1.7 General Terminology: Glossary

Costas Loizou; Shuwei Chen; Deak Kirkham

1.7 General Terminology: Glossary

The pronunciation guide given in these glossaries draws on the symbols International Phonetic Alphabet. We have made two adaptations to the IPA system. Firstly, we use slanted lines as opposed to square brackets to enclose pronunciation material. We do this as the slanted line convention is more common in language teaching contexts. Secondly, we use the symbol <r> to denote the voiced alveolar approximant sound which appears in British English at the beginning of the each word in the phrase 'red rhombus'; the IPA would instead use an inverted 'r' symbols namely <ɹ>. We have done this to avoid unnecessary complication to the visuals of the glossaries.

In addition to these conventions, please also note that the pronunciations given are indicative of contemporary Standard British English pronunciation and should not be read as normative, nor as a target pronunciation that you must aim to produce. The pronunciation of English varies widely across World Englishes; part of becoming fluent in the academic English of mathematics is developing an awareness of, and sensitivity to, the variation in pronunciation that you will encounter.

The terminology in the first table of this section covers mathematical writing and reasoning vocabulary, including definitions and setup, proof and argumentation, precision and comparison, algebraic or procedural operations, relationships and conditions, and logical quantifiers, connectives, and proof phrases.

Word	Pronunciation	Part of speech词性	Related terminology	Chinese meaning	Example sentence 1	Example sentence 2
accurate	/ˈækjərət/	adj.		准确的（允许存在误差）	This result is accurate to two decimal places. 该结果精确到小数点后两位。	This formula gives an accurate expression. 该公式可给出精准表达式。
analogous	/əˈnæləɡəs/	adj.		类似的，类比的	This formula is analogous to the area formula. 该公式与面积公式形式相似。	The property is analogous for all odd numbers. 所有奇数都具备类似性质。
apply	/əˈplaɪ/	v.		应用，运用	Apply the chain rule to differentiate \(y=\sin(x^2)\). 应用链式法则求导 \(y=\sin(x^2)\)。	To simplify the proof, apply the Mean Value Theorem. 为了简化证明，应用中值定理。
approximate	/əˈprɒksɪmeɪt/	v.	approximation n.; approximate adj.	近似的，近似	The approximate value of \(\sqrt{2}\) is \(1.414\). \(\sqrt{2}\) 的近似值为 \(1.414\)。	We approximate the curve with a straight line segment. 我们用一段直线来近似这条曲线。
argue	/ˈɑːɡjuː/	v.	argument n. ; justify 证明合理性；说明……成立	论证说明	We argue that \(x \neq 1\) in this equation. 我们论证该方程中 \(x \neq 1\)。	Argue why \(y = x^2\) has no negative minimum. 论证 \(y = x^2\) 不存在负的最小值。
assume	/əˈsuːm/	v.	suppose 假设（讨论某种情况）	假定，假设（作为前提条件)	Assume that function \(h (x)\) is continuous on interval \([0,1]\). 假设函数 \(h (x)\) 在区间 \([0,1]\) 上连续。	Assume two lines \(l_1\) and \(l_2\) are parallel to each other. 假设直线 \(l_1\) 与 \(l_2\) 互相平行。
be proportional to	/biː prəˈpɔːʃənl tuː/	phr.		与……成正比	The area of a circle is proportional to \(r^2\). 圆的面积与 \(r^2\) 成正比。	The force is proportional to the acceleration. 力与加速度成正比。
belong to	/bɪˈlɒŋ tuː/	phr.		属于	The point \((0,0)\) belongs to the graph of \(y=x^2\). 点 \((0,0)\) 属于 \(y=x^2\) 的图像。	The number \(\pi\) belongs to the set of irrational numbers. \(\pi\) 属于无理数集合。
calculate	/ˈkælkjuleɪt/	v.	calculation n. ；compute 计算（算法化）； evaluate 求值；计算	计算	Calculate the derivative of \(f(x)=x^3\) at \(x=2\). 计算函数 \(f(x)=x^3\) 在 \(x=2\) 处的导数。	To find the slope, calculate \(f'(3)\). 为了求斜率，计算 \(f'(3)\)。
claim	/kleɪm/	v./n.	assert 断言（比 claim 更正式、更强）	断言，提出（待证明命题）	We claim that \(x=3\) is the unique solution. 我们断言 \(x=3\) 是唯一解。	The author claims this formula holds for all \(n\). 该作者称此公式对所有 \(n\) 都成立。
coincide with	/ˌkəʊɪnˈsaɪd wɪð/	phr.		与……一致，重合	The two graphs coincide with each other on \([0,1]\). 两条图像在 \([0,1]\) 上重合。	The limit coincides with the function value. 极限值与函数值一致。
conclude	/kənˈkluːd/	v.		得出结论，推知	We conclude \(x \lt 0\) after analyzing the inequality. 分析不等式后，我们得出 \(x \lt 0\)。	Conclude that the equation has two distinct real roots. 判定该方程有两个不等实数根。
confirm	/kənˈfɜːm/	v.		证实，确认（检验后支持结论）	Confirm \(x = 6\) satisfies \(4x - 5 = 19\). 确认 \(x = 6\) 满足 \(4x - 5 = 19\)。	Confirm the final answer of the integral. 确认积分运算的最终答案。
