Year 10 CAIE Further Mathematics: Rapid Memorisation Guide to Vocabulary & Terminology | Year 10 CAIE 进阶数学：词汇术语速记指南

📚 Year 10 CAIE Further Mathematics: Rapid Memorisation Guide to Vocabulary & Terminology | Year 10 CAIE 进阶数学：词汇术语速记指南

In Year 10 CAIE Further Mathematics, mastering the precise vocabulary is not just about passing exams—it is about learning to think like a mathematician. Terminology such as ‘differentiate’, ‘integrate’, ‘complex conjugate’, and ‘vector cross product’ names powerful concepts that form the building blocks of advanced problem-solving. This guide breaks down the essential terms you need to know, pairing clear English definitions with concise Chinese translations and memorable explanations, so you can recall them quickly under exam pressure.

在 Year 10 CAIE 进阶数学中，掌握精确的词汇不仅是为了通过考试，更是为了学会像数学家一样思考。’differentiate’（求导）、’integrate’（积分）、’complex conjugate’（共轭复数）、’vector cross product’（向量叉积）等术语，命名了构成高阶问题解决能力的基石概念。本指南将你需要掌握的核心术语逐一分解，将清晰的英文定义与简洁的中文翻译配对，并提供易记的说明，帮助你在考试压力下快速回忆。

1. Differentiation & Derivatives | 微分与导数

‘Differentiation’ is the process of finding the rate at which one quantity changes with respect to another. The result of this process is called the ‘derivative’ of a function. If you have a function y = f(x), its derivative is often written as dy/dx or f'(x). It measures the gradient of the tangent to the curve at any given point. A ‘turning point’ occurs where the derivative equals zero.

‘Differentiation’（微分）是求一个量相对于另一个量变化率的过程。这个过程得出的结果称为函数的 ‘derivative’（导数）。如果有一个函数 y = f(x)，其导数通常写作 dy/dx 或 f'(x)。它衡量的是曲线上任意给定点处切线的斜率。当导数等于零时，就出现了 ‘turning point’（驻点）。

Derivative (导数): The instantaneous rate of change; gradient function. / 瞬时变化率；斜率函数。
Turning point (驻点): A point where dy/dx = 0, and the tangent is horizontal. / dy/dx = 0 的点，切线为水平线。
Second derivative (二阶导数): The derivative of the derivative, written as d²y/dx² or f”(x); used to determine the nature of a turning point. / 导数的导数，记作 d²y/dx² 或 f”(x)；用于判断驻点的性质。

If f(x) = xⁿ, then f'(x) = nxⁿ⁻¹

For a quick recall, think of differentiation as ‘slope-finding’. In Chinese, the term 导数 directly implies the leading number that guides the function’s direction. Remember the second derivative test: if f”(x) > 0 at a turning point, the point is a ‘local minimum’; if f”(x) < 0, it is a 'local maximum'.

为了快速记忆，可以把 differentiation 想象成“找坡度”。中文里，’导数’ 直接暗示了引导函数方向的领头数字。记住二阶导数检验：如果在驻点处 f”(x) > 0，该点是 ‘local minimum’（局部极小值）；如果 f”(x) < 0，则是 'local maximum'（局部极大值）。

2. Integration & Antiderivatives | 积分与原函数

‘Integration’ is the reverse process of differentiation. It is often described as finding the ‘antiderivative’. The result of an indefinite integral is a family of functions, plus an arbitrary constant C. A ‘definite integral’ evaluates the area under a curve between two limits a and b, denoted by ∫ₐᵇ f(x) dx. The Fundamental Theorem of Calculus links differentiation and integration as inverse operations.

