📚 CIE Additional Mathematics Glossary: Quick Memory Guide for Year 10 | CIE 进阶数学词汇速记指南
Mastering CIE Additional Mathematics (often known as IGCSE 0606) requires more than just problem-solving skills – it demands a deep and effortless recall of precise mathematical vocabulary. This guide provides a structured glossary with memory hooks, linking each English term to its Chinese equivalent so that Year 10 learners can build a bilingual mathematical mindset. Read through the paired explanations and you will find that terminology becomes a strength rather than a hurdle.
掌握 CIE 进阶数学（通常指 IGCSE 0606）不仅需要解题技巧，更需要对精确数学词汇的深刻且流畅的记忆。本指南提供了一个结构化的词汇表，并配有记忆钩子，将每个英文术语与其中文对应词连接起来，帮助十年级学生建立双语数学思维。仔细阅读这些成对的解释，你会发现术语将成为你的优势，而非障碍。
1. Sets and Notations | 集合与符号
A set is a well-defined collection of distinct objects. When you see curly braces { }, think 'a box of items' – all the elements belong to that box. The symbol ∈ means 'is an element of' and ∉ means 'is not an element of'. The universal set ξ contains every element under consideration, while the empty set ∅ is a set with nothing inside.
集合是明确定义的互异对象的全体。看到花括号 { } 时，想象一个装着物品的盒子——所有元素都属于这个盒子。符号 ∈ 表示“属于”，而 ∉ 表示“不属于”。全集 ξ 包含所讨论的全部元素，空集 ∅ 则是里面什么都没有的集合。
A subset B ⊆ A means every element of B is also in A. A proper subset B ⊂ A requires B to be strictly smaller than A. The complement of A, written A', contains everything in ξ that is not in A. The union A ∪ B resembles 'adding' both sets together, while the intersection A ∩ B is the 'overlap'.
子集 B ⊆ A 表示 B 的每个元素都在 A 中。真子集 B ⊂ A 要求 B 严格小于 A。A 的补集写作 A'，包含全集 ξ 中所有不在 A 里的元素。并集 A ∪ B 类似于将两个集合“相加”，交集 A ∩ B 则是“重叠”部分。
Memory tip: The union symbol ∪ looks like a U (Union), and the intersection symbol ∩ looks like an upside-down U or a bridge where two roads meet. For complement, think of 'everything outside the set'.
记忆技巧：并集符号 ∪ 看起来像字母 U（Union），交集符号 ∩ 像倒置的 U 或两条道路交汇的桥梁。对于补集，只需思考“集合之外的所有东西”。
2. Functions and Relations | 函数与关系
A function f maps each input x to exactly one output f(x). Imagine a vending machine: you press a button (input) and exactly one drink comes out (output). The domain is the set of all possible inputs, and the range is the set of all actual outputs produced by those inputs. A relation that gives more than one output for the same input is not a function.
函数 f 将每个输入 x 映射到恰好一个输出 f(x)。想象一台自动售货机：你按下一个按钮（输入），恰好有一种饮料出来（输出）。定义域是所有可能输入的集合，值域是这些输入所产生的所有实际输出的集合。如果对于同一个输入，关系给出了多个输出，那么它就不是一个函数。
Composite functions combine two processes: gf(x) means first apply f, then apply g. An inverse function f⁻¹(x) 'undoes' what f does – swapping the role of input and output. For f⁻¹ to exist, f must be one-to-one (each output comes from exactly one input). The vertical line test helps check whether a graph represents a function: any vertical line must cut the graph at most once.
复合函数将两个过程组合起来：gf(x) 表示先作用 f，再作用 g。反函数 f⁻¹(x)“撤销” f 的效果——把输入和输出的角色互换。要使 f⁻¹ 存在，f 必须是一对一的（每个输出来自恰好一个输入）。垂直线检验有助于判断一个图像是否表示函数：任何一条垂直线与图像最多只能有一个交点。
Memory tip: Think of f as 'feeder' and g as 'gatherer'. Composite gf(x) is 'feed then gather'. For inverse, pronounce f⁻¹ as 'f inverse' and recall it like an 'undo' button.
记忆技巧：把 f 想成“feeder”（喂养者），g 想成“gatherer”（收集者）。复合 gf(x) 就是“先喂后收”。对于反函数，将 f⁻¹ 读作“f inverse”，回想它就像一个“撤销”按钮。
3. Quadratic Functions | 二次函数
x + c, where a ≠ 0. The graph is a parabola: it opens upward if a > 0 (smile) and downward if a < 0 (frown). The vertex is the turning point – a. Completing the square rewrites ax² + bx + c into the form a(x − h)² + k, where (h, k) is the vertex.