consequently	/ˈkɒnsɪkwəntli/	adv.		结果是	The function fails the continuity test, consequently it is not differentiable. 该函数不满足连续性条件，因此不可导。	The domain is not connected, consequently the function has segmented properties. 定义域不连通，因此函数具有分段特性。
consider	/kənˈsɪdə(r)/	v.		考虑	Consider the interval \([1,5]\) and discuss function monotonicity. 考察区间 \([1,5]\)，讨论函数单调性。	Consider the series \(1+2+3+ \cdots +n\) and calculate its sum. 考察数列 \(1+2+3+ \cdots +n\)，计算其和。
construct	/kənˈstrʌkt/	v.	construction n.	构造，构建	We construct a tangent line at \(x=2\) to analyze the function trend. 我们在 \(x=2\) 处构造切线来分析函数变化趋势。	We construct a piecewise function \(f (x) = x\) \((x\geq 0)\), \(f (x) = -x\) \((x \lt 0)\) for discussion. 我们构造分段函数 \(f (x) = x\) \((x\geq 0)\), \(f (x) = -x\) \((x \lt 0)\) 用于分析讨论。
contain	/kənˈteɪn/	v.	consist of 由…组成	包含，含有	The polynomial contains a quadratic term. 该多项式包含一个二次项。	This set contains infinitely many elements. 该集合包含无穷多个元素。
convert	/kənˈvɜːt/	v.	conversion n.	转换，换算	Convert the angle from degrees to radians. 将角度从度数转换为弧度。	The expression can be converted into a logarithmic form. 该表达式可以转换为对数形式。
correspond to	/ˌkɒrɪˈspɒnd tuː/	phr.		对应于，与……相对应	Each coefficient corresponds to a basis vector. 每个系数对应一个基向量。	The angle \(\theta\) corresponds to a point on the unit circle. 角 \(\theta\) 对应单位圆上的一个点。
deduce	/dɪˈdjuːs/	v.	deduce from 由...推得；infer 推断（比 deduce 弱）	逻辑推导	Deduce the formula for the area of a trapezoid. 推导出梯形的面积公式。	Deduce the range of variable \(t\) from given constraints. 根据已知约束条件推导出变量 \(t\) 的取值范围。
define	/dɪˈfaɪn/	v.	well-defined 良定义的	定义	Define the set \(A = \{1,2,3 \}\) for the following calculation. 为后续计算定义集合 \(A = \{1,2,3 \}\)。	Define vector \({\bf a} = (3,4)\) in the two-dimensional coordinate system. 在二维坐标系中定义向量 \({\bf a} = (3,4)\)。
demonstrate	/ˈdemənstreɪt/	v.	illustrate 举例说明、画图说明；exemplify 作为……的例子	论证，证明（强调展示过程）	This example demonstrates how to solve a quadratic equation using the formula. 这个例子展示了如何使用公式求解二次方程。	The graph clearly demonstrates that the function is increasing on the interval \([0,1]\). 该图像清楚地表明该函数在区间 \([0,1]\) 上是递增的。
denote	/dɪˈnəʊt/	v.		表示，记作	The symbol \(\mathbb{N}\) denotes the set of all natural numbers. 符号 \(\mathbb{N}\) 表示全体自然数集合。	We denote the solution of equation \(x^2-4=0\) as \( x_1\) and \(x_2\). 我们用 \(x_1\) 和 \(x_2\) 表示方程 \(x^2-4=0\) 的解。
depend on	/dɪˈpend ɒn/	phr.	depend upon	依赖于，取决于	The convergence depends on the choice of \(n\). 收敛性取决于 \(n\) 的选择。	The result depends on whether the function is continuous. 结果取决于函数是否连续。
determine	/dɪˈtɜːmɪn/	v.	uniquely determine	决定，确定	The initial condition determines the unique solution. 初始条件决定唯一解。	The sign of \(f'(x)\) determines whether the function is increasing. \(f'(x)\) 的符号决定函数是否单调递增。
equate	/ɪˈkweɪt/	v.	equation n.	使相等，列等式	Equate the coefficients of \(x^2\) on both sides. 令等式两边 \(x^2\) 的系数相等。	By equating the two expressions, we obtain \(a=3\). 通过令两个表达式相等，我们得到 \(a=3\)。
equivalent	/ɪˈkwɪvələnt/	adj.	equivalence n.	等价，等同	The two equations are equivalent. 这两个方程是等价的。	The inequality is equivalent to \(x \gt 2\). 该不等式等价于 \(x \gt 2\)。
exact	/ɪɡˈzækt/	adj.		精确的	Calculate the exact area of the circle. 计算圆的精确面积。	Find the exact coordinate of the intersection point. 求出交点的精确坐标。
examine	/ɪɡˈzæmɪn/	v.		考察，检验	Examine whether \(x = 2\) satisfies \(5x - 1 = 9\). 检验 \(x = 2\) 是否满足 \(5x - 1 = 9\)。	Examine if two lines \(y=x\) and \(y=x+1\) intersect. 检验直线 \(y=x\) 与 \(y=x+1\) 是否相交。
exclusive	/ɪkˈskluːsɪv/	adj.	see also inclusive 相容的	互斥的	In mathematics, "\(x \gt 0\)" and "\(x \lt 0\)" are exclusive. 在数学中，"\(x \gt 0\)" 和 "\(x \lt 0\)" 是互斥的。	The events of rolling an even number and rolling an odd number are mutually exclusive. 掷出偶数和掷出奇数这两个事件是互斥事件。
express	/ɪkˈspres/	v.		表示，表达	Express \(y\) in terms of \(x\). 用 \(x\) 表示 \(y\)。	The solution can be expressed as a power series. 解可以表示为幂级数。
fix	/fɪks/	v.		令，在证明中固定一个对象	Fix \(n = 10\), calculate the sum from \(i=1\) to \(n\). 固定 \(n = 10\)，计算 \(i\) 从 \(1\) 到 \(n\) 的累加和。	Fix the side length of square to \(7\), compute its area. 固定正方形边长为 \(7\)，计算其面积。
