantor proved rationals can be paired one-to-one with integers. Arrange fractions in a table ordered by sum of numerator and denominator and walk through it along diagonals; every fraction gets a unique label with none missing. Sets pairable with the positive integers are countable — integers, evens, and rationals all have “just as many” elements.
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《什么是数学》(柯朗)（第8章：无穷集合）
What Is Mathematics? (Courant), ch. 8 on infinite sets
#康托尔 #可数无穷 #一一对应
p 进数（p-adic numbers）是怎么用“被 p 整除的程度”来重新定义“距离”的？
How do p-adic numbers redefine 'distance' using 'how divisible by p' instead of size?
p 进数把两个数 x, y 的距离定义为 |x−y|_p = p^(−k)，其中 k 是 x−y 含有因子 p 的幂次；越能被 p 整除，距离越近。这给出了与实数截然不同的“拓扑”，使整除性成为邻近性的度量，在数论与代数几何（如费马大定理的模形式证明）中极为关键。
The p-adic distance sets |x−y|_p = p^(−k), where k is the power of p dividing x−y: the more divisible by p, the closer. This yields a topology opposite to the reals', making divisibility the measure of nearness, and is central to number theory and arithmetic geometry (e.g., the modular proof of Fermat's Last Theorem).
难度 进阶 · 适合 高中(15-18)
CC BY 4.0 · 来源待核验 · Source pending review
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出处备注：著录含书名，但缺版次 / 出版社 / 年份中的一部分，需编辑补全后才能做独立比对。
Gouvêa, F. Q., 《p进数导论》(第2版), Springer
Gouvêa, F. Q., 'p-adic Numbers: An Introduction' (2nd ed.), Springer
#p进数 #p进赋值 #数论拓扑 #非阿基米德
为什么“负数不能开平方”的禁忌可以被打破？虚数有用吗？
How was the taboo against square roots of negatives broken, and are imaginary numbers useful?
约定 i²=−1 并不产生矛盾，而是把数系从实数扩充到复数 z=a+bi，加减乘除都定义齐全。复数给了方程 x²+1=0 以根，更关键的是把旋转几何化——乘以 i 就是逆时针旋转 90°。傅里叶变换、交流电路和量子力学都离不开复数，量子态的叠加本质上就是复系数组合。
Postulating i²=−1 causes no contradiction; it extends the real numbers to the complex numbers z=a+bi with full arithmetic. Complex numbers give x²+1=0 a root and, crucially, encode rotation geometrically—multiplying by i rotates by 90° counterclockwise. Fourier transforms, AC circuits, and quantum mechanics all depend on them; superposition of quantum states is literally complex-linear combination.
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《什么是数学》(柯朗)（复数章）
What Is Mathematics? (Courant), chapter on complex numbers
#虚数 #复数 #数系扩充
康托尔如何用对角线论证证明区间 [0,1] 内的实数不可能与全体自然数一一配对?
How does Cantor’s diagonal argument show that the real numbers in the interval [0,1] cannot be paired one-to-one with the natural numbers?
先假设 [0,1] 中所有实数能排成一列 x1, x2, x3,…。然后构造新数 y:让 y 的第 n 位小数与 xn 的第 n 位小数不同,同时避开数字 0 和 9,以免落入两种小数表示的歧义。这样 y 与名单上每个数至少有一位相异,却不在名单之中,与假设矛盾。所以 [0,1] 是不可数集合,其基数严格大于自然数的基数 ℵ₀。
Assume first that all reals in [0,1] can be listed as a sequence x1, x2, x3, … Now build a new number y whose n-th decimal digit differs from the n-th digit of xn, avoiding the digits 0 and 9 to dodge ambiguous dual expansions. Then y differs from every listed number in at least one digit yet appears nowhere on the list—a contradiction. Hence [0,1] is uncountable, with cardinality strictly greater than the natural numbers’ ℵ₀.
难度 进阶 · 适合 大众
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Cantor, G.(1891). Über eine elementare Frage der Mannigfaltigkeitslehre, Jahresbericht der Deutschen Mathematiker-Vereinigung 1, S.75-78
Cantor, G. (1891). Über eine elementare Frage der Mannigfaltigkeitslehre, Jahresbericht der Deutschen Mathematiker-Vereinigung 1, pp. 75-78
#集合论 #不可数性
为什么“乘以 0 一定得 0”，而“除以 0”却不允许？
Why does 'multiplying by 0 always gives 0', yet 'dividing by 0' is not allowed?