‘Integration’（积分）是微分的逆过程，常被描述为求 ‘antiderivative’（原函数）。不定积分的结果是一族函数，外加一个任意常数 C。’definite integral’（定积分）计算曲线在两点 a 和 b 之间下方的面积，记作 ∫ₐᵇ f(x) dx。微积分基本定理将微分和积分联系为逆运算。

Antiderivative (原函数): A function F(x) such that F'(x) = f(x). / 一个函数 F(x)，满足 F'(x) = f(x)。
Constant of integration (积分常数): The arbitrary constant C added to an indefinite integral. / 不定积分中加上的任意常数 C。
Definite integral (定积分): An integral with upper and lower limits, giving a numerical value representing area. / 带有上下限的积分，给出表示面积的数值。

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ -1

To memorise integration, think of it as ‘area-accumulation’. The symbol ∫ is an elongated S, standing for ‘sum’. In Chinese, 积分 literally means ‘accumulated parts’, which perfectly captures the idea of summing infinitely many infinitesimally small slices to find an area.

为了记住积分，可以把它想成“面积累积”。符号 ∫ 是一个拉长的 S，代表 ‘sum’（求和）。中文里，’积分’ 字面意思是“累积的部分”，完美地捕捉了将无穷多个无穷小的薄片相加以求面积的思想。

3. Trigonometric Functions & Identities | 三角函数与恒等式

‘Trigonometry’ in further mathematics extends beyond right-angled triangles to the unit circle, where the sine, cosine, and tangent functions are defined for all real angles, measured in radians. A ‘radian’ is the angle subtended at the centre of a circle by an arc equal in length to the radius. Key identities such as sin²θ + cos²θ ≡ 1 and tanθ ≡ sinθ/cosθ are essential tools for simplifying expressions and solving equations.

进阶数学中的 ‘Trigonometry’（三角学）从直角三角形扩展到单位圆，在单位圆上，正弦、余弦和正切函数定义为所有实数角，以弧度度量。’radian’（弧度）是圆心角所对的弧长等于半径时的角度。sin²θ + cos²θ ≡ 1 和 tanθ ≡ sinθ/cosθ 等关键恒等式是化简表达式和求解方程的基础工具。

Radian (弧度): The angle where the arc length equals the radius; π radians = 180°. / 弧长等于半径时所对应的角度；π 弧度 = 180°。
Cotangent (cot, 余切): cotθ = 1/tanθ = cosθ/sinθ. / 余切。
Secant (sec, 正割) and cosecant (cosec, 余割): secθ = 1/cosθ; cosecθ = 1/sinθ. / 正割和余割。

sin²θ + cos²θ ≡ 1

A powerful memory trick is the ‘CAST diagram’, showing which trigonometric functions are positive in each quadrant. Starting from the fourth quadrant (bottom right) and moving anticlockwise: C for Cosine positive, A for All positive, S for Sine positive, T for Tangent positive. This diagram is your quickest tool for solving trigonometric equations within a given domain.

一个强大的记忆技巧是 ‘CAST 图’，它显示了每个象限中哪些三角函数为正。从第四象限（右下）开始逆时针移动：C 代表余弦为正，A 代表全部为正，S 代表正弦为正，T 代表正切为正。这张图是你在给定范围内求解三角方程的最快工具。

4. Vectors: Direction & Magnitude | 向量：方向与大小

A ‘vector’ is a quantity that has both magnitude and direction, unlike a ‘scalar’, which has only magnitude. Vectors are represented geometrically by directed line segments and algebraically by column vectors, such as (x, y, z) or xi + yj + zk. The ‘position vector’ of a point A is the vector from the origin O to A, denoted by OA⃗ or a. The ‘magnitude’ of vector a is written as |a| or ||a||, calculated using Pythagoras’ theorem.

