The prime distribution Theorem

黎曼猜想文献黎曼猜想最早由德国数学家贝尔纳德·黎曼在1859年的一篇题为《关于函数的复变量》的论文中提出。

这篇论文发表在《哥廷根科学与文学学报》（Göttingische Gelehrte Anzeigen）上。

然而，黎曼在这篇论文中并没有直接论述黎曼猜想，而是提出了一些相关的概念和定理，为后来对黎曼猜想的研究奠定了基础。

黎曼猜想的原文（德文）可以在以下链接中找到：https://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZe ta.html除了黎曼的原始论文，还有许多对黎曼猜想的研究文献。

以下是一些关于黎曼猜想的重要文献：1. Hardy, G.H., & Littlewood, J.E. (1914). "Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes." Acta Mathematica, 37, 193-239.2. Newman, M. (1984). "The Riemann Hypothesis." Bulletin of the American Mathematical Society, 1(2), 367-399.3. Conrey, J. B., & Ghosh, A. (1999). "An estimate for the remainder term in the prime number theorem." Journal of the London Mathematical Society, 60(2), 257-266.4. Odlyzko, A. M. (1987). "The 10^20-th zero of the Riemann zeta function and 175 million of its neighbors." AT&T Bell Laboratories Technical Journal, 66(6), 673-686.这些文献提供了从不同方面对黎曼猜想进行研究的观点和方法。

质数与合数的英语Primes and CompositesThe world of numbers is a fascinating realm, filled with intricate patterns and captivating relationships. At the heart of this numerical landscape lie two distinct categories: primes and composites. These two groups, though seemingly simple, hold the key to unlocking the deeper mysteries of mathematics.Primes, the building blocks of the number system, are those positive integers greater than 1 that are only divisible by 1 and themselves. These special numbers, such as 2, 3, 5, 7, 11, and 13, possess a unique and unwavering property – they cannot be expressed as the product of two smaller positive integers. This fundamental characteristic sets them apart from their counterparts, the composites.Composites, on the other hand, are positive integers greater than 1 that can be expressed as the product of two or more smaller positive integers. These numbers, such as 4, 6, 8, 9, and 10, are not as "pure" as primes, as they can be broken down into smaller factors. Composites can be further divided into two subcategories: squarenumbers, which are the product of two equal integers, and non-square numbers, which are the product of two distinct integers.The relationship between primes and composites is a delicate and intricate dance, with each group playing a vital role in the grand scheme of mathematics. Primes, being the building blocks of the number system, serve as the foundation upon which all other numbers are constructed. They are the atoms of the numerical world, the indivisible entities that cannot be broken down any further. Composites, on the other hand, are the molecules – the complex structures formed by the combination of these prime "atoms."The study of primes and composites has captivated mathematicians for centuries, leading to the development of numerous fascinating theories and applications. One of the most intriguing aspects of this field is the distribution of primes throughout the number system. While primes may appear to be randomly scattered, mathematicians have discovered patterns and regularities in their occurrence, leading to the formulation of the Prime Number Theorem and the Riemann Hypothesis, two of the most significant unsolved problems in mathematics.The importance of primes and composites extends far beyond the realm of pure mathematics. These numerical categories have found applications in various fields, including cryptography, where theunique properties of primes are exploited to create secure communication systems. Primes are also crucial in the study of number theory, a branch of mathematics that delves into the intricate relationships between different types of numbers and their underlying structures.In the realm of computer science, the study of primes and composites has given rise to efficient algorithms for factoring large numbers, a crucial task in the field of cryptography. The ability to quickly identify the prime factors of a composite number has significant implications for the security of digital communication and the protection of sensitive information.Moreover, the exploration of primes and composites has led to the development of fascinating mathematical concepts, such as the Goldbach Conjecture, which states that every even integer greater than 2 can be expressed as the sum of two primes. This conjecture, which remains unproven despite its apparent simplicity, continues to captivate mathematicians and stimulate further research.In conclusion, the world of primes and composites is a rich and captivating domain, filled with intricate patterns, profound insights, and far-reaching applications. As we delve deeper into the study of these numerical categories, we uncover the hidden beauty and complexity that lie at the heart of the mathematical universe. Theexploration of primes and composites promises to yield new discoveries, challenging our understanding of the numerical landscape and pushing the boundaries of human knowledge.。

Prime number 质数Consecutive number 连续整数 Consecutive even integer 连续偶数Cross multiply 交叉相乘 Coefficient 系数Complete quadratic equation 完全二次方程 Complementary function 余函数Constant 常数 Coordinate system 坐标系 Decimal 小数Decimal point 小数点 Decimal fraction 纯小数Decimal arithmetic 十进制运算 Decimal system/decimal scale 十进制Denominator 分母 Difference 差 Direct proportion 正比Divide 除 Divided evenly 被整除 Differential 微分Distinct 不同的 Dividend 被除数，红利 Division 除法Division sign 除号 Divisor 因子，除数 Divisible 可被整除的Equivalent fractions 等值分数 Equivalent equation 等价方程式Equivalence relation 等价关系 Even integer/number 偶数Exponent指数，幂Equation方程Equationofthefirstdegree一次方程Endpoint端点Estimation近似Factor因子Factorablequadraticequation可因式分解的二次方程Incompletequadraticequation不完全二次方程Factorial阶乘Factorization因式分解Geometricmean几何平均数Graphtheory图论Inequality不等式Improperfraction假分数Infinitedecimal无穷小数Inverseproportion反比Irrationalnumber无理数Infinitesimalcalculus微积分Infinity无穷大Infinitesimal无穷小Integerable可积分的Integral积分Integraldomain整域Integrand被积函数Integratingfactor积分因子Inversefunction反函数Inverse/reciprocal倒数 Leastcommondenominator最小公分母Leastcommonmultiple最小公倍数 Literalcoefficient字母系数Liketerms同类项 Linear线的 Minuend被减数Subtrahend被减数 Mixeddecimal混合小数 Mixednumber带分数Minor子行列式 Multiplicand被乘数 Multiplication乘法Multiplier乘数 Monomial单项式 Mean平均数Mode众数 Median中数 Negativenumber负数 Numerator分子Nullset空集 Numbertheory数论 Numberline数轴Numericalanalysis数值分析 Naturallogarithm自然对数Naturalnumber自然数 Nonnegative非负数 Originalequation原方程Ordinaryscale十进制 Ordinal序数 Percentage百分比Parentheses括号Polynomial多项式Power乘方Product积Properfraction真分数Proportion比例Permutation排列Propersubset真子集Primefactor质因子Progression数列Quadrant象限Quadraticequation二次方程Quarter四分之一Ratio比率Realnumber实数Roundoff四舍五入Roundto四舍五入Root根Radicalsign根号Rootsign根号Recurringdecimal循环小数Sequence数列Similarterms同类项Tens十位Tenths十分位Trinomial三相式Units个位Unit单位Weightedaverage加权平均值Union并集Yard码Wholenumber整数Mutuallyexclusive互相排斥Independentevents相互独立事件Probability概率Combination组合Standarddeviation标准方差Range值域Frequencydistribution频率分布Domain定义域Bargraph柱图Geometryterms：Anglebisector角平分线Adjacentangle邻角Alternateangel内错角Acuteangle锐角Obtuseangle钝角Bisect角平分线Adjacentvertices相邻顶点Arc弧Altitude高Arm直角三角形的股Complexplane复平面Convexpolygon凸多边形Complementaryangle余角Cube立方体Centralangle圆心角Circle圆Clockwise顺时钟方向Counterclockwise逆时钟方向Chord弦Circularcylinder圆柱体Congruent全等的Correspondingangle同位角Circumference周长Concentriccircles同心圆Circlegraph扇面图Cone圆锥Circumscribe外切Inscribe内切Diagonal对角线Decagon十边形Hexagon六边形Nonagon九边形Octagon八边形Pentagon五边形Quadrilateral四边形Polygon多边形Diameter直径Edge棱Equilateraltriangle等边三角形Exteriorangle外角/内角Extent维数Exterioranglesonthesamesideofthetransversal同旁外角Hypotenuse三角形的斜边Intercept截距Includedangle夹角Intersect相交Inscribedtriangle内接三角形Isoscelestriangle等腰三角形Midpoint中点Minoraxis短轴Origin原点Oblique斜三角形Planegeometry平面几何Oblateness椭圆Parallelogram平行四边形Parallellines平行线Perpendicular垂直的Pythagoreantheorem勾股定理Piechart扇图Quadrihedron三角锥Radius半径Rectangle长方形Regularpolygon正多边形Rhombus菱形Rightcircularcylinder直圆柱体Righttriangle直角三角形Rightangle直角Rectangularsolid正多面体Regularprism正棱柱Regularpyramid正棱锥Regularsolid/polyhedron正多面体Slope斜率SphereSide边长Segmentofacircle弧形Semicircle半圆Solid立体 Square正方形，平方GMAT考试的数学部分包括37个多项选择题，要求在75分钟内完成。

