什么是环（ring）？
What is a ring?
环是一个带加法和乘法（加法成交换群、乘法结合且对加法分配律成立）的集合，但乘法不一定有逆元或交换。整数集在加减乘下就是一个环。
A ring has addition forming a commutative group and associative multiplication distributing over addition; integers form a ring.
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
#代数 #抽象代数
模（module）是什么？
What is a module?
模是向量空间在环上的推广：标量来自环而非域，因而不一定能作除法。阿贝尔群、理想都是模的特例，模论是研究环结构的主要方式。
A module generalizes a vector space by allowing scalars from a ring, not necessarily a field. Abelian groups and ideals are examples; modules are the main way to study rings.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
《近世代数》刘绍学
Abstract Algebra by Dummit and Foote
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780135693025 (2004)
#代数 #模 #环
为什么有理数上的多项式环是一个欧几里得整环？
Why is the polynomial ring over the rationals a Euclidean domain?
因为对任意两个有理系数多项式都可以做带余除法（反复消去首项），且余式的次数严格小于除式的次数。这样“次数”就充当了欧几里得函数，反复做带余除法等价于辗转相除求最大公因式。因此 Q[x] 是欧几里得整环，从而是主理想整环和唯一分解整环：每个多项式都能唯一分解成不可约多项式之积。
Because any two rational polynomials admit division with remainder (repeatedly cancelling leading terms), the remainder having strictly smaller degree than the divisor. The degree then serves as a Euclidean function, and iterated division is exactly the Euclidean algorithm computing GCDs. Hence Q[x] is a Euclidean domain, therefore a principal ideal domain and a unique factorization domain: every polynomial factors uniquely into irreducibles.
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
#多项式环 #欧几里得整环 #唯一分解
诺特环是什么？
What is a Noetherian ring?
诺特环是满足“升链条件”的环：任何理想升链最终稳定。埃米·诺特的工作表明这一性质极大简化了代数与代数几何，是现代交换代数的基础。
A Noetherian ring satisfies the ascending chain condition on ideals, so any chain stabilizes. Emmy Noether showed this simplifies algebra and algebraic geometry profoundly.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Dummit & Foote《抽象代数》第3版，Wiley, 2004（自由群一节）
R. S. Dummit & R. M. Foote, Abstract Algebra, 3rd ed., Wiley, 2004 (section on free groups)
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780135693025 (2004)
#代数 #环 #诺特
奇异值分解（SVD）是什么？
What is the singular value decomposition (SVD)?
SVD 把任意矩阵 A 分解为 A = U Σ Vᵀ，其中 U、V 是正交矩阵，Σ 是对角线上非负“奇异值”的对角阵。它用于降噪、压缩、推荐系统和求解最小二乘。
SVD writes A=UΣVᵀ with orthogonal U,V and diagonal Σ of nonnegative singular values; used in denoising, compression, recommender systems and least squares.
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
#代数 #奇异值分解
自由群是什么？
What is a free group?
自由群由一组生成元构成，除群公理外不含任何额外关系，可看作“最自由”地组合生成元及其逆。它在群论中扮演类似于向量空间中基的角色。
A free group is generated by symbols with no relations beyond the group axioms, freely combining generators and their inverses. It plays a role like a basis in vector spaces.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Dummit & Foote《抽象代数》第3版，Wiley, 2004（自由群一节）
R. S. Dummit & R. M. Foote, Abstract Algebra, 3rd ed., Wiley, 2004 (section on free groups)
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780135693025 (2004)
#代数 #群 #生成元
矩阵指数是什么？
What is the matrix exponential?
矩阵指数 e^{At} 是把标量指数函数推广到矩阵，由幂级数定义。它在解线性微分方程组 x'=Ax 时给出解 x(t)=e^{At}x(0)，也用于量子力学与控制系统。
The matrix exponential e^{At} generalizes e^x via power series; it solves linear systems x'=Ax as x(t)=e^{At}x(0), used in quantum mechanics and control.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Higham, N. J.，《Functions of Matrices: Theory and Computation》，SIAM，2008
Higham, N. J., Functions of Matrices: Theory and Computation, SIAM, 2008
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780898716467 (2008)
#代数 #矩阵指数
秩-零化度定理说的是什么？
What does the rank-nullity theorem say?
对 m×n 矩阵 A（即线性映射 V→W），秩（像空间的维数）与零化度（核的维数）之和等于 n：rank(A)+nullity(A)=n。它表达了一种“守恒”：被映射保留的独立方向与被压到零的方向加起来必然等于输入空间的维数。例如三阶线性映射的秩为 2 时，齐次方程 Ax=0 必有 1 维解空间。
For an m×n matrix A (a linear map V→W), the rank (dimension of the image) plus the nullity (dimension of the kernel) equals n: rank(A)+nullity(A)=n. It expresses a conservation law: the independent directions preserved by the map plus the directions collapsed to zero must total the dimension of the input space. A rank-2 linear map on 3D space therefore has a 1-dimensional solution space for Ax=0.
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
#秩-零化度定理 #核与像 #线性映射
为什么有限域的元素个数总是某个素数的幂？
Why must a finite field have a prime-power number of elements?
有限域必有素数特征 p：把 1 自加 p 次就得到 0，这 p 个元素构成素域 F_p。整个有限域可以看作 F_p 上的向量空间，设其维数为 n，元素个数就是 pⁿ。反过来，对任意素数 p 和正整数 n，同构意义下恰好存在一个 pⁿ 元有限域。有限域是纠错码（如里德-所罗门码）、密码学和组合设计的重要工具。
A finite field has prime characteristic p: adding 1 to itself p times gives 0, and these p elements form the prime field F_p. The whole field is a vector space over F_p; if its dimension is n, the field has pⁿ elements. Conversely, for every prime p and positive integer n there is exactly one field of order pⁿ up to isomorphism. Finite fields power error-correcting codes (e.g., Reed–Solomon), cryptography, and combinatorial designs.