‘vector’（向量）是一个既有大小又有方向的量，不像 ‘scalar’（标量）只有大小。向量在几何上用有向线段表示，在代数上用列向量表示，如 (x, y, z) 或 xi + yj + zk。点 A 的 ‘position vector’（位置向量）是从原点 O 到 A 的向量，记作 OA⃗ 或 a。向量 a 的 ‘magnitude’（模）写作 |a| 或 ||a||，用勾股定理计算。

Scalar product (dot product, 数量积/点积): a · b = |a||b|cosθ, resulting in a scalar. / a · b = |a||b|cosθ，结果是一个标量。
Vector product (cross product, 向量积/叉积): a × b = |a||b|sinθ n̂, resulting in a vector perpendicular to both a and b. / a × b = |a||b|sinθ n̂，结果是一个垂直于 a 和 b 的向量。
Unit vector (单位向量): A vector with magnitude 1, often denoted with a hat, e.g. i, j, k. / 模为 1 的向量，常带帽子符号，如 i, j, k。

To remember the difference between dot and cross products, use the mnemonic: ‘Dot gives a dot (number), Cross gives a cross (new vector).’ In Chinese, 点积 and 叉积 visually suggest their operations: a dot (·) yields a scalar point, while a cross (×) yields a new directional line in space.

要记住点积和叉积的区别，可以用这个口诀：’Dot 出点（数），Cross 出叉（新向量）。’ 中文里，’点积’ 和 ‘叉积’ 在视觉上就暗示了它们的运算：点 (·) 产出一个标量点，叉 (×) 产出一个空间中新的有向线段。

5. Complex Numbers: Real & Imaginary Parts | 复数：实部与虚部

‘Complex numbers’ extend the real number system by introducing the ‘imaginary unit’ i, defined as i² = -1. A complex number is written in the form z = a + bi, where a is the ‘real part’ Re(z) and b is the ‘imaginary part’ Im(z). The ‘complex conjugate’ of z is z̄ = a – bi. Complex numbers can be plotted on an ‘Argand diagram’, where the x-axis is the real axis and the y-axis is the imaginary axis.

‘complex numbers’（复数）通过引入 ‘imaginary unit’（虚数单位）i（定义为 i² = -1）来扩展实数系统。复数写作 z = a + bi 的形式，其中 a 是 ‘real part’（实部）Re(z)，b 是 ‘imaginary part’（虚部）Im(z)。z 的 ‘complex conjugate’（共轭复数）是 z̄ = a – bi。复数可以绘制在 ‘Argand diagram’（阿根图）上，其中 x 轴是实轴，y 轴是虚轴。

Modulus (模): The distance of z from the origin on the Argand diagram, |z| = √(a² + b²). / 复数 z 在阿根图上到原点的距离。
Argument (辐角): The angle θ made by the line representing z with the positive real axis, arg(z). / 表示 z 的线段与正实轴形成的角度 θ。
Complex conjugate (共轭复数): Reflection of z across the real axis; z̄ = a – bi. / z 关于实轴的反射。

A vital rule to remember: when solving polynomial equations with real coefficients, complex roots always occur in ‘conjugate pairs’. If 2 + 3i is a root, then 2 – 3i must also be a root. This is a favourite exam point.

一条要记住的关键规则：在求解实系数多项式方程时，复根总是以 ‘conjugate pairs’（共轭对）的形式出现。如果 2 + 3i 是一个根，那么 2 – 3i 必定也是根。这是考试中常考的一点。

6. Matrices: Rows, Columns & Determinants | 矩阵：行、列与行列式

A ‘matrix’ is a rectangular array of numbers arranged in rows and columns. The order of a matrix is given as m × n, where m is the number of rows and n the number of columns. A ‘square matrix’ has the same number of rows and columns. The ‘determinant’ of a 2×2 matrix [[a, b], [c, d]] is ad – bc, and it determines whether the matrix is ‘singular’ (no inverse) or ‘non-singular’.

