📚 Year 8 CAIE Further Mathematics: Vocabulary & Terminology Quick Reference Guide | Year 8 CAIE 进阶数学：词汇术语速记指南
Mastering the language of mathematics is the first step to excelling in problem solving. This guide provides clear definitions, memory tricks, and bilingual explanations for the key terms you will meet in the Year 8 CAIE Further Mathematics syllabus. Keep it handy for homework, revision, and building confidence in your mathematical communication.
掌握数学语言是解题出色的第一步。这份指南为你提供Year 8 CAIE进阶数学大纲中关键术语的清晰定义、记忆技巧和双语解释。将它放在手边，用于作业、复习，并建立你在数学表达上的信心。
1. Number Types and Sets | 数字类型与集合
Integer: A whole number that can be positive, negative, or zero, with no fractional or decimal part. Examples: –3, 0, 7.
整数：可以正、负或零的整数，没有分数或小数部分。例如：–3, 0, 7。
Rational number: Any number that can be written as a fraction a/b, where a and b are integers and b ≠ 0. All terminating and recurring decimals are rational.
有理数：任何能写成 a/b 分数形式的数，其中 a 和 b 为整数且 b ≠ 0。所有有限小数和循环小数都是有理数。
Irrational number: A number that cannot be expressed as a simple fraction. Its decimal expansion is non-terminating and non-repeating. Key examples include π and √2.
无理数：无法表示为简单分数的数。其小数展开无限不循环。典型例子包括 π 和 √2。
Real numbers: The set of all rational and irrational numbers. Every point on the number line corresponds to a real number.
实数：所有有理数与无理数组成的集合。数轴上的每一个点都对应一个实数。
Set notation { }: Used to list a collection of objects or numbers. For instance, the set of even numbers less than 10 can be written as {2, 4, 6, 8}.
集合表示法 { }：用于列出一组对象或数字。例如，小于10的偶数集合可以写作 {2, 4, 6, 8}。
2. Powers, Roots and Surds | 幂、根与根式
Index (power/exponent) how many times a number is multiplied by itself. In 5³, the index is 3, meaning 5 × 5 × 5.
指数（幂）：表示一个数乘以自身多少次。在 5³ 中，指数是 3，表示 5 × 5 × 5。
Base: The number that is being raised to a power. In 2⁴, the base is 2.
底数：被乘方运算的那个数。在 2⁴ 中，底数是 2。
Square root (√): A value that, when multiplied by itself, gives the original number. √25 = 5 because 5 × 5 = 25.
平方根 (√)：一个值，与自身相乘后得到原来的数。√25 = 5，因为 5 × 5 = 25。
Surd: An irrational root, often written with the radical symbol, that cannot be simplified to a whole number or a simple fraction. √3 and 2+√5 are surds.
根式（无理根）：一个无理根，通常用根号书写，无法化简为整数或简单分数。例如 √3 和 2+√5 都是根式。
Standard form (scientific notation) very small numbers as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. 6.4 × 10⁵ = 640 000.
标准形式（科学记数法）：一种书写极大或极小数的方法，形式为 a × 10ⁿ，其中 1 ≤ a < 10 且 n 为整数。6.4 × 10⁵ = 640 000。
3. Algebraic Expressions and Identities | 代数表达式与恒等式
Variable: A symbol, usually a letter, used to represent an unknown or changeable number. In 3x + y, x and y are variables.
变量：一个符号，通常是字母，用来表示未知或可变的数。在 3x + y 中，x 和 y 是变量。
Coefficient: The number multiplying a variable. In 4m, the coefficient of m is 4.
系数：乘以变量的数字。在 4m 中，m 的系数是 4。
Constant: A fixed numerical value that does not change. In the expression 7t + 9, 9 is a constant term.
常数：一个固定不变的数值。在表达式 7t + 9 中，9 是常数项。
Like terms: Terms that contain exactly the same variable(s) raised to the same power(s). 3ab and –5ab are like terms; 2x² and 4x are not.
同类项：包含完全相同的变量且各变量次数相同的项。3ab 和 –5ab 是同类项；2x² 和 4x 不是。
Expand: To remove brackets by multiplying each term inside the bracket by the term outside. 2(a + 3) expands to 2a + 6.
展开：通过将括号外的项乘以括号内的每一项来去掉括号。2(a + 3) 展开得 2a + 6。
Factorise: The reverse of expanding; rewriting an expression as a product of its factors. 3x + 12 factorises to 3(x + 4).