一般二次函数是 f(x) = ax² + bx + c，其中 a ≠ 0。其图像是一条抛物线：若 a > 0 则开口向上（微笑），若 a < 0 则开口向下（皱眉）。顶点即转折点——抛物线上的最大值或最小值。配方法将 ax² + bx + c 重写为 a(x − h)² + k 的形式，其中 (h, k) 为顶点坐标。
axis); Δ < 0, no real roots. The roots themselves are given by the quadratic formula: x = (−b ± √b² − 4ac) / (2a). The axis of symmetry is the vertical line x = −b/(2a).
判别式 Δ = b² − 4ac 能告诉你根（x 轴交点）的情况。若 Δ > 0，有两个相异实根；Δ = 0，一个重根（与 x 轴相切）；Δ < 0，无实根。根本身由求根公式给出：x = (−b ± √b² − 4ac) / (2a)。对称轴是垂直线 x = −b/(2a)。
Memory tip: The discriminant 'discriminates' ny suspects (roots) exist. For vertex, h = −b/(2a) is halfway between the two roots, so it's the 'heart' of the parabola.
记忆技巧：判别式会“区分”（discriminate）根的类型。想象一位侦探（判别式）判断有多少个嫌疑人（根）。对于顶点，h = −b/(2a) 正好在两个根的中间，因此它是抛物线的“心脏”。
4. Equations and Inequalities | 方程与不等式
x. Simultaneous linear equations can be tackled by elimination or substitution. When you multiply an inequality by a negative number, the inequality sign must be reversed – a classic pitfall to remember.
一元一次方程的形式为 px + q = 0，通过移项求解 x。联立一次方程组可通过消元法或代入法求解。若将不等式乘以一个负数，不等号必须反转——这是需要牢记的经典陷阱。
s into intervals; test a value from each interval to determine where the inequality holds. 'Open' circles are used for strict inequalities (<, >) and 'closed' circles for inclusive ones (≤, ≥) on a number line.
像 ax² + bx + c > 0 这样的二次不等式通常通过绘制抛物线来求解。根将 x 轴分成若干区间；从每个区间选一个值代入检验，即可判断不等式在哪里成立。在数轴上，空心圆圈用于严格不等式（<, >），实心圆圈用于含有等号的不等式（≤, ≥）。
Memory tip: Reversing the inequality sign when multiplying by a negative is like the 'negative flip'. Think: 'If you times by a minus, the sign must swinus!' For intervals, imagine the parabola as a bridge: above the x-axis means > 0, below means < 0.
记忆技巧：乘以负数时反转不等号，就像“负号翻转”。这样想：'If you times by a minus, the sign must swinus!' 对于区间，把抛物线想象成一座桥：在 x 轴上方表示 > 0，下方表示 < 0。
5. Indices and Surds | 指数与根式
0), and negative indices represent reciprocals: a⁻ⁿ = 1/aⁿ. Fractional indices link to roots: a1/n = ⁿ√a, am/n = (ⁿ√a)ᵐ.
指数（幂）用于计数一个底数被自乘的次数。主要法则：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ。零指数得 1（a⁰ = 1, a ≠ 0），负指数表示倒数：a⁻ⁿ = 1/aⁿ。分数指数与根式相关联：a1/n = ⁿ√a，am/n = (ⁿ√a)ᵐ。
A surd is an irrational root that cannot be simplified to an integer or fraction, such as √2 or ∛5. Surds can be simplified by factoring out perfect squares (√a²b = a√b) and rationalising denominators (multiplying top and bottom by the conjugate to remove the root from the denominator).
根式是无法化简为整数或分数形式的无理根，例如 √2 或 ∛5。根式可通过提取完全平方数来化简（√a²b = a√b），并进行分母有理化（上下同时乘以共轭式，以消去分母中的根号）。
Memory tip: Index laws are like a ladder: multiplication climbs up (add powers), division steps down (subtract powers). For surds, think of the root as a 'home' you need to move out of the denominator – rationalising is like building a new foundation.