‘matrix’（矩阵）是按行和列排列的数字矩形阵列。矩阵的阶记作 m × n，其中 m 是行数，n 是列数。’square matrix’（方阵）的行数和列数相同。2×2 矩阵 [[a, b], [c, d]] 的 ‘determinant’（行列式）是 ad – bc，它决定了矩阵是 ‘singular’（奇异矩阵，无逆矩阵）还是 ‘non-singular’（非奇异矩阵）。

Identity matrix (I, 单位矩阵): A square matrix with 1s on the leading diagonal and 0s elsewhere; acts like 1 in multiplication. / 主对角线上为 1，其余位置为 0 的方阵；在乘法中的作用类似于 1。
Inverse matrix (A⁻¹, 逆矩阵): A matrix such that A × A⁻¹ = I = A⁻¹ × A. / 满足 A × A⁻¹ = I = A⁻¹ × A 的矩阵。
Singular matrix (奇异矩阵): A square matrix with determinant 0; has no inverse. / 行列式为 0 的方阵；没有逆矩阵。

A = [[a, b], [c, d]]	det(A) = ad – bc
If det(A) ≠ 0:	A⁻¹ = 1/(ad-bc) × [[d, -b], [-c, a]]

Think of the determinant as a ‘gatekeeper’: if it is zero, the inverse is ‘locked’ and inaccessible. The formula for the inverse swaps the diagonal elements a and d, and negates the off-diagonal elements b and c. In Chinese, 行列式 suggests an operation on rows and columns combined, while 奇异 means ‘strange’ or ‘exceptional’, hinting that the matrix behaves abnormally by lacking an inverse.

可以把行列式想象成一个“守门人”：如果它为零，逆矩阵就被“锁住”而无法获取。求逆公式将对角线元素 a 和 d 交换，并将非对角线元素 b 和 c 变号。中文里，’行列式’ 暗示着对行和列的组合操作，而 ‘奇异’ 意为“奇怪的”或“异常的”，暗示该矩阵因缺少逆矩阵而表现反常。

7. Sequences & Series | 数列与级数

A ‘sequence’ is an ordered list of numbers following a specific pattern, such as an ‘arithmetic progression’ (AP) where the difference between consecutive terms is constant. A ‘series’ is the sum of the terms of a sequence. The ‘common difference’ d in an AP and the ‘common ratio’ r in a ‘geometric progression’ (GP) are the defining parameters. A GP converges to a finite sum to infinity if |r| < 1.

‘sequence’（数列）是一个按特定模式排列的有序数字列表，比如 ‘arithmetic progression’（等差数列），其中连续两项的差为常数。’series’（级数）是数列各项的和。等差数列中的 ‘common difference’（公差）d 和 ‘geometric progression’（等比数列）中的 ‘common ratio’（公比）r 是定义性的参数。如果 |r| < 1，等比级数收敛于一个有限的和。

Arithmetic progression (等差数列): nth term = a + (n-1)d; sum to n terms = n/2 × [2a + (n-1)d]. / 第 n 项；前 n 项和。
Geometric progression (等比数列): nth term = arⁿ⁻¹; sum to n terms = a(1 – rⁿ)/(1 – r) for r ≠ 1. / 第 n 项；前 n 项和（r ≠ 1）。
Sum to infinity (无穷级数和): S∞ = a/(1 – r), valid only for |r| < 1. / 仅在 |r| < 1 时成立。

The notation Σ (capital sigma) stands for ‘summation’. The expression Σₖ₌₁ⁿ uₖ means you sum the terms uₖ from k = 1 to k = n. To memorise the formulas, keep a clear distinction: an arithmetic sequence involves adding the same number, while a geometric sequence involves multiplying by the same number. The Chinese terms 等差 (equal difference) and 等比 (equal ratio) encode this perfectly.