abbreviation 简写符号；简写abscissa 横坐标absolute complement 绝对补集absolute error 绝对误差absolute inequality 绝不等式absolute maximum 绝对极大值absolute minimum 绝对极小值absolute monotonic 绝对单调absolute value 绝对值accelerate 加速acceleration 加速度acceleration due to gravity 重力加速度; 地心加速度accumulation 累积accumulative 累积的accuracy 准确度act on 施于action 作用; 作用力acute angle 锐角acute-angled triangle 锐角三角形add 加addition 加法addition formula 加法公式addition law 加法定律addition law(of probability) （概率）加法定律additive inverse 加法逆元; 加法反元additive property 可加性adjacent angle 邻角adjacent side 邻边adjoint matrix 伴随矩阵algebra 代数algebraic 代数的algebraic equation 代数方程algebraic expression 代数式algebraic fraction 代数分式；代数分数式algebraic inequality 代数不等式algebraic number 代数数algebraic operation 代数运算algebraically closed 代数封闭algorithm 算法系统; 规则系统alternate angle （交）错角alternate segment 内错弓形alternating series 交错级数alternative hypothesis 择一假设; 备择假设; 另一假设altitude 高；高度；顶垂线；高线ambiguous case 两义情况；二义情况amount 本利和；总数analysis 分析；解析analytic geometry 解析几何angle 角angle at the centre 圆心角angle at the circumference 圆周角angle between a line and a plane 直与平面的交角angle between two planes 两平面的交角angle bisection 角平分angle bisector 角平分线；分角线angle in the alternate segment 交错弓形的圆周角angle in the same segment 同弓形内的圆周角angle of depression 俯角angle of elevation 仰角angle of friction 静摩擦角; 极限角angle of greatest slope 最大斜率的角angle of inclination 倾斜角angle of intersection 相交角；交角angle of projection 投射角angle of rotation 旋转角angle of the sector 扇形角angle sum of a triangle 三角形内角和angles at a point 同顶角angular displacement 角移位angular momentum 角动量angular motion 角运动angular velocity 角速度annum(X% per annum) 年（年利率X%）anti-clockwise direction 逆时针方向；返时针方向anti-clockwise moment 逆时针力矩anti-derivative 反导数; 反微商anti-logarithm 逆对数；反对数anti-symmetric 反对称apex 顶点approach 接近；趋近approximate value 近似值approximation 近似；略计；逼近Arabic system 阿刺伯数字系统arbitrary 任意arbitrary constant 任意常数arc 弧arc length 弧长arc-cosine function 反余弦函数arc-sin function 反正弦函数arc-tangent function 反正切函数area 面积Argand diagram 阿根图, 阿氏图argument (1)论证; (2)辐角argument of a complex number 复数的辐角argument of a function 函数的自变量arithmetic 算术arithmetic mean 算术平均；等差中顶；算术中顶arithmetic progression 算术级数；等差级数arithmetic sequence 等差序列arithmetic series 等差级数arm 边array 数组; 数组arrow 前号ascending order 递升序ascending powers of X X 的升幂assertion 断语; 断定associative law 结合律assumed mean 假定平均数assumption 假定；假设asymmetrical 非对称asymptote 渐近asymptotic error constant 渐近误差常数at rest 静止augmented matrix 增广矩阵auxiliary angle 辅助角auxiliary circle 辅助圆auxiliary equation 辅助方程average 平均；平均数；平均值average speed 平均速率axiom 公理axiom of existence 存在公理axiom of extension 延伸公理axiom of inclusion 包含公理axiom of pairing 配对公理axiom of power 幂集公理axiom of specification 分类公理axiomatic theory of probability 概率公理论axis 轴axis of parabola 抛物线的轴axis of revolution 旋转轴axis of rotation 旋转轴axis of symmetry 对称轴back substitution 回代bar chart 棒形图；条线图；条形图；线条图base （1）底；（2）基；基数base angle 底角base area 底面base line 底线base number 底数；基数base of logarithm 对数的底basis 基Bayes' theorem 贝叶斯定理bearing 方位（角）；角方向（角）bell-shaped curve 钟形图belong to 属于Bernoulli distribution 伯努利分布Bernoulli trials 伯努利试验bias 偏差；偏倚biconditional 双修件式; 双修件句bijection 对射; 双射; 单满射bijective function 对射函数; 只射函数billion 十亿bimodal distribution 双峰分布binary number 二进数binary operation 二元运算binary scale 二进法binary system 二进制binomial 二项式binomial distribution 二项分布binomial expression 二项式binomial series 二项级数binomial theorem 二项式定理bisect 平分；等分bisection method 分半法；分半方法bisector 等分线；平分线Boolean algebra 布尔代数boundary condition 边界条件boundary line 界（线）；边界bounded 有界的bounded above 有上界的；上有界的bounded below 有下界的；下有界的bounded function 有界函数bounded sequence 有界序列brace 大括号bracket 括号breadth 阔度broken line graph 折线图calculation 计算calculator 计算器；计算器calculus (1) 微积分学; (2) 演算cancel 消法；相消canellation law 消去律canonical 典型; 标准capacity 容量cardioid 心脏Cartesian coordinates 笛卡儿坐标Cartesian equation 笛卡儿方程Cartesian plane 笛卡儿平面Cartesian product 笛卡儿积category 类型；范畴catenary 悬链Cauchy sequence 柯西序列Cauchy's principal value 柯西主值Cauchy-Schwarz inequality 柯西- 许瓦尔兹不等式central limit theorem 中心极限定理central line 中线central tendency 集中趋centre 中心；心centre of a circle 圆心centre of gravity 重心centre of mass 质量中心centrifugal force 离心力centripedal acceleration 向心加速度centripedal force force 向心力centroid 形心；距心certain event 必然事件chain rule 链式法则chance 机会change of axes 坐标轴的变换change of base 基的变换change of coordinates 坐标轴的变换change of subject 主项变换change of variable 换元；变量的换characteristic equation 特征(征)方程characteristic function 特征(征)函数characteristic of logarithm 对数的首数; 对数的定位部characteristic root 特征(征)根chart 图；图表check digit 检验数位checking 验算chord 弦chord of contact 切点弦circle 圆circular 圆形；圆的circular function 圆函数；三角函数circular measure 弧度法circular motion 圆周运动circular permutation 环形排列; 圆形排列; 循环排列circumcentre 外心；外接圆心circumcircle 外接圆circumference 圆周circumradius 外接圆半径circumscribed circle 外接圆cissoid 蔓叶class 区；组；类class boundary 组界class interval 组区间；组距class limit 组限；区限class mark 组中点；区中点classical theory of probability 古典概率论classification 分类clnometer 测斜仪clockwise direction 顺时针方向clockwise moment 顺时针力矩closed convex region 闭凸区域closed interval 闭区间coaxial 共轴coaxial circles 共轴圆coaxial system 共轴系coded data 编码数据coding method 编码法co-domain 上域coefficient 系数coefficient of friction 摩擦系数coefficient of restitution 碰撞系数; 恢复系数coefficient of variation 变差系数cofactor 余因子; 余因式cofactor matrix 列矩阵coincide 迭合；重合collection of terms 并项collinear 共线collinear planes 共线面collision 碰撞column (1)列；纵行；(2) 柱column matrix 列矩阵column vector 列向量combination 组合common chord 公弦common denominator 同分母；公分母common difference 公差common divisor 公约数；公约common factor 公因子；公因子common logarithm 常用对数common multiple 公位数；公倍common ratio 公比common tangent 公切commutative law 交换律comparable 可比较的compass 罗盘compass bearing 罗盘方位角compasses 圆规compasses construction 圆规作图compatible 可相容的complement 余；补余complement law 补余律complementary angle 余角complementary equation 补充方程complementary event 互补事件complementary function 余函数complementary probability 互补概率complete oscillation 全振动completing the square 配方complex conjugate 复共轭complex number 复数complex unmber plane 复数平面complex root 复数根component 分量component of force 分力composite function 复合函数; 合成函数composite number 复合数；合成数composition of mappings 映射构合composition of relations 复合关系compound angle 复角compound angle formula 复角公式compound bar chart 综合棒形图compound discount 复折扣compound interest 复利；复利息compound probability 合成概率compound statement 复合命题; 复合叙述computation 计算computer 计算机；电子计算器concave 凹concave downward 凹向下的concave polygon 凹多边形concave upward 凹向上的concentric circles 同心圆concept 概念conclusion 结论concurrent 共点concyclic 共圆concyclic points 共圆点condition 条件conditional 条件句；条件式conditional identity 条件恒等式conditional inequality 条件不等式conditional probability 条件概率cone 锥；圆锥（体）confidence coefficient 置信系数confidence interval 置信区间confidence level 置信水平confidence limit 置信极限confocal section 共焦圆锥曲congruence (1)全等；(2)同余congruence class 同余类congruent 全等congruent figures 全等图形congruent triangles 全等三角形conic 二次曲; 圆锥曲conic section 二次曲; 圆锥曲conical pendulum 圆锥摆conjecture 猜想conjugate 共轭conjugate axis 共轭conjugate diameters 共轭轴conjugate hyperbola 共轭(直)径conjugate imaginary / complex number 共轭双曲conjugate radical 共轭虚/复数conjugate surd 共轭根式; 共轭不尽根conjunction 合取connective 连词connector box 捙接框consecutive integers 连续整数consecutive numbers 连续数；相邻数consequence 结论；推论consequent 条件；后项conservation of energy 能量守恒conservation of momentum 动量守恒conserved 守恒consistency condition 相容条件consistent 一贯的；相容的consistent estimator 相容估计量constant 常数constant acceleration 恒加速度constant force 恒力constant of integration 积分常数constant speed 恒速率constant term 常项constant velocity 怛速度constraint 约束；约束条件construct 作construction 作图construction of equation 方程的设立continued proportion 连比例continued ratio 连比continuity 连续性continuity correction 连续校正continuous 连续的continuous data 连续数据continuous function 连续函数continuous proportion 连续比例continuous random variable 连续随机变量contradiction 矛盾converge 收敛convergence 收敛性convergent 收敛的convergent iteration 收敛的迭代convergent sequence 收敛序列convergent series 收敛级数converse 逆(定理)converse of a relation 逆关系converse theorem 逆定理conversion 转换convex 凸convex polygon 凸多边形convexity 凸性coordinate 坐标coordinate geometry 解析几何；坐标几何coordinate system 坐标系系定理；系；推论coplanar 共面coplanar forces 共面力coplanar lines 共面co-prime 互质; 互素corollary 系定理; 系; 推论correct to 准确至；取值至correlation 相关correlation coefficient 相关系数correspondence 对应corresponding angles (1)同位角；(2)对应角corresponding element 对应边corresponding sides 对应边cosecant 余割cosine 余弦cosine formula 余弦公式cost price 成本cotangent 余切countable 可数countable set 可数集countably infinite 可数无限counter clockwise direction 逆时针方向；返时针方向counter example 反例counting 数数；计数couple 力偶Carmer's rule 克莱玛法则criterion 准则critical point 临界点critical region 临界域cirtical value 临界值cross-multiplication 交叉相乘cross-section 横切面；横截面；截痕cube 正方体；立方；立方体cube root 立方根cubic 三次方；立方；三次(的)cubic equation 三次方程cubic roots of unity 单位的立方根cuboid 长方体；矩体cumulative 累积的cumulative distribution function 累积分布函数cumulative frequecy 累积频数；累积频率cumulative frequency curve 累积频数曲cumulative frequcncy distribution 累积频数分布cumulative frequency polygon 累积频数多边形；累积频率直方图curvature of a curve 曲线的曲率curve 曲线curve sketching 曲线描绘(法)curve tracing 曲线描迹(法)curved line 曲线curved surface 曲面curved surface area 曲面面积cyclic expression 输换式cyclic permutation 圆形排列cyclic quadrilateral 圆内接四边形cycloid 旋输线; 摆线cylinder 柱；圆柱体cylindrical 圆柱形的damped oscillation 阻尼振动data 数据De Moivre's theorem 棣美弗定理De Morgan's law 德摩根律decagon 十边形decay 衰变decay factor 衰变因子decelerate 减速decelaration 减速度decile 十分位数decimal 小数decimal place 小数位decimal point 小数点decimal system 十进制decision box 判定框declarative sentence 说明语句declarative statement 说明命题decoding 译码decrease 递减decreasing function 递减函数；下降函数decreasing sequence 递减序列；下降序列decreasing series 递减级数；下降级数decrement 减量deduce 演绎deduction 推论deductive reasoning 演绎推理definite 确定的；定的definite integral 定积分definition 定义degenerated conic section 降级锥曲线degree (1) 度; (2) 次degree of a polynomial 多项式的次数degree of accuracy 准确度degree of confidence 置信度degree of freedom 自由度degree of ODE 常微分方程次数degree of precision 精确度delete 删除; 删去denary number 十进数denominator 分母dependence (1)相关; (2)应变dependent event(s) 相关事件; 相依事件; 从属事件dependent variable 应变量; 应变数depreciation 折旧derivable 可导derivative 导数derived curve 导函数曲线derived function 导函数derived statistics 推算统计资料; 派生统计资料descending order 递降序descending powers of x x的降序descriptive statistics 描述统计学detached coefficients 分离系数(法)determinant 行列式deviation 偏差; 变差deviation from the mean 离均差diagonal 对角线diagonal matrix 对角矩阵diagram 图; 图表diameter 直径diameter of a conic 二次曲线的直径difference 差difference equation 差分方程difference of sets 差集differentiable 可微differential 微分differential coefficient 微商; 微分系数differential equation 微分方程differential mean value theorem 微分中值定理differentiate 求...的导数differentiate from first principle 从基本原理求导数differentiation 微分法digit 数字dimension 量; 量网; 维(数)direct impact 直接碰撞direct image 直接像direct proportion 正比例direct tax, direct taxation 直接税direct variation 正变(分)directed angle 有向角directed line 有向直线directed line segment 有向线段directed number 有向数direction 方向; 方位direction angle 方向角direction cosine 方向余弦direction number 方向数direction ratio 方向比directrix 准线Dirichlet function 狄利克来函数discontinuity 不连续性discontinuous 间断(的);连续(的); 不连续(的) discontinuous point 不连续点discount 折扣discrete 分立; 离散discrete data 离散数据; 间断数据discrete random variable 间断随机变数discrete uniform distribution 离散均匀分布discriminant 判别式disjoint 不相交的disjoint sets 不相交的集disjunction 析取dispersion 离差displacement 位移disprove 反证distance 距离distance formula 距离公式distinct roots 相异根distincr solution 相异解distribution 公布distributive law 分配律diverge 发散divergence 发散(性)divergent 发散的divergent iteration 发散性迭代divergent sequence 发散序列divergent series 发散级数divide 除dividend (1)被除数；(2)股息divisible 可整除division 除法division algorithm 除法算式divisor 除数；除式；因子divisor of zero 零因子dodecagon 十二边形domain 定义域dot 点dot product 点积double angle 二倍角double angle formula 二倍角公式double root 二重根dual 对偶duality (1)对偶性; (2) 双重性due east/ south/ west /north 向东/ 南/ 西/ 北dynamics 动力学eccentric angle 离心角eccentric circles 离心圆eccentricity 离心率echelon form 梯阵式echelon matrix 梯矩阵edge 棱；边efficient estimator 有效估计量effort 施力eigenvalue 本征值eigenvector 本征向量elastic body 弹性体elastic collision 弹性碰撞elastic constant 弹性常数elastic force 弹力elasticity 弹性element 元素elementary event 基本事件elementary function 初等函数elementary row operation 基本行运算elimination 消法elimination method 消去法；消元法ellipse 椭圆ellipsiod 椭球体elliptic function 椭圆函数elongation 伸张；展empirical data 实验数据empirical formula 实验公式empirical probability 实验概率；经验概率empty set 空集encoding 编码enclosure 界限end point 端点energy 能; 能量entire surd 整方根epicycloid 外摆线equal 相等equal ratios theorem 等比定理equal roots 等根equal sets 等集equality 等(式)equality sign 等号equation 方程equation in one unknown 一元方程equation in two unknowns(variables) 二元方程equation of a straight line 直线方程equation of locus 轨迹方程equiangular 等角(的)equidistant 等距(的)equilateral 等边(的)equilateral polygon 等边多边形equilateral triangle 等边三角形equilibrium 平衡equiprobable 等概率的equiprobable space 等概率空间equivalence 等价equivalence class 等价类equivalence relation 等价关系equivalent 等价(的)error 误差error allowance 误差宽容度error estimate 误差估计error term 误差项error tolerance 误差宽容度escribed circle 旁切圆estimate 估计；估计量estimator 估计量Euclidean algorithm 欧几里德算法Euclidean geometry 欧几里德几何Euler's formula 尤拉公式；欧拉公式evaluate 计值even function 偶函数even number 偶数evenly distributed 均匀分布的event 事件exact 真确exact differential form 恰当微分形式exact solution 准确解；精确解；真确解exact value 法确解；精确解；真确解example 例excentre 外心exception 例外excess 起exclusive 不包含exclusive disjunction 不包含性析取exclusive events 互斥事件exercise 练习exhaustive event(s) 彻底事件existential quantifier 存在量词expand 展开expand form 展开式expansion 展式expectation 期望expectation value, expected value 期望值；预期值experiment 实验；试验experimental 试验的experimental probability 实验概率explicit function 显函数exponent 指数exponential function 指数函数exponential order 指数阶; 指数级express…in terms of… 以………表达expression 式；数式extension 外延；延长；扩张；扩充extension of a function 函数的扩张exterior angle 外角external angle bisector 外分角external point of division 外分点extreme point 极值点extreme value 极值extremum 极值face 面factor 因子；因式；商factor method 因式分解法factor theorem 因子定理；因式定理factorial 阶乘factorization 因子分解；因式分解factorization of polynomial 多项式因式分解fallacy 谬误FALSE 假(的)falsehood 假值family 族family of circles 圆族family of concentric circles 同心圆族family of straight lines 直线族feasible solution 可行解；容许解Fermat's last theorem 费尔马最后定理Fibonacci number 斐波那契数；黄金分割数Fibonacci sequence 斐波那契序列fictitious mean 假定平均数figure (1)图(形)；(2)数字final velocity 末速度finite 有限finite dimensional vector space 有限维向量空间finite population 有限总体finite probability space 有限概率空间finite sequence 有限序列finite series 有限级数finite set 有限集first approximation 首近似值first derivative 一阶导数first order differential equation 一阶微分方程first projection 第一投影; 第一射影first quartile 第一四分位数first term 首项fixed deposit 定期存款fixed point 定点fixed point iteration method 定点迭代法fixed pulley 定滑轮flow chart 流程图focal axis 焦轴focal chord 焦弦focal length 焦距focus(foci) 焦点folium of Descartes 笛卡儿叶形线foot of perpendicular 垂足for all X 对所有Xfor each /every X 对每一Xforce 力forced oscillation 受迫振动form 形式；型formal proof 形式化的证明format 格式；规格formula(formulae) 公式four leaved rose curve 四瓣玫瑰线four rules 四则four-figure table 四位数表fourth root 四次方根fraction 分数；分式fraction in lowest term 最简分数fractional equation 分式方程fractional index 分数指数fractional inequality 分式不等式free fall 自由下坠free vector 自由向量; 自由矢量frequency 频数；频率frequency distribution 频数分布；频率分布frequency distribution table 频数分布表frequency polygon 频数多边形；频率多边形friction 摩擦; 摩擦力frictionless motion 无摩擦运动frustum 平截头体fulcrum 支点function 函数function of function 复合函数；迭函数functional notation 函数记号fundamental theorem of algebra 代数基本定理fundamental theorem of calculus 微积分基本定理gain 增益；赚；盈利gain perent 赚率；增益率；盈利百分率game （1）对策；（2）博奕Gaussian distribution 高斯分布Gaussian elimination 高斯消去法general form 一般式；通式general solution 通解；一般解general term 通项generating function 母函数; 生成函数generator (1)母线; (2)生成元geoborad 几何板geometric distribution 几何分布geometric mean 几何平均数；等比中项geometric progression 几何级数；等比级数geometric sequence 等比序列geometric series 等比级数geometry 几何；几何学given 给定；已知global 全局; 整体global maximum 全局极大值; 整体极大值global minimum 全局极小值; 整体极小值golden section 黄金分割grade 等级gradient (1)斜率；倾斜率；(2)梯度grand total 总计graph 图像；图形；图表graph paper 图表纸graphical method 图解法graphical representation 图示；以图样表达graphical solution 图解gravitational acceleration 重力加速度gravity 重力greatest term 最大项greatest value 最大值grid lines 网网格线group 组；grouped data 分组数据；分类数据grouping terms 并项；集项growth 增长growth factor 增长因子half angle 半角half angle formula 半角公式half closed interval 半闭区间half open interval 半开区间harmonic mean (1) 调和平均数; (2) 调和中项harmonic progression 调和级数head 正面（钱币）height 高（度）helix 螺旋线hemisphere 半球体；半球heptagon 七边形Heron's formula 希罗公式heterogeneous (1)参差的; (2)不纯一的hexagon 六边形higher order derivative 高阶导数highest common factor(H.C.F) 最大公因子；最高公因式；最高公因子Hindu-Arabic numeral 阿刺伯数字histogram 组织图；直方图；矩形图Holder's Inequality 赫耳德不等式homogeneous 齐次的homogeneous equation 齐次方程Hooke's law 虎克定律horizontal 水平的；水平horizontal asymptote 水平渐近线horizontal component 水平分量horizontal line 横线；水平线horizontal range 水平射程hyperbola 双曲线hyperbolic function 双曲函数hypergeometric distribution 超几何分布hypocycloid 内摆线hypotenuse 斜边hypothesis 假设hypothesis testing 假设检验hypothetical syllogism 假设三段论hypotrochoid 次内摆线idempotent 全幂等的identical 全等；恒等identity 等（式）identity element 单位元identity law 同一律identity mapping 恒等映射identity matrix 恒等矩阵identity relation 恒等关系式if and only if/iff 当且仅当；若且仅若if…, then 若….