📚 Year 10 Edexcel Further Maths: Glossary & Terminology Quick-Memorisation Guide | Year 10 Edexcel 进阶数学：词汇术语速记指南
Mastering the language of mathematics is the first step towards fluency in problem-solving. This guide breaks down the essential terminology you will encounter in the Year 10 Edexcel Further Maths curriculum, pairing precise definitions with memory hacks and contextual examples. From the precise notation of set theory to the geometric intuition behind vector operations, each term is presented as a building block for deeper mathematical reasoning.
掌握数学语言是熟练解题的第一步。本指南分解了你在 Year 10 Edexcel 进阶数学课程中将遇到的核心术语，将精确定义与记忆技巧和情境示例配对。从集合论的精巧符号到向量运算背后的几何直觉，每个术语都是构建更深层次数学推理的基石。
1. Algebra & Expressions | 代数与表达式
An expression ÷) but without an equals sign. For example, 3x² + 5x − 7 is an expression, while an equation sets two expressions equal to each other. Like terms have identical variable parts and can be combined: 4xy and −2xy are like terms, but 3x² and 5x are not.
表达式是通过数学运算（+、−、×、÷）组合在一起的一组项，但没有等号。例如，3x² + 5x − 7 是一个表达式，而方程是将两个表达式设为相等。同类项具有相同的变量部分并且可以合并：4xy 和 −2xy 是同类项，但 3x² 和 5x 不是。
The coefficient is the numerical factor multiplying a variable. In 7a³b, 7 is the coefficient. Expanding brackets uses the distributive law: a(b + c) = ab + ac. Factorising is the reverse process, rewriting an expression as a product of its factors by identifying the highest common factor (HCF).
系数是乘以变量的数值因子。在 7a³b 中，7 是系数。展开括号使用分配律：a(b + c) = ab + ac。因式分解是相反的过程，通过找出最高公因式 (HCF) 将表达式重写为因式的乘积。
- Monomial — one term (5x³) | 单项式 — 一项 (5x³)
- Binomial — two terms (x + y) | 二项式 — 两项 (x + y)
- Trinomial — three terms (x² + 5x + 6) | 三项式 — 三项 (x² + 5x + 6)
- Polynomial — many terms, general form aₙxⁿ + ... + a₀ | 多项式 — 多项，一般形式 aₙxⁿ + ... + a₀
Memory tip: the prefix tells the story — mono (one), bi (two), tri (three), poly (many). This same root system appears across mathematics and science.
记忆技巧：前缀讲述故事——mono（一）、bi（二）、tri（三）、poly（多）。这一相同的词根系统贯穿数学和科学。
2. Equations & Inequalities | 方程与不等式
An equation states that two expressions are equal, using the symbol =. A linear equation has the variable raised only to the power 1, such as 2x + 3 = 11. The solution or root is the value of the variable that makes the equation true. Simultaneous equations share two or more variables and must be solved together — typical methods include elimination and substitution.
方程用符号 = 表示两个表达式相等。线性方程的变量指数仅为 1，例如 2x + 3 = 11。解或根是使方程成立的变量值。联立方程共享两个或多个变量，必须一起求解——典型方法包括加减消元法和代入法。
Inequalities use symbols <, >, ≤, ≥ to compare expressions. The solution is a range of values rather than a single number. When multiplying or dividing by a negative number, the inequality sign must be reversed. Quadratic equations d can be solved by factorising, completing the square, or using the quadratic formula. The discriminant (Δ = b² − 4ac) determines the nature of the roots.
不等式使用符号 <、>、≤、≥ 来比较表达式。解是一个范围的值，而不是一个单一的数。当乘以或除以一个负数时，不等号必须反转。二次方程的形式为 ax² + bx + c = 0 (a ≠ 0)，可以通过因式分解、配方法或使用二次公式求解。判别式 (Δ = b² − 4ac) 决定根的性质。
| Discriminant | 判别式 | Nature of Roots | 根的性质 |
|---|---|
| b² − 4ac > 0 | Two distinct real roots | 两个不等实根 |
| b² − 4ac = 0 | One repeated real root | 一个重实根 |
| b² − 4ac < 0 | No real roots | 无实根 |
3. Functions & Graphs | 函数与图像
A function is a special relationship where each input (x-value) maps to exactly one output (y-value). The notation f(x) is read as 'f of x'. The set of all possible inputs is the domain, and the set of all possible outputs is the range. A function is one-to-one if each y-value corresponds to exactly one x-value; it fails the vertical line test if a vertical line intersects its graph more than once.