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
#有限域 #素域 #纠错码
伽罗瓦理论如何解释为什么五次以上方程没有一般根式解？
How does Galois theory explain why equations of degree five or higher lack a general radical solution?
伽罗瓦理论把多项式方程的根对称性与一个群（伽罗瓦群）对应，根式可解等价于该群可解。一般五次方程的伽罗瓦群是对称群S5，它不可解，因此不存在通用的根式公式。阿廷与斯图尔特的教材把这一对应讲得极清楚。
Galois theory associates the symmetries among a polynomial's roots with a group, its Galois group, and radical solvability is equivalent to that group being solvable. The generic quintic has Galois group S5, which is not solvable, so no general radical formula exists. Artin and Stewart present this correspondence clearly.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-11-03
来源 · Source
Artin《Galois Theory》；Stewart《Galois Theory》
Artin, Galois Theory; Stewart, Galois Theory
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL1207126W：Galois theory（首版 1942）按
#伽罗瓦理论 #根式解 #可解群
凯莱定理是什么？
What is Cayley's theorem?
凯莱定理说：任一有限群都同构于某个置换群（对称群的子群）。这意味着群论本质上都可视为对“排列”的研究。
Cayley's theorem states every group is isomorphic to a subgroup of some symmetric group of permutations, so group theory is essentially the study of arrangements.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
《近世代数》刘绍学
Abstract Algebra by Dummit and Foote
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780135693025 (2004)
#代数 #群 #置换
什么是张量？
What is a tensor?
张量是向量与矩阵的推广，可看作多维数组，并能在坐标变换下按特定规则变化。它在物理（应力、曲率）和深度学习（权重、激活）中无处不在。
A tensor generalizes vectors and matrices as a multi-dimensional array transforming by definite rules under coordinate changes; central to physics and deep learning.
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
#代数 #张量
线性规划里为什么最优解一定出现在可行域的“顶点”上？
In linear programming, why must an optimal solution occur at a vertex of the feasible region?
目标函数是线性函数，其水平线（等值线）是一族平行直线，函数值沿一个方向单调变化。可行域由线性不等式围成，是凸多边形（高维下是凸多面体）。若最优在区域内部或边上，沿垂直于等值线的方向移动（或沿边滑动）总能让值继续增大或不变，直到撞到顶点——除非最优在整条边上取得，而那样的边两端仍是顶点。所以只需检查顶点，单纯形法正是从一个顶点沿边跳到更优顶点的算法，实际规模问题的求解都建立在这个几何事实上。
The objective is linear, so its level sets are parallel lines and the function changes monotonically in one direction. The feasible region, cut out by linear inequalities, is a convex polygon (polytope in higher dimensions). If an optimum lay strictly inside or on an edge, moving perpendicular to the level lines (or sliding along the edge) would keep improving or at least not decrease the value, until a vertex is hit — unless the optimum is attained along an entire edge, whose endpoints are vertices anyway. Hence only vertices need checking: the simplex method walks from vertex to vertex along improving edges, and virtually all practical LP solvers rest on this geometric fact.
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
#线性规划 #凸多面体 #顶点最优
希尔伯特零点定理是什么？
What is Hilbert's Nullstellensatz?
它建立多项式方程组的“公共零点”与多项式理想之间的对应：若一组多项式没有公共零点，则它们的某个组合等于常数1。它是代数几何的基石。
It links common zeros of polynomial equations to polynomial ideals: if polynomials share no zero, some combination equals 1. It is a cornerstone of algebraic geometry.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Cox、Little 与 O'Shea《Ideals, Varieties, and Algorithms》第 8 章关于射影空间的齐次坐标讨论
D. Cox, J. Little and D. O'Shea, Ideals, Varieties, and Algorithms, Chapter 8 on homogeneous coordinates in projective space
独立核验依据 · Independent verification: 核验依据 1 ISBN 9783540978473 (2016)
#代数 #代数几何 #多项式
把 n 个符号重新排列的所有方式组成什么结构？它的“奇偶性”又是什么？
What structure do all rearrangements of n symbols form, and what is their “parity”?
所有排列在“先做排列 A 再做排列 B”的复合下构成对称群 Sₙ，共有 n! 个元素。任何排列都能写成若干次两两交换（对换）的乘积，虽然写法不唯一，但所需对换次数的奇偶性是确定的：偶数次叫偶排列，奇数次叫奇排列。偶排列全体构成正规子群 Aₙ（交错群），它有 n!/2 个元素；符号函数把群映到 {±1}，是理解置换群表示和五次方程不可解的关键工具。
All permutations under composition form the symmetric group Sₙ, with n! elements. Every permutation factors into transpositions (swaps of two symbols); the factorization is not unique, but its length's parity is: even permutations need an even number, odd ones an odd number. The even permutations form a normal subgroup Aₙ, the alternating group, of size n!/2. The sign homomorphism mapping Sₙ onto {±1} is the key tool behind representation theory and the unsolvability of the quintic.
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
#对称群 #排列 #交错群
群表示论（group representation）为什么能把抽象的“对称”翻译成具体的矩阵？
Why can group representation theory translate abstract 'symmetry' into concrete matrices?
