Year 13 SQA Mathematics: Quick Vocabulary and Terminology Memorisation Guide | SQA 数学：词汇术语速记指南

📚 Year 13 SQA Mathematics: Quick Vocabulary and Terminology Memorisation Guide | SQA 数学：词汇术语速记指南

Building a solid mathematical vocabulary is like learning the grammar of a new language — it gives you the power to read questions with confidence and articulate solutions clearly. In SQA Advanced Higher Mathematics, precise terminology appears in every exam paper, from the phrasing of a proof to the description of a geometric transformation. This guide groups essential terms by topic, presents them in clear English with matched Chinese explanations, and highlights memory aids so you can recognise, recall, and use them correctly under pressure.

建立扎实的数学词汇就像学习一门新语言的语法——它能让你自信地阅读题目并清晰地表达解答。在 SQA 高等数学（Advanced Higher）考试中，精准的术语出现在每一道题里，从证明的措辞到几何变换的描述。本指南按主题分组核心术语，用清晰的英文与中文解释配对呈现，并突出记忆诀窍，帮助你在考场上准确识别、回忆并正确使用它们。

1. Foundation Terms in Algebra and Functions | 代数与函数基础术语

A ‘polynomial’ is an expression of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where n is a non‑negative integer and the coefficients aᵢ are real numbers. The highest power of x is called the ‘degree’ of the polynomial.

“多项式”是形如 aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀ 的表达式，其中 n 为非负整数，系数 aᵢ 为实数。x 的最高次幂称为多项式的“次数”。

The ‘domain’ of a function is the set of all possible input values, while the ‘range’ is the set of all possible output values. For f(x) = √x, the domain is x ≥ 0 because you cannot take the square root of a negative number in the real number system.

函数的“定义域”是所有可能输入值的集合，而“值域”是所有可能输出值的集合。对于 f(x) = √x，定义域为 x ≥ 0，因为在实数系中不能对负数开平方。

A ‘root’ (or ‘zero’) of a function is a value of x that makes f(x) = 0. Graphically, roots correspond to x‑intercepts. The ‘remainder theorem’ states that when a polynomial f(x) is divided by (x − a), the remainder is f(a).

函数的“根”（或“零点”）是使 f(x) = 0 的 x 值。图像上，根对应 x 轴截距。“余式定理”指出，当多项式 f(x) 除以 (x − a) 时，余式为 f(a)。

Composite function (f ∘ g)(x) = f(g(x)) | 复合函数
Inverse function f⁻¹(x) | 反函数
Odd / Even function | 奇函数 / 偶函数
Asymptote | 渐近线

Memory tip: Think of ‘degree’ as the ‘power summit’ of the polynomial. The inverse function ‘undoes’ the original — swapping x and y is the key step.

记忆诀窍：把“次数”想象成多项式的“最高权力”。反函数“撤销”原函数的作用——交换 x 和 y 是关键步骤。

2. Calculus Vocabulary | 微积分词汇

The ‘derivative’ of a function f(x), denoted f'(x) or dy/dx, measures the instantaneous rate of change — it is the slope of the tangent line at a point. The process of finding a derivative is called ‘differentiation’.

函数 f(x) 的“导数”，记作 f'(x) 或 dy/dx，衡量瞬时变化率——即某点处切线的斜率。求导数的过程称为“微分”（differentiation）。

‘Integration’ is the reverse process of differentiation. The ‘indefinite integral’ ∫ f(x) dx gives a family of functions plus an arbitrary constant C. The ‘definite integral’ ∫ₐᵇ f(x) dx calculates the signed area between the curve and the x‑axis from x = a to x = b.

