📚 Year 9 CIE Further Maths Vocabulary Memorization Guide | Year 9 CIE 进阶数学词汇术语速记指南
Mastering the language of mathematics is half the battle in further maths. This guide groups essential Year 9 CIE terms by topic, explains their meanings clearly, and shares memory tricks to help you recall them quickly and confidently in exams.
掌握数学语言是学好进阶数学的关键。本指南按知识点分类整理了 Year 9 CIE 的核心术语，给出清晰释义，并分享速记技巧，帮助你在考试中快速、自信地回忆起这些词汇。
1. Number and Arithmetic Terms | 数与算术术语
Integer refers to any whole number, positive, negative, or zero. Remember ‘integer’ by thinking of ‘intact’ – it is a complete number with no fractional parts.
整数 (Integer) 指任何正整数、负整数或零。“整数” 这个词可以联想 “完整”，因为它没有小数部分。
Prime number is a natural number greater than 1 that has exactly two distinct factors: 1 and itself. A quick mental trick: if it is even and greater than 2, it cannot be prime.
质数 (Prime number) 是大于 1 的自然数，且恰好只有两个不同的因数：1 和它本身。记住一个诀窍：大于 2 的偶数绝不可能是质数。
Surd is an irrational root of a rational number, like √2 or ∛5. Think ‘sur-d’ as ‘square-root unsimplified radical digit’ – it is a root that cannot be expressed as a simple fraction.
无理根式 (Surd) 是有理数的无理根，如 √2 或 ∛5。可以把 “surd” 想象成 “square-root unsimplified radical digit”（未化简的根式数字），它无法写成简单的分数。
Significant figures are the meaningful digits in a number, starting from the first non-zero digit. Remember ‘sig figs’ as ‘significant fingers’ counting from the leftmost non-zero digit.
有效数字 (Significant figures) 是一个数中从第一个非零数字开始的有意义的数字。可以将 “sig figs” 想象成 “重要的手指”，从左开始数第一个非零数字。
2. Algebraic Expressions and Equations | 代数表达式与方程
Coefficient is the numerical factor in a term. In 5x², 5 is the coefficient. You can pair ‘co-‘ (together) and ‘efficient’ – it works together with the variable.
系数 (Coefficient) 是项中的数值因数。在 5x² 中，5 就是系数。可以将 “co-”（共同）和 “efficient”（效率）结合来看 —— 它与变量共同作用。
Variable is a symbol, usually a letter, representing an unknown value. Visualize it as ‘varying-able’ – its value can change.
变量 (Variable) 是一个符号，通常用字母表示，代表未知的值。可以把它看作 “varying-able”（能变化的）—— 它的值会变。
Expression is a combination of terms without an equals sign, such as 3x + 2. An equation contains an equals sign, like 3x + 2 = 11. ‘Ex-press-ion’ is just a press of terms; ‘e-qua-tion’ has ‘equal’.
表达式 (Expression) 是不带等号的项的组合，如 3x + 2。方程 (Equation) 含有等号，如 3x + 2 = 11。“Expression” 只是项的拼凑，而 “equation” 里有 “equal”（相等）。
Quadratic comes from ‘quad’ meaning square – an expression of degree 2, e.g., ax² + bx + c. Think of a square having four sides, linking to ‘quad’ in ‘quadratic’.
二次式 (Quadratic) 来自 “quad” 表示平方 —— 指次数为 2 的表达式，如 ax² + bx + c。可以联想正方形有四条边，而 “quad” 表示四，记在 “quadratic” 里。
3. Geometry: Shapes and Angles | 几何：形状与角度
Acute angle is an angle less than 90°. Think of a ‘cute’ little angle – it is small and sharp.
锐角 (Acute angle) 是小于 90° 的角。可以想象一个 “可爱的小角” (a cute little angle)，它小而尖。
Obtuse angle lies between 90° and 180°. The word sounds like ‘obtuse’ – it is ‘obviously huge’, but not yet a straight line. Or think of ‘obese’ – overweight but not flat.
钝角 (Obtuse angle) 在 90° 到 180° 之间。这个词听起来像 “obviously huge”（明显很大），但还不是直线。或者联想 “obese”（肥胖），超重但还没躺平。
Perpendicular lines intersect at 90°. ‘Per-pen-dicular’ – the ‘pen’ drops straight down, forming a right angle. A bisector divides an angle or segment into two equal parts. ‘Bi-‘ means two, and ‘-sector’ means cut.