群表示把一个群的元素映射到可逆矩阵，使群的乘法对应矩阵乘法，于是抽象的对称操作（旋转、置换）变成可计算的线性变换。表示论是连接代数、几何与物理（如量子力学的角动量）的桥梁，傅里叶分析也可视为圆周群表示的特例。
A group representation maps each group element to an invertible matrix so the group product becomes matrix multiplication, turning abstract symmetries (rotations, permutations) into computable linear maps. Representation theory bridges algebra, geometry, and physics (e.g., quantum angular momentum), and Fourier analysis is a special case of representing the circle group.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
Serre, J.-P., 《有限群的线性表示》, Springer
Serre, J.-P., 'Linear Representations of Finite Groups', Springer
独立核验依据 · Independent verification: 核验依据 1 ISBN 9787040220407（豆瓣书目条目: 有限群的线性表示）
#群表示 #对称 #矩阵 #傅里叶分析
二次、三次、四次方程都有求根公式，为什么五次方程没有？
Quadratic, cubic, and quartic equations have root formulas — why not the quintic?
关键不在“难”，而在于结构。求根公式本质上是把系数进行加减乘除和开方运算；阿贝尔 1824 年证明一般五次方程做不到这一点。伽罗瓦给出了更深刻的判据：方程能用根式求解当且仅当其伽罗瓦群的置换结构“可解”。五次方程的伽罗瓦群是 S5（所有 5 个根的置换），它不可解，因此一般五次没有根式通解。
The obstruction is structural, not difficulty. A root formula is built from the coefficients by arithmetic plus root extraction; in 1824 Abel proved this is impossible for the general quintic. Galois gave a deeper criterion: an equation is solvable by radicals exactly when its Galois group is solvable. The quintic's Galois group is S5, all permutations of five roots, which is not solvable — hence no general radical formula exists.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-10-03 · 2 条独立证据
来源 · Source
《什么是数学》(柯朗)（方程论）
What Is Mathematics? (Courant), theory of equations
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
#五次方程 #伽罗瓦理论 #不可解性
李群是什么？
What are Lie groups?
李群既是群又是光滑流形，群运算连续可微，如旋转群 SO(3)。它把对称性与微积分结合，是粒子物理和广义相对论的基本数学框架。
A Lie group is both a group and a smooth manifold with differentiable operations, e.g. the rotation group SO(3). It links symmetry and calculus, underpinning particle physics and relativity.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-11-03
来源 · Source
Hall, B. C., 《李群、李代数及其表示》, Springer
Hall, B. C., 'Lie Groups, Lie Algebras, and Representations', Springer
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL20671601W：Lie Groups, Lie Algebr
#代数 #李群 #对称性
什么是群论？
What is group theory?
群论是研究“群”这一代数结构的数学分支，群由一组元素和一个满足结合律、有单位元且每个元素有逆元的运算组成。群论是现代代数、对称研究和粒子物理的基础工具。
Group theory is the branch of mathematics studying groups, algebraic structures consisting of a set and an operation that is associative, has an identity element, and gives every element an inverse. It is a fundamental tool in modern algebra, the study of symmetry, and particle physics.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
《近世代数》刘绍学
Abstract Algebra by Dummit and Foote
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780135693025 (2004)
#群论 #抽象代数 #对称
什么是复数？
What is a complex number?
复数是形如 a + bi 的数，其中 i 是满足 i² = −1 的虚数单位，a 和 b 为实数。复数让所有二次方程都有解，并广泛用于电气工程和量子力学。
A complex number has the form a + bi, where i is the imaginary unit satisfying i² = −1, and a and b are real numbers. Complex numbers ensure every quadratic equation has a solution and are widely used in electrical engineering and quantum mechanics.
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
Axler,《Linear Algebra Done Right》第3版,Springer,2015;中译《线性代数应该这样学》人民邮电出版社
S. Axler, Linear Algebra Done Right, 3rd ed., Springer, 2015; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 CROSSREF 标识 10.1007/978-3-319-11080-6：Linear Algebra Don
#复数 #虚数 #代数
李代数（Lie algebra）怎样用“无穷小生成元”描述连续对称（如旋转）？
How does a Lie algebra describe continuous symmetries (like rotations) via 'infinitesimal generators'?
李代数是一个带反对称括号 [x,y] 的向量空间，它对应李群的“切空间”，刻画群在恒等元附近的无穷小变换。旋转群 SO(3) 的李代数由角动量算子组成，括号反映“先绕 x 再绕 y 转”与反向的差；这为粒子物理与微分方程对称性提供了统一语言。
A Lie algebra is a vector space with an antisymmetric bracket [x,y]; it is the tangent space of a Lie group and captures the group's infinitesimal transformations near the identity. The Lie algebra of SO(3) consists of angular-momentum operators, whose bracket reflects the mismatch between rotating about x then y versus the reverse—a unified language for symmetries in physics and differential equations.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-11-03
来源 · Source
Hall, B. C., 《李群、李代数及其表示》, Springer
Hall, B. C., 'Lie Groups, Lie Algebras, and Representations', Springer
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL20671601W：Lie Groups, Lie Algebr
#李代数 #李群 #无穷小生成元 #对称性
怎样求矩阵 [[2,1],[1,2]] 的特征值与特征向量？
How do you find the eigenvalues and eigenvectors of [[2,1],[1,2]]?
先由 det(A−λI)=0 求特征值：(2−λ)²−1=0，得 λ=3 和 λ=1。对 λ=3，解 (A−3I)v=0，即 (−1,1;1,−1)v=0，取 v=(1,1)；对 λ=1，解 (A−I)v=0，即 (1,1;1,1)v=0，取 v=(1,−1)。几何上，这个对称矩阵把 (1,1) 方向拉伸 3 倍、保持 (1,−1) 方向不动，且两个特征向量互相垂直。
First eigenvalues from det(A−λI)=0: (2−λ)²−1=0 gives λ=3 and λ=1. For λ=3, solve (A−3I)v=0, i.e., (−1,1;1,−1)v=0, take v=(1,1). For λ=1, solve (A−I)v=0, i.e., (1,1;1,1)v=0, take v=(1,−1). Geometrically this symmetric matrix stretches the (1,1) direction by 3, leaves (1,−1) fixed, and the two eigenvectors are perpendicular.