for all	/fɔː(r) ɔːl/	phr.		对所有，任意	This inequality holds for all real numbers \(x\) in the domain of \(f(x)\). 该不等式对\(f(x)\)定义域内的所有实数\(x\)成立。	For all \(x\in R\), \(x^2 \geq 0\). 对所有属于实数集的 \(x\)，都有 \(x^2 \geq 0\)。
fundamentals	/ˌfʌndəˈmentlz/	n.		基本原理	These rules belong to the fundamentals of algebra. 这些法则属于代数学的基础内容。	Review the fundamentals before complex calculation. 复杂运算前复习基础知识点。
generalise	/ˈdʒenrəlaɪz/	v.	generalisation n.	推广，一般化（从特殊到一般）	Generalise the formula from special cases to all \(n\). 将特例公式推广到全体 \(n\)。	We can generalise this method to similar problems. 我们可将此方法推广到同类题型。
generate	/ˈdʒenəreɪt/	v.		生成，产生	This mapping rule generates a function \(y=\sqrt{x+1}\) defined on \(x\geq -1\). 该对应法则生成定义域为 \(x\geq -1\) 的函数 \(y=\sqrt{x+1}\)。	Recursive formulas generate infinite sequences \(\{a_n\}\) in limit analysis. 在极限分析中，递推公式可以生成无穷数列 \(\{a_n\}\)。
given	/ˈɡɪvən/	adj.		已知，给定	Given \(a = 6\), compute the value of \(4a - 1\). 已知 \(a = 6\)，计算 \(4a - 1\) 的值。	Given set \(B = \{2,4,6\}\), find all its subsets. 已知集合 \(B = \{2,4,6\}\)，求出它的所有子集。
if ... then...	/ɪf ... ðen/	phr.		如果……那么……	If \(x=0\), then \(xy=0\). 如果\(x=0\)，则\(xy=0\)。	If \(f(x)\) is differentiable, then it is continuous. 如果\(f(x)\)可导，则它连续。
if and only if	/ɪf ænd ˈəʊnli ɪf/	phr.	缩写 iff	当且仅当	The series converges if and only if its partial sum sequence is convergent. 级数收敛当且仅当其部分和数列收敛。	\(x^2 = 1\) if and only if \(x = 1\) or \(x = -1\). \(x^2 = 1\) 当且仅当 \(x = 1\) 或 \(x = -1\)。
imply	/ɪmˈplaɪ/	v.	implication n.	蕴含，意味着	Parallel lines imply equal corresponding angles. 两直线平行则同位角相等。	\(f (x) = 0\) implies \(x\) is a root of the equation. \(f (x) = 0\) 说明 \(x\) 是方程的根。
in particular	/ɪn pəˈtɪkjələ(r)/	phr.		特别地	In particular, constant functions have a zero derivative for all real \(x\). 特别地，常函数在全体实数域上的导数恒为零。	In particular, linear functions maintain constant first derivatives. 特别地，一次函数的一阶导数为常数。
in terms of	/ɪn tɜːmz ɒv/	phr.		用……来表示	Express \(y\) in terms of \(x\). 用 \(x\) 表示 \(y\)。	Rewrite the integral in terms of \(u\). 用 \(u\) 改写该积分。
inclusive	/ɪnˈkluːsɪv/	adj.	see also exclusive 互斥的	相容的	In mathematics, the word "or" is usually inclusive, meaning that one or both conditions may hold. 在数学中，“或”通常是包含性的，即一个条件成立或两个条件同时成立都可以。	The inequality \(x \geq 1\) is inclusive because it includes the endpoint \(1\). 不等式 \(x \geq 1\) 是包含性的，因为它包含端点 \(1\)。
indicate	/ˈɪndɪkeɪt/	v.		表明，指示	The calculation result indicates \(a = 7\). 计算结果表明 \(a = 7\)。	The graph indicates \(y = x\) is a straight line. 图像显示 \(y = x\) 是一条直线。
intuitively	/ɪnˈtjuːɪtɪvli/	adv.		直觉上地	We can intuitively judge \(x \gt y\) from the graph. 我们可从图像直观判断 \(x \gt y\)。	Intuitively, two parallel lines never intersect. 直观来看，两条平行线永不相交。
is equal to	/ɪz ˈiːkwəl tuː/	phr.		等于	Five plus three is equal to eight. 五加三等于八。	The length of side \(A\) is equal to the length of side \(B\). 边 \(A\) 的长度等于边 \(B\) 的长度。
it follows that	/ɪt ˈfɒləʊz ðæt/	phr.		因此可得	It follows that the series diverges from the failure of the necessary condition. 由必要条件不成立可推得该级数发散。	It follows that the definite integral is positive for positive integrands. 由此可知，正被积函数的定积分结果为正。
lead to	/liːd tuː/	phr.	result in 导致；引起（强调结果）	导致，通向（强调推理过程）	A small perturbation of the independent variable will lead to a tiny change of \(f(x)\). 自变量的微小扰动会导致\(f(x)\)产生微小变化。	Infinite oscillation near \(x=0\) leads to the inexistence of the function limit. 函数在\(x=0\)处无限振荡导致极限不存在。
let	/let/	v.		设，令（引入变量）	Let \(t = 2k\), simplify the formula \(3t + 4\). 令 \(t = 2k\)，化简式子 \(3t + 4\)。	Let \(D\) represent the diameter of the circle. 令 \(D\) 表示圆的直径。
let us consider	/let əs kənˈsɪdə(r)/	phr.		我们考虑	Let us consider the piecewise property of the function at \(x=1\). 我们来分析函数在\(x=1\)处的分段性质。	Let us consider the convergence of the positive term infinite series. 我们来探讨正项无穷级数的收敛性。
logical proposition	/ˈlɒdʒɪkl ˌprɒpəˈzɪʃn/	n. phr.		逻辑命题	The statement "\(2\) is an even number" is a logical proposition because it has a definite truth value. “\(2\) 是偶数”这一陈述是一个逻辑命题，因为它具有确定的真假值。	Every logical proposition in a proof must be either true or false. 证明中的每个逻辑命题都必须为真或假。