符号 Σ（大写西格玛）代表 ‘summation’（求和）。表达式 Σₖ₌₁ⁿ uₖ 意味着将各项 uₖ 从 k = 1 加到 k = n。要记住公式，需保持清晰区分：等差数列涉及加上同一个数，而等比数列涉及乘以同一个数。中文术语 ‘等差’ 和 ‘等比’ 完美地编码了这一点。

8. Polynomials & Factor Theorem | 多项式与因式定理

A ‘polynomial’ is an expression of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where n is a non-negative integer called the ‘degree’. The ‘Factor Theorem’ states that (x – p) is a factor of f(x) if and only if f(p) = 0. The ‘Remainder Theorem’ states that when f(x) is divided by (x – p), the remainder is f(p). Together, these theorems allow you to factorise higher-degree polynomials and solve equations.

‘polynomial’（多项式）是形如 aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀ 的表达式，其中 n 是一个非负整数，称为 ‘degree’（次数）。’Factor Theorem’（因式定理）指出，(x – p) 是 f(x) 的因式当且仅当 f(p) = 0。’Remainder Theorem’（余式定理）指出，当 f(x) 除以 (x – p) 时，余数为 f(p)。这两个定理结合起来，允许你对高次多项式进行因式分解并求解方程。

Degree (次数): The highest power of the variable x in a polynomial. / 多项式中变量 x 的最高次幂。
Root of a polynomial (多项式的根): A value r such that f(r) = 0; also called a zero. / 满足 f(r) = 0 的值 r；也称为零点。
Repeated root (重根): A root that appears more than once, e.g. (x – 2)² has a repeated root at x = 2. / 出现不止一次的根。

To apply the Factor Theorem efficiently, test integer factors of the constant term a₀ as potential roots. Once you find one root p, divide the polynomial by (x – p) to reduce its degree, and repeat the process. This is often called ‘polynomial long division’ or ‘synthetic division’. The Chinese phrase 因式定理 directly links 因式 (factor) to the theorem, making the connection between roots and factors explicit.

要高效应用因式定理，可以测试常数项 a₀ 的整数因子作为潜在的根。一旦找到一个根 p，将多项式除以 (x – p) 以降低次数，然后重复这个过程。这通常称为 ‘polynomial long division’（多项式长除）或 ‘synthetic division’（综合除法）。中文短语 ‘因式定理’ 直接将 ‘因式’ 与定理联系起来，使根与因式之间的关联变得明确。

9. Functions: Domain, Range & Composition | 函数：定义域、值域与复合

A ‘function’ f maps each element x from its ‘domain’ to exactly one element f(x) in its ‘range’. A function must be ‘well-defined’: one input cannot produce two different outputs. ‘Composition’ of functions, written as fg(x) or (f ∘ g)(x), means applying g first, then f. An ‘inverse function’ f⁻¹ reverses the effect of f, so that f⁻¹(f(x)) = x for all x in the domain of f. Not all functions have inverses; f must be ‘one-to-one’ (bijective) on its domain.

‘function’（函数）f 将其 ‘domain’（定义域）中的每个元素 x 映射到其 ‘range’（值域）中唯一的元素 f(x)。函数必须是 ‘well-defined’（良定义的）：一个输入不能产生两个不同的输出。函数的 ‘composition’（复合），写作 fg(x) 或 (f ∘ g)(x)，意味着先应用 g，再应用 f。’inverse function’（反函数）f⁻¹ 逆转 f 的效果，使得对于 f 定义域中的所有 x，都有 f⁻¹(f(x)) = x。并非所有函数都有反函数；f 在其定义域上必须是 ‘one-to-one’（单射）的。

Domain (定义域): The set of all possible input values (x-values) for which the function is defined. / 函数有定义的所有可能输入值（x 值）的集合。
Range (值域): The set of all possible output values (f(x)) the function can produce. / 函数能产生的所有可能输出值（f(x)）的集合。
One-to-one function (单射函数): A function where each y-value comes from exactly one x-value; passes the ‘horizontal line test’. / 每个 y 值恰好来自一个 x 值的函数；通过“水平线检验”。

To find the inverse of a function, write y = f(x), swap x and y, then solve for y. The domain of f⁻¹ is the range of f. The graphs of f and f⁻¹ are symmetric about the line y = x. In Chinese, 定义域 literally means ‘the defined region’, and 值域 means ‘the value region’, which helpfully anchors the concept of domain as the input space and range as the output space.

