Year 9 Cambridge Advanced Mathematics: Quick Vocabulary Guide | Year 9 剑桥进阶数学：词汇术语速记指南

📚 Year 9 Cambridge Advanced Mathematics: Quick Vocabulary Guide | Year 9 剑桥进阶数学：词汇术语速记指南

Mastering the precise vocabulary of Cambridge Advanced Mathematics is the foundation for success in Year 9 and beyond. This guide collects essential terms from algebra, functions, calculus, trigonometry and more, offering clear English definitions followed by Chinese translations and memory aids. Use it to build confidence when reading exam questions, writing solutions, and explaining your reasoning.

掌握剑桥进阶数学的精准词汇是 Year 9 学生迈向成功的基础。本指南收集了来自代数、函数、微积分、三角学等领域的核心术语，每个术语先给出清晰的英文定义，紧跟着中文翻译和记忆诀窍。反复使用它，你就能自信地阅读考题、书写解答并解释推理过程。

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1. Number Systems and Set Notation | 数系与集合符号

Natural numbers (ℕ): The set of positive whole numbers {1, 2, 3, ...}. Sometimes 0 is included.

自然数 (ℕ)： 正整数的集合 {1, 2, 3, ...}。有时包含 0。

Integers (ℤ): All whole numbers, positive, negative, and zero: {..., −2, −1, 0, 1, 2, ...}. ℤ comes from the German word 'Zahlen'.

整数 (ℤ)： 所有正整数、负整数和零：{..., −2, −1, 0, 1, 2, ...}。ℤ 源自德语 “Zahlen”。

Rational numbers (ℚ): Numbers that can be written as a fraction a/b where a and b are integers and b ≠ 0. ℚ reminds us of 'quotient'.

有理数 (ℚ)： 可以写成分数 a/b 的数，其中 a、b 为整数且 b ≠ 0。ℚ 让人联想到 “商”。

Irrational numbers: Numbers that cannot be expressed as a simple fraction, e.g. √2, π. Their decimal expansions are non‑terminating and non‑repeating.

无理数： 不能表示为简单分数的数，例如 √2、π。其小数形式无限不循环。

Real numbers (ℝ): The set of all rational and irrational numbers. Every point on the number line corresponds to a real number.

实数 (ℝ)： 所有有理数和无理数的集合。数轴上的每一个点都对应一个实数。

Set notation: x ∈ A means 'x is an element of A'; x ∉ A means 'x is not in A'. A ⊂ B means A is a subset of B. A ∪ B is the union, A ∩ B is the intersection. ∅ is the empty set.

集合符号： x ∈ A 表示 “x 是 A 的元素”；x ∉ A 表示 “x 不在 A 中”。A ⊂ B 表示 A 是 B 的子集。A ∪ B 是并集，A ∩ B 是交集。∅ 表示空集。

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2. Algebraic Expressions and Identities | 代数表达式与恒等式

Term, coefficient, variable, constant: In 5x² − 3x + 7, the terms are 5x², −3x, and 7; 5 and −3 are coefficients; x is the variable; 7 is the constant.

项、系数、变量、常数： 在 5x² − 3x + 7 中，项为 5x²、−3x 和 7；5 和 −3 是系数；x 是变量；7 是常数。

Like terms: Terms that contain exactly the same variable(s) raised to the same power. 4x² and −x² are like terms; 4x² and 3x are not.

同类项： 所含变量及对应指数完全相同的项。例如 4x² 和 −x² 是同类项；4x² 和 3x 不是同类项。

Expanding brackets: Using the distributive law to remove parentheses, e.g. 2(x + 3) = 2x + 6.
(x + a)(x + b) = x² + (a + b)x + ab.

展开括号： 使用分配律去掉括号，例如 2(x + 3) = 2x + 6。
(x + a)(x + b) = x² + (a + b)x + ab。

Factorisation: Writing an expression as a product of its factors. Common examples include taking out the highest common factor and recognising the difference of two squares:
a² − b² = (a − b)(a + b).

因式分解： 将表达式写成因式的乘积。常见方法包括提取公因式和识别平方差：
a² − b² = (a − b)(a + b)。

Identity (≡): A statement that is true for all values of the variable, e.g. x² − 1 ≡ (x − 1)(x + 1). The triple bar emphasises the equality holds identically.