manipulate	/məˈnɪpjəleɪt/	v.	manipulation n.	运算，处理	We manipulate both sides of the inequality by adding \(1\). 我们在不等式两边同时加\(1\)进行变形。	The formula can be manipulated into a more useful form. 该公式可以变形成更有用的形式。
necessary	/ˈnesəsəri/	adj.	necessarily adv.；necessary and sufficient condition 充要条件	必要的	Existence of derivative is a necessary condition for function continuity. 可导是函数连续的必要条件。	Closed interval domain is a necessary condition for the extreme value theorem. 闭区间定义域是极值定理成立的必要条件。
note that	/nəʊt ðæt/	phr.	notice that；observe that	注意到	Note that differentiability implies continuity, but the converse is not true. 注意：可导可以推出连续，反之不成立。	Note that infinite series convergence does not guarantee term convergence. 注意：无穷级数收敛不代表通项单调收敛。
observe	/əbˈzɜːv/	v.	note 注意；指出； remark 备注；指出	注意到	Observe that \(x^2 \geq 0\) for all real \(x\). 可注意到对任意实数 \(x\)，都有 \(x^2 \geq 0\)。	Observe the change of \(y\) when \(x\) increases. 注意到 \(x\) 增大时 \(y\) 的变化规律。
obtain	/əbˈteɪn/	v.	derive 推导得到； yield 得出，产生； produce 产生；给出； give 给出；得到	得到，获得	We obtain the domain of the function by solving the inequality \(x-2>0\). 通过求解不等式 \(x-2>0\)，我们得到该函数的定义域。	One can obtain the integral \(\int_0^{\pi} \sin x dx\) via the Newton-Leibniz formula. 借助牛顿 - 莱布尼茨公式可以算出积分\(\int_0^{\pi} \sin x dx\)。
otherwise	/ˈʌðəwaɪz/	adv./conj.		否则	We take \(x \gt 0\), otherwise the logarithmic function is undefined. 我们取\(x \gt 0\)，否则对数函数无定义。	Assume the function is differentiable; otherwise the derivative does not exist. 假设函数可导，否则导数不存在。
preserve	/prɪˈzɜːv/	v.		保持，保留	Addition preserves the inequality \(a \lt b\). 加法运算不改变不等式 \(a \lt b\) 的方向。	Rotation preserves the shape and size of figures. 旋转变换保持图形形状与大小不变。
proof by contradiction	/pruːf baɪ ˌkɒntrəˈdɪkʃn/	phr.		反证法	We use proof by contradiction to prove that \(\sqrt{2}\) is an irrational number. 我们用反证法证明 \(\sqrt{2}\) 是无理数。	Proof by contradiction is widely used in number theory and geometry proofs. 反证法广泛应用于数论与几何证明。
prove	/pruːv/	v.	proof n. ; show 证明；表明（没有prove 正式）；verify 验证，检验（已知条件或结果）	证明	Prove the sum of interior angles of a triangle is \(180^\circ\). 证明三角形内角和为 \(180^\circ\)。	Prove \(n^2+n\) is always an even number for integer \(n\). 证明对于整数\( n\)，\(n^2+n\) 恒为偶数。
provided that	/prəˈvaɪdɪd ðæt/	conj.		只要，给定	Provided that \(x\neq 0\), the function \(f(x)=\frac{1}{x}\) is well-defined and differentiable. 若\(x\neq 0\)，函数\(f(x)=\frac{1}{x}\)有定义且可导。	\(f(x)\) is continuous provided that \(x \neq 0\). \(f(x)\) 连续，条件是 \(x \neq 0\)。
rearrange	/ˌriːəˈreɪndʒ/	v.	rearrangement n.	重新整理，移项	Rearrange the equation \(x+3=7\) to solve for \(x\). 重排方程 \(x+3=7\) 以求出 \(x\)。	The formula can be rearranged to express \(y\) in terms of \(x\). 该公式可以重新整理为用 \(x\) 表示 \(y\)。
recall that	/rɪˈkɔːl ðæt/	phr.		回忆，复习	Recall that polynomial functions are infinitely differentiable on real numbers. 回顾知识点，多项式函数在实数域上无穷次可导。	Recall that the chain rule is essential for composite function differentiation. 回顾可知，链式法则是复合函数求导的核心法则。
reduce	/rɪˈdjuːs/	v.	cancel 消去；约去	约简，化简	Reduce the fraction \(\frac{12}{18}\) to its lowest terms. 将分数 \(\frac{12}{18}\) 约分到最简形式。	The substitution reduces the complexity of the integral. 该换元降低了积分的复杂程度。
relate to	/rɪˈleɪt tuː/	phr.	associate with 与……关联	与……相关	This theorem relates to differentiable functions. 该定理与可导函数有关。	The derivative relates to the slope of the tangent line. 导数与切线斜率有关。
require	/rɪˈkwaɪə(r)/	v.	requirement n.	要求，需要	This method requires solving a quadratic equation first. 该方法要求先求解一个二次方程。	The theorem requires several assumptions. 该定理需要若干假设条件。
rewrite	/ˌriːˈraɪt/	v.		改写，重写	Rewrite the integral in terms of the variable \(u\). 将积分用变量 \(u\) 改写。	Rewrite \(x^2+2x+1\) as \((x+1)^2\). 将 \(x^2+2x+1\) 改写为 \((x+1)^2\)。
satisfy	/ˈsætɪsfaɪ/	v.	fulfill 满足条件（比 satisfy 弱）； obey 满足，遵守规则	满足，成立	Find all \(x\) that satisfy \(x^2−4=0\). 求所有满足 \(x^2−4=0\) 的 \(x\)。	The solution satisfies the boundary condition. 该解满足边界条件。
set	/set/	v.		设定，设定变量	Set the range of variable \(z\) as \(-2 \lt z \lt 6\). 设定变量 \(z\) 的取值范围为 \(-2 \lt z \lt 6\)。	Set the initial value of the sequence \(a_1 = 1\). 设定数列首项 \(a_1 = 1\)。
simplify	/ˈsɪmplɪfaɪ/	v.		化简，简化	The expression simplifies to \(2x+3\). 该表达式化简为 \(2x+3\)。	Simplify the radical before evaluating the limit. 求极限前先化简根式。