记忆技巧：指数法则像一架梯子：乘法向上爬（指数相加），除法向下走（指数相减）。对于根式，把根号想象成一个需要从分母中“搬出去”的家——有理化就是重新打地基。
6. Logarithmic Functions | 对数函数
A logarithm answers the question: 'To what power must the base be raised to yield a given number?' If bˣ m. Key laws: log₆(xy) = log₆ x + log₆ y, log₆(x/y) = log₆ x − log₆ y, log₆(xⁿ) = n log₆ x.
对数回答这样一个问题：“底数必须被提升到多少次方，才能得到某个给定的数？”若 bˣ = y，则 x = log₆ y。自然对数 ln x 以 e ≈ 2.718 为底，而 log₁₀ x 是常用对数。关键法则：log₆(xy) = log₆ x + log₆ y，log₆(x/y) = log₆ x − log₆ y，log₆(xⁿ) = n log₆ x。
through (1, 0), have the y-axis as a vertical asymptote, and are the inverse of exponential functions aˣ ing the mirror relationship between exponentials and logarithms is a powerful memory tool.
换底公式：logₐ y = log₆ y / log₆ a。y = logₐ x 的图像经过 (1, 0)，以 y 轴为垂直渐近线，并且是指数函数 aˣ 的反函数。理解指数函数与对数函数之间的镜像关系，是一个强大的记忆工具。
Memory tip: The log is the exponent. Say 'log base b of y equals x' while pointing to x in bˣ = y – the log is just the exponent in disguise. For ln, remember 'natural' lives in nature, and e comes from growth processes.
记忆技巧：对数就是指数。指着 bˣ = y 中的 x 说“以 b 为底 y 的对数等于 x”——对数只是伪装起来的指数。对于 ln，记住“自然”对数存在于自然界中，而 e 来自连续增长过程。
7. Coordinate Geometry | 坐标几何
oordinates: ((x₁+x₂)/2, (y₁+y₂)/2). The gradient (slope) of a line is m = (y₂ − y₁)/(x₂ − x₁).
两点 (x₁, y₁) 和 (x₂, y₂) 之间的距离是 √[(x₂ − x₁)² + (y₂ − y₁)²]，它源自勾股定理。中点就是坐标的平均值：((x₁+x₂)/2, (y₁+y₂)/2)。一条直线的斜率（坡度）为 m = (y₂ − y₁)/(x₂ − x₁)。
The equation of a line can be written in gradient–intercept form y = mx + c, point– −1 (m₁ × m₂ = −1). The angle a line makes with the positive x-axis is related to the gradient by m = tan θ.
直线方程可写成斜截式 y = mx + c、点斜式 y − y₁ = m(x − x₁)，或一般式 ax + by + c = 0。平行线的斜率相等；垂直线的斜率乘积为 −1（m₁ × m₂ = −1）。直线与 x 轴正半轴之间的夹角 θ 与斜率的关系为 m = tan θ。
Memory tip: Distance formula: 'difference in x squared plus difference in y squared, then root it all' – like a square root sandwich. For perpendicular gradients, think 'flip and negate' the fraction to get the negative reciprocal.
记忆技巧：距离公式：“x 差平方加 y 差平方，再整体开方”——像一个平方根三明治。对于垂直直线的斜率，想象“将分数翻转并取相反数”，就得到负倒数。
8. Trigonometry | 三角学
os θ = adjacent/hypotenuse, tan θ = opposite/adjacent. The mnemonic SOH CAH TOA is universally useful: Sine Opposite Hypotenuse, Cosine Adjacent Hypotenuse, Tangent Opposite Adjacent.
直角三角形中三个主要的三角比是正弦（sin）、余弦（cos）和正切（tan）。sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。记忆口诀 SOH CAH TOA 全球通用：正弦即对斜（对边比斜边），余弦即邻斜（邻边比斜边），正切即对邻（对边比邻边）。
act values for 30°, 45°, 60° should be memorised. For example, sin 30° = 1/2, cos 45° = 1/√2 = √2/2, tan 60° = √3. The unit circle extends definitions to all angles, and the CAST diagram indicates which ratios are positive in each quadrant.
30°、45°、60° 的精确值需要牢记。例如 sin 30° = 1/2，cos 45° = 1/√2 = √2/2，tan 60° = √3。单位圆将这些定义扩展到任意角，而 CAST 图用于判断各象限中三角比的正负。
c² − 2bc cos A are used for non-right-angled triangles. The area of a triangle is ½ ab sin C.