“积分”是微分的逆过程。“不定积分” ∫ f(x) dx 得到一族函数加上任意常数 C。“定积分” ∫ₐᵇ f(x) dx 计算曲线与 x 轴之间从 x = a 到 x = b 的有符号面积。

A ‘stationary point’ occurs where f'(x) = 0. It can be a ‘local maximum’, ‘local minimum’, or a ‘point of inflection’. The ‘second derivative’ f”(x) helps classify stationary points: if f”(x) > 0, the point is a local minimum; if f”(x) < 0, it is a local maximum.

“驻点”发生在 f'(x) = 0 处。它可以是“局部极大值”、“局部极小值”或“拐点”。“二阶导数” f”(x) 有助于对驻点进行分类：若 f”(x) > 0，该点为局部极小值；若 f”(x) < 0，则为局部极大值。

Advanced Higher extends these ideas to ‘parametric differentiation’ (where x and y are both given in terms of a parameter t) and ‘implicit differentiation’ (where the equation is not solved for y).

SQA 高等数学将这些概念扩展到“参数微分”（x 和 y 都用参数 t 表示）和“隐函数微分”（方程未解出 y 的情形）。

English Term	中文术语	Notation
Derivative	导数	dy/dx, f'(x)
Integral	积分	∫ f(x) dx
Limit	极限	limₓ→ₐ f(x)
Chain rule	链式法则	dy/dx = (dy/du)(du/dx)
Product rule	乘法法则	(uv)’ = u’v + uv’

Tip: For the chain rule, imagine peeling an onion — differentiate the outer function, then multiply by the derivative of the inner function.

提示：链式法则好比剥洋葱——先对外层函数求导，再乘以内层函数的导数。

3. Vectors and Matrices | 向量与矩阵

A ‘vector’ is a quantity with both magnitude and direction. In three dimensions it is written as ai + bj + ck or as a column vector. The ‘magnitude’ (or ‘modulus’) of a vector v = ai + bj + ck is √(a² + b² + c²).

“向量”是具有大小和方向的量。在三维空间里，写作 ai + bj + ck 或列向量形式。向量 v = ai + bj + ck 的“模”（大小）为 √(a² + b² + c²)。

The ‘dot product’ (or scalar product) of two vectors a and b is defined as a·b = |a||b|cos θ, where θ is the angle between them. It is used to test orthogonality: if a·b = 0, the vectors are perpendicular.

两个向量 a 和 b 的“点积”（或标量积）定义为 a·b = |a||b|cos θ，其中 θ 是两向量之间的夹角。它用于检验正交性：若 a·b = 0，则两向量垂直。

The ‘cross product’ a × b gives a vector perpendicular to both a and b, with magnitude |a||b|sin θ. It is only defined in three dimensions.

“叉积” a × b 给出一个同时垂直于 a 和 b 的向量，大小为 |a||b|sin θ。它仅在三维空间中定义。

A ‘matrix’ is a rectangular array of numbers. The ‘determinant’ of a 2×2 matrix [a, b; c, d] is ad − bc. For a 3×3 matrix, the determinant relates to volume scaling and invertibility. The ‘inverse’ of a matrix A, written A⁻¹, satisfies A A⁻¹ = I.

“矩阵”是一个数字矩形阵列。2×2 矩阵 [a, b; c, d] 的“行列式”为 ad − bc。对于 3×3 矩阵，行列式与体积缩放和可逆性相关。矩阵 A 的“逆矩阵”记作 A⁻¹，满足 A A⁻¹ = I。

Remember: Cross product → vector output; dot product → scalar output. Determinant = 0 means the matrix is singular (no inverse).

记忆：叉积 → 向量输出；点积 → 标量输出。行列式 = 0 意味着矩阵是奇异的（无逆矩阵）。

4. Sequences and Series | 数列与级数

A ‘sequence’ is an ordered list of numbers u₁, u₂, u₃, … A ‘series’ is the sum of the terms of a sequence. An ‘arithmetic sequence’ has a constant difference d between consecutive terms, while a ‘geometric sequence’ has a constant ratio r.