垂线 (Perpendicular) 相交成 90° 的直线。可以想象笔直落下，形成直角。平分线 (Bisector) 把一个角或线段分成相等的两部分。“Bi-” 表示二，“-sector” 表示切割。
Polygon is a closed 2D shape with straight sides. ‘Poly’ means many, ‘gon’ means angle. So many angles. Regular polygon has all sides and angles equal.
多边形 (Polygon) 是由直线围成的封闭二维图形。“Poly” 表示多，“gon” 表示角，所以是多角形。正多边形所有边和角都相等。
4. Coordinate Geometry Basics | 坐标几何基础
Gradient (or slope) measures steepness, calculated as rise over run. A handy memory: ‘Gradient’ = ‘grade’ + ‘ent’, like measuring the grade of a hill.
斜率 (Gradient) 衡量倾斜程度，计算公式为纵轴变化量除以横轴变化量。可以这么记：“gradient” 就像测量山坡的 “grade”（坡度）。
Intercept is where a line crosses an axis. The y-intercept is where x = 0. ‘Intercept’ = ‘inter’ (between) + ‘cept’ (taken) – the point where the line is taken by the axis.
截距 (Intercept) 是直线与坐标轴的交点。y 轴截距即 x = 0 时的点。“Intercept” 可拆为 “inter”（在……之间）和 “cept”（被拿取）—— 直线被坐标轴截取的点。
Midpoint of a segment is the average of the endpoints’ coordinates. The formula ((x₁+x₂)/2, (y₁+y₂)/2) uses the idea of ‘middle’ point by averaging.
中点 (Midpoint) 是线段端点的坐标平均值。公式 ((x₁ + x₂)/2, (y₁ + y₂)/2) 利用了平均来求 “中间” 点。
Distance formula uses Pythagoras: √[(x₂−x₁)² + (y₂−y₁)²]. The square root of the sum of the squares of the differences – a direct application of a² + b² = c².
距离公式 (Distance formula) 利用勾股定理：√[(x₂−x₁)² + (y₂−y₁)²]。两个方向差的平方和的平方根，正是 a² + b² = c² 的直接应用。
5. Sets and Venn Diagrams | 集合与韦恩图
Set is a collection of distinct objects. The notation { } encloses elements. Remember a set as a ‘set of items in a bag’.
集合 (Set) 是不同对象的全体。花括号 { } 里放置元素。可以把集合想象成 “袋子里的一套物品”。
Union (A ∪ B) combines all elements from both sets. The symbol ∪ looks like a cup holding all the elements. Intersection (A ∩ B) contains only elements common to both sets; ∩ resembles an arch where only shared elements pass through.
并集 (Union) (A ∪ B) 包含两个集合中所有的元素。符号 ∪ 像一个杯子，容纳所有元素。交集 (Intersection) (A ∩ B) 只包含两个集合都有的元素；符号 ∩ 像一座桥，只有共同元素才能通过。
Complement (A’) is all elements not in set A, relative to the universal set. It ‘completes’ the set. Universal set (ξ) is the entire background set.
补集 (Complement) (A’) 是相对于全集，不在集合 A 中的所有元素。它“补全”了集合。全集 (Universal set) (ξ) 是总体的背景集合。
Venn diagram uses overlapping circles to show logical relationships. Imagine a ‘Venn’ as a ‘window’ onto sets’ members.
韦恩图 (Venn diagram) 用交叠的圆圈表示集合间的逻辑关系。可以把 “Venn” 想象成一扇观察集合成员的小窗。
6. Functions and Relations | 函数与关系
A function is a rule that assigns exactly one output for each input. f(x) = 2x + 3 is a function. ‘One input, one output’ – think of a machine that gives one candy per coin.
函数 (Function) 是一种规则，每个输入恰好对应一个输出。f(x) = 2x + 3 就是一个函数。“一个输入，一个输出”——就像一台机器，投一个硬币出一颗糖。
Domain is the set of all possible inputs. ‘Domain’ sounds like ‘do-main’ – the main inputs you are allowed to use. Range is the set of all possible outputs – the ‘range’ of results.
定义域 (Domain) 是所有可能输入的集合。“Domain” 可以拆成 “do” 和 “main”——你被允许使用的主要输入。值域 (Range) 是所有可能输出的集合 —— 结果的范围。
Inverse function, written f⁻¹(x), reverses the original function. The ‘inverse’ undoes what f does. If f adds 2, f⁻¹ subtracts 2.