关键恒等式：tan θ = sin θ / cos θ，sin² θ + cos² θ = 1。正弦定理 a/sin A = b/sin B = c/sin C 和余弦定理 a² = b² + c² − 2bc cos A 用于非直角三角形。三角形的面积公式为 ½ ab sin C。
Memory tip: SOH CAH TOA – pretend you're a Japanese karate student: 'SOH-CAH-TOA!' bowing each time. For exact values, draw a 30-60-90 triangle (sides 1, √3, 2) and a 45-45-90 triangle (sides 1, 1, √2) to read off ratios.
记忆技巧：SOH CAH TOA——假装自己是一名日本空手道学员，每喊一次便鞠躬一次：“SOH-CAH-TOA！”对于精确值，画一个 30-60-90 三角形（边长 1, √3, 2）和一个 45-45-90 三角形（边长 1, 1, √2），就可以直接读出比值。
9. Differentiation | 微分
hange or gradient of a function. The derivative of y = xⁿ is dy/dx = nxⁿ⁻¹. The notation f'(x) or dy/dx both denote the derivative. The process 'brings down the power and reduces it by one'. For a constant c, d(c)/dx = 0.
微分用于求函数的瞬时变化率或斜率。y = xⁿ 的导数是 dy/dx = nxⁿ⁻¹。记号 f'(x) 或 dy/dx 都表示导数。求导的过程是“将指数移下来，并将指数减一”。对于常数 c，d(c)/dx = 0。
ing x into dy/dx. The tangent line at (x₁, y₁) has equation y − y₁ = m(x − x₁) where m = f'(x₁). A normal line is perpendicular to the tangent, so its gradient is −1/m.
和/差的导数等于导数的和/差。曲线上某点的斜率通过将 x 代入 dy/dx 求得。点 (x₁, y₁) 处的切线方程为 y − y₁ = m(x − x₁)，其中 m = f'(x₁)。法线垂直于切线，因此其斜率为 −1/m。
Second derivative d²y/dx² gives the rate of change of the gradient. When f'(x) = 0, we find stationary points; the sign of f''(x) tells whether it is a maximum (f''(x) < 0), minimum (f'' nt of inflection. Connected rates of change use the chain rule: dy/dx = (dy/du) × (du/dx).
二阶导数 d²y/dx² 给出斜率的变化率。当 f'(x) = 0 时，我们找到驻点；f''(x) 的正负告诉我们该点是极大值点（f''(x) < 0）、极小值点（f''(x) > 0）还是拐点。相关变化率使用链式法则：dy/dx = (dy/du) × (du/dx)。
Memory tip: Differentiation is 'power drop' – the power jumps down as a multiplier and the new power is one less. Think of f'(x) as the 'slope finder'. For max/min, remember: second derivative negative means a 'frowning' maximum, positive means a 'smiling' minimum.
记忆技巧：微分就是“指数降落”——指数跳下来作为乘数，新指数减一。把 f'(x) 想作“斜率探测器”。至于极大/极小值，记住：二阶导数为负意味着“皱眉”的极大值，为正意味着“微笑”的极小值。
10. Integration | 积分
ntiation. The indefinite integral ∫ f(x) dx yields a family of functions F(x) + C, where F' and C is the constant of integration. For xⁿ (n ≠ −1), ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C. The power 'goes up by one' and you divide by the new power.
积分是微分的逆运算。不定积分 ∫ f(x) dx 产生一族函数 F(x) + C，其中 F'(x) = f(x)，C 为积分常数。对于 xⁿ（n ≠ −1），∫ xⁿ dx = xⁿ⁺¹/(n+1) + C。指数“增加 1”，并除以新的指数。
is evaluated using the fundamental theorem: F(b) − F(a). Areas below the x-axis give a negative contribution, so for total area you split the interval where the curve crosses the x-axis and take absolute values.
定积分 ∫ₐᵇ f(x) dx 表示从 x = a 到 x = b 曲线与 x 轴之间的带符号面积。其计算使用基本定理：F(b) − F(a)。x 轴下方的区域贡献负值，因此求总面积时需要以曲线与 x 轴交点为界拆分区间，并取绝对值。
rate velocity over time, you get the change in displacement; integrating acceleration gives change in velocity.