则；如果…..则illustration 例证；说明image 像点；像image axis 虚轴imaginary circle 虚圆imaginary number 虚数imaginary part 虚部imaginary root 虚根imaginary unit 虚数单位impact 碰撞implication 蕴涵式；蕴含式implicit definition 隐定义implicit function 隐函数imply 蕴涵；蕴含impossible event 不可能事件improper fraction 假分数improper integral 广义积分; 非正常积分impulse 冲量impulsive force 冲力incentre 内力incircle 内切圆inclination 倾角；斜角inclined plane 斜面included angle 夹角included side 夹边inclusion mapping 包含映射inclusive 包含的；可兼的inclusive disjunction 包含性析取；可兼析取inconsistent 不相的(的)；不一致(的) increase 递增；增加increasing function 递增函数increasing sequence 递增序列increasing series 递增级数increment 增量indefinite integral 不定积分idenfinite integration 不定积分法independence 独立；自变independent equations 独立方程independent event 独立事件independent variable 自变量；独立变量indeterminate (1)不定的；(2)不定元；未定元indeterminate coefficient 不定系数；未定系数indeterminate form 待定型；不定型index,indices 指数；指index notation 指数记数法induced operation 诱导运算induction hypothesis 归纳法假设inelastic collision 非弹性碰撞inequality 不等式；不等inequality sign 不等号inertia 惯性；惯量infer 推断inference 推论infinite 无限；无穷infinite dimensional 无限维infinite population 无限总体infinite sequence 无限序列；无穷序列infinite series 无限级数；无穷级数infinitely many 无穷多infinitesimal 无限小；无穷小infinity 无限(大)；无穷(大)inflection (inflexion) point 拐点；转折点inherent error 固有误差initial approximation 初始近似值initial condition 原始条件；初值条件initial point 始点；起点initial side 始边initial value 初值；始值initial velocity 初速度initial-value problem 初值问题injection 内射injective function 内射函数inner product 内积input 输入input box 输入inscribed circle 内切圆insertion 插入insertion of brackets 加括号instantaneous 瞬时的instantaneous acceleration 瞬时加速度instantaneous speed 瞬时速率instantaneous velocity 瞬时速度integer 整数integrable 可积integrable function 可积函数integral 积分integral index 整数指数integral mean value theorem 积数指数integral part 整数部份integral solution 整数解integral value 整数值integrand 被积函数integrate 积；积分；......的积分integrating factor 积分因子integration 积分法integration by parts 分部积分法integration by substitution 代换积分法；换元积分法integration constant 积分常数interaction 相互作用intercept 截距；截段intercept form 截距式intercept theorem 截线定理interchange 互换interest 利息interest rate 利率interest tax 利息税interior angle 内角interior angles on the same side of the transversal 同旁内角interior opposite angle 内对角intermediate value theorem 介值定理internal bisector 内分角internal division 内分割internal energy 内能internal force 内力internal point of division 内分点interpolating polynomial 插值多项式interpolation 插值inter-quartile range 四分位数间距intersect 相交intersection (1)交集；(2)相交；(3)交点interval 区间interval estimation 区间估计；区域估计intuition 直观invalid 失效；无效invariance 不变性invariant (1)不变的；(2)不变量；不变式inverse 反的；逆的inverse circular function 反三角函数inverse cosine function 反余弦函数inverse function 反函数；逆函数inverse cosine function 反三角函数inverse function 反函数；逆映射inverse mapping 反向映射；逆映射inverse matrix 逆矩阵inverse problem 逆算问题inverse proportion 反比例；逆比例inverse relation 逆关系inverse sine function 反正弦函数inverse tangent function 反正切函数inverse variation 反变(分)；逆变(分)invertible 可逆的invertible matrix 可逆矩阵irrational equation 无理方程irrational number 无理数irreducibility 不可约性irregular 不规则isomorphism 同构isosceles triangle 等腰三角形iterate (1)迭代值; (2)迭代iteration 迭代iteration form 迭代形iterative function 迭代函数iterative method 迭代法jet propulsion 喷气推进joint variation 联变(分)；连变(分)kinetic energy 动能kinetic friction 动摩擦known 己知L.H.S. 末项L'Hospital's rule 洛必达法则Lagrange interpolating polynomial 拉格朗日插值多项代Lagrange theorem 拉格朗日定理Lami's law 拉密定律Laplace expansion 拉普拉斯展式last term 末项latent root 本征根; 首通径lattice point 格点latus rectum 正焦弦; 首通径law 律；定律law of conservation of momentum 动量守恒定律law of indices 指数律；指数定律law of inference 推论律law of trichotomy 三分律leading coefficient 首项系数leading diagonal 主对角线least common multiple, lowest common multiple (L.C.M) 最小公倍数；最低公倍式least value 最小值left hand limit 左方极限lemma 引理lemniscate 双纽线length 长(度)letter 文字；字母like surd 同类根式like terms 同类项limacon 蜗牛线limit 极限limit of sequence 序列的极限limiting case 极限情况limiting friction 最大静摩擦limiting position 极限位置line 线；行line of action 作用力线line of best-fit 最佳拟合line of greatest slope 最大斜率的直线line of intersection 交线line segment 线段linear 线性；一次linear convergence 线性收敛性linear differeantial equation 线性微分方程linear equation 线性方程；一次方程linear equation in two unknowns 二元一次方程；二元线性方程linear inequality 一次不等式；线性不等式linear momentum 线动量linear programming 线性规划linearly dependent 线性相关的linearly independent 线性无关的literal coefficient 文字系数literal equation 文字方程load 负荷loaded coin 不公正钱币loaded die 不公正骰子local maximum 局部极大(值)local minimum 局部极小(值)locus, loci 轨迹logarithm 对数logarithmic equation 对数方程logarithmic function 对数函数logic 逻辑logical deduction 逻辑推论；逻辑推理logical step 逻辑步骤long division method 长除法loop 回路loss 赔本；亏蚀loss per cent 赔率；亏蚀百分率lower bound 下界lower limit 下限lower quartile 下四分位数lower sum 下和lower triangular matrix 下三角形矩阵lowest common multiple(L.C.M) 最小公倍数machine 机械Maclaurin expansion 麦克劳林展开式Maclaurin series 麦克劳林级数magnitude 量；数量；长度；大小major arc 优弧；大弧major axis 长轴major sector 优扇形；大扇形major segment 优弓形；大弓形mantissa 尾数mantissa of logarithm 对数的尾数；对数的定值部many to one 多个对一个many-sided figure 多边形many-valued 多值的map into 映入map onto 映上mapping 映射marked price 标价Markov chain 马可夫链mass 质量mathematical analysis 数学分析mathematical induction 数学归纳法mathematical sentence 数句mathematics 数学matrix 阵; 矩阵matrix addition 矩阵加法matrix equation 矩阵方程matrix multiplication 矩阵乘法matrix operation 矩阵运算maximize 极大maximum absolute error 最大绝对误差maximum point 极大点maximum value 极大值mean 平均(值)；平均数；中数mean deviation 中均差；平均偏差mean value theorem 中值定理measure of dispersion 离差的量度measurement 量度mechanical energy 机械能median (1)中位数；(2)中线meet 相交；相遇mensuration 计量；求积法method 方法method of completing square 配方法method of interpolation 插值法; 内插法method of least squares 最小二乘法; 最小平方法method of substitution 代换法；换元法method of successive substitution 逐次代换法; 逐次调替法method of superposition 迭合法metric unit 十进制单位mid-point 中点mid-point formula 中点公式mid-point theorem 中点定理million 百万minimize 极小minimum point 极小点minimum value 极小值Minkowski Inequality 闵可夫斯基不等式minor (1)子行列式；(2)劣；较小的minor arc 劣弧；小弧minor axis 短轴minor of a determinant 子行列式minor sector 劣扇形；小扇形minor segment 劣弓形；小弓形minus 减minute 分mixed number(fraction) 带分数modal class 众数组mode 众数model 模型modulo (1)模; 模数; (2)同余modulo arithmetic 同余算术modulus 模; 模数modulus of a complex number 复数的模modulus of elasticity 弹性模(数)moment arm (1)矩臂; (2)力臂moment of a force 力矩moment of inertia 贯性矩momentum 动量monomial 单项式monotone 单调monotonic convergence 单调收敛性monotonic decreasing 单调递减monotonic decreasing function 单调递减函数monotonic function 单调函数monotonic increasing 单调递增monotonic increasing function 单调递增函数motion 运动movable pulley 动滑轮multinomial 多项式multiple 倍数multiple angle 倍角multiple-angle formula 倍角公式multiple root 多重根multiplicand 被乘数multiplication 乘法multiplication law (of probability) (概率)乘法定律multiplicative inverse 乘法逆元multiplicative property 可乘性multiplicity 重数multiplier 乘数；乘式multiply 乘multi-value 多值的mulually disjoint 互不相交mutually exclusive events 互斥事件mutually independent 独立; 互相独立mutually perpendicular lines 互相垂直n factorial n阶乘n th derivative n阶导数n th root n次根；n次方根n the root of unity 单位的n次根Napierian logarithm 纳皮尔对数; 自然对数natural logarithm 自然对数natural number 自然数natural surjection 自然满射necessary and sufficient condition 充要条件necessary condition 必要条件negation 否定式negative 负negative angle 负角negative binomial distribution 负二项式分布negative index 负指数negative integer 负整数negative number 负数negative vector 负向量; 负矢量neighborhood 邻域net 净(值)net force 净力Newton-Cote's rule 牛顿- 高斯法则Newton-Raphson's method 牛顿- 纳逊方法Newton's formula 牛顿公式Newton's law of motion 牛顿运动定律Newton's method 牛顿方法n-gon n边形nonagon 九边形non-collinear 不共线non-commutative 非交换的non-linear 非线性non-linear equation 非线性方程non-negative 非负的non-reflexive 非自反的。