恒等式 (≡)： 对变量的所有取值都成立的等式，例如 x² − 1 ≡ (x − 1)(x + 1)。三横线用以强调恒等关系。

Completing the square: Rewriting a quadratic expression in the form a(x − p)² + q. For x² + bx, we add and subtract (b/2)².

配方法： 将二次式改写为 a(x − p)² + q 的形式。对于 x² + bx，我们加上并减去 (b/2)²。

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3. Equations and Inequalities | 方程与不等式

Linear equation: rm y = mx + c, where the highest power of the variable is 1. Its graph is a straight line.

线性方程： 形如 y = mx + c 的方程，其中变量的最高次数为 1。其图像是一条直线。

Quadratic equation: An equation: n of the form ax² + bx + c = 0, where a ≠ 0. Solutions are given by the quadratic formula:
x = [−b ± √b² − 4ac] / (2a).

二次方程： 形如 ax² + bx + c = 0 的方程，a ≠ 0。解由二次公式给出：
x = [−b ± √b² − 4ac] / (2a)。

Discriminant (Δ): c. If Δ > 0, two distinct real roots; Δ = 0, one repeated real root; Δ < 0, no real roots.

判别式 (Δ)： Δ = b² − 4ac。若 Δ > 0，有两个相异实根；Δ = 0，有一个重根；Δ < 0，无实根。

Inequality symbols: r than or equal to). Solving inequalities follows similar steps to equations, but multiply/divide by a negative number reverses the sign.

不等式符号： < (小于), > (大于), ≤ (小于等于), ≥ (大于等于)。解不等式与解方程类似，但乘以或除以负数时要改变方向。

Absolute value: |x| = distance from 0 on the number line. |x| = a gives x = a or x = −a; |x| < k gives −k < x < k.

绝对值： |x| 表示数轴上 x 到原点的距离。|x| = a 的解为 x = a 或 x = −a；|x| < k 的解为 −k < x < k。

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4. Functions and Graphs | 函数与图像

Function f(x): A rule that assigns each input x exactly one output f(x). 'f(x)' is read 'f of x'.

函数 f(x)： 一种规则，使每一个输入 x 都对应唯一的输出 f(x)。f(x) 读作 “f of x”。

Domain and range: The domain is the set of all allowed input values (x). The range is the set of all possible output values (f(x)).

定义域与值域： 定义域是所有允许的输入值 (x) 的集合。值域是所有可能的输出值 (f(x)) 的集合。

One‑one function: A function where each output comes from exactly one input. Passes the horizontal line test. Only one‑one functions have inverses.

一一映射函数： 每个输出都唯一对应一个输入的函数。可通过水平线检验。只有一一映射函数才有反函数。

Inverse function f⁻¹: . If f(a) = b, then f⁻¹(b) = a. The graph of f⁻¹ is the reflection of f in the line y = x.

反函数 f⁻¹： 反转 f 的作用。如果 f(a) = b，则 f⁻¹(b) = a。f⁻¹ 的图像是 f 关于直线 y = x 的对称图形。

Composite function f(g(x)): Apply g first, then apply f to the result. Written as (f ∘ g)(x) = f(g(x)).

复合函数 f(g(x))： 先施加 g，再将 f 作用于结果。记作 (f ∘ g)(x) = f(g(x))。

Transformations of graphs: y = f(x) + k shifts up by k; y = f(x + k) shifts left by k; y = −f(x) reflects in the x‑axis; y = f(−x) reflects in the y‑axis.

图像变换： y = f(x) + k 向上平移 k 个单位；y = f(x + k) 向左平移 k 个单位；y = −f(x) 关于 x 轴对称；y = f(−x) 关于 y 轴对称。

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5. Exponents and Logarithms | 指数与对数

Exponent laws: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, a⁰ = 1 (a ≠ 0), a⁻ⁿ = 1/aⁿ.

指数定律： aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，a⁰ = 1 (a ≠ 0)，a⁻ⁿ = 1/aⁿ。

Rational exponents: a¹/ⁿ = ⁿ√a, and aᵐ/ⁿ = ⁿ√aᵐ = (ⁿ√a)ᵐ. For example, 8²/³ = (³√8)² = 2² = 4.