“数列”是一组有序的数字 u₁, u₂, u₃, …。 “级数”是数列各项之和。“等差数列”相邻项之差为常数 d，“等比数列”相邻项之比为常数 r。

The sum of the first n terms of an arithmetic series is Sₙ = n/2 (2a + (n−1)d), and for a geometric series Sₙ = a(1 − rⁿ)/(1 − r), where a is the first term. If |r| < 1, the infinite geometric series converges to a/(1 − r).

等差数列前 n 项和公式为 Sₙ = n/2 (2a + (n−1)d)，等比级数前 n 项和为 Sₙ = a(1 − rⁿ)/(1 − r)，其中 a 为首项。若 |r| < 1，无穷等比级数收敛于 a/(1 − r)。

‘Maclaurin series’ is a power series expansion of a function about x = 0: f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + … It is a key topic in Advanced Higher, used to approximate functions.

“麦克劳林级数”是函数在 x = 0 处的幂级数展开：f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + …。这是 SQA 高等数学的重点，用于逼近函数。

Convergence | 收敛
Divergence | 发散
Recurrence relation | 递推关系
Binomial theorem | 二项式定理

Visualise Maclaurin as building a function from its derivatives at a single point — the more terms you add, the better the approximation near zero.

把麦克劳林级数想象成从函数在一点的各阶导数“建造”原函数——添加的项越多，零点附近的逼近就越精确。

5. Complex Numbers | 复数

A ‘complex number’ is of the form z = a + bi, where a and b are real numbers and i² = −1. The real part is Re(z) = a, and the imaginary part is Im(z) = b.

“复数”具有 z = a + bi 的形式，其中 a 和 b 为实数，i² = −1。实部 Re(z) = a，虚部 Im(z) = b。

The ‘complex conjugate’ of z = a + bi is z* = a − bi. The modulus |z| = √(a² + b²) represents the distance from the origin in the complex plane. The argument arg(z) is the angle measured from the positive real axis.

z = a + bi 的“共轭复数”为 z* = a − bi。模 |z| = √(a² + b²) 表示复平面上点到原点的距离。辐角 arg(z) 是从正实轴测量的角度。

‘Polar form’ writes z = r(cos θ + i sin θ), where r = |z| and θ = arg(z). Euler’s formula gives the compact exponential form z = reⁱᶿ. This is essential for multiplication, division, and finding powers/roots.

“极坐标形式”写作 z = r(cos θ + i sin θ)，其中 r = |z| 且 θ = arg(z)。欧拉公式给出紧凑的指数形式 z = reⁱᶿ。这在乘除运算和求幂/根时至关重要。

De Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ). It helps compute powers and nth roots of complex numbers.

棣莫弗定理：(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)。它有助于计算复数的幂和 n 次方根。

Key memory: i² = −1, i³ = −i, i⁴ = 1. The complex plane is called the Argand diagram.

关键记忆：i² = −1，i³ = −i，i⁴ = 1。复平面称为阿甘图（Argand diagram）。

6. Differential Equations | 微分方程

A ‘differential equation’ involves an unknown function and its derivatives. A ‘first‑order’ equation contains only the first derivative dy/dx. ‘Separable’ equations can be rewritten so that all y‑terms are on one side with dy and all x‑terms with dx.

“微分方程”包含未知函数及其导数。“一阶”方程仅包含一阶导数 dy/dx。“可分离变量”方程可以改写，使所有含 y 的项与 dy 在一边，所有含 x 的项与 dx 在另一边。

A ‘general solution’ contains an arbitrary constant; a ‘particular solution’ is found by applying an initial condition, e.g. y(0) = 1. SQA problems often involve contexts such as cooling, population growth, or mixing.

“通解”包含任意常数；“特解”通过施加初始条件（例如 y(0) = 1）求得。SQA 题目常涉及冷却、种群增长或混合等背景。

The ‘integrating factor’ method solves linear first‑order equations of the form dy/dx + P(x)y = Q(x). The integrating factor is e^(∫ P(x) dx).