反函数 (Inverse function) 写作 f⁻¹(x)，它逆转原函数的运算。“反函数” 撤销了 f 所做的运算。如果 f 加 2，f⁻¹ 就减 2。
Composite function f(g(x)) means apply g first, then f. ‘Composite’ is like composite material – layers of functions applied one after another.
复合函数 (Composite function) f(g(x)) 意味着先用 g，再用 f。“复合” 就像复合材料一样，一层层地应用函数。
7. Sequences and Series | 数列与级数
An arithmetic sequence has a common difference between consecutive terms, e.g., 3, 7, 11, 15… with d = 4. Think ‘arithmetic’ = ‘add repeatedly’ – the difference is added.
等差数列 (Arithmetic sequence) 前后项之间有一个共同的差，如 3, 7, 11, 15… 公差 d = 4。可以把 “arithmetic” 看成 “add repeatedly”（重复加）——每次加上公差。
A geometric sequence has a common ratio, multiplying by the same number each time, e.g., 2, 6, 18, 54… with r = 3. ‘Geo-metric’ – ‘geo’ means earth, but think ‘multiply’, like ‘growth’ multiples.
等比数列 (Geometric sequence) 有公比，每次都乘以相同的数，如 2, 6, 18, 54… 公比 r = 3。“Geometric” 可以联想到 “grow by multiples”（成倍增长）。
The nth term formula for arithmetic is uₙ = a + (n-1)d. ‘uₙ’ is the term in position n; a is the first term.
等差数列的通项公式 (nth term) 是 uₙ = a + (n-1)d。uₙ 是位于第 n 位的项，a 是首项。
A series is the sum of the terms of a sequence. So a sequence lists numbers, a series adds them. ‘Series’ sounds like ‘sum-eries’.
级数 (Series) 是数列各项之和。数列列出数字，级数则是把它们加起来。“Series” 听起来像 “sum-eries”（求和系列）。
8. Vectors in Two Dimensions | 二维向量
A vector has both magnitude (length) and direction. Represented as a column matrix or with i, j notation. ‘Vector’ sounds like ‘director’ – it gives direction plus size.
向量 (Vector) 既有大小又有方向。可以用列矩阵或 i, j 表示。“Vector” 发音类似 “director”（导演），它给出方向加上大小。
Magnitude of vector v = √(x² + y²). The ‘magnitude’ is the ‘magnifying’ size – calculated by Pythagoras.
向量 v 的模 (Magnitude) 是 √(x² + y²)。“Magnitude” 就是 “magnifying” 的量，用勾股定理计算。
Unit vector has magnitude 1. The standard unit vectors are i (1,0) and j (0,1). ‘Unit’ means one – a vector of length one.
单位向量 (Unit vector) 模为 1。标准单位向量是 i (1,0) 和 j (0,1)。“Unit” 表示 1，长度为 1 的向量。
Resultant vector is the sum of two or more vectors. The ‘resultant’ is the result of adding them tip-to-tail. Triangle law or parallelogram law.
合向量 (Resultant vector) 是两个或多个向量的总和。“Resultant” 是它们首尾相连相加的结果，可用三角形法则或平行四边形法则。
9. Probability and Statistics Terminology | 概率与统计术语
Probability measures the chance of an event, ranging from 0 (impossible) to 1 (certain). ‘Prob-ability’ – think ‘probably able to happen’.
概率 (Probability) 衡量事件发生的可能性，范围从 0（不可能）到 1（必然）。“Probability” 可拆为 “probably able”（很可能能够发生）。
Sample space is the set of all possible outcomes of an experiment. It is the ‘space’ of all ‘samples’. Often listed as a table or set.
样本空间 (Sample space) 是某个试验所有可能结果的集合。它是所有“样本”的“空间”，常用表格或集合列出。
Mean is the average: sum of values divided by number of values. Median is the middle value when ordered. Mode is the most frequent value. Memory aid: mode = ‘most often’.
平均数 (Mean) 是总和除以个数。中位数 (Median) 是把数据排序后中间的那个值。众数 (Mode) 是出现次数最多的值。记忆法：mode = “most often”（最常出现）。
Range is the difference between the largest and smallest data values, measuring spread. ‘Range’ = how far the data reaches.