有理指数： a¹/ⁿ = ⁿ√a，且 aᵐ/ⁿ = ⁿ√aᵐ = (ⁿ√a)ᵐ。例如 8²/³ = (³√8)² = 2² = 4。

Logarithm definition: logₐ y = x ⇔ aˣ = y. This means 'the logarithm to base a of y equals x'.

对数定义： logₐ y = x ⇔ aˣ = y。读作 “以 a 为底 y 的对数等于 x”。

Natural logarithm (ln): ln x = logₑ x, where e ≈ 2.71828. It is the inverse function of eˣ.

自然对数 (ln)： ln x = logₑ x，其中 e ≈ 2.71828。它是 eˣ 的反函数。

Change of base: logₐ b = (logₓ b) / (logₓ a), commonly using base 10 or base e for calculation.

换底公式： logₐ b = (logₓ b) / (logₓ a)，计算时常取以 10 或 e 为底。

Log laws: logₐ (MN) = logₐ M + logₐ N; logₐ (M/N) = logₐ M − logₐ N; logₐ (Mᵏ) = k logₐ M.

对数运算法则： logₐ (MN) = logₐ M + logₐ N；logₐ (M/N) = logₐ M − logₐ N；logₐ (Mᵏ) = k logₐ M。

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6. Trigonometric Ratios and Identities | 三角比与恒等式

Right‑angled triangle ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

直角三角形边比： sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。

Special angles (degrees): Memorise exact values: sin 30° = ½, sin 45° = √2/2, sin 60° = √3/2. cos 30° = √3/2, cos 45° = √2/2, cos 60° = ½. tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.

特殊角（度数）： 牢记精确值：sin 30° = ½，sin 45° = √2/2，sin 60° = √3/2。cos 30° = √3/2，cos 45° = √2/2，cos 60° = ½。tan 30° = 1/√3，tan 45° = 1，tan 60° = √3。

Radian measure: d. 1 rad ≈ 57.3°. Radian is the SI unit for angle; arc length s = rθ when θ is in radians.

弧度制： 180° = π rad。1 rad ≈ 57.3°。弧度是角的国际单位制单位；当 θ 以弧度为单位时，弧长 s = rθ。

Pythagorean identity: is is derived from the unit circle and leads to tan²θ + 1 = sec²θ (where sec θ = 1/cos θ).

毕达哥拉斯恒等式： sin²θ + cos²θ = 1。它来自单位圆的定义，并可衍生出 tan²θ + 1 = sec²θ（其中 sec θ = 1/cos θ）。

Sine and cosine rules: gle, a / sin A = b / sin B = c / sin C (sine rule). Cosine rule: a² = b² + c² − 2bc cos A.

正弦与余弦定理： 对于任意三角形，a / sin A = b / sin B = c / sin C（正弦定理）。余弦定理：a² = b² + c² − 2bc cos A。

Periodic graphs: y = cos x have period 2π, amplitude 1. y = tan x has period π and asymptotes at x = ±π/2.

周期图像： y = sin x 和 y = cos x 的周期为 2π，振幅为 1。y = tan x 的周期为 π，在 x = ±π/2 处有渐近线。

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7. Differentiation: Rates of Change | 微分：变化率

Derivative f'(x) or dy/dx: Measures the instantaneous rate of change of y with respect to x. It is the limit of the difference quotient.

导数 f'(x) 或 dy/dx： 度量 y 关于 x 的瞬时变化率。它是差商的极限。

Power rule: If f(x) = xⁿ, then f'(x) = n xⁿ⁻¹. This holds for any real constant n.

幂法则： 若 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹。对于任意实常数 n 均成立。

Sum/difference and constant multiple rules: d/dx [u ± v] = du/dx ± dv/dx, and d/dx [c u] = c du/dx, where c is constant.

和差与常数倍法则： d/dx [u ± v] = du/dx ± dv/dx，且 d/dx [c u] = c du/dx，其中 c 为常数。

Tangent and normal: e gradient m of the tangent. The normal is perpendicular to the tangent (gradient = −1/m).

切线与法线： 某点的导数给出切线的斜率 m。法线与切线垂直（斜率为 −1/m）。

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