“积分因子”法求解形如 dy/dx + P(x)y = Q(x) 的一阶线性方程。积分因子为 e^(∫ P(x) dx)。

Second‑order homogeneous linear equations with constant coefficients, ay” + by’ + c = 0, are solved using the ‘auxiliary equation’ am² + bm + c = 0. The nature of the roots (real distinct, repeated, or complex conjugate) determines the form of the solution.

常系数二阶齐次线性方程 ay” + by’ + c = 0 用“辅助方程”am² + bm + c = 0 求解。根的特性（相异实根、重根、共轭复根）决定了解的形式。

Think of the auxiliary equation as the ‘DNA’ of the solution — it tells you whether the behaviour is exponential growth, decay, or oscillation.

把辅助方程想象成解的“基因”——它告诉你函数是指数增长、衰减还是振荡。

7. Proof and Logic | 证明与逻辑

Mathematical ‘proof’ is a logical argument that establishes the truth of a statement beyond doubt. Common methods in Advanced Higher include ‘direct proof’, ‘proof by contradiction’, ‘proof by induction’, and ‘proof by contrapositive’.

数学“证明”是一种逻辑论证，无庸置疑地确立命题的真假。SQA 高等数学中常见的方法包括“直接证明”、“反证法”、“数学归纳法”和“逆否命题证明”。

In a proof by contradiction, you assume the negation of what you want to prove and show it leads to an impossible conclusion — a classic example is the proof that √2 is irrational.

在反证法中，你假设要证明的命题的否定成立，然后推导出不可能的结果——经典例子是证明 √2 为无理数。

‘Proof by induction’ has two steps: the base case (usually n = 1) and the inductive step (assume true for n = k, then prove for n = k + 1). This method is heavily tested on summation formulas, divisibility, and inequalities.

“数学归纳法”有两个步骤：基础情况（通常 n = 1）和归纳步骤（假设 n = k 时成立，进而证明 n = k + 1 时成立）。该方法在求和公式、整除性和不等式证明中频繁考查。

The ‘disproof’ of a statement can be achieved by providing a single ‘counter‑example’. Logical connectives like ‘if … then’, ‘if and only if’ (⇔), and ‘implies’ (⇒) must be used with precision.

对一个命题的“否证”可以通过提供一个“反例”来实现。逻辑连接词如“如果……则”、“当且仅当”（⇔）和“蕴含”（⇒）必须精确使用。

Memorise the standard induction script: “Assume true for n = k, then show true for n = k + 1” — write it out in full sentences in the exam.

记住归纳法的标准句式：“假设 n = k 时成立，然后证明 n = k + 1 时成立”——在考试中用完整的句子写出来。

8. Geometry and Trigonometry | 几何与三角学

‘Radians’ are the standard unit of angle measure in Advanced Higher: π radians = 180°. Most calculus formulas for trig functions require angles in radians.

“弧度”是 SQA 高等数学中角度的标准单位：π 弧度 = 180°。涉及三角函数的多数微积分公式都要求角度以弧度为单位。

The ‘compound angle formulas’ are sin(A ± B) = sin A cos B ± cos A sin B, and cos(A ± B) = cos A cos B ∓ sin A sin B. These are essential for differentiating and integrating trig functions.

“复合角公式”为 sin(A ± B) = sin A cos B ± cos A sin B，以及 cos(A ± B) = cos A cos B ∓ sin A sin B。这些对于三角函数的微分和积分至关重要。

The identities sin²θ + cos²θ ≡ 1, 1 + tan²θ ≡ sec²θ, and 1 + cot²θ ≡ cosec²θ appear constantly. The ‘double‑angle’ identities follow directly: sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 2 cos²θ − 1 = 1 − 2 sin²θ.