极差 (Range) 是最大值减最小值，衡量数据的散布程度。“Range” 代表着数据延伸的范围。
10. Trigonometry Introduction | 三角学入门
Hypotenuse is the longest side of a right-angled triangle, opposite the right angle. The word sounds like ‘hi-pot-en-use’ – imagine a hippopotamus stretching across the longest side.
斜边 (Hypotenuse) 是直角三角形中最长的边，对着直角。这个词听起来像 “hippo-ten-use”（河马在拉伸），想象河马拉长了最长的那条边。
Opposite and adjacent are relative to a given angle. The opposite side faces the angle; the adjacent is next to it (not the hypotenuse).
对边 (Opposite) 和 邻边 (Adjacent) 是相对于某个给定角而言的。对边正对着角，邻边紧挨着角（排除斜边）。
Sine, cosine, tangent (sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj). The mnemonic SOH CAH TOA is a classic: ‘Some Old Humans Can Always Have Their Old Apples’.
正弦、余弦、正切 (sin θ = 对/斜, cos θ = 邻/斜, tan θ = 对/邻)。记忆口诀 SOH CAH TOA 很经典：’Some Old Humans Can Always Have Their Old Apples’。
Angle of elevation is measured upwards from the horizontal; angle of depression downwards. Elevation lifts your eyes up; depression drops them down.
仰角 (Angle of elevation) 是从水平线向上测量；俯角 (Angle of depression) 是向下测量。仰角让你抬起视线，俯角让你垂下目光。
11. Matrices | 矩阵
A matrix (plural: matrices) is a rectangular array of numbers, arranged in rows and columns. The order is rows × columns, e.g., a 2×3 matrix has 2 rows and 3 columns.
矩阵 (Matrix)（复数 : matrices）是一个按行和列排列的数字矩形阵列。阶 (Order) 是行数 × 列数，如 2×3 矩阵有 2 行 3 列。
Element is an individual entry, labelled as aᵢⱼ, where i is the row and j is the column. Think ‘i’ for ‘row i’ and ‘j’ for ‘jolumn’ – silly but memorable.
元素 (Element) 是单个项，记作 aᵢⱼ，其中 i 是行，j 是列。可以联想 “i” 代表 “row i”，而 “j” 像是 “jolumn” 的开头，虽然滑稽但好记。
Equal matrices have the same order and identical corresponding elements. Zero matrix has all entries 0.
相等矩阵 (Equal matrices) 阶相同且对应元素全相等。零矩阵 (Zero matrix) 所有元素都是 0。
Matrix addition works element-wise; only matrices of the same order can be added. Multiplication by a scalar multiplies every element by that number.
矩阵加法 (Addition) 是对应元素相加；只有同阶矩阵才能相加。标量乘法 (Scalar multiplication) 将每个元素乘以该数。
12. Logical Reasoning | 逻辑推理
Converse, inverse, and contrapositive are transformations of an implication ‘If p, then q’. Contrapositive is ‘If not q, then not p’, which is logically equivalent to the original.
逆命题 (Converse)、否命题 (Inverse) 与 逆否命题 (Contrapositive) 是蕴含命题 “若 p 则 q” 的变形。逆否命题 “若非 q 则非 p” 与原命题逻辑等价。
Proof by contradiction assumes the opposite of what you want to prove, then finds a contradiction. ‘Contra-diction’ – speaking against yourself to reveal truth.
反证法 (Proof by contradiction) 先假设要证明的结论的反面成立，然后推出矛盾。“Contradiction” 就是 “与自己所说的对抗” 从而揭示真理。
The universal quantifier ∀ means ‘for all’, and the existential quantifier ∃ means ‘there exists’. ‘All’ looks like an upside-down A; ‘Exists’ looks like an E reversed.
全称量词 (Universal quantifier) ∀ 表示 “对所有”，存在量词 (Existential quantifier) ∃ 表示 “存在”。符号 ∀ 像倒过来的 A（All），∃ 像反过来的 E（Exists）。
Necessary condition: q is necessary for p if p → q is true. ‘Necessary’ = needed. Sufficient condition: p is sufficient for q if p → q is true. ‘Sufficient’ = enough.
必要条件 (Necessary condition)：若 p → q 真，则 q 是 p 的必要条件。“必要” 就是必需的。充分条件 (Sufficient condition)：若 p → q 真，则 p 是 q 的充分条件。“充分” 就是足够了。
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