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
#特征值 #特征向量 #对称矩阵
伽罗瓦理论解决什么问题？
What problem does Galois theory solve?
伽罗瓦理论用“群”来描述多项式方程能否用根式求解，并证明了五次及以上的一般方程没有求根公式。
Galois theory uses groups to characterize solvability by radicals, proving no general formula exists for degree-5+ equations.
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
#代数 #伽罗瓦
在 Hölder 不等式的经典证明中，Young 不等式起什么作用？请说明从逐点的 ab ≤ a^p/p + b^q/q（其中 1/p + 1/q = 1）出发，如何得到 Σ|x_i y_i| ≤ ||x||_p ||y||_q。
What role does Young’s inequality play in the classical proof of Hölder’s inequality? Starting from the pointwise bound ab ≤ a^p/p + b^q/q (with 1/p + 1/q = 1), how does one obtain Σ|x_i y_i| ≤ ||x||_p ||y||_q?
先归一化：两边除以||x||_p||y||_q，只需证Σaᵢbᵢ≤1，其中Σaᵢ^p=Σbᵢ^q=1。逐项用Young不等式aᵢbᵢ≤aᵢ^p/p+bᵢ^q/q，求和得右端为1/p+1/q=1。归一化吸收尺度自由度，使初等估计升级为全局最优界；等号在|yᵢ|∝|xᵢ|^(p−1)时成立，常数不可改进。
Normalize first: divide both sides by ||x||_p ||y||_q and rescale the sequences, reducing matters to proving Σ a_i b_i ≤ 1 with Σ a_i^p = Σ b_i^q = 1. Apply Young’s inequality termwise, a_i b_i ≤ a_i^p/p + b_i^q/q, and sum: the right side is at most 1/p + 1/q = 1. Normalization absorbs the scale freedom, promoting an elementary pointwise estimate to a sharp global bound; equality holds when |y_i| is proportional to |x_i|^(p−1), so the constant cannot be improved.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 来源待核验 · Source pending review
来源 · Source
出处备注：作者 / 书名 / 版次 / 出版社 / 年份齐备，仅缺独立比对证据；已在可信度报告单列披露，不再重复核验。
Hardy, G. H., Littlewood, J. E., Pólya, G.，《不等式》(Inequalities)，第2版，Cambridge University Press，1952年。
Hardy, G. H., Littlewood, J. E., Pólya, G., Inequalities, 2nd ed., Cambridge University Press, 1952.
#赫尔德不等式 #杨氏不等式 #Lp范数
什么是多项式？
What is a polynomial?
多项式是由变量和常数通过加法、减法和乘法（非负整数次幂）组合而成的代数式，例如 3x² + 2x − 5。多项式的次数由最高次项的指数决定。
A polynomial is an algebraic expression made of variables and constants combined by addition, subtraction, and multiplication with non-negative integer exponents, such as 3x² + 2x − 5. The degree of a polynomial is given by the highest exponent appearing in it.
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
Axler,《Linear Algebra Done Right》第3版,Springer,2015;中译《线性代数应该这样学》人民邮电出版社
S. Axler, Linear Algebra Done Right, 3rd ed., Springer, 2015; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 CROSSREF 标识 10.1007/978-3-319-11080-6：Linear Algebra Don
#多项式 #代数式 #次数
一元二次方程的求根公式是什么？
What is the quadratic formula?
对于一元二次方程 ax² + bx + c = 0（a≠0），其解可由公式 x = (−b ± √(b²−4ac)) / (2a) 给出。根号里的 b²−4ac 称为判别式，决定方程有几个实根。
For a quadratic equation ax² + bx + c = 0 with a not equal to 0, its solutions are given by x = (−b ± √(b²−4ac)) / (2a). The expression b²−4ac under the square root is called the discriminant and determines how many real roots the equation has.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-11-03
来源 · Source
Beckmann, S., 《中小学数学教师数学》(第4版), Pearson
Beckmann, S., 'Mathematics for Elementary Teachers' (4th ed.), Pearson
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL18708410W：Mathematics for elemen
#二次方程 #求根公式 #判别式
RSA 公钥加密系统为什么依赖数论中的大数分解难题？
Why does RSA public-key encryption rely on the difficulty of factoring large numbers?
RSA 由 Rivest、Shamir 和 Adleman 于 1977 年提出，其公钥是两个大素数的乘积 n，私钥依赖把 n 分解回这两个素数。目前没有已知的高效算法能在经典计算机上分解足够大的 n，因此加密可行而解密（无密钥时）极难。它每天保护着互联网通信，但其安全性正受到量子算法（Shor 算法）的潜在威胁。
Proposed by Rivest, Shamir, and Adleman in 1977, RSA's public key is the product n of two large primes, while the private key depends on factoring n back into those primes. No efficient classical algorithm is known for factoring sufficiently large n, so encryption is easy but decryption without the key is hard. It secures everyday internet communications, though quantum algorithms (Shor's) threaten this security.
难度 进阶 · 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
#RSA #公钥密码 #大数分解 #素数
什么是不等式？
What is an inequality?
不等式是用大于（>）、小于（<）等符号表示两个量不相等关系的式子，例如 x > 3。解不等式是找出使不等式成立的所有可能取值，注意乘以负数时要改变不等号方向。
An inequality is a statement that two quantities are not equal, written with symbols such as greater-than or less-than, for example x > 3. Solving an inequality means finding all values that make it true, and one must reverse the inequality sign when multiplying by a negative number.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-11-03
来源 · Source
Beckmann, S., 《中小学数学教师数学》(第4版), Pearson
Beckmann, S., 'Mathematics for Elementary Teachers' (4th ed.), Pearson
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL18708410W：Mathematics for elemen
#不等式 #比较 #代数
什么是线性无关？
What is linear independence?