运动学应用：速度 v = ds/dt，加速度 a = dv/dt，位移 s = ∫ v dt。对速度随时间积分得到位移的变化量；对加速度积分得到速度的变化量。
Memory tip: Integration 'raises the power and divides'. Think of differentiation as shrinking (powers down) and integration as growing (powers up). For definite integrals, visualise the area 'from start to end' – the F(b) − F(a) pattern is like the final position minus the initial position.
记忆技巧：积分“增加指数并除以新指数”。把微分想象成缩小（指数下降），积分想象成扩大（指数上升）。对于定积分，想象“从起点到终点”的面积——F(b) − F(a) 的模式就像终点位置减去起点位置。
11. Vectors | 向量
itten in column form (x, y) or as xi + yj. The magnitude (length) of vector a = xi + yj is |a| = √x² + y². A unit vector has magnitude 1; to find the unit vector in the direction of a, divide a by |a|.
向量是既有大小又有方向的量，常写为列向量 (x, y) 或 xi + yj 的形式。向量 a = xi + yj 的大小（长度）为 |a| = √x² + y²。单位向量的大小为 1；要得到 a 方向上的单位向量，用 a 除以 |a| 即可。
ion vector of a point is its coordinates relative to the origin. The vector from point A to point B is AB = OB − OA.
向量加法通过分量分别相加：(a₁, a₂) + (b₁, b₂) = (a₁+b₁, a₂+b₂)。标量乘法缩放长度：k × a = (kx, ky)。点的位置向量是该点相对于原点的坐标。点 A 到点 B 的向量为 AB = OB − OA。
The dot product (scalar product) a · b = |a||b| d to find the angle between two vectors and to test perpendicularity (if a · b = 0, then a ⟂ b). Vectors are parallel if one is a scalar multiple of the other.
点积（标量积）a · b = |a||b| cos θ = a₁b₁ + a₂b₂。它用于求两个向量之间的夹角，以及判断垂直关系（若 a · b = 0，则 a ⟂ b）。若一个向量是另一个的标量倍数，则它们平行。
Memory tip: A vector is a 'directed arrow'. Magnitude comes from Pythagoras. For dot product, the formula a₁b₁ + a₂b₂ is like 'multiply and add horizontally'. The cosine formula link helps with angle problems: cos θ = (a · b) / (|a||b|).
记忆技巧：向量是一支“有方向的箭”。其大小来自勾股定理。对于点积，公式 a₁b₁ + a₂b₂ 就像“水平相乘再相加”。余弦公式的联系有助于角度问题：cos θ = (a · b) / (|a||b|)。
12. Permutations and Combinations | 排列与组合
The factorial n! line. Permutations consider order; the number of ways to arrange r items from n is ⁿPᵣ = n!/(n−r)!. Combinations ignore order; the number of ways to select r items from n is ⁿCᵣ = n!/[r!(n−r)!].
阶乘 n! = n × (n−1) × ... × 2 × 1，表示将 n 个不同物体排成一列的方法数。排列考虑顺序；从 n 个中选取 r 个进行排列的方法数为 ⁿPᵣ = n!/(n−r)!。组合忽略顺序；从 n 个中选取 r 个的方法数为 ⁿCᵣ = n!/[r!(n−r)!]。
The terms 'permutation' and 'combination' roup matters (like a team of 3 from 10). The symmetry ⁿCᵣ = ⁿCₙ₋ᵣ helps reduce arithmetic.
术语“排列”与“组合”常常混淆。当位置重要时使用排列（如赛跑奖牌：金、银、铜），而当只关心小组本身时使用组合（如从 10 人中选 3 人组队）。对称性 ⁿCᵣ = ⁿCₙ₋ᵣ 有助于简化计算。
Arrangements with repeated items: if there are n items with p of one type, q of another, etc., the number of distinct permutations is n!/(p! q! ...). This is a powerful tool for word shuffling problems.
含有重复元素的排列：若有 n 个物品，其中一类有 p 个相同、另一类有 q 个相同……那么相异排列数为 n!/(p! q! ...)。这是解决字母重排问题的有力工具。
Memory tip: Permutation = Position matters; Combination = Choice (no order). 'P' before 'C' – Order is higher priority in Permutation. The formula ⁿCᵣ divides by an extra r! to 'kill order'. The nCr button is often 'choose'.
记忆技巧：Permutation（排列）中位置重要；Combination（组合）仅看选择（无顺序）。字母顺序上“P”先于“C”——排列中顺序优先级更高。公式 ⁿCᵣ 额外除以 r! 以“消除顺序”。计算器上的 nCr 键常读作“choose”。
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