函数是一种特殊关系，其中每个输入（x 值）只对应一个输出（y 值）。符号 f(x) 读作“f of x”。所有可能输入的集合是定义域，所有可能输出的集合是值域。如果每个 y 值只对应一个 x 值，函数是一一对应的；如果一条垂直线与图像相交多于一次，它就无法通过垂直线检验。
Composite functions combine two functions: (g ∘ f)(x) = g(f(x)) means apply f first, then g. The inverse function f f(x); graphically, it is a reflection in the line y = x. Not all functions have inverses — only one-to-one functions do.
复合函数将两个函数结合：(g ∘ f)(x) = g(f(x)) 表示先应用 f，再应用 g。反函数 f⁻¹(x) 逆转 f(x) 的效果；在图像上，它是关于直线 y = x 的反射。并非所有函数都有反函数——只有一一对应的函数才有。
Key function transformations: f(x + a) shifts the graph left by a; f(x) + a shifts it up by a; f(−x) reflects in the y-axis; −f(x) reflects in the x-axis; af(x) stretches vertically by factor a; f(ax) stretches horizontally by factor 1/a.
关键函数变换：f(x + a) 将图像向左平移 a；f(x) + a 向上平移 a；f(−x) 关于 y 轴反射；−f(x) 关于 x 轴反射；af(x) 垂直拉伸 a 倍；f(ax) 水平拉伸 1/a 倍。
4. Sequences & Series | 数列与级数
A sequence is an ordered list of numbers following a rule. An arithmetic sequence (or arithmetic progression, AP) has a constant common difference d between consecutive terms: uₙ = a + (n − 1)d, where a is the first term. A geometric sequence (GP) has a constant common ratio r: uₙ = arⁿ⁻¹. A series is the sum of the terms of a sequence.
数列是按规则排列的有序数字列表。等差数列 (AP) 的相邻项之间有恒定的公差 d：uₙ = a + (n − 1)d，其中 a 是首项。等比数列 (GP) 有恒定的公比 r：uₙ = arⁿ⁻¹。级数是数列各项之和。
The Fibonacci sequence is defined recursively: F₁ = 1, F₂ = 1, Fₙ = Fₙ₋₁ + Fₙ₋₂. Recurrence relations define each term using previous terms, and you must be able to generate terms and analyse long-term behaviour, including approaching a limit.
斐波那契数列通过递推定义：F₁ = 1, F₂ = 1, Fₙ = Fₙ₋₁ + Fₙ₋₂。递推关系用前一项定义后一项，你必须能够生成项并分析长期行为，包括趋近于一个极限。
ₙ = n/2 [2a + (n − 1)d] = n/2 (a + l), where l is the last term. For geometric series with |r| < 1, the sum to infinity exists: S∞ = a / (1 − r).
要记住的求和公式：等差数列求和 Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l)，其中 l 是末项。对于 |r| < 1 的等比数列，无穷和存在：S∞ = a / (1 − r)。
5. Geometry & Measure | 几何与测量
Congruent shapes are identical in form and size (symbol ≅); similar shapes have the same angles but proportional side lengths. The scale factor k enlarges lengths by k, areas by k², and volumes by k³. A circle theorem is a geometric rule linking angles, chords, tangents, and radii — the angle in a semicircle is always 90°, and the angle at the centre is twice the angle at the circumference.
全等图形形状和大小完全相同（符号 ≅）；相似图形对应角相等但边长成比例。比例因子 k 将长度放大 k 倍，面积放大 k² 倍，体积放大 k³ 倍。圆定理是联系角、弦、切线和半径的几何规则——半圆中的圆周角始终为 90°，圆心角是圆周角的二倍。
Loci (singular: locus) are sets of points satisfying a given condition, such as the perpendicular bisector of a line segment, which is the locus of points equidistant from two fixed points. Bearings are measured clockwise from North, always given as three-digit angles (e.g., 045°).
轨迹（单数：locus）是满足给定条件的点集，例如线段的垂直平分线是到两个固定点距离相等的点的轨迹。方位角从正北顺时针测量，始终表示为三位数的角度（例如 045°）。
6. Trigonometry | 三角学
The three primary trigonometric ratios are defined on a right-angled triangle: sine sin θ = opposite / hypotenuse, cosine cos θ = adjacent / hypotenuse, tangent tan θ = opposite / adjacent. Memorise SOH CAH TOA. The exact values for 0°, 30°, 45°, 60°, 90° must be known without a calculator.
三个主要的三角比在直角三角形上定义：正弦 sin θ = 对边 / 斜边，余弦 cos θ = 邻边 / 斜边，正切 tan θ = 对边 / 邻边。熟记 SOH CAH TOA。0°、30°、45°、60°、90° 的精确值必须在不使用计算器的情况下掌握。
The sine rule (a / sin A = b / sin B = c / sin C) works for any triangle, as does the cosine rule (a² = b² + c² − 2bc cos A). The area formula using sine is Area = ½ ab sin C. For angles beyond 90°, the unit circle extends definitions: the CAST diagram shows which ratios are positive in each quadrant.