乘法是“重复相加”，任何数加 0 次仍是 0，所以 a×0=0。但除法是乘法的逆：若 x = a÷0，则应满足 x×0 = a；可任何数乘 0 都得 0，无法等于非零的 a，而若 a=0 又会有无穷多解，所以除以 0 没有确定答案，被禁止。
Multiplication is repeated addition, and adding anything zero times gives 0, so a×0=0. Division is the inverse: x = a÷0 would require x×0 = a, but anything times 0 is 0—never a nonzero a—and if a=0 every x works, so division by 0 has no definite answer and is forbidden.
难度 入门 · 适合 小学低段(6-8)
CC BY 4.0 · 已核验 · Verified 2026-11-02
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Stewart, I. 与 Tall, D., 《数学的基础》, Oxford University Press
Stewart, I. & Tall, D., 'The Foundations of Mathematics', Oxford University Press
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL5838165W：The foundations of math
#零的乘法 #除以零 #逆运算 #数系
什么是分数？
What is a fraction?
分数表示把一个整体平均分成若干份后取其中几份，写作分子/分母。例如 3/4 表示四份中取三份。分数和除法、小数、百分数本质上都是同一种“比例”的不同写法。
A fraction represents taking some of the equal parts of a whole, written as numerator/denominator. For example, 3/4 means three of four equal parts. Fractions, division, decimals, and percentages are essentially different notations for the same proportion.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
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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
#分数 #比例 #除法
什么是整数？
What are integers?
整数包括正整数、负整数和零，如 …−3、−2、−1、0、1、2、3…。在自然数基础上加入负数，我们就能表示欠债、下降等“相反方向”的量。整数的加减乘结果仍是整数，但除法一般不是。
Integers include positive numbers, negative numbers, and zero, like ...−3, −2, −1, 0, 1, 2, 3... Adding negatives to the naturals lets us represent opposite directions such as debt or drop. The sum, difference, and product of integers are integers, but division generally is not.
难度 入门 · 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
#整数 #负数 #数系
什么是偶数和奇数？
What are even and odd numbers?
能被 2 整除的整数是偶数（如 0、2、4），除以 2 余 1 的是奇数（如 1、3、5）。偶数加偶数仍是偶数，奇数加奇数是偶数，奇偶规律在整除和密码学中很有用。判断一个数奇偶只需看它个位（十进制下）是奇是偶。
Integers divisible by 2 are even (0, 2, 4), and those leaving remainder 1 are odd (1, 3, 5). Even plus even is even, odd plus odd is even; parity rules are useful in divisibility and cryptography. In decimal, you only need the last digit to tell parity.
难度 入门 · 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
#偶数 #奇数 #整除
什么是小数？
What is a decimal?
小数是用十进制表示的数，小数点右边依次是十分位、百分位、千分位……例如 0.25 表示 25/100。有限小数可以化成分数，而像 1/3=0.333… 这样无限循环的小数也能化成分数。
A decimal is a number in base ten, with places to the right of the point being tenths, hundredths, thousandths, and so on; for example, 0.25 means 25/100. Terminating decimals convert to fractions, and repeating ones like 1/3 = 0.333... also do.
难度 入门 · 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
#小数 #十进制 #分数
负数有什么用？
What are negative numbers used for?
负数用来表示与正数相反方向的量，比如欠钱、温度低于零、海平面以下高度。引入负数后，减法不再受限，方程也更容易求解。它在记账、物理和坐标系统中都不可或缺。
Negative numbers represent quantities opposite in direction to positives, such as debt, temperatures below zero, or heights below sea level. With negatives, subtraction is unrestricted and equations become easier to solve. They are indispensable in accounting, physics, and coordinate systems.
难度 入门 · 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
#负数 #相反量 #数系
什么是自然数？
What are natural numbers?
自然数通常指用来计数的数：1、2、3、4……（有时也把 0 包括在内）。它们是最基本的数系，是算术和整个数学的起点。自然数可以进行加减乘，但除法不一定得到自然数。
Natural numbers are the counting numbers 1, 2, 3, 4, ... (sometimes including 0). They are the most basic number system and the starting point of arithmetic and all of mathematics. They support addition, subtraction, and multiplication, but division need not yield a natural number.
难度 入门 · 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
#自然数 #计数 #数系