恒等式 sin²θ + cos²θ ≡ 1、1 + tan²θ ≡ sec²θ 和 1 + cot²θ ≡ cosec²θ 频繁出现。“倍角”恒等式直接可得：sin 2θ = 2 sin θ cos θ；cos 2θ = cos²θ − sin²θ = 2 cos²θ − 1 = 1 − 2 sin²θ。

In ‘vectors in 3D’, the equation of a line can be expressed as r = a + λb, where a is a position vector on the line and b is the direction vector. The angle between a line and a plane is found using the dot product.

在“三维向量”中，直线方程可表示为 r = a + λb，其中 a 为直线上一点的位置向量，b 为方向向量。直线与平面之间的夹角通过点积求得。

Important distinction: cosecant (cosec) = 1/sin, secant (sec) = 1/cos, cotangent (cot) = 1/tan. Many students mix these up.

重要区分：余割 (cosec) = 1/sin，正割 (sec) = 1/cos，余切 (cot) = 1/tan。许多学生容易混淆。

9. Statistics and Probability | 统计与概率

A ‘random variable’ X assigns a numerical value to each outcome of a random experiment. A ‘discrete’ random variable takes specific values, while a ‘continuous’ one takes values in an interval.

“随机变量” X 为随机实验的每个结果赋予一个数值。“离散”随机变量取特定值，而“连续”随机变量在某一区间内取值。

The ‘expected value’ (or mean) E(X) of a discrete random variable is Σ x·P(X = x). The ‘variance’ Var(X) = E(X²) − [E(X)]² measures spread.

离散随机变量的“期望值”（或均值）E(X) = Σ x·P(X = x)。“方差” Var(X) = E(X²) − [E(X)]² 衡量离散程度。

In probability, events A and B are ‘independent’ if P(A ∩ B) = P(A)P(B). They are ‘mutually exclusive’ if they cannot happen at the same time, so P(A ∩ B) = 0.

在概率中，若 P(A ∩ B) = P(A)P(B)，则事件 A 与 B “独立”。若它们不能同时发生，则“互斥”，此时 P(A ∩ B) = 0。

‘Binomial distribution’ B(n, p) models the number of successes in n independent trials each with probability p. The mean is np and variance is np(1 − p). The ‘Poisson distribution’ Po(λ) models rare events occurring at a constant average rate λ.

“二项分布” B(n, p) 描述在 n 次独立试验中每次成功概率为 p 的成功次数，均值为 np，方差为 np(1 − p)。“泊松分布” Po(λ) 对以恒定平均速率 λ 发生的罕见事件建模。

For a continuous random variable, probabilities are given by areas under the ‘probability density function’ (pdf). The total area under the pdf must equal 1.

对于连续随机变量，概率由“概率密度函数” (pdf) 下的面积给出。pdf 下的总面积必须等于 1。

Memory aid: “Binom = Two outcomes (success/failure), Poisson = Rare events. Variance formula Var(X) = E(X²) − μ² is universal.”

记忆帮手：“二项 = 两种结果（成功/失败），泊松 = 稀有事件。方差公式 Var(X) = E(X²) − μ² 普遍适用。”

10. Common Command Words and Exam Phrases | 常见指令词与考试用语

Understanding what the exam question is asking is half the battle. ‘Hence’ means you must use the previous result. ‘Hence or otherwise’ allows a different method, but the previous result often gives the quickest path.

理解考题要求是成功的一半。“Hence（据此）”意味着你必须使用前面的结果。“Hence or otherwise（据此或其他方法）”允许不同方法，但前面的结果通常提供最快捷的路径。

‘Determine’, ‘Find’ and ‘Calculate’ all require a numerical or algebraic answer, while ‘Prove’, ‘Show that’ and ‘Verify’ require a logical argument or demonstration. ‘State’ means you can write the answer without showing working.