要求函数的反函数，写出 y = f(x)，交换 x 和 y，然后解出 y。f⁻¹ 的定义域是 f 的值域。f 和 f⁻¹ 的图像关于直线 y = x 对称。中文里，’定义域’ 字面意思是“被定义的区域”，’值域’ 是“值的区域”，这有助于将定义域锚定为输入空间、值域锚定为输出空间。

10. Logarithms & Exponentials | 对数与指数

‘Logarithms’ are the inverse operations of exponentiation. The statement y = bˣ is equivalent to log_b(y) = x, where b is the ‘base’. The ‘natural logarithm’, written as ln x, has base e (approximately 2.71828), and the ‘common logarithm’, written as log x or lg x, has base 10. The function eˣ is its own derivative, making it uniquely important in calculus and in modelling continuous growth and decay processes.

‘logarithms’（对数）是指数运算的逆运算。y = bˣ 等价于 log_b(y) = x，其中 b 是 ‘base’（底数）。’natural logarithm’（自然对数），写作 ln x，以 e（约等于 2.71828）为底；’common logarithm’（常用对数），写作 log x 或 lg x，以 10 为底。函数 eˣ 是它自己的导数，这使得它在微积分以及连续增长和衰减过程的建模中具有独特的重要性。

Base (底数): The number b in an exponential expression bˣ or logarithm log_b(x). / 指数表达式 bˣ 或对数 log_b(x) 中的数字 b。
Natural exponential function (自然指数函数): eˣ, where e ≈ 2.71828; d/dx(eˣ) = eˣ. / 以 e 为底的指数函数；其导数是它本身。
Change of base formula (换底公式): log_a(x) = log_b(x)/log_b(a). / 对数从一个底数转换到另一个底数的公式。

ln(eˣ) = x e^(ln x) = x

A powerful memory aid: ‘Logarithms count the number of factors.’ The logarithm log_b(N) answers the question: ‘To what power must I raise b to get N?’ The change of base formula is essential when your calculator only has ln and log₁₀ buttons. In Chinese, 对数 originally means ‘pairing numbers’, reflecting how logarithms pair a base and a result to find the exponent.

一个强大的记忆辅助：’对数计算的是因子的个数。’ 对数 log_b(N) 回答了这个问题：’我必须把 b 升到多少次幂才能得到 N？’ 当你的计算器只有 ln 和 log₁₀ 按钮时，换底公式至关重要。中文里，’对数’ 原意是“配对数字”，反映了对数如何将底数和结果配对以求指数。

11. Probability & Combinatorics | 概率与组合数学

In further maths probability, ‘combinations’ and ‘permutations’ are fundamental counting tools. A ‘permutation’ is an arrangement of objects where order matters, while a ‘combination’ is a selection where order does not matter. The number of ways to choose r objects from n distinct objects is given by the ‘binomial coefficient’ ⁿCᵣ = n!/(r!(n-r)!). The ‘binomial theorem’ expands (a + b)ⁿ into a sum of terms involving these coefficients.

在进阶数学的概率部分，’combinations’（组合）和 ‘permutations’（排列）是基本的计数工具。’permutation’ 是顺序重要的对象排列，而 ‘combination’ 是顺序不重要的对象选择。从 n 个不同对象中选择 r 个的方式数由 ‘binomial coefficient’（二项式系数）ⁿCᵣ = n!/(r!(n-r)!) 给出。’binomial theorem’（二项式定理）将 (a + b)ⁿ 展开为涉及这些系数的各项之和。