solve	/sɒlv/	v.	solution n.	求解，解出	Solve the equation \(x^2−5x+6=0\). 求解方程 \(x^2−5x+6=0\)。	Solve for \(y\) in terms of \(x\). 求出 \(y\) 关于 \(x\) 的表达式。
specific case	/spəˈsɪfɪk keɪs/	n.	general case 一般情况	特例，具体情形	Constant functions serve as a specific case of bounded continuous functions on \(\mathbb{R}\). 常函数是实数集\(\mathbb{R}\)上有界连续函数的一类特例。	Monotonic increasing functions are a specific case of monotonic functions. 单调递增函数是单调函数的一类特例。
specify	/ˈspesɪfaɪ/	v.		指定，限定	Specify the condition \(a \geq b\) in this proof. 在本证明中明确条件 \(a \geq b\)。	Specify the upper limit 10 for the summation. 指定求和运算的上限为 10。
step by step	/step baɪ step/	phr.		逐步地	Step by step simplification leads to the final answer. 逐步化简得到最终答案。	We solve the equation step by step. 我们一步一步解方程。
substitute	/ˈsʌbstɪtjuːt/	v.	substitution n.；insert 插入；代入	代入，替换	Substitute \(x=2\) into the equation \(y=x^2+1\). 将 \(x=2\) 代入方程 \(y=x^2+1\)。	By substituting the boundary condition, we obtain \(C=0\). 通过代入边界条件，我们得到 \(C=0\)。
such that	/sʌtʃ ðæt/	phr.	s.t.	使得	There exists a constant \(M\) such that \(\|f(x)\|\leq M\) for all \(x\in [a,b]\). 存在常数 \(M\), 使得对所有\(x\in [a,b]\)，均有\(\|f(x)\|\leq M\)。	We construct a function such that its derivative equals the given integrand. 我们构造函数使其导数等于已知被积函数。
sufficient	/səˈfɪʃənt/	adj.	sufficiently adv.；necessary and sufficient condition 充要条件	充分的	Continuity is a sufficient condition for the integrability of bounded functions. 连续性是有界函数可积的充分条件。	Monotonic boundedness is a sufficient condition for sequence convergence. 单调有界是数列收敛的充分条件。
there exists	/ðeə ɪɡˈzɪst/	phr.		存在	There exists \(x \in \mathbb{R}\) such that \(x^2 = 2\). 存在一个实数 \(x\)，使得 \(x^2 = 2\)。	There exists a unique solution for the linear differential equation under initial conditions. 初值条件下该线性微分方程存在唯一解。
therefore	/ˈðeəfɔː(r)/	adv.	hence，thus	因此	The function is differentiable everywhere, therefore it is continuous everywhere. 该函数处处可导，因此它处处连续。	The series satisfies the convergence test, therefore it converges absolutely. 该级数满足收敛判别条件，因此绝对收敛。
transform	/trænsˈfɔːm/	v.	transformation n.	变换，转化	The substitution transforms the integral into a simpler form. 该换元将积分变换成更简单的形式。	The graph is transformed by reflecting it across the \(x\)-axis. 图像通过关于 \(x\) 轴对称而发生变换。
trivial	/ˈtrɪviəl/	adj.		平凡的，显而易见的	The solution of \(x = 0\) is a trivial case. \(x = 0\) 是一个平凡解。	The trivial condition does not affect the result. 该无关紧要的条件不影响结果。
validity	/vəˈlɪdəti/	n.	valid v.	有效性，正确性	Check the validity of equation \(x^2 + 1 = 0\) in real numbers. 检验方程 \(x^2 + 1 = 0\) 在实数范围内是否成立。	Discuss the validity of this proof method. 讨论该证明方法的有效性。
vary	/veəri/	v.	variation n.; variable n.	变化，改变	As \(n\) increases, the sequence varies more slowly. 随着 \(n\) 增大，数列变化得更缓慢。	The value of the function varies with \(x\). 函数值随 \(x\) 的变化而变化。
we have shown that	/wiː hæv ʃəʊn ðæt/	phr.	As shown above 如上所示； It can be shown that 可以证明……	我们已证明	We have shown that continuous functions on closed intervals are bounded. 我们已证得闭区间上的连续函数必有界。	We have shown that odd symmetric integrals equal zero over symmetric domains. 我们已证明奇函数在对称区间的积分为零。
whenever	/wenˈevə(r)/	conj.		每当，在……情况下	Whenever the derivative equals zero, the function may attain an extremum. 当导数为零时，函数可能取得极值。	Whenever the integrand is odd, the symmetric integral equals zero. 当被积函数为奇函数时，对称区间积分结果为零。
with respect to	/wɪð rɪˈspekt tuː/	phr.	缩写 w.r.t.	关于，就……而言	Differentiate with respect to \(x\). 对\(x\)求导。	Area changes with respect to time. 面积随时间变化。
without loss of generality	/wɪˈðaʊt lɒs ɒv ˌdʒenəˈræləti/	phr.	wlog	不失一般性	Without loss of generality, assume \(a \geq b\); we only need to prove \(a^2 \geq ab\). 不失一般性，设 \(a \geq b\)，我们只需证明 \(a^2 \geq ab\)。	Without loss of generality, take \(x_0 = 0\), then the function simplifies to \(f(t) = t^3 + t\). 不失一般性，令 \(x_0 = 0\)，则该函数化简为 \(f(t) = t^3 + t\)。

The terminology in the second table of this section covers foundational mathematical vocabulary, including arithmetic operations, number types and properties, proportional reasoning, probability and measurement, set theory, intervals and ordered pairs, variables and mappings, function types, equations, and inequalities.