一组向量若不能由其余向量的线性组合表示（即仅当系数全为0时组合才为0），则称线性无关。它是定义维数与基的基础。
Vectors are linearly independent if none is a linear combination of the others; this defines dimension and bases.
难度 进阶 · 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
#代数 #线性代数
拉格朗日定理在群论中说明了子群阶数与群阶数有什么关系？
What relation between a subgroup's order and the group's order does Lagrange's theorem state in group theory?
拉格朗日定理指出有限群的子群阶数必整除整个群的阶数，因此元素的阶也整除群阶。它直接推出费马小定理等结论，是群论的基本约束。杜米特与富特以及罗特曼的抽象代数教材都把它作为早期核心定理。
Lagrange's theorem states that the order of a subgroup of a finite group divides the order of the whole group, so an element's order also divides the group order. It immediately yields results like Fermat's little theorem and is a basic constraint in group theory. Dummit & Foote and Rotman present it as an early core theorem.
难度 进阶 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Dummit 与 Foote《Abstract Algebra》；Rotman《An Introduction to the Theory of Groups》
Dummit & Foote, Abstract Algebra; Rotman, An Introduction to the Theory of Groups
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780135693025 (2004)
#拉格朗日定理 #群论 #子群阶数
行列式是什么？
What is a determinant?
行列式是把方阵映射为一个标量的函数，反映该矩阵所表示线性变换对体积的缩放比例。行列式为0意味着矩阵不可逆。
The determinant maps a square matrix to a scalar giving the volume-scaling factor of its linear map; zero means non-invertible.
难度 进阶 · 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
#代数 #矩阵
工厂要决定每种产品做多少，才能让利润最大？这种问题数学怎么解？
How does mathematics decide how many units of each product a factory should make to maximize profit?
把产量设为变量，利润写成这些变量的线性表达式，机器工时、原料等限制写成不等式，问题就变成“在线性约束下最大化线性目标”。这类问题叫线性规划：可行方案构成一个多边形区域，最优解总出现在区域的顶点上，单纯形算法就是沿着边从一个顶点走到更好的顶点。
Set output levels as variables, write profit as a linear expression in them, and turn limits like machine hours and raw materials into inequalities — the problem becomes maximizing a linear objective subject to linear constraints. These are linear programs: feasible plans form a polygonal region, the optimum always sits at a vertex, and the simplex algorithm walks from vertex to vertex along edges, always improving.
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
#线性规划 #最优化 #单纯形法
代数基本定理是什么？
What is the fundamental theorem of algebra?
它断言任意非常数复系数多项式都至少有一个复根，因此 n 次多项式恰有 n 个复根（计重数）。高斯给出了首个严格证明。
It states every nonconstant polynomial with complex coefficients has a complex root, so a degree-n polynomial has exactly n complex roots counting multiplicity. Gauss gave the first rigorous proof.
难度 进阶 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
Ahlfors《Complex Analysis》；Conway《Functions of One Complex Variable》
Ahlfors, Complex Analysis; Conway, Functions of One Complex Variable
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL2816395W：Functions of one comple
#代数 #多项式 #复数
什么是方程？
What is an equation?
方程是含有等号、并且包含未知数的数学陈述，表示两边的值相等。解方程是找出使等式成立的未知数的值，例如 x + 2 = 5 的解是 x = 3。
An equation is a mathematical statement containing an equals sign and one or more unknowns, asserting that two expressions have the same value. Solving an equation means finding the value of the unknown that makes the statement true, for example x + 2 = 5 gives x = 3.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-11-03
来源 · Source
Beckmann, S., 《中小学数学教师数学》(第4版), Pearson
Beckmann, S., 'Mathematics for Elementary Teachers' (4th ed.), Pearson
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL18708410W：Mathematics for elemen
#方程 #未知数 #代数
矩阵乘以向量到底在“做”什么？为什么说矩阵是变换的化身？
What does multiplying a matrix by a vector actually do, and why is a matrix a transformation in disguise?
把矩阵作用在向量上，就是把平面或空间按某个规则重新摆布：比如旋转矩阵把每个向量转过一个角度，对角矩阵沿坐标轴伸缩，反射矩阵把图形翻到另一侧。这些操作都是线性的——直线保持为直线，原点保持不动。矩阵的行列式还告诉你变换把面积或体积放大了多少倍。
Applying a matrix to a vector rearranges the plane or space by a fixed rule: a rotation matrix turns every vector through an angle, a diagonal matrix stretches along axes, and a reflection flips figures to the other side. All are linear — straight lines stay straight and the origin stays put. The determinant even tells you how much the transformation scales areas or volumes.
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
#线性变换 #矩阵 #行列式
哈密顿发明的「四元数」是什么？它有什么用？
What are the quaternions invented by Hamilton, and what are they used for?
四元数是形如 a + bi + cj + dk 的超复数，其中 i、j、k 满足 i² = j² = k² = ijk = −1，由哈密顿于 1843 年刻在都柏林桥上。与实数和复数不同，四元数乘法不可交换（ij ≠ ji），使其特别适合无奇点地描述三维旋转。如今它是计算机图形学、机器人姿态控制和航天姿态确定中的标准工具。
Quaternions are hypercomplex numbers of the form a + bi + cj + dk with i² = j² = k² = ijk = −1, carved by Hamilton on a Dublin bridge in 1843. Unlike real or complex numbers, their multiplication is non-commutative (ij ≠ ji), which makes them ideal for describing 3D rotations without singularities. They are now standard in computer graphics, robotics, and spacecraft attitude control.
难度 进阶 · 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
#四元数 #哈密顿 #非交换代数 #三维旋转
什么是矩阵？
What is a matrix?