代数部分1.有关基本运算：add, plus加subtract减difference差multiply, times乘product积divide除divisible可被整除的divided evenly被整除dividend被除数divisor因子,除数quotient商remainder余数factorial阶乘power乘方radical sign, root sign根号round to四舍五入to the nearest四舍五入2.有关集合union并集proper subset真子集solution set解集3.有关代数式、方程和不等式algebraic term代数项like terms, similar terms同类项numerical coefficient数字系数literal coefficient字母系数inequality不等式triangle inequality三角不等式range值域original equation原方程equivalent equation同解方程等价方程linear equation线性方程 x +6=224.有关分数和小数proper fraction真分数improper fraction假分数mixed number带分数vulgar fraction, common fraction普通分数simple fraction简分数complex fraction繁分数numerator分子denominator分母leastcommon denominator最小公分母quarter四分之一decimal fraction纯小数infinite decimal无穷小数recurring decimal循环小数tenths unit十分位5.基本数学概念arithmetic mean算术平均值weighted average加权平均值geometric mean几何平均数exponent指数,幂base乘幂的底数,底边cube立方数,立方体square root平方根cube root立方根common logarithm常用对数digit数字constant常数variable变量inverse function反函数complementary function余函数linear一次的,线性的factorization因式分解absolute value绝对值,.｜-32｜=32round off四舍五入6.有关数论natural number自然数positive number正数negative number负数odd integer, odd number奇数even integer, even number偶数integer, whole number整数positive whole number正整数negative whole number负整数consecutive number连续整数real number, rational number实数,有理数irrationalnumber无理数composite number合数……prime number质数……reciprocal倒数common divisor公约数multiple 倍数leastcommon multiple最小公倍数primefactor质因子commonfactor公因子ordinary scale, decimal scale十进制nonnegative非负的tens十位units个位mode众数median中数common ratio公比7.数列arithmetic progressionsequence等差数列geometric progressionsequence等比数列8.其它approximate近似anticlockwise 逆顺时针方向cardinal基数ordinal序数direct proportion正比distinct不同的estimation估计,近似parentheses括号proportion 比例permutation排列combination组合table表格trigonometric function三角函数unit单位,位几何部分1.所有的角alternate angle内错角corresponding angle 同位角vertical angle对顶角central angle圆心角interior angle 内角exterior angle外角supplementary angles补角complementary angle余角adjacent angle邻角acute angle锐角obtuse angle钝角right angle直角round angle周角straight angle平角included angle 夹角2.所有的三角形equilateral triangle等边三角形scalene triangle不等边三角形isosceles triangle等腰三角形right triangle 直角三角形oblique斜三角形inscribed triangle内接三角形3.有关收敛的平面图形,除三角形外semicircle半圆concentric circles同心圆quadrilateral四边形pentagon五边形hexagon六边形heptagon七边形octagon八边形nonagon九边形decagon十边形polygon 多边形parallelogram平行四边形equilateral等边形plane平面square正方形,平方rectangle长方形regular polygon正多边形rhombus菱形trapezoid梯形4.其它平面图形arc弧line, straightline直线line segment线段parallel lines平行线segment of a circle 弧形5.有关立体图形7.有关坐标coordinate system坐标系rectangular coordinate直角坐标系origin原点abscissa横坐标ordinate纵坐标number line数轴quadrant象限slope斜率complex plane复平面8. 其它plane geometry 平面几何trigonometry三角学bisect平分circumscribe外切inscribe 内切intersect相交perpendicular垂直Pythagorean theorem勾股定理congruent全等的multilateral多边的一般词汇数学 mathematics, mathsBrE, mathAmE公理 axiom定理 theorem计算 calculation运算 operation证明 prove假设 hypothesis, hypothesespl.命题 proposition算术 arithmetic加 plusprep., addv., additionn.被加数 augend, summand加数 addend和sum减minusprep., subtractv., subtractionn.被减数 minuend减数 subtrahend差remainder乘timesprep., multiplyv., multiplicationn.被乘数 multiplicand,faciend乘数 multiplicator积product除divided byprep., dividev., divisionn.被除数 dividend除数 divisor商quotient等于 equals, is equalto, is equivalent to 大于 is greater than小于 is lesser than大于等于 is equal or greater than小于等于 is equal or lesser than运算符 operator数字 digit数number自然数 naturalnumber整数 integer小数 decimal小数点 decimalpoint分数 fraction分子 numerator分母 denominator比ratio正positive负negative零null, zero, nought, nil十进制 decimalsystem二进制 binary system十六进制 hexadecimalsystem权weight, significance进位 carry截尾 truncation四舍五入 round下舍入 rounddown上舍入 roundup有效数字 significant digit无效数字 insignificantdigit代数 algebra公式 formula,formulaepl.单项式 monomial多项式 polynomial, multinomial系数 coefficient未知数 unknown,x-factor, y-factor, z-factor 等式,方程式 equation一次方程 simple equation二次方程 quadratic equation三次方程 cubic equation四次方程 quartic equation不等式 inequation阶乘 factorial对数 logarithm指数,幂 exponent乘方 power二次方,平方 square三次方,立方 cube四次方 thepowerof four, thefourth powern 次方 thepowerof n, the nthpower开方evolution, extraction二次方根,平方根 square root三次方根,立方根 cube root四次方根 the root of four, the fourth root n 次方根 theroot of n, the nth root集合 aggregate元素 element空集 void子集 subset交集 intersection并集 union补集 complement映射 mapping函数 function定义域 domain, field of definition 值域 range常量 constant变量 variable单调性 monotonicity奇偶性 parity周期性 periodicity图象 image数列,级数 series微积分 calculus微分 differential导数 derivative极限 limit无穷大 infinitea. infinityn.无穷小 infinitesimal积分 integral定积分 definite integral不定积分 indefinite integral 有理数 rationalnumber无理数 irrationalnumber实数 realnumber虚数 imaginarynumber复数 complex number矩阵 matrix行列式 determinant几何 geometry点 point线 line面 plane体 solid线段 segment射线 radial平行 parallel相交 intersect角 angle角度 degree弧度 radian锐角 acute angle直角 rightangle钝角 obtuseangle平角 straight angle周角 perigon底 base边 side高 height三角形 triangle锐角三角形 acute triangle直角三角形 righttriangle直角边 leg斜边 hypotenuse勾股定理 Pythagorean theorem钝角三角形 obtusetriangle不等边三角形 scalenetriangle等腰三角形 isosceles triangle等边三角形 equilateral triangle四边形 quadrilateral平行四边形 parallelogram矩形 rectangle长 length宽 width菱形 rhomb, rhombus, rhombipl., diamond 正方形 square梯形 trapezoid直角梯形 righttrapezoid等腰梯形 isosceles trapezoid 五边形 pentagon六边形 hexagon七边形 heptagon八边形 octagon九边形 enneagon十边形 decagon十一边形 hendecagon十二边形 dodecagon多边形 polygon正多边形 equilateral polygon 圆 circle圆心 centreBrE, centerAmE半径 radius直径 diameter圆周率 pi弧 arc半圆 semicircle扇形 sector环 ring椭圆 ellipse圆周 circumference周长 perimeter面积 area轨迹 locus, locapl.相似 similar全等 congruent四面体 tetrahedron五面体 pentahedron六面体 hexahedron平行六面体 parallelepiped 立方体 cube七面体 heptahedron八面体 octahedron九面体 enneahedron十面体 decahedron十一面体 hendecahedron十二面体 dodecahedron二十面体 icosahedron多面体 polyhedron棱锥 pyramid棱柱 prism棱台 frustum of a prism 旋转 rotation轴 axis圆锥 cone圆柱 cylinder圆台 frustum of a cone球 sphere半球 hemisphere底面 undersurface表面积 surfacearea体积 volume空间 space坐标系 coordinates坐标轴 x-axis, y-axis, z-axis 横坐标 x-coordinate纵坐标 y-coordinate原点 origin双曲线 hyperbola抛物线 parabola三角 trigonometry正弦 sine余弦 cosine正切 tangent余切 cotangent正割 secant余割 cosecant反正弦 arc sine反余弦 arc cosine反正切 arc tangent反余切 arc cotangent反正割 arc secant反余割 arc cosecant相位 phase周期 period振幅 amplitude内心 incentreBrE, incenterAmE外心 excentreBrE,excenterAmE旁心 escentreBrE, escenterAmE垂心 orthocentreBrE, orthocenterAmE重心 barycentreBrE, barycenterAmE内切圆 inscribed circle外切圆 circumcircle统计 statistics平均数 average加权平均数 weighted average方差 variance标准差 root-mean-square deviation, standard deviation 比例 propotion百分比 percent百分点 percentage百分位数 percentile排列 permutation组合 combination概率,或然率 probability分布 distribution正态分布 normal distribution非正态分布 abnormaldistribution 图表 graph条形统计图 bar graph柱形统计图 histogram折线统计图 broken linegraph曲线统计图 curve diagram扇形统计图 piediagram其他数学词汇：mathematics数学theorem定理calculation计算operation运算arithmetic算术,四则运算addition加法subtraction减法multiplication乘法division除法sum和remainder余product积quotient商2 plus 1 equals3二加一等于三4 minus2 equals 2四减二等于二4 multiplied by 5, 4 times 5四乘以五addend加数plus sign加号minussign减号multiplicand被乘数multiplier乘数divisor除数dividend被除数decimal point小数点noughtpoint four零点四fraction分数ratio比proportion比例numerator分子common denominator公分母exponent指数differentialcalculus微分学integral calculus积分slide rule计算尺function函数derivative导数power幂,乘方to raise to the power of five 使乘五次方x squared某数的平方cube三次方threecubed三次方的to the fourth power,to the power of four 乘四次方square root平方根cube root立方根rule of three比例法logarithm对数logarithmtable对数表algebra代数equation等式,方程式unknown未知数simpleequation一次方程quadratic equation二次方程cubic equation三次方程微积分反函数: inverse function.幂函数: power function. 指数函数: exponential function. 对数函数: logarithmic function. 三角函数: trigonometric function. 反三角函数: anti-trigonometric function. 复合函数: compound function.初等函数: elementary function. 双曲函数: hyperbolic function.反双曲函数: anti-hyperbolic function. 无穷小:infinitesimal. 无穷大:infinity.连续性:continuity. 间断点:discontinuous point.介值定理:intermediate value theorem. 导数:derivative. 微分:differential. 函数的单调性:monotonicity of function. 曲线:curve. 曲线的凹凸性:concavity of curve. 曲线的拐点: keen point of curve.曲率:curvature. 不定积分: indefinite integralindeterminate integral. 定积分: definite integral. 广义积分:improper integral. 空间解析几何:space analytic geometry. 向量代数:vector algebra. 空间直角坐标系:space rectangular coordinate system. 数量积:scalar product. 向量积:vector product. 混合积:triple product. 曲面:surface. 二次曲面:second-degree surface.线性代数Aadjontadjugate of matrix A A 的伴随矩阵augmented matrix A 的增广矩阵Bblock diagonal matrix 块对角矩阵block matrix 块矩阵basic solution set 基础解系CCauchy-Schwarz inequality 柯西 - 许瓦兹不等式characteristic equation 特征方程characteristic polynomial 特征多项式coffcient matrix 系数矩阵cofactor 代数余子式cofactor expansion 代数余子式展开column vector 列向量commuting matrices 交换矩阵consistent linear system 相容线性方程组Cramer's rule 克莱姆法则Cross- product term 交叉项DDeterminant 行列式Diagonal entries 对角元素Diagonal matrix 对角矩阵Dimension of a vector space V 向量空间 V 的维数Eechelon matrix 梯形矩阵eigenspace 特征空间eigenvalue 特征值eigenvector 特征向量eigenvector basis 特征向量的基elementary matrix 初等矩阵elementary row operations 行初等变换Ffull rank 满秩fundermental set of solution 基础解系Ggrneral solution 通解Gram-Schmidt process 施密特正交化过程Hhomogeneous linear equations 齐次线性方程组Iidentity matrix 单位矩阵inconsistent linear system 不相容线性方程组indefinite matrix 不定矩阵indefinit quatratic form 不定二次型infinite-dimensional space 无限维空间inner product 内积inverse of matrix A 逆矩阵JKLlinear combination 线性组合linearly dependent 线性相关linearly independent 线性无关linear transformation 线性变换lower triangular matrix 下三角形矩阵Mmain diagonal of matrix A 矩阵的主对角matrix 矩阵Nnegative definite quaratic form 负定二次型negative semidefinite quadratic form 半负定二次型nonhomogeneous equations 非齐次线性方程组nonsigular matrix 非奇异矩阵nontrivial solution 非平凡解norm of vector V 向量 V 的范数normalizing vector V 规范化向量Oorthogonal basis 正交基orthogonal complemen t 正交补orthogonal decomposition 正交分解orthogonally diagonalizable matrix 矩阵的正交对角化orthogonal matrix 正交矩阵orthogonal set 正交向量组orthonormal basis 规范正交基orthonomal set 规范正交向量组Ppartitioned matrix 分块矩阵positive definite matrix 正定矩阵positive definite quatratic form 正定二次型positive semidefinite matrix 半正定矩阵positive semidefinite quadratic form 半正定二次型Qquatratic form 二次型rank of matrix A 矩阵 A 的秩 rAreduced echelon matrix 最简梯形阵row vector 行向量Sset spanned by { } 由向量 { } 所生成similar matrices 相似矩阵similarity transformation 相似变换singular matrix 奇异矩阵solution set 解集合standard basis 标准基standard matrix 标准矩阵 Isubmatrix 子矩阵subspace 子空间symmetric matrix 对称矩阵Ttrace of matrix A 矩阵 A 的迹 tr Atranspose of A 矩阵 A 的转秩triangle inequlity 三角不等式trivial solution 平凡解Uunit vector 单位向量upper triangular matrix 上三角形矩阵vandermonde matrix 范得蒙矩阵vector 向量vector space 向量空间WZzero subspace 零子空间zero vector 零空间概率与统计Probability theory 概率论mathematical statistics数理统计deterministic phenomenon确定性现象random phenomenon随机现象sample space样本空间random occurrence随机事件fundamental event基本事件certain event必然事件impossible event不可能事件random test随机试验incompatible events互不相容事件frequency 频率classical probabilistic model 古典概型geometric probability 几何概率conditional probability条件概率multiplication theorem乘法定理Bayes's formula贝叶斯公式Prior probability先验概率Posterior probability后验概率Independent events相互独立事件Bernoulli trials 贝努利试验random variable随机变量probability distribution概率分布distribution function分布函数discrete random variable 离散随机变量distribution law分布律hypergeometric distribution超几何分布random sampling model 随机抽样模型binomial distribution二项分布Poisson distribution泊松分布geometric distribution几何分布probability density概率密度continuous random variable 连续随机变量uniformly distribution均匀分布exponential distribution 指数分布numerical character数字特征mathematical expectation数学期望variance 方差moment 矩central moment中心矩n-dimensional random variable n-维随机变量two-dimensionalrandom variable 二维离散随机变量joint probability distribution 联合概率分布joint distribution law 联合分布律joint distribution function 联合分布函数boundary distribution law 边缘分布律boundary distribution function 边缘分布函数exponential distribution 二维指数分布continuous random variable 二维连续随机变量joint probability density 联合概率密度boundary probability density 边缘概率密度conditional distribution条件分布conditional distribution law 条件分布律conditional probability density 条件概率密度covariance 协方差dependency coefficient相关系数normal distribution 正态分布limit theorem 极限定理standard normal distribution 标准正态分布logarithmic normal distribution 对数正态分布covariance matrix协方差矩阵central limit theorem 中心极限定理Chebyshev'sinequality 切比雪夫不等式Bernoulli's lawoflarge numbers 贝努利大数定律statistics 统计量simple random sample 简单随机样本sample distribution function 样本分布函数sample mean样本均值sample variance样本方差sample standard deviation 样本标准差sample covariance 样本协方差sample correlation coefficient 样本相关系数order statistics 顺序统计量sample median样本中位数sample fractiles样本极差sampling distribution 抽样分布parameter estimation 参数估计estimator 估计量estimate value 估计值unbiased estimator 无偏估计unbiassedness 无偏性biased error 偏差mean square error 均方误差relative efficient相对有效性minimum variance 最小方差asymptotic unbiased estimator 渐近无偏估计量uniformly estimator一致性估计量moment method of estimation 矩法估计maximum likelihood methodofestimation 极大似然估计法likelihood function 似然函数maximum likelihood estimator 极大似然估计值interval estimation 区间估计hypothesis testing 假设检验statistical hypothesis 统计假设simple hypothesis 简单假设composite hypothesis复合假设rejection region 拒绝域acceptance domain 接受域test statistics 检验统计量linear regression analysis 线性回归分析。