正弦定理 (a / sin A = b / sin B = c / sin C) 适用于任何三角形，余弦定理 (a² = b² + c² − 2bc cos A) 同样适用。使用正弦的面积公式是 Area = ½ ab sin C。对于超过 90° 的角，单位圆扩展了定义：CAST 图显示每个象限中哪些比值为正。
- Trigonometric identities: sin²θ + cos²θ ≡ 1, tan θ ≡ sin θ / cos θ | 三角恒等式：sin²θ + cos²θ ≡ 1，tan θ ≡ sin θ / cos θ
- Period of sin and cos is 360°; period of tan is 180° | sin 和 cos 的周期是 360°；tan 的周期是 180°
7. Vectors | 向量
A vector has both magnitude (size) and direction, unlike a scalar which has only magnitude. Vectors are written as bold letters or with an arrow: a or a⃗, or as a column vector. The magnitude of vector (x, y) is √x² + y², found using Pythagoras' theorem.
向量既有大小又有方向，不同于只有大小的标量。向量用粗体字母或带箭头书写：a 或 a⃗，或作为列向量。向量 (x, y) 的模是 √x² + y²，用勾股定理求得。
Vector addition is done component-wise or using the triangle law (tip-to-tail). Scalar multiplication ka changes the magnitude by factor |k| and reverses direction if k is negative. Unit vectors have magnitude 1; i and j are the standard unit vectors along the x- and y-axes. Parallel vectors are scalar multiples of each other. The position vector of a point is its displacement from the origin.
向量加法按分量进行或使用三角形法则（首尾相接）。标量乘法 ka 将模改变 |k| 倍，如果 k 为负则反转方向。单位向量的模为 1；i 和 j 是沿 x 轴和 y 轴的标准单位向量。平行向量互为标量倍数。点的位置向量是它从原点的位移。
Collinear points raight line; algebraically, their position vectors satisfy AB = k AC for some scalar k. In geometric proof using vectors, you express relationships without coordinates and show parallelism, midpoints, or ratios.
共线点位于同一直线上；在代数上，它们的位置向量满足 AB = k AC（对于某个标量 k）。在使用向量的几何证明中，你无需坐标即可表达关系，并证明平行、中点或比例。
8. Calculus Basics | 微积分基础
Differentiation finds the gradient function (derivative) of a curve. For f(x) = xⁿ, the derivative is f'(x) = nxⁿ⁻¹. The derivative f'(x) gives the gradient of the tangent at any point. A stationary point occurs where f'(x) = 0; its nature (maximum, minimum, or point of inflection) is determined by the second derivative f''(x) or by examining the sign change of f'(x).
微分找出曲线的梯度函数（导数）。对于 f(x) = xⁿ，导数是 f'(x) = nxⁿ⁻¹。导数 f'(x) 给出任意点处切线的梯度。驻点出现在 f'(x) = 0 处；其性质（极大值、极小值或拐点）由二阶导数 f''(x) 或通过检查 f'(x) 的符号变化来确定。
Integration is the reverse of differentiation (anti-differentiation). The indefinite integral of xⁿ is (xⁿ⁺¹)/(n+1) + C, where C is the constant of integration. Definite integration between limits a and b gives the area under the curve between x = a and x = b. The Fundamental Theorem of Calculus links differentiation and integration as inverse processes.
积分是微分的逆运算（反微分）。xⁿ 的不定积分是 (xⁿ⁺¹)/(n+1) + C，其中 C 是积分常数。在界限 a 与 b 之间的定积分给出 x = a 和 x = b 之间曲线下的面积。微积分基本定理将微分和积分连接为逆过程。
Memory aid: 'Differentiate down, Integrate up' — differentiation drops the power by 1 and multiplies by the old power; integration raises the power by 1 and divides by the new power.
记忆口诀：“微分向下，积分向上”——微分将幂降低 1 并乘以原幂；积分将幂升高 1 并除以新幂。
9. Probability & Statistics | 概率与统计
Probability impossible) to 1 (certain). For mutually exclusive events A and B, P(A or B) = P(A) + P(B) — the addition rule. For independent events, P(A and B) = P(A) × P(B) — the multiplication rule. Conditional probability P(A|B) is the probability of A given that B has occurred, calculated as P(A ∩ B) / P(B).