Word	Pronunciation	Part of speech词性	Related terminology	Chinese meaning	Example sentence 1	Example sentence 2
arithmetic operation	/əˈrɪθmətɪk ˌɒpəˈreɪʃn/	n. phr.	opposite operation 逆运算	算术运算	Arithmetic operations include addition and subtraction. 算术运算包括加减法。	We need to finish all arithmetic operations inside brackets first according to calculation rules. 根据运算规则，我们需要先完成括号内的所有算术运算。
addition	/əˈdɪʃn/	n.	plus 加（介词，加号读法）； sum 和（加法结果）	加法	Addition is the operation of combining two or more numbers into one total. 加法是把两个或多个数字合并成总和的运算。	We use the sign plus to stand for addition in basic maths. 在基础数学里我们用加号代表加法。
subtraction	/səbˈtrækʃn/	n.	minus 减（介词，减号读法）, difference 差（减法结果）	减法	Subtraction finds the difference between two given numbers. 减法用来求出两个给定数字的差值。	Subtraction is the inverse operation of addition. 减法是加法的逆运算。
multiplication	/ˌmʌltɪplɪˈkeɪʃn/	n.	times 乘（介词，乘号口语读法）；product 乘积（乘法结果）	乘法	Multiplication means repeatedly adding the same number several times. 乘法表示把同一个数字重复相加若干次。	Multiplication has higher priority than addition and subtraction in calculation. 计算时乘法优先级高于加法和减法。
division	/dɪˈvɪʒn/	n.	divided by 除以， dividend 被除数， divisor 除数， quotient 商, remainder 余数	除法	Division splits a number into equal smaller parts. 除法将一个数分成若干相等的小部分。	Division by zero is undefined in all real number calculations. 在所有实数运算中，零不能作为除数。
summation	/sʌˈmeɪʃn/	n.	summation symbol 求和符号 \(\sum\)	求和	Summation of the series gives a finite value. 级数求和得到有限值。	The summation sign represents repeated addition. 求和符号表示重复加法。
increment	/ˈɪŋkrəmənt/	n.		增加量，增量	Increment of \(x\) is small. \(x\) 的增量很小。	The function increases by a constant increment. 函数以恒定增量增加。
decimal	/ˈdesɪml/	n.	小数；decimal places 小数位数	小数	The decimal representation of \(\frac{1}{2}\) is \(0.5\). \(\frac{1}{2}\)的小数表示是\(0.5\)。	A repeating decimal occurs when division does not terminate. 除法不终止时会出现循环小数。
fraction	/ˈfrækʃn/	n.	numerator 分子; denominator 分母; over 除以，比上; reciprocal 倒数	分数	A fraction represents a ratio. 分数表示一个比值。	The fraction simplifies to \(\frac{1}{2}\). 该分数化简为\(\frac{1}{2}\)。
ratio	/ˈreɪʃiəʊ/	n.	proportion 比例	比	The ratio is \(2:3\). 比值是\(2:3\)。	The ratio of \(x\) to \(y\) is constant. \(x\)与\(y\)的比是常数。
percent	/pəˈsent/	n.	percentage 比例/百分比（作为整体概念）	百分之	\(50\) percent of \(x\) is \(y\). \(x\)的百分之\(50\)是\(y\)。	Increase by \(10\) percent. 增加百分之\(10\)。
natural number	/ˈnætʃrəl ˈnʌmbə(r)/	n. phr.	integer 整数; rational number 有理数; rational function 有理函数; irrational number 无理数	自然数	\(n\) is a natural number. \(n\)是自然数。	Natural numbers start from \(1\). 自然数从\(1\)开始。
even	/ˈiːvn/	adj.		偶数	\(2\) is an even number. \(2\)是偶数。	Even numbers are divisible by \(2\). 偶数可被\(2\)整除。
odd	/ɒd/	adj.		奇数	\(3\) is an odd number. \(3\)是奇数。	Odd numbers are not divisible by \(2\). 奇数不能被\(2\)整除。
multiple	/ˈmʌltɪpl/	n.	multiply v.	倍数	\(6\) is a multiple of \(3\). \(6\)是\(3\)的倍数。	Multiples of \(5\) include \(10, 15\). \(5\)的倍数包括\(10, 15\)。
factor	/ˈfæktə(r)/	n.	factorise v.; common factor 公因数 / 公因式	因子，因式	The greatest common factor of \(12\) and \(18\) is \(6\), which simplifies the fraction. \(12\)和\(18\)的最大公因数是\(6\)，用于约分分数。	\(3\) is a factor of \(12\). \(3\) 是 \(12\) 的一个因子。
power	/paʊə(r)/	n.		幂	\(2^3\) is a power of \(2\). \(2^3\)是\(2\)的幂。	Power grows exponentially. 幂增长是指数型的。
square	/skweə(r)/	n./v.	square root 平方根	平方	\(x^2\) is the square of \(x\). \(x\)的平方是\(x^2\)。	Square of \(3\) is \(9\). \(3\)的平方是\(9\)。
cube	/kjuːb/	n./v.	cube root 立方根	立方	\(x^3\) is the cube of \(x\). \(x\)的立方是\(x^3\)。	Cube of \(2\) is \(8\). \(2\)的立方是\(8\)。