数学专业用词的英文翻译数学的英文篇一：数学英文词汇大全微积分第一章函数与极限Chapter1FunctionandLimit集合set元素element子集subset空集emptyset并集union交集intersection差集differenceofset基本集basicset补集complementset直积directproduct笛卡儿积Cartesianproduct开区间openinterval闭区间closedinterval半开区间halfopeninterval有限区间finiteinterval区间的长度lengthofaninterval无限区间infiniteinterval领域neighborhood领域的中心centreofaneighborhood领域的半径radiusofaneighborhood左领域leftneighborhood右领域rightneighborhood映射mappingX到Y的映射mappingofXontoY满射surjection单射injection一一映射one-to-onemapping 双射bijection算子operator变化transformation函数function逆映射inversemapping复合映射compositemapping自变量independentvariable因变量dependentvariable定义域domain函数值valueoffunction函数关系functionrelation值域range自然定义域naturaldomain单值函数singlevaluedfunction 多值函数multiplevaluedfunction 单值分支one-valuedbranch函数图形graphofafunction绝对值函数absolutevalue符号函数sighfunction整数部分integralpart阶梯曲线stepcurve当且仅当ifandonlyif(iff)分段函数piecewisefunction上界upperbound下界lowerbound有界boundedness无界unbounded函数的单调性monotonicityofafunction 单调增加的increasing单调减少的decreasing单调函数monotonefunction函数的奇偶性parity(odevity)ofafunction 对称symmetry偶函数evenfunction奇函数oddfunction函数的周期性periodicityofafunction周期period反函数inversefunction直接函数directfunction复合函数compositefunction中间变量intermediatevariable函数的运算operationoffunction基本初等函数basicelementaryfunction 初等函数elementaryfunction幂函数powerfunction指数函数exponentialfunction对数函数logarithmicfunction三角函数trigonometricfunction反三角函数inversetrigonometricfunction 常数函数constantfunction双曲函数hyperbolicfunction双曲正弦hyperbolicsine双曲余弦hyperboliccosine双曲正切hyperbolictangent反双曲正弦inversehyperbolicsine反双曲余弦inversehyperboliccosine反双曲正切inversehyperbolictangent极限limit数列sequenceofnumber收敛convergence收敛于aconvergetoa发散divergent极限的唯一性uniquenessoflimits收敛数列的有界性boundednessofaconvergentsequence 子列subsequence函数的极限limitsoffunctions函数当x趋于x0时的极限limitoffunctionsasxapproachesx0 左极限leftlimit右极限rightlimit单侧极限one-sidedlimits水平渐近线horizontalasymptote无穷小infinitesimal无穷大infinity铅直渐近线verticalasymptote夹逼准则squeezerule单调数列monotonicsequence高阶无穷小infinitesimalofhigherorder低阶无穷小infinitesimaloflowerorder同阶无穷小infinitesimalofthesameorder等阶无穷小equivalentinfinitesimal函数的连续性continuityofafunction增量increment函数在x0连续thefunctioniscontinuousatx0左连续leftcontinuous右连续rightcontinuous区间上的连续函数continuousfunction函数在该区间上连续functioniscontinuousonaninterval不连续点discontinuitypoint第一类间断点discontinuitypointofthefirstkind第二类间断点discontinuitypointofthesecondkind初等函数的连续性continuityoftheelementaryfunctions定义区间definedinterval最大值globalmaximumvalue(absolutemaximum)最小值globalminimumvalue(absoluteminimum)零点定理thezeropointtheorem介值定理intermediatevaluetheorem第二章导数与微分Chapter2DerivativeandDifferential速度velocity匀速运动uniformmotion平均速度averagevelocity瞬时速度instantaneousvelocity圆的切线tangentlineofacircle切线tangentline切线的斜率slopeofthetangentline位置函数positionfunction导数derivative可导derivable函数的变化率问题problemofthechangerateofafunction导函数derivedfunction左导数left-handderivative右导数right-handderivative单侧导数one-sidedderivatives在闭区间【a,b】上可导isderivableontheclosedinterval[a,b] 切线方程tangentequation角速度angularvelocity成本函数costfunction边际成本marginalcost链式法则chainrule隐函数implicitfunction显函数explicitfunction二阶函数secondderivative三阶导数thirdderivative高阶导数nthderivative莱布尼茨公式Leibnizformula对数求导法log-derivative参数方程parametricequation相关变化率correlativechangerata微分differential可微的differentiable函数的微分differentialoffunction自变量的微分differentialofindependentvariable微商differentialquotient间接测量误差indirectmeasurementerror绝对误差absoluteerror相对误差relativeerror第三章微分中值定理与导数的应用Chapter3MeanValueTheoremofDifferentialsandtheApplicati onofDerivatives罗马定理Rolle’stheorem费马引理Fermat’slemma拉格朗日中值定理Lagrange’smeanvaluetheorem驻点stationarypoint稳定点stablepoint临界点criticalpoint辅助函数auxiliaryfunction拉格朗日中值公式Lagrange’smeanvalueformula柯西中值定理Cauchy’smeanvaluetheorem洛必达法则L’Hospital’sRule0/0型不定式indeterminateformoftype0/0不定式indeterminateform泰勒中值定理Taylor’smeanvaluetheorem泰勒公式Taylorformula余项remainderterm拉格朗日余项Lagrangeremainderterm麦克劳林公式Maclaurin’sformula佩亚诺公式Peanoremainderterm凹凸性concavity凹向上的concaveupward,cancaveup凹向下的，向上凸的concavedownward’concavedown拐点inflectionpoint函数的'极值extremumoffunction极大值local(relative)maximum最大值global(absolute)mximum极小值local(relative)minimum最小值global(absolute)minimum目标函数objectivefunction曲率curvature弧微分arcdifferential平均曲率averagecurvature曲率园circleofcurvature曲率中心centerofcurvature曲率半径radiusofcurvature渐屈线evolute渐伸线involute根的隔离isolationofroot隔离区间isolationinterval切线法tangentlinemethod第四章不定积分Chapter4IndefiniteIntegrals原函数primitivefunction(antiderivative) 积分号signofintegration被积函数integrand数学的英文篇二：数学英文词汇代数部分1.有关基本运算：add,plus加subtract减difference差multiply,times乘product积divide除divisible可被整除的dividedevenly被整除dividend被除数divisor因子，除数quotient商remainder余数factorial阶乘power乘方radicalsign,rootsign根号roundto四舍五入tothenearest四舍五入2.有关集合union并集propersubset真子集solutionset解集3.有关代数式、方程和不等式algebraicterm代数项liketerms,similarterms同类项numericalcoefficient数字系数literalcoefficient字母系数inequality不等式triangleinequality三角不等式range值域originalequation原方程equivalentequation同解方程等价方程linearequation线性方程(e.g.5x+6=22)4.有关分数和小数properfraction真分数improperfraction假分数mixednumber带分数vulgarfraction,commonfraction普通分数simplefraction简分数complexfraction繁分数numerator分子denominator分母(least)commondenominator(最小)公分母quarter四分之一decimalfraction纯小数infinitedecimal无穷小数recurringdecimal循环小数tenthsunit十分位5.基本数学概念arithmeticmean算术平均值weightedaverage加权平均值geometricmean几何平均数exponent指数，幂base乘幂的底数,底边cube立方数，立方体squareroot平方根cuberoot立方根commonlogarithm常用对数digit数字constant常数variable变量inversefunction反函数complementaryfunction余函数linear一次的，线性的factorization因式分解absolutevalue绝对值，e.g.｜-32｜=32roundoff四舍五入6.有关数论naturalnumber自然数positivenumber正数negativenumber负数oddinteger,oddnumber奇数eveninteger,evennumber偶数integer,wholenumber整数positivewholenumber正整数negativewholenumber负整数consecutivenumber连续整数realnumber,rationalnumber实数,有理数irrational(number)无理数compositenumber合数e.g.4,6,8,9,10,12,14,15……primenumber质数e.g.2,3,5,7,11,13,15……reciprocal倒数commondivisor公约数multiple倍数(least)commonmultiple(最小)公倍数(prime)factor(质)因子commonfactor公因子ordinaryscale,decimalscale十进制nonnegative非负的tens十位units个位mode众数median中数commonratio公比7.数列arithmeticprogression(sequence)等差数列geometricprogression(sequence)等比数列8.其它approximate近似(anti)clockwise(逆)顺时针方向cardinal基数ordinal序数directproportion正比distinct不同的estimation估计，近似parentheses括号proportion比例数学的英文篇三：数学英文代数ALGEBRA1.数论naturalnumber自然数positivenumber正数negativenumber 负数oddinteger,oddnumber奇数eveninteger,evennumber偶数integer,wholenumber整数positivewholenumber正整数negativewholenumber负整数consecutivenumber连续整数realnumber,rationalnumber实数,有理数irrational（number）无理数inverse倒数compositenumber合数 e.g.4,6,8,9,10,12,14,15…primenumber质数e.g.2,3,5,7,11,13,15…reciprocal倒数commondivisor公约数multiple倍数(minimum)commonmultiple(最小)公倍数(prime)factor(质)因子commonfactor公因子ordinaryscale,decimalscale十进制nonnegative非负的tens十位units个位mode众数mean平均数median中值commonratio公比2.基本数学概念arithmeticmean算术平均值weightedaverage加权平均值geometricmean几何平均数exponent指数，幂base乘幂的底数,底边cube立方数，立方体squareroot 平方根cuberoot立方根commonlogarithm常用对数digit数字constant常数variable变量inversefunction 反函数complementaryfunction余函数linear一次的，线性的factorization因式分解absolutevalue绝对值，e.g.｜-32｜=32roundoff四舍五入数学3.基本运算add，plus加subtract减difference差multiply,times乘product积divide除divisible可被整除的dividedevenly被整除dividend被除数，红利divisor因子，除数，公约数quotient商remainder余数factorial阶乘power乘方radicalsign,rootsign根号roundto四舍五入tothenearest四舍五入4.代数式，方程，不等式algebraicterm代数项liketerms,similarterms同类项numericalcoefficient数字系数literalcoefficient字母系数inequality不等式triangleinequality三角不等式range值域originalequation原方程equivalentequation同解方程，等价方程linearequation线性方程(e.g.5x+6=22)5.分数，小数properfraction真分数improperfraction假分数mixednumber 带分数vulgarfraction，commonfraction普通分数simplefraction简分数complexfraction繁分数numerator分子denominator分母(least)commondenominator（最小）公分母quarter四分之一decimalfraction纯小数infinitedecimal无穷小数recurringdecimal循环小数tenthsunit十分位6.集合union并集propersubset真子集solutionset解集7.数列arithmeticprogression(sequence)等差数列geometricprogression(sequence)等比数列8.其它approximate近似(anti)clockwise(逆)顺时针方向cardinal基数ordinal序数directproportion正比distinct不同的estimation估计，近似parentheses括号proportion比例permutation排列combination组合table表格trigonometricfunction三角函数unit单位,位几何GEOMETRY1.角alternateangle内错角correspondingangle同位角verticalangle对顶角centralangle圆心角interiorangle内角exteriorangle外角supplementaryangles补角complementaryangle余角adjacentangle邻角acuteangle锐角obtuseangle 钝角rightangle直角roundangle周角straightangle平角includedangle夹角2.三角形equilateraltriangle等边三角形scalenetriangle不等边三角形isoscelestriangle等腰三角形righttriangle直角三角形oblique斜三角形inscribedtriangle内接三角形3.收敛的平面图形，除三角形外semicircle半圆concentriccircles同心圆quadrilateral四边形pentagon五边形hexagon六边形heptagon七边形octagon八边形nonagon九边形decagon十边形polygon多边形parallelogram平行四边形equilateral等边形plane平面square 正方形，平方rectangle长方形regularpolygon正多边形rhombus菱形trapezoid梯形4.其它平面图形arc弧line,straightline直线linesegment线段parallellines平行线segmentofacircle弧形5.立体图形cube立方体，立方数rectangularsolid长方体regularsolid/regularpolyhedron正多面体circularcylinder圆柱体cone圆锥sphere球体solid立体的6.图形的附属概念planegeometry平面几何trigonometry三角学bisect平分circumscribe外切inscribe内切intersect相交perpendicular垂直Pythagoreantheorem 勾股定理（毕达哥拉斯定理）congruent全等的multilateral多边的altitude高depth深度side边长circumference,perimeter周长radian弧度surfacearea表面积volume体积arm直角三角形的股crosssection横截面centerofacircle圆心chord弦diameter直径radius半径anglebisector角平分线diagonal对角线化edge棱faceofasolid立体的面hypotenuse 斜边includedside夹边leg三角形的直角边median（三角形的）中线base底边，底数（e.g.2的5次方，2就是底数）opposite直角三角形中的对边midpoint中点endpoint端点vertex(复数形式vertices)顶点tangent切线的transversal截线intercept截距7.坐标coordinatesystem坐标系rectangularcoordinate直角坐标系origin原点abscissa横坐标ordinate纵坐标numberline数轴quadrant象限slope斜率complexplane复平面【关于数学专业用词的英文翻译】。