因式分解：展开的逆运算；将一个表达式改写为因式的乘积。3x + 12 分解为 3(x + 4)。
Identity (≡): An equation that is true for all values of the variable(s). For example, (x + 2)² ≡ x² + 4x + 4.
恒等式 (≡)：对所有变量值都成立的等式。例如 (x + 2)² ≡ x² + 4x + 4。
4. Equations and Inequalities | 方程与不等式
Equation qual, containing an equals sign (=). It is only true for certain values. e.g., 2x + 1 = 7.
方程：说明两个表达式相等的数学陈述，含有等号 (=)。仅对某些数值成立。例如，2x + 1 = 7。
Solve find the value(s) of the variable that make an equation true. Solving 3y = 12 gives y = 4.
求解：找到使方程成立的变量值。解方程 3y = 12 得 y = 4。
Simultaneous equations: A set of two or more equations with the same variables that must all be satisfied at the same time. Solved by elimination or substitution.
联立方程组：含有相同变量的两个或多个方程，必须同时满足。可通过消元法或代入法求解。
Inequality (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to).
不等式：使用 < (小于)、> (大于)、≤ (小于等于) 或 ≥ (大于等于) 等符号比较两个表达式的陈述。
Solution set: The range of values that satisfy an inequality. For x > 3, the solution set is all numbers greater than 3.
解集：满足不等式的所有数值的范围。对于 x > 3，解集是所有大于 3 的数。
5. Sequences and Patterns | 数列与规律
Sequence: An ordered list of numbers following a rule. Each number in the sequence is called a term.
数列：按照某种规则排列的一列有序数字。数列中的每个数字称为一项。
Term: An individual number in a sequence. The first term is often denoted by U₁ or T₁.
项：数列中的单个数字。第一项通常记作 U₁ 或 T₁。
nth term: A formula that allows you to find any term in a sequence without listing all the previous terms. For the sequence 5, 8, 11, 14, ... the nth term is 3n + 2.
第 n 项（通项公式）：一个允许你不必列出前面所有项就能找到数列中任意项的公式。对于数列 5, 8, 11, 14, ...，第 n 项是 3n + 2。
Arithmetic sequence: A sequence where the difference between consecutive terms is constant. This difference is called the common difference, d.
等差数列：相邻项之间的差恒定的数列。这个差称为公差，记作 d。
Term-to-term rule: A rule that tells you how to get from one term to the next, e.g. 'add 4'.
项与项之间的递推规则：告诉你如何从一项得到下一项的规则，例如“加 4”。
6. Fractions, Decimals and Percentages | 分数、小数与百分比
Numerator / Denominator: In the fraction 3/8, the numerator is 3 (parts we have) and the denominator is 8 (total equal parts).
分子 / 分母：在分数 3/8 中，分子是 3（我们拥有的部分），分母是 8（总共的等分）。
Equivalent fractions value but have different numerators and denominators. 1/2 = 2/4 = 3/6 are all equivalent.
等值分数：表示相同值但分子和分母不同的分数。1/2 = 2/4 = 3/6 都是等值分数。
Improper fraction: A fraction where the numerator is greater than or equal to the denominator, such as 9/4.
假分数：分子大于或等于分母的分数，例如 9/4。
Mixed number: A number consisting of an integer and a proper fraction, such as 2 1/4.
带分数：由一个整数和一个真分数组成的数，例如 2 1/4。
Percentage increase / decrease: Change expressed as a fraction of the original amount multiplied by 100. A rise from £50 to £60 is a 20% increase.
百分比增减：用变化量除以原始数量再乘以 100% 来表示的变化。从 £50 涨到 £60 是增加了 20%。
7. Ratio, Proportion and Rates | 比、比例与率
Ratio: A way of comparing quantities of the same kind. The ratio of boys to girls 2:3 means for every 2 boys there are 3 girls.
比：比较同一种类数量的方式。男女比 2:3 表示每 2 个男生对应 3 个女生。
Proportion: A statement that two ratios are equal. Two quantities are in direct proportion when their ratio stays constant.
比例：描述两个比相等的陈述。当两个量的比值保持不变时，它们成正比。
Unit rate: The amount per one unit of another quantity. Speed in km/h, price per litre, and 60 words per minute are all unit rates.
单位率：每单位另一量的数量。速度 千米/时、每升价格以及每分钟 60 个单词都是单位率。
Scale factor: The multiplier used to enlarge or reduce lengths in similar shapes. If a shape is enlarged by scale factor 3, all sides become 3 times longer.