prime	/praɪm/	n./adj.		质数，素数	\(2\) is a prime number. \(2\)是质数。	Prime numbers have exactly two divisors. 质数只有两个因数。
coprime	/kəʊˈpraɪm/	n./adj.		互质数	\(5\) and \(9\) are coprime. \(5\)和\(9\)互质。	Coprime numbers share no common factors. 互质数没有公因子。
value	/væljuː/	n.	absolute value 绝对值	值	The value of the function at \(x=2\) is \(5\). 函数在\(x=2\)时的值为\(5\)。	The absolute value of a negative number is always positive. 负数的绝对值总是正数。
non-zero	/nɒn ˈzɪərəʊ/	adj.		非零的	Let \(x\) be non-zero. 设\(x\)非零。	Divide by a non-zero number. 除以非零数。
finite	/ˈfaɪnaɪt/	adj.	finitely adv. 有限地	有限的	The set is finite. 该集合是有限的。	Finite sequence has limited terms. 有限序列有有限项。
infinite	/ˈɪnfaɪnaɪt/	adj.	infinitely adv. 无限地； infinity n. 无限	无限的	The set of integers is infinite. 整数集是无限的。	Infinite series may converge. 无穷级数可能收敛。
probability	/ˌprɒbəˈbɪləti/	n.		概率	The probability of rolling a six is \(\frac{1}{6}\). 掷出6的概率是\(\frac{1}{6}\)。	The probability approaches zero as \(n\) increases. 当\(n\)增加时概率趋近于\(0\)。
factorial	/fækˈtɔːriəl/	n.		阶乘	The factorial of \(5\) is \(120\). \(5\)的阶乘是\(120\)。	Factorial grows very rapidly as \(n\) increases. \(n\) 增大时阶乘增长非常快。
measure	/ˈmeʒə(r)/	n.		度量，测量	We measure the length using a ruler. 我们用尺子测量长度。	Angle measure is given in degrees or radians. 角度可以用度或弧度表示。
set	/set/	n.	element n.元素； empty set 空集	集合	We define a set containing only three elements \(\{1,2,3\}\). 我们定义一个仅含三个元素 \(\{1,2,3\}\) 的集合。	The set of all positive integers is written as \(\mathbb{N}^+\). 全体正整数构成的集合记作 \(\mathbb{N}^+\)。
subset	/ˈsʌbset/	n.	proper subset 真子集; superset 超集	子集	\(\{2,4\}\) is a subset of the set \(\{1,2,3,4\}\). \(\{2,4\}\) 是集合 \(\{1,2,3,4\}\) 的子集。	Every interval is a subset of the real number set \(\mathbb{R}\). 任意区间都是实数集 \(\mathbb{R}\) 的子集。
union	/ˈjuːniən/	n.		并集	The union of set \(A\) and set \(B\) includes all elements from \(A\) or \(B\). 集合 \(A\) 与集合 \(B\) 的并集包含所有属于 \(A\) 或 \(B\) 的元素。	We calculate the union of two intervals \([0,1] \cup (1,3)\). 我们计算区间 \([0,1]\) 与 \((1,3)\) 的并集。
intersection	/ˌɪntəˈsekʃn/	n.		交集	The intersection of two sets only keeps elements shared by both sets. 两个集合的交集只保留两者共有的元素。	The intersection of \([1,5]\) and \([3,7]\) equals \([3,5]\). 区间 \([1,5]\) 与 \([3,7]\) 的交集为 \([3,5]\)。
complement	/ˈkɒmplɪmənt/	n.	universal set 全集	补集	The complement of set \(A\) contains every element in the universal set not in \(A\). 集合 \(A\) 的补集包含全集中所有不在 \(A\) 里的元素。	If the universal set is \(\mathbb{R}\), the complement of positive real numbers is non-positive real numbers. 若全集为实数集 \(\mathbb{R}\)，正实数的补集是非正实数。
interval	/ˈɪntəvl/	n.	closed interval 闭区间； open interval 开区间	区间	An interval represents a continuous range of real numbers on the number line. 区间代表数轴上一段连续的实数范围。	The solution of this inequality forms a closed interval \([-2,4]\). 该不等式的解构成闭区间 \([-2,4]\)。
half-open	/hɑːf ˈəʊpən/	adj.	half-closed 半闭的	半开的	\((0,1]\) is a half-open interval, which includes \(1\) but excludes \(0\). \((0,1]\) 是半开区间，包含 \(1\) 但不包含 \(0\)。	This half-open interval starts at \(2\) and ends at \(6\), only containing the upper endpoint. 这个半开区间起点为 \(2\), 终点为 \(6\)，仅包含上端点。
endpoint	/ˈɛndpɔɪnt/	n.		端点	The endpoints of the closed interval \([a,b]\) are \(a\) and \(b\). 闭区间 \([a,b]\) 的两个端点是 \(a\) 和 \(b\)。	A parenthesis means the endpoint is not included in the interval. 圆括号代表该端点不包含在区间内。
ordered pair	/ˈɔːdəd peə/	n.	ordered triple 有序三元组	有序数对	Each point on the Cartesian plane matches one unique ordered pair \((x,y)\). 直角坐标系内每个点对应唯一一组有序数对 \((x,y)\)。	The ordered pair \((2,5)\) is different from \((5,2)\). 有序数对 \((2,5)\) 和 \((5,2)\) 不相等。