解析数论是使用数学分析作为工具来解决数论问题的分支。

微积分和复变函数论发展以后，产生了解析数论。

该学科的第一个主要成就是狄利克雷用解析方法证明了Dirichlet's theorem on arithmetic progressions。

依靠黎曼zeta函数对素数定理的证明是另一个里程碑。

解析数论是解决数论中艰深问题的重要工具，数论中有些问题必须由解析方法才能提出或解决。

中国的华罗庚、王元、陈景润等人在“哥德巴赫猜想”、“华林问题”等解析数论问题上取得世界公认的成就。

黎曼ζ函数Riemann zeta functionIn mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of great significance in number theory because of its relation to the distribution of prime numbers. It also has applications in other areas such as physics, probability theory, and applied statistics. The Riemann hypothesis, a conjecture about the distribution of the zeros of the Riemann zeta function, is considered by many mathematicians to be the most important unsolved problem in pure mathematics.[1]DefinitionThe Riemann zeta-function ζ(s) is the function of a complex variable s initially defined by the following infinite series:As a Dirichlet series with bounded coefficient sequence this series converges absolutely to an analytic function on the open half-plane of s such that Re(s) > 1 and diverges on the open half-plane of s such that Re(s) < 1. The function defined by the series on the half-plane of convergence can however be continued analytically to all complex s≠ 1. For s= 1 the series is formally identical to the harmonic series which diverges to infinity. As a result, the zeta function becomes a meromorphic function of the complex variable s, which is holomorphic in the region {s∈ C : s≠1} of the complex plane and has a simple pole at s= 1 with residue 1.Specific valuesThe values of the zeta function obtained from integral arguments are called zeta constants. The following are the most commonly used values of the Riemann zeta function.this is the harmonic series.this is employed in calculating the critical temperature for a Bose–Einstein condensate in physics, and for spin-wave physics in magnetic systems.the demonstration of this equality is known as the Basel problem. The reciprocal of this sum answers the question: What is the probability that two numbers selected at random are relatively prime? [2]this is called Apéry's constant.Stefan–Boltzmann law and Wien approximation in physics.Euler product formulaThe connection between the zeta function and prime numberswas discovered by Leonhard Euler, who proved the identitywhere, by definition, the left hand side is ζ(s) and the infiniteproduct on the right hand side extends over all prime numbers p(such expressions are called Euler products):Both sides of the Euler product formula converge for Re(s) > 1. The proof of Euler's identity uses only the formula for the geometric series and the fundamental theorem of arithmetic. Since the harmonic series, obtained when s= 1, diverges, Euler's formula implies that there are infinitely many primes. For s an integer number, the Euler product formula can be used to calculate the probability that s randomly selected integers are relatively prime. It turns out that this probability is indeed 1/ζ(s). The functional equationThe Riemann zeta function satisfies the functional equationvalid for all complex numbers s, which relates its values at points s and 1 −s. Here, Γ denotes the gamma function. This functional equation was established by Riemann in his 1859 paper On the Number of Primes Less Than a Given Magnitude and used to construct the analytic continuation in the first place. An equivalent relationship was conjectured by Euler in 1749 for the functionAccording to André Weil, Riemann seems to have been very familiar with Euler's work on the subject.[3]The functional equation given by Riemann has to be interpreted analytically if any factors in the equation have a zero or pole. For instance, when s is 2, the right side has a simple zero in the sine factor and a simple pole in the Gamma factor, which cancel out and leave a nonzero finite value. Similarly, when s is 0, the right side has a simple zero in the sine factor and a simple pole in the zeta factor, which cancel out and leave a finite nonzero value. When s is 1, the right side has a simple pole in the Gamma factor that is not cancelled out by a zero in any other factor, which is consistent with the zeta-function on the left having a simple pole at 1.There is also a symmetric version of the functional equation, given by first definingThe functional equation is then given by(Riemann defined a similar but different function which he called ξ(t).)The functional equation also gives the asymptotic limit(GergőNemes, 2007)Zeros, the critical line, and the Riemann hypothesisThe functional equation shows that the Riemann zeta function has zeros at −2, −4, ... . These are called the trivial zeros. They are trivial in the sense that their existence is relatively easy to prove, for example, from sin(πs/2) being 0 in the functional equation. The non-trivial zeros have captured far more attention because their distribution not only is far less understood but, more importantly, their study yields impressive results concerning prime numbers and related objects in number theory. It is known that any non-trivial zero lies in the open strip {s∈ C: 0 < Re(s) < 1}, which is called the critical strip. The Riemann hypothesis, considered to be one of the greatest unsolved problems in mathematics, asserts that any non-trivial zero s has Re(s) = 1/2. In the theory of the Riemann zeta function, the set{s∈ C: Re(s) = 1/2} is called the critical line. For the Riemann zeta function on the critical line, see Z-function.The location of the Riemann zeta function's zeros is of great importance in the theory of numbers. From the fact that allnon-trivial zeros lie in the critical strip one can deduce the prime number theorem. A better result[4]is that ζ(σ+ i t) ≠ 0 whenever |t| ≥ 3 andThe strongest result of this kind one can hope for is the truth of the Riemann hypothesis, which would have many profound consequences in the theory of numbers.It is known that there are infinitely many zeros on the critical line. Littlewood showed that if the sequence (γn) contains the imaginary parts of all zeros in the upper half-plane in ascending order, thenThe critical line theorem asserts that a positive percentage of the nontrivial zeros lies on the critical line.In the critical strip, the zero with smallest non-negative imaginary part is 1/2 + i14.13472514... Directly from the functional equation one sees that the non-trivial zeros are symmetric about the axis Re(s) = 1/2. Furthermore, the fact that ζ(s) = ζ(s*)* for all complex s≠ 1 (* indicating complex conjugation) implies that the zeros of the Riemann zeta function are symmetric about the real axis.The statistics of the Riemann zeta zeros are a topic of interest to mathematicians because of their connection to big problems like the Riemann hypothesis, distribution of prime numbers, etc. Through connections with random matrix theory and quantum chaos, the appeal is even broader. The fractal structure of the Riemann zeta zero distribution has been studied using rescaled range analysis.[5] The self-similarity of the zero distribution is quite remarkable, and is characterized by a large fractal dimension of 1.9. This rather large fractal dimension is found over zeros covering at least fifteen orders of magnitude, and also for the zeros of other L-functions.The properties of the Riemann zeta function in the complex plane, specifically along parallels to the imaginary axis, has also been studied, by the relation to prime numbers, in recentphysical interference experiments, by decomposing the sum into two parts with opposite phases, ψ and ψ*, which then are brought to interference. [6]For sums involving the zeta-function at integer and half-integer values, see rational zeta series.[O] ReciprocalThe reciprocal of the zeta function may be expressed as a Dirichlet series over the Möbius functionμ(n):for every complex number s with real part > 1. There are a number of similar relations involving various well-known multiplicative functions; these are given in the article on the Dirichlet series.The Riemann hypothesis is equivalent to the claim that this expression is valid when the real part of s is greater than 1/2. [O] UniversalityThe critical strip of the Riemann zeta function has the remarkable property of universality. This zeta-function universality states that there exists some location on the critical strip that approximates any holomorphic function arbitrarily well. Since holomorphic functions are very general, this property is quite remarkable.[O] Representations[O] Mellin transformThe Mellin transform of a function f(x) is defined asin the region where the integral is defined. There are various expressions for the zeta-function as a Mellin transform. If the real part of s is greater than one, we havewhere Γ denotes the Gamma function. By subtracting off the first terms of the power series expansion of 1/(exp(x) −1) around zero, we can get the zeta-function in other regions. In particular, in the critical strip we haveand when the real part of s is between −1 and 0,We can also find expressions which relate to prime numbers and the prime number theorem. If π(x) is the prime-counting function, thenfor values with We can relate this to the Mellin transform of π(x) bywhereconverges forA similar Mellin transform involves the Riemann prime-counting function J(x), which counts prime powers p n with a weight of 1/n,so that Now we haveThese expressions can be used to prove the prime number theorem by means of the inverse Mellin transform. Riemann's prime-counting function is easier to work with, and π(x) can be recovered from it by Möbius inversion.Also, from the above (specifically, the second equation in this section), we can write the zeta function in the commonly seen form:[O] Laurent seriesThe Riemann zeta function is meromorphic with a single pole of order one at s= 1. It can therefore be expanded as a Laurent series about s= 1; the series development then isThe constants γn here are called the Stieltjes constants and can be defined by the limitThe constant term γ0 is the Euler-Mascheroni constant.[O] Rising factorialAnother series development valid for the entire complex plane iswhere is the rising factorial This can be used recursively to extend the Dirichlet series definition to all complex numbers.The Riemann zeta function also appears in a form similar to the Mellin transform in an integral over the Gauss-Kuzmin-Wirsing operator acting on x s−1; that context gives rise to a series expansion in terms of the falling factorial.[O] Hadamard productOn the basis of Weierstrass's factorization theorem, Hadamard gave the infinite product expansionwhere the product is over the non-trivial zeros ρ of ζ and the letter γ again denotes the Euler-Mascheroni constant. A simpler infinite product expansion isThis form clearly displays the simple pole at s = 1, the trivial zeros at −2, −4, ... due to the gamma function term in the denominator, and the non-trivial zeros at s = ρ.[O] Globally convergent seriesA globally convergent series for the zeta function, valid for all complex numbers s except s = 1, was conjectured by Konrad Knopp and proved by Helmut Hasse in 1930:The series only appeared in an Appendix to Hasse's paper, and did not become generally known until it was rediscovered more than 60 years later (see Sondow, 1994).Peter Borwein has shown a very rapidly convergent series suitable for high precision numerical calculations. The algorithm, making use of Chebyshev polynomials, is described in the article on the Dirichlet eta function.[O] ApplicationsAlthough mathematicians regard the Riemann zeta function as being primarily relevant to the "purest" of mathematical disciplines, number theory, it also occurs in applied statistics (see Zipf's law and Zipf-Mandelbrot law), physics, and the mathematical theory of musical tuning.During several physics-related calculations, one must evaluate the sum of the positive integers; paradoxically, on physical grounds one expects a finite answer. When this situation arises, there is typically a rigorous approach involving much in-depth analysis, as well as a "short-cut" solution relying on the Riemann zeta-function. The argument goes as follows: we wish to evaluate the sum 1 + 2 + 3 + 4 + · · ·, but we can rewrite it as a sum of reciprocals:The sum S appears to take the form of However, −1 lies outside of the domain for which the Dirichlet series for thezeta-function converges. However, a divergent series of positive terms such as this one can sometimes be represented in a reasonable way by the method of Ramanujan summation (see Hardy, Divergent Series.) Ramanujan summation involves an application of the Euler–Maclaurin summation formula, and when applied to the zeta-function, it extends its definition to the whole complex plane. In particularwhere the notation indicates Ramanujan summation.[7]For even powers we have:and for odd powers we have a relation with the Bernoulli numbers:Zeta function regularization is used as one possible means of regularization of divergent series in quantum field theory. In one notable example, the Riemann zeta-function shows up explicitly in the calculation of the Casimir effect.[O] GeneralizationsThere are a number of related zeta functions that can be considered to be generalizations of Riemann's zeta-function. These include the Hurwitz zeta functionwhich coincides with Riemann's zeta-function when q = 1 (note that the lower limit of summation in the Hurwitz zeta function is 0, not 1), the Dirichlet L-functions and the Dedekind zeta-function. For other related functions see the articles Zeta function andL-function.The polylogarithm is given bywhich coincides with Riemann's zeta-function when z = 1.The Lerch transcendent is given bywhich coincides with Riemann's zeta-function when z = 1 and q = 1 (note that the lower limit of summation in the Lerch transcendent is 0, not 1).The Clausen function that can be chosen as the real orimaginary part ofThe multiple zeta functions are defined byOne can analytically continue these functions to then-dimensional complex space. The special values of these functions are called multiple zeta values by number theorists and have been connected to many different branches in mathematics and physics.[O] Zeta-functions in fiction。