Permutation (排列): An ordered arrangement; ⁿPᵣ = n!/(n-r)!. / 有序排列。
Combination (组合): An unordered selection; ⁿCᵣ = n!/(r!(n-r)!). / 无序选择。
Binomial expansion (二项展开式): (a + b)ⁿ = Σₖ₌₀ⁿ ⁿCₖ aⁿ⁻ᵏ bᵏ. / 二项式幂的展开。

To distinguish permutations from combinations quickly: ‘Permutation = Position matters (P for Position). Combination = Choice only (C for Choice).’ For probability problems, always consider whether the events are ‘independent’ (P(A ∩ B) = P(A) × P(B)) or ‘mutually exclusive’ (P(A ∪ B) = P(A) + P(B)). The Chinese terms 排列 (arranging) and 组合 (grouping) graphically encode their meanings.

快速区分排列和组合：’Permutation = 位置重要（P 代表 Position）。Combination = 只是选择（C 代表 Choice）。’ 对于概率问题，始终要考虑事件是 ‘independent’（独立的，P(A ∩ B) = P(A) × P(B)）还是 ‘mutually exclusive’（互斥的，P(A ∪ B) = P(A) + P(B)）。中文术语 ‘排列’ 和 ‘组合’ 形象地编码了它们的含义。

12. Proof & Mathematical Language | 证明与数学语言

Year 10 further mathematics introduces formal proof language. ‘Proof by induction’ is a technique for proving that a statement holds for all positive integers n. It involves a ‘base case’, an ‘inductive hypothesis’, and an ‘inductive step’. ‘Proof by contradiction’ assumes the negation of what you want to prove, and then logically derives a contradiction. ‘Proof by exhaustion’ checks all possible cases in a finite set. Understanding the precise meaning of terms like ‘iff’ (if and only if), ‘implies’ (⇒), and ‘equivalent’ (⇔) is crucial.

Year 10 进阶数学引入了形式化的证明语言。’Proof by induction’（归纳法证明）是一种证明某个命题对所有正整数 n 成立的技术。它涉及 ‘base case’（基础情况）、’inductive hypothesis’（归纳假设）和 ‘inductive step’（归纳步骤）。’Proof by contradiction’（反证法）假设你要证明的命题的否定成立，然后逻辑推导出一个矛盾。’Proof by exhaustion’（穷举法证明）检查有限集合中的所有可能情况。理解 ‘iff’（当且仅当）、’implies’（蕴含，⇒）和 ‘equivalent’（等价，⇔）等术语的精确含义至关重要。

If and only if (iff, 当且仅当): A ⇔ B means A and B are logically equivalent; both true or both false. / A 和 B 逻辑等价；同真同假。
Implies (蕴含): A ⇒ B means if A is true, then B must be true. / 如果 A 为真，那么 B 必然为真。
Counterexample (反例): A single example that disproves a claim; vital in refuting false statements. / 反驳一个断言的一个例子；在否定错误命题时至关重要。

When writing a proof by induction, always structure it clearly: Step 1: Prove true for n = 1. Step 2: Assume true for n = k. Step 3: Prove true for n = k + 1 using the assumption. Step 4: Conclude by induction. The phrase ‘QED’ (quod erat demonstrandum) traditionally marks the end of a proof. In Chinese, 证明 is the general term for proof, while 归纳法 explicitly mentions the method of including observed cases to infer a general rule.

在撰写归纳法证明时，务必结构清晰：第一步：证明 n = 1 时命题成立。第二步：假设 n = k 时命题成立。第三步：利用该假设证明 n = k + 1 时命题成立。第四步：通过归纳得出结论。短语 ‘QED’（证明完毕）传统上标志着证明的结束。中文里，’证明’ 是证明的通用术语，而 ‘归纳法’ 明确提到了通过纳入观察到的案例来推断一般规则的方法。

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Year 10 CAIE Further Mathematics: Rapid Memorisation Guide to Vocabulary & Terminology | Year 10 CAIE 进阶数学：词汇术语速记指南