概率在 0（不可能）到 1（必然）的尺度上衡量可能性。对于互斥事件 A 和 B，P(A 或 B) = P(A) + P(B)——加法法则。对于独立事件，P(A 和 B) = P(A) × P(B)——乘法法则。条件概率 P(A|B) 是在 B 已发生的情况下 A 发生的概率，计算为 P(A ∩ B) / P(B)。
Venn diagrams and tree diagrams are essential tools for visualising combined events. A sample space is the set of all possible outcomes. The expected frequency of an event is its probability multiplied by the number of trials. In Further Maths, you also encounter the binomial distribution: X ~ B(n, p), where n is the number of trials and p the probability of success in each.
维恩图和树状图是可视化组合事件的重要工具。样本空间是所有可能结果的集合。事件的期望频数是其概率乘以试验次数。在进阶数学中，你还会遇到二项分布：X ~ B(n, p)，其中 n 是试验次数，p 是每次成功的概率。
10. Set Theory & Notation | 集合论与符号
Sets are collections of distinct objects called elements. The universal set E (or U) contains all elements under consideration. Intersection A ∩ B contains elements in both A and B. Union A ∪ B contains elements in A or B or both. The complement A' contains all elements not in A.
集合是称为元素的不同对象的聚集。全集 E（或 U）包含所考虑的所有元素。交集 A ∩ B 包含同时在 A 和 B 中的元素。并集 A ∪ B 包含在 A 或 B 或两者之中的元素。补集 A' 包含不在 A 中的所有元素。
Set-builder notation describes a set by a rule: {x | x > 3, x ∈ R} reads 'the set of all x such that x is greater than 3, where x is a real number'. The number of elements in set A is denoted n(A). The empty set ∅ has no elements.
集合构造式记号通过规则描述集合：{x | x > 3, x ∈ R} 读作“所有 x 的集合，使得 x 大于 3，其中 x 是实数”。集合 A 中元素的个数记为 n(A)。空集 ∅ 没有元素。
Membership symbols: ∈ means 'is an element of', ∉ means 'is not an element of'. Subset symbols: A ⊆ B means A is a subset of B (every element of A is also in B); A ⊂ B means A is a proper subset of B (A is contained in B but not equal to B).
属于符号：∈ 表示“是……的元素”，∉ 表示“不是……的元素”。子集符号：A ⊆ B 表示 A 是 B 的子集（A 的每个元素都在 B 中）；A ⊂ B 表示 A 是 B 的真子集（A 包含在 B 中但不等于 B）。
11. Proof & Logic | 证明与逻辑
Proof is a logical argument that establishes the truth of a mathematical statement. Direct proof starts from known facts and uses logical steps to reach a conclusion. Proof by exhaustion checks every possible case individually — practical only for small finite sets. Proof by contradiction assumes the statement is false and derives a logical impossibility, forcing the original statement to be true.
证明是确立数学陈述为真的逻辑论证。直接证明从已知事实出发，使用逻辑步骤得出结论。穷举证明逐个检查每种可能情况——仅对小的有限集实用。反证法假设陈述为假并推导出逻辑上的不可能性，从而迫使原陈述为真。
Identities (symbol ≡) are true for all values of the variable, unlike equations which hold only for specific values. Counterexamples disprove a conjecture by providing a single case where it fails. In algebraic proof, you manipulate expressions using valid operations while maintaining equivalence at each step.
恒等式（符号 ≡）对变量的所有值都成立，不同于仅在特定值处成立的方程。反例通过提供一个失败的单一案例来反驳猜想。在代数证明中，你使用有效运算操作表达式，同时在每一步保持等价性。
12. Rapid Memory Techniques | 快速记忆技巧
Create acronyms for sequential steps: SOHCAHTOA for trig ratios, BIDMAS/BODMAS for order of operations. Use visual associations: imagine the CAST diagram as a compass quadrant, or picture the quadratic formula sung to 'Pop Goes the Weasel'. Build flashcards with the term on one side and definition plus example on the reverse — test yourself in both directions (term → definition, definition → term).
为连续步骤创建首字母缩略词：SOHCAHTOA 用于三角比，BIDMAS/BODMAS 用于运算顺序。使用视觉联想：将 CAST 图想象为指南针象限，或将二次公式想象成用“Pop Goes the Weasel”的曲调唱出来。制作闪卡，一面写术语，另一面写定义和示例——双向测试自己（术语 → 定义，定义 → 术语）。
Group related terms into concept clusters: keep all function vocabulary (domain, range, inverse, composite, one-to-one) together. Teach the concept aloud to an imaginary student — verbalisation consolidates memory. Finally, practise active recall by writing down everything you remember about a topic before checking your notes, rather than passively re-reading.
将相关术语归入概念集群：将所有函数词汇（定义域、值域、反函数、复合函数、一一对应）放在一起。向想象中的学生大声讲解概念——语言化巩固记忆。最后，在检查笔记之前写下你记得的关于某个主题的所有内容，通过主动回忆来练习，而不是被动重读。
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