independent variable	/ˌɪndɪˈpendənt ˈveəriəbl/	n. phr.	dependent variable 因变量	自变量	We usually place the independent variable on the horizontal axis when drawing a graph. 绘制函数图像时，我们通常把自变量放在横轴上。	In the function \(y = x^2 + 1\), \(x\) is the independent variable and \(y\) depends on \(x\). 在函数 \(y = x^2 + 1\) 中，\(x\) 是自变量，而 \(y\) 依赖于\( x\)。
domain	/ˈdɒmeɪn/	n.		定义域	The domain of \(f(x)\) excludes \(x=0\). 函数定义域不包括\(x=0\)。	Domain determines valid input values. 定义域决定合法输入。
codomain	/ˌkəʊdəˈmeɪn/	n.		陪域	The codomain specifies possible outputs. 陪域表示可能输出集合。	Codomain may be larger than range. 陪域可能大于值域。
range	/reɪndʒ/	n.		值域	The range is the set of all outputs. 值域是所有输出的集合。	Range depends on the function behavior. 值域依赖函数行为。
output	/ˈaʊtpʊt/	n.		输出	The output of \(f(2)\) is \(5\). \(f(2)\)的输出是\(5\)。	Output depends on input values. 输出依赖输入值。
input	/ˈɪnpʊt/	n.		输入	The input value determines the output. 输入值决定输出。	Changing input changes the function result. 改变输入会改变函数结果。
polynomial function	/pəˈnɒmiəl ˈfʌŋkʃn/	n.		多项式函数	A polynomial function has integer exponents. 多项式函数的指数为整数。	Polynomial functions are continuous everywhere. 多项式函数处处连续。
trigonometric function	/ˌtrɪɡənəˈmetrɪk ˈfʌŋkʃn/	n. phr.	sine正弦； cosine余弦； tangent正切； cotangent余切； secant正割； cosecant余割	三角函数	Sine is a trigonometric function. 正弦是三角函数。	Trigonometric functions describe periodic motion. 三角函数描述周期运动。
exponential function	/ˌekspəˈnenʃl ˈfʌŋkʃn/	n. phr.	exponent 指数	指数函数	The exponential function grows rapidly. 指数函数增长迅速。	Exponential function models population growth. 指数函数用于人口增长模型。
logarithmic function	/ˌlɒɡəˈrɪðmɪk ˈfʌŋkʃn/	n. phr.	logarithm 对数	对数函数	The logarithmic function is the inverse of exponential function. 对数函数是指数函数的反函数。	Logarithmic function transforms multiplication into addition. 对数函数将乘法转为加法。
power function	/paʊə ˈfʌŋkʃn/	n. phr.		幂函数	A power function has the form \(x^n\). 幂函数形式为\(x^n\)。	Power functions describe scaling behavior. 幂函数描述尺度变化。
constant function	/ˈkɒnstənt ˈfʌŋkʃn/	n. phr.		常值函数	A constant function has the same output for all inputs. 常函数对所有输入输出相同值。	Constant function graphs are horizontal lines. 常函数图像是水平直线。
identity function	/aɪˈdentəti ˈfʌŋkʃn/	n. phr.		恒等函数	The identity function satisfies \(f(x)=x\). 恒等函数满足\(f(x)=x\)。	Identity function leaves values unchanged. 恒等函数不改变数值。
piecewise function	/ˌpiːswaɪz ˈfʌŋkʃn/	n. phr.		分段函数	A piecewise function is defined by different rules. 分段函数由不同规则定义。	Piecewise function changes behavior at boundaries. 分段函数在分界点改变规则。
inverse function	/ˈɪnvɜːs ˈfʌŋkʃn/	n. phr.	arcsine 反正弦； arctangent 反正切； arcsecant 反正割	反函数	The inverse function reverses the original mapping. 反函数逆转原函数映射。	Inverse function satisfies \(f(f^{-1} (x))=x\). 反函数满足复合为\(f(f^{-1} (x))=x\)。
implicit function	/ɪmˈplɪsɪt ˈfʌŋkʃn/	n. phr.	explicit function 显函数； implicitly adv. 显式地	隐函数	An implicit function is not solved for \(y\) explicitly. 隐函数未显式解出\(y\)。	Implicit function defines relations between variables. 隐函数定义变量关系。
composite function	/ˈkɒmpəzɪt ˈfʌŋkʃn/	n. phr.	basic function 基本函数	复合函数	The composite function is \(f(g(x))\). 复合函数是\(f(g(x))\)。	Composite functions combine two mappings. 复合函数结合两个映射。
quadratic	/kwəˈdrætɪk/	adj./n.		二次的	Quadratic equations have degree \(2\). 二次方程次数为\(2\)。	Quadratic functions form parabolas. 二次函数形成抛物线。
inequality	/ˌɪnɪˈkwɒləti/	n.		不等式	Solving the inequality gives \(x\) between \(0\) and \(5\). 解不等式得到\(x\)在\(0\)到\(5\)之间。	The inequality \(x \gt 3\) defines a range of values. 不等式\(x \gt 3\)定义一个取值范围。