缩放因子：用于放大或缩小相似形边长的乘数。若形状以缩放因子 3 放大，则所有边长变为原来的 3 倍。
8. Geometry: Angles, Polygons and Circles | 几何：角度、多边形与圆
Acute / Obtuse / Reflex: An acute angle is less than 90°, an obtuse angle is between 90° and 180°, a reflex angle is greater than 180°.
锐角 / 钝角 / 优角：锐角小于 90°，钝角在 90° 至 180° 之间，优角大于 180°。
Polygon: A closed 2D shape with straight sides. A regular polygon has all sides and angles equal.
多边形：由直线边围成的封闭二维图形。正多边形的所有边长和角度均相等。
Interior and exterior angles: The interior angle is inside the polygon at a vertex; the exterior angle is formed between a side and an extension of an adjacent side. Their sum is 180°.
内角与外角：内角是多边形内部顶点处的角；外角是由多边形的一条边与相邻边的延长线所形成的角。它们之和为 180°。
Congruent: Shapes that are identical in size and shape. They can be exactly superimposed by translation, rotation, or reflection.
全等：大小和形状完全相同的图形。它们可通过平移、旋转或镜像反射完全重合。
Similar: Shapes that have the same shape but not necessarily the same size. Corresponding angles are equal and side lengths are in proportion.
相似：形状相同但大小不一定相同的图形。对应角相等，边长成比例。
Circumference The perimeter of a circle. C = πd or C = 2πr, where r is the radius and d is the diameter.
圆周：圆的周长。C = πd 或 C = 2πr，其中 r 是半径，d 是直径。
9. Measures, Perimeter, Area and Volume | 测量、周长、面积与体积
Perimeter: The total distance around the outside of a 2D shape. Add all side lengths.
周长：围绕二维形状外部的总距离。将所有边长相加。
Area ce a 2D shape covers, measured in square units (cm², m²). Rectangle area = length × width.
面积：二维形状覆盖的表面量，以平方单位 (cm², m²) 计量。矩形面积 = 长 × 宽。
Compound shape: A shape made up of two or more simpler shapes. To find its area, split it into rectangles, triangles, etc., and sum the areas.
组合图形：由两个或多个简单图形组成的形状。求其面积时，可将其拆分为矩形、三角形等，再对面积求和。
Surface area he : The total area of all faces of a 3D solid. For a cube of side length s, surface area = 6s².
表面积：三维立体所有面的总面积。边长为 s 的立方体，表面积为 6s²。
Volume ccupies, measured in cubic units (cm³, m³). Prism volume = area of cross-section × length.
体积：三维物体占据的空间量，以立方单位 (cm³, m³) 计量。棱柱的体积 = 横截面积 × 长。
Capacity : The maximum amount a container can hold, often measured in litres or millilitres. 1 litre = 1000 cm³.
容量：容器所能容纳的最大量，通常以升或毫升计。1 升 = 1000 cm³。
10. Statistics and Probability | 统计与概率
Mean, Median, Mode, Range ÷ number of items. Median is the middle value. Mode is the most frequent. Range = maximum – minimum.
平均数、中位数、众数、极差：描述平均与分散程度的度量。平均数是总和÷数据个数。中位数是中间值。众数是最常出现的值。极差 = 最大值 − 最小值。
Frequency table: A table that lists categories or intervals and how often each occurs (frequency). Useful for organising raw data.
频数表：列出类别或区间及每个类别出现次数（频数）的表格。有助于整理原始数据。
Scatter graph: A plot of points showing the relationship between two variables. If points follow an upward trend, there is positive correlation.
散点图：用点的分布展示两个变量关系的一种图。如果点呈上升趋势，则存在正相关。
Correlation: Describes the strength and direction of a linear relationship. It can be positive, negative, or none.
相关性：描述线性关系的强度和方向。可以为正相关、负相关或无相关。
Probability scale: A number line from 0 to 1 where 0 means impossible and 1 means certain. An event with probability 0.5 is equally likely.
概率尺度：从 0 到 1 的数轴，0 表示不可能，1 表示必然发生。概率为 0.5 的事件表示等可能。
Sample space: The set of all possible outcomes of an experiment. For tossing a coin, the sample space is {Heads, Tails}.
样本空间：一个试验所有可能结果的集合。对于掷硬币，样本空间是 {正面, 反面}。
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