矩阵是按行、列排成的矩形数表，真正的身份却是线性变换的坐标。乘法定义成行乘列再相加，是为了让两次连续变换合成一个矩阵，先做 B 再做 A 等于做 AB；相乘也因此要求前者的列数与后者的行数相等。两点易混：它一般不满足交换律，AB 与 BA 常不相等，顺序必须交代；矩阵也不是行列式，后者只是从方阵算出的一个数。
A matrix is a rectangular table of numbers arranged in rows and columns, and its real identity is the coordinate description of a linear transformation. Multiplication is defined as row against column, summed, precisely so that two transformations in a row collapse into a single matrix, doing B and then A being the same as doing AB, which is also why a product needs the columns of the first factor to match the rows of the second. Keep two things apart: the multiplication is not commutative in general, since AB and BA often differ, so the order has to be stated; and a matrix is not a determinant, the latter being just a single number computed from a square matrix.
难度 进阶 · 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
#代数 #矩阵
矩阵的特征向量在几何上意味着什么？
What does an eigenvector of a matrix mean geometrically?
把矩阵看成对空间的线性变换，特征向量就是变换后方向不变、只被伸缩（甚至反向）的向量，伸缩倍数就是特征值。这个几何观点让我们直接看懂旋转、缩放、投影等变换，也是主成分分析、结构振动分析以及谷歌 PageRank 排序背后的思想。
Viewing a matrix as a linear transformation, an eigenvector is a direction that is only stretched (or flipped) by the transformation, never rotated; the stretch factor is the eigenvalue. This geometric view makes rotations, scalings, and projections transparent, and underlies PCA, structural vibration analysis, and Google's PageRank.
难度 进阶 · 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
Axler,《Linear Algebra Done Right》第3版,Springer,2015;中译《线性代数应该这样学》人民邮电出版社
S. Axler, Linear Algebra Done Right, 3rd ed., Springer, 2015; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 CROSSREF 标识 10.1007/978-3-319-11080-6：Linear Algebra Don
#特征向量 #特征值 #线性变换
群（group）在数学里是什么？
What is a group in mathematics?
群是一个带二元运算的集合，满足封闭性、结合律、有单位元且每个元素有逆元。例如整数在加法下构成群，它是现代代数研究对称性的基本语言。
A group is a set with a binary operation that is closed, associative, has an identity and inverses. Integers under addition form a group; groups are the basic language for symmetry.
难度 进阶 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
《近世代数》刘绍学
Abstract Algebra by Dummit and Foote
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780135693025 (2004)
#代数 #群 #对称性
向量空间是什么？
What is a vector space?
向量空间是一组对象（向量），在加法和数乘下封闭并满足八条公理，如平面上的箭头或线性方程组的解集合。它是线性代数的核心舞台。
A vector space is a set of vectors closed under addition and scalar multiplication, obeying eight axioms, like arrows in the plane. It is the central setting of linear algebra.
难度 进阶 · 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
Axler,《Linear Algebra Done Right》第3版,Springer,2015;中译《线性代数应该这样学》人民邮电出版社
S. Axler, Linear Algebra Done Right, 3rd ed., Springer, 2015; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 CROSSREF 标识 10.1007/978-3-319-11080-6：Linear Algebra Don
#代数 #向量空间 #线性代数
复数和虚数单位 i 是什么？
What are complex numbers and the unit i?
虚数单位 i 满足 i² = −1。复数写成 a + b i（a、b 为实数），拓展了数的范围，使任意多项式都有根。
The unit i satisfies i²=−1; a complex number a+bi extends reals so every polynomial has roots.
难度 进阶 · 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
#代数 #复数
二次方程的求根公式是什么？
What is the quadratic formula?
对方程 ax²+bx+c=0，求根公式为 x = (−b ± √(b²−4ac)) / (2a)。
For ax²+bx+c=0, the roots are x=(−b±√(b²−4ac))/(2a).
难度 进阶 · 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
#代数 #二次方程
什么是比例？
What is a proportion?
比例表示两个比值相等的关系，如 a/b = c/d，常用于按相同倍数缩放数量。例如地图比例尺就是把实际距离按比例缩小。
A proportion states that two ratios are equal, such as a/b = c/d, and is often used to scale quantities by the same factor. For example, a map's scale shrinks real distances by a fixed proportion.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-11-03
来源 · Source
Beckmann, S., 《中小学数学教师数学》(第4版), Pearson
Beckmann, S., 'Mathematics for Elementary Teachers' (4th ed.), Pearson
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL18708410W：Mathematics for elemen
#比例 #比值 #代数
为什么两个矩阵相乘，交换顺序后结果往往不同？
Why does swapping the order of two matrix products usually change the result?
矩阵代表对空间施加的变换，比如旋转、缩放或镜像。两个矩阵相乘就是“先做一个变换、再做另一个”的合成。先旋转再平移和先平移再旋转，得到的效果显然不一样，所以矩阵乘法一般不满足交换律，相乘时顺序很重要。
A matrix encodes a transformation of space, such as a rotation, a scaling, or a reflection. Multiplying matrices means composing transformations: first apply one, then the other. Rotating and then translating generally gives a different effect than translating and then rotating, so matrix multiplication is usually non-commutative — order matters.
难度 进阶 · 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
#矩阵 #线性变换 #交换律
矩阵的秩怎么帮助我们判断线性方程组有没有解？
How does the rank of a matrix help decide whether a linear system has solutions?