附4.GOGO的数学词汇mapimagefixed pointcomposite functionone to one / injectiveonto / surjectivebijectiveinverse functionreciprocaldenominatorsymmetric with respect to the y axis / the origin / the line y=xabscissax/y interceptordinateanalytic geometryparabolahyperbolaconic sections 二次曲线系coefficientspoint-slope formuladirectrix 准线 vertex focuslatus rectum 过焦点平行于准线的弦radius, centertangent linedegenerate 退化major axis / minor axiseccentricity e=c/adifference 差branch 支asymptotes 渐近线focal axispolynomial equationsquadratic polynomial 二次多项式quadratic formuladiscriminant判别式division algorithm / remainder theorem quotientfundamental theorem of algebra multiplicity 重根conjugate radical 共轭根the complex conjugate 共轭复数‘monic (an=1)the rational roots theoremlogarithm 对数GRE用log x表示lnx trigonometry 三角几何complementary 互余cos = complementary sinetangent / cotangent / secant / cosecant terminal side 终边quadrant 象限arbitrary angle 钝角trig function 三角函数periodicity 周期性periodequidistant 等距sequence 序列convergent / divergent(minus) infinitymonotonic 单调bounded 有界the sandwich (or squeeze) theorem approach A from above (右逼近A) continuous functionThe Intermediate Value Theorem 中值定理derivative 导数secant line 割线tangent line 切线normal line 法线differential 微分的linear appropriationimplicit differentiation 隐函数求导concave up (convex) f’’>0concave down (concave) f’’<0inflection point 拐点local minimum / absolute minimum critical point / stationary point (f’=0)nth-derivative test fn>0 极小，反之极大adjacent sides 毗连的边relate rates dy/dt= r dx/dtindefinite integration 不定积分intersect 曲线相交rectangular (or Cartesian) coordinates polar coordinatescardioids 心形线r=2a(1+cos(sita)) solids of revolution 旋转体infinite series 无穷极数harmonic seriesp-seriesalternation seriespower seriesthe radius of convergencethe interval of convergencearccosine functionarcsine functiondomainadjoint 伴随阵determinantexpected valueprobability density functionderivativeinflection pointrankeigenvalueeigenvetoreigenspacesubsetpolyhedron 多面体vertices / vertexinverse of the matrixorthogonal 正交height 多项式系数绝对值和+最高次tracepolynomialidempotent 幂等A2=Anilpotent 幂零scalar 数量阵fixed pointthe qth roots of unitycoset 陪集dot/scalar product 点积projAB B在A的投影cross producttriple scalar product （A*B）•Cmagnitude 模parametric equation 参数方程symmetric equation （直线）对称式generator, elements cracking p111arbitrarylevel curve of height 等高线contour curve 轮廓线（被平面截的截面）hyperboloid 双曲面circular poraboloidcylindrical coordinatesspherical coordinatespartial derivativedirectional derivativesgradientsaddle point cracking p131 Hessian matrixline integralThe Fundamental Theorem of Calculus for Line Integral 势场内线积分只与起止点有关gradient field My=Nx conservative 值与路值无关Green Theorem cracking p152ordinary/partial differential equation (ODE/PDE)homogeneous of degree n n阶齐次exact differentialintegrating function 积分因子inconsistent （线性方程组）无解commutative 交换的invertible 可逆的associative 传递性coefficient matrixaugmented matrix 增广矩阵Gaussian Eliminationechelon formparameter 参数nullspacelinear combinationspan 几个向量的所有线性组合trivial combination即linearly independent basis a minimal spanning set for avector spacedimension 基中向量数normal vector 法向量column space / row spaceLaplace expansions 即按某行/列展开adjugate matrix 共轭矩阵Cramer’s Rule克莱莫法则scalar 数乘linear operation=linear transformation kernel / nullity / range / rankRank plus Nullity TheoremCayley-HamiltonTheorem p(A)=0 divisibility, factor, multipleprime number, compositegreatest common divisor (gcd)least common multiple (lcm)the congruence equation ax=b(mod n)the Euclidean Algorithm 欧氏算法cracking p222congruence 余数binary operation on S S*S S associate ：a•(b•c)=(a•b)•c semigroup条件identity 单位元semigroup+identity=monoidmonoid+inverse=groupabelian groupgeneral/special linear groupSn symmetric group对称群（阶为n!）S3为最小的6阶非阿贝尔对称群alternating group 置换群（同上）polygon 多边形equilateral triangle 等边三角形isosceles triangle 等腰三角形Dn nth dihedral group :order(Dn)=2n additive group of integers modulo n multiplicative group of integers modulo p cyclic groupKlein four-group, or viergruppeproper subgroupnontrivial subgroupgenerators 生成元finitely generatedisomorphism 同构homomorphism 同态monomorphism 单同态epimorphism 满同态endomorphism 自同态automorphism 自同构direct product (a, b) cracking p237 direct sum 同上if abelian elementary divisors/ invariant factors cracking p238normal subgruoup 正规子群inner automorphism induced by aunity 环乘法单位元unit 存在乘法逆的非零元素ring with unity 幺环commutative ringsubringcharacteristicring of integersr ing of integers modulo n (Zn, +, •)ring of Gaussian integers Z(i)ring of polynomials in x over R R[x] ring of real-valued functions on R RR 交换幺环evaluation (or substitution) homomorphism at a cracking p249Frobenius endomorphism f(a)=app is a prime numberbinomial theorem 二项式定理integral domain 整环left/right zero divisor 零因子cancellation law a!=0, ac=ab, them c=bdivision ring 无零因子的环field= commutative division ring又，有限整环是域strictly-skew field= noncommutative division ring体real quatenion 四元素体Boolean ring 该环中元素idempotentsubset > supersetuniversal setcomplement of B relative to A A-Bunion / intersectionsymmetric difference (A-B)U(B-A) Cartesian product 笛卡尔积open / closed intervalcardinality (cardinal number) 元素数countably infinitealgebraic numbers cracking p267power set of Alevels of infinitycardinal number of continuum transcendental numberscombination, permutationbinomial coefficientpigeonhole principle 抽屉原理probabilityBoolean algebra (or algebra) of sets on S: E 指the power set of S 的子集probability measure on E cracking p274 distribution functionvariance, standard deviationthe normal distribution 正态分布standard normal distributionbinomial distribution 二项分布imaginary unit iprinciple argument 幅角主值sample space (S), outcomes (S中元素), events（E中元素，S的子集）independent独立, mutually exclusive相斥Bernoulli trialspolar form, exponential formprincipal logarithmprincipal value of zwhyperbolic function 双曲函数Laplace equation / harmonic uxx+uyy=0 entire function 在复平面内解析disk of convergencepunctured open disk cracking p312 singularity, isolated singularitypole of order nsimple pole (n=1) double pole (n=2) essential singularityannulus 环面singular (or principal ) part / analytic part residueHausdorff spaceindiscrete / trivial topologyinterior, exterior, boundary, limit point, closureinterior+boundary=closurelower-limit topology B=[a,b) connectedcovering, open coveringcompactnessnorm of a point cracking p290 Euclidean metric 欧氏度量square metricopen map != continuous 一来一去，方向反homeomorphism = continuous + open map upper bound, bounded abovelub=suremum (sup)glb=infimum (inf)complete space = no holesLebesgue measurable setssingletonLebesgue measurable function Lebsegue integrablestep function cracking p296 index of a group 子群中元素的最小公共阶tangent 相切loop 循环inradius 内径trajectory 轨迹centroid 重心clusterpartial fraction expansionsufficient / necessary condition。

高斯(Gauss 1777~1855)生于Brunswick，位于现在德国中北部。

他的祖父是农民，父亲是泥水匠，母亲是一个石匠的女儿，有一个很聪明的弟弟，高斯这位舅舅，对小高斯很照顾，偶而会给他一些指导，而父亲可以说是一名「大老粗」，认为只有力气能挣钱，学问这种劳什子对穷人是没有用的。

高斯很早就展现过人才华，三岁时就能指出父亲帐册上的错误。

七岁时进了小学，在破旧的教室里上课，老师对学生并不好，常认为自己在穷乡僻壤教书是怀才不遇。

高斯十岁时，老师考了那道著名的「从一加到一百」，终于发现了高斯的才华，他知道自己的能力不足以教高斯，就从汉堡买了一本较深的数学书给高斯读。

同时，高斯和大他差不多十岁的助教Bartels变得很熟，而Bartels的能力也比老师高得多，后来成为大学教授，他教了高斯更多更深的数学。

老师和助教去拜访高斯的父亲，要他让高斯接受更高的教育，但高斯的父亲认为儿子应该像他一样，作个泥水匠，而且也没有钱让高斯继续读书，最后的结论是－－去找有钱有势的人当高斯的赞助人，虽然他们不知道要到哪里找。

经过这次的访问，高斯免除了每天晚上织布的工作，每天和Bartels讨论数学，但不久之后，Bartels也没有什么东西可以教高斯了。

1788年高斯不顾父亲的反对进了高等学校。

数学老师看了高斯的作业后就要他不必再上数学课，而他的拉丁文不久也凌驾全班之上。

高斯1777年4月30日生于不伦瑞克的一个工匠家庭，1855年2月23日卒于哥廷根。

幼时家境贫困，但聪敏异常，受一贵族资助才进学校受教育。

1795～1798年在哥廷根大学学习，1798年转入黑尔姆施泰特大学，翌年因证明代数基本定理获博士学位。

从1807年起担任格丁根大学教授兼格丁根天文台台长直至逝世。

高斯的成就遍及数学的各个领域，在数论、非欧几何、微分几何、超几何级数、复变函数论以及椭圆函数论等方面均有开创性贡献。

他十分注重数学的应用，并且在对天文学、大地测量学和磁学的研究中也偏重于用数学方法进行研究。