秩是行向量（或列向量）中线性无关的最大个数，也是消元后非零行的数目。方程组 Ax=b 有解的充要条件是 A 的秩等于增广矩阵 [A|b] 的秩；若这个秩再等于未知量个数则解唯一，小于它则有无穷多解（出现自由变量）。零行对应 0=非零 的矛盾，就是无解的情形。
The rank is the largest number of linearly independent rows (or columns), i.e., the count of nonzero rows after elimination. The system Ax=b is solvable iff rank(A) equals the rank of the augmented matrix [A|b]; if that rank also equals the number of unknowns the solution is unique, otherwise infinitely many (free variables) appear. A zero row reading 0=nonzero is exactly the inconsistent case.
难度 进阶 · 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
#矩阵秩 #线性方程组 #增广矩阵
什么是因式分解？
What is factoring?
因式分解是把一个多项式写成几个更简单因式的乘积，例如 x²−1 = (x−1)(x+1)。
Factoring rewrites a polynomial as a product of simpler factors, e.g. x²−1=(x−1)(x+1).
难度 进阶 · 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
#代数 #因式分解
数学里的“群”到底是什么？它为什么能统一描述对称？
What exactly is a group in mathematics, and why does it unify the description of symmetry?
群是“一个集合 + 一种运算”的代数结构，要满足四条规则：运算封闭、满足结合律、有单位元、每个元素有逆元。比如整数在加法下成群，单位元是 0。对称操作的集合也成群——先做哪个对称再哪个，顺序不同结果可能不同，但四个条件依然成立。
A group is an algebraic structure: a set together with an operation obeying four rules — closure, associativity, an identity element, and an inverse for every element. The integers under addition form a group with identity 0. Sets of symmetry operations form groups too: order of composition may matter, but all four conditions still hold.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-03
来源 · Source
《什么是数学》(柯朗)（变换与群）
What Is Mathematics? (Courant), transformations and groups
#群论 #代数结构 #对称
凯莱-哈密顿定理是什么？
What is the Cayley–Hamilton theorem?
它说每个方阵都满足它自己的特征方程：把矩阵代入其特征多项式会得到零矩阵。例如对 2×2 矩阵 A，有 A² − tr(A)A + det(A)I = 0。
Cayley–Hamilton states every square matrix satisfies its own characteristic equation; e.g. for 2×2 A, A²−tr(A)A+det(A)I=0.
难度 进阶 · 适合 小学高段(9-12)
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
#代数 #凯莱哈密顿
预测传染病传播、估算湖水污染扩散，数学家是怎么把现实“翻译”成数学的？
To forecast disease spread or pollution in a lake, how do mathematicians translate reality into mathematics?
建模分几步：先做假设、挑出关键量，比如易感人数、传染率；再把它们的关系写成方程，比如差分或微分方程；然后求解、用真实数据校验，最后根据结果修正模型。模型永远是对现实的简化，它的价值在于“预测与比较”，而建立模型的每一步都建立在严谨的数学推理之上。
Modeling proceeds in stages: make assumptions and identify key quantities such as susceptible individuals and infection rate; write their relations as equations, e.g., difference or differential equations; solve them, validate against real data, then refine. A model is always a simplification of reality; its value lies in prediction and comparison, and each step rests on rigorous mathematical reasoning.
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：此条未指向带版次的可引用著作（通识性内容、机构文件或未展开的英文书名），本库不提供独立书目出处；如需精确引用，需由编辑补全具体篇目与版次。
NCTM（美国数学教师理事会，数学建模）
NCTM (National Council of Teachers of Mathematics), mathematical modeling
#数学建模 #方程 #模型验证
线性代数中的「特征值」和「特征向量」有什么实际意义？
What is the practical meaning of eigenvalues and eigenvectors in linear algebra?
对一个线性变换（矩阵），特征向量是在该变换下只被拉伸或压缩、方向不变的特殊向量，对应的拉伸倍数就是特征值。特征值揭示了系统的主轴、振动频率和稳定性：例如谷歌网页排名（PageRank）就归结为求一个巨大矩阵的主特征向量。它们也是主成分分析（PCA）和数据降维的核心。
For a linear transformation (matrix), an eigenvector is a special vector whose direction is unchanged, only scaled, by the transformation, and the scaling factor is its eigenvalue. Eigenvalues reveal a system's principal axes, vibration frequencies, and stability—Google's PageRank, for instance, reduces to finding the leading eigenvector of a huge matrix. They are also central to principal component analysis (PCA) and data reduction.
难度 进阶 · 适合 小学高段(9-12)
CC BY 4.0 · 已核验 · Verified 2026-11-03
来源 · Source
吉尔伯特·斯特朗《线性代数及其应用》（MIT）；美国数学学会（AMS）
Gilbert Strang, Linear Algebra and Its Applications (MIT); AMS
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL3285496W：Linear algebra and its
#特征值 #特征向量 #线性变换 #主成分分析
多项式因式分解在代数中为何重要，它与根的存在有何联系？
Why is polynomial factorization important in algebra, and how is it tied to the existence of roots?
因式分解把多项式拆成低次不可约因子的乘积，每一个一次因子对应一个根，由代数基本定理保证复数域上完全可分解。它把求根问题与结构分析联系起来，是编码与方程求解的基础。杜米特与富特以及范德瓦尔登的著作都系统论述。
Factorization splits a polynomial into products of lower-degree irreducible factors, each linear factor corresponding to a root, fully guaranteed over the complex numbers by the fundamental theorem of algebra. It links root-finding with structural analysis and underlies coding and equation solving. Dummit & Foote and van der Waerden treat it systematically.
难度 进阶 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Dummit 与 Foote《Abstract Algebra》；van der Waerden《Algebra》
Dummit & Foote, Abstract Algebra; van der Waerden, Algebra
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780135693025 (2004)
#多项式因式分解 #代数基本定理 #不可约因子
什么是向量？
What is a vector?
向量既有大小又有方向，可表示位移、速度等。在坐标中常用带箭头的数列表示，如 (3, 4)。
A vector has magnitude and direction, representing displacement or velocity; written as an arrow like (3,4).
难度 进阶 · 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
#代数 #向量
多项式里，什么时候 (x-r) 是 f(x) 的一个因式？
When is (x-r) a factor of a polynomial f(x)?
当且仅当把 r 代入多项式得到 f(r)=0。比如 f(x)=x^2-5x+6，代入 x=2 得 0，所以 (x-2) 是它的因式。这个结论叫因式定理，它把“求根”和“因式分解”连在了一起：找到了根，就找到了因式。
Exactly when substituting r gives f(r)=0. For f(x)=x^2-5x+6, plugging in 2 yields 0, so (x-2) is a factor. This is the Factor Theorem, and it links roots to factorization: find a root and you have found a factor.
难度 进阶 · 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
#因式定理 #多项式 #求根
什么是函数？
What is a function?
函数是把每个输入值对应到唯一输出值的规则。例如 f(x)=2x 把输入3变成输出6。
A function maps each input to exactly one output, e.g. f(x)=2x sends 3 to 6.
难度 进阶 · 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
#代数 #函数
多项式的次数指什么？
What is the degree of a polynomial?
多项式的次数是其最高次项的指数。例如 3x² + 2x + 1 的次数是2。它决定图像大致形状与根的个数上界。
The degree is the highest exponent; 3x²+2x+1 has degree 2, bounding the number of roots and shaping the graph.
难度 进阶 · 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
#代数 #多项式
算术—几何平均值不等式（AM-GM）为什么成立？等号何时取到？
Why does the arithmetic–geometric mean inequality (AM-GM) hold, and when is equality attained?
因对数严格凹，詹森不等式给出 ln[(x_1+…+x_n)/n] ≥ [ln x_1+…+ln x_n]/n = ln((x_1⋯x_n)^(1/n))，取指数即知算术平均不小于几何平均。凹性严格，等号当且仅当诸数全相等。此式是极值问题的基石：总和给定时均分使乘积最大，乘积给定时均分使总和最小。
Since the logarithm is strictly concave, Jensen's inequality applied to positive numbers x_1,…,x_n gives ln[(x_1+…+x_n)/n] ≥ (ln x_1+…+ln x_n)/n = ln((x_1⋯x_n)^(1/n)); exponentiating both sides yields AM ≥ GM. Strict concavity forces equality if and only if all the numbers are equal. The inequality is a cornerstone of extremal problems: for a fixed sum, equal numbers maximize the product, and for a fixed product, equal numbers minimize the sum.
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：作者 / 书名 / 版次 / 出版社 / 年份齐备，仅缺独立比对证据；已在可信度报告单列披露，不再重复核验。
G. H. Hardy、J. E. Littlewood 与 G. Pólya《不等式》第2版 (Inequalities)，剑桥大学出版社，1952。
G. H. Hardy, J. E. Littlewood & G. Pólya, Inequalities, 2nd edition, Cambridge University Press, 1952.
#均值不等式 #詹森不等式 #凹性 #极值问题
为什么分数乘分数，等于“分子乘分子、分母乘分母”？
Why does multiplying fractions equal 'numerator times numerator, denominator times denominator'?
把 1/2 看成“一半”，则 (1/2)×(1/3) 是“一半的三分之一”，即整体分成 2×3=6 份取 1 份，得 1/6。代数上 a/b · c/d = (ac)/(bd) 正是“把两块各均分后再交叉拼成 b×d 格”的结果，所以分子分母分别相乘。
See 1/2 as 'half'; then (1/2)×(1/3) is 'a third of a half'—the whole split into 2×3=6 equal parts, taking 1, gives 1/6. Algebraically a/b·c/d=(ac)/(bd) is exactly what you get by cross-combining the b and d equal divisions, so numerators and denominators multiply separately.
难度 入门 · 适合 小学低段(6-8)
CC BY 4.0 · 已核验 · Verified 2026-11-03
来源 · Source
Beckmann, S., 《中小学数学教师数学》(第4版), Pearson
Beckmann, S., 'Mathematics for Elementary Teachers' (4th ed.), Pearson
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL18708410W：Mathematics for elemen
#分数乘法 #分子分母 #等分 #代数
方程 x+3=5 中 x 是几？
In the equation x+3=5, what is x?
方程两边同减3，得到 x = 5−3 = 2。
Subtract 3 from both sides to get x = 5−3 = 2.
难度 入门 · 适合 小学低段(6-8)
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
#代数 #方程
什么是倒数？
What is a reciprocal (multiplicative inverse)?
一个数的倒数是与它相乘得1的数。例如 3 的倒数是 1/3，因为 3 × 1/3 = 1。0没有倒数。
The reciprocal of x is 1/x, since x·(1/x)=1; 0 has no reciprocal.
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
#代数 #倒数
什么是数轴？
What is a number line?
数轴是一条画了刻度的直线，用点表示数：右边的数总比左边的大。它是理解加减和大小的好工具。
A number line is a straight line with marks; each point is a number, right means greater. It helps visualize addition and order.
难度 入门 · 适合 小学低段(6-8)
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
#代数 #数轴
什么是等号？
What is the equals sign?
等号“=”表示它两边的数量或表达式相等，是数学中表达平衡与等价关系的基本符号。等式可以传递：若 a = b 且 b = c，则 a = c。
The equals sign = indicates that the quantities or expressions on its two sides are equal, a basic symbol for balance and equivalence in mathematics. Equality is transitive: if a = b and b = c, then a = c.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Enderton, H. B., 《逻辑的数学导引》(第2版), Academic Press
Enderton, H. B., 'A Mathematical Introduction to Logic' (2nd ed.), Academic Press
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780122384523 (2001)
#等号 #等式 #代数