“Determine（确定）”“Find（求）”和“Calculate（计算）”都要求给出数值或代数答案，而“Prove（证明）”“Show that（证明）”和“Verify（验证）”要求逻辑论证或演示。“State（陈述）”意味着你可以直接写出答案而无需展示过程。

‘Write down’ implies the answer is short and often comes directly from the formula sheet or a simple observation. ‘Sketch’ means produce a graph showing key features such as intercepts, turning points and asymptotes — precise scales are not always required.

“Write down（写下）”意味着答案简短，通常直接来源于公式表或简单的观察。“Sketch（草图）”意味着画出图像并展示截距、转折点、渐近线等关键特征——不总是要求精确刻度。

‘Evaluate’ means substitute values and simplify to a number. ‘Simplify’ means reduce an expression to its most compact form. ‘Express … in the form …’ means rewrite the expression to match the given structure.

“Evaluate（求值）”意味着代入数值并简化为一个数。“Simplify（化简）”意味着将表达式化为最简形式。“Express … in the form …（将…表示为…形式）”意味着改写表达式以符合给定的结构。

When you see ‘exact value’, never give a decimal approximation — leave your answer in surd form or with π.

当你看到“exact value（精确值）”，绝不要给出小数近似值——答案保留根号形式或含有 π。

Finally, ‘justify’ and ‘explain’ both require a written reason, often a sentence referencing a theorem or property.

最后，“justify（说明理由）”和“explain（解释）”都要求书写理由，通常是一句引用定理或性质的话。

11. Quick Visual Mnemonics | 速记视觉口诀

三角函数的导数口诀：“Sin → Cos，Cos → − Sin”——对 cos 求导时加一个负号。积分口诀：“Sin 积分得 − Cos，Cos 积分得 Sin。”记住 ∫ tan x dx = ln|sec x| + C，而 ∫ sec x dx = ln|sec x + tan x| + C，这是常见的混淆点。

For complex numbers: ‘Modulus first, then argument’ when converting to polar form. Draw the Argand diagram to check the quadrant of the angle.

复数技巧：转换为极坐标形式时“先求模，再求辐角”。画出阿甘图检查角所在的象限。

For vectors: “Dot gives scalar, cross gives vector.” “a × b is perpendicular to both — use the right‑hand rule to check direction.”

向量技巧：“点乘得标量，叉乘得向量。”“a × b 垂直于 a 和 b——用右手定则检验方向。”

For sequences: “Arithmetic = add, Geometric = multiply.” The word ‘ratio’ in geometric signals multiplication.

数列技巧：“等差 = 加，等比 = 乘。”“等比”中的“比”字暗示乘法。

Rehearse these mnemonics aloud while writing out a formula or two every day — the vocabulary will become second nature by exam day.

每天边写一两个公式边大声复述这些口诀——到考试那天，这些术语将成为你的第二本能。

12. Practical Study Strategies for Terminology | 术语实用学习策略

Create bilingual flashcards with the English term on one side and the Chinese term plus a short definition in both languages on the other. Review them using active recall, not just passive reading.

制作双语抽认卡，一面写英文术语，另一面写中文术语及双语简短定义。用主动回忆而非被动阅读来复习它们。

Whenever you solve a past paper question, underline every command word and technical term, then say it aloud in both languages. This trains your brain to process exam language instantly without translation delays.

每当你做一道历年真题时，划出每个指令词和技术术语，然后用两种语言大声读出来。这能训练你的大脑即时处理考试语言，而无需翻译延迟。

Keep a ‘terminology log’ organised by topic, adding new words as you encounter them in class or revision. Write an example sentence for each term — using it in context improves retention far more than memorising a dictionary definition.

保留一本按主题整理的“术语日志”，在课堂或复习中遇到新词时随时添加。为每个术语写一个例句——在语境中使用术语比记忆字典定义更能提高记忆力。

Explaining a concept to a friend in both English and Chinese forces you to truly understand it. Try it on topics like the chain rule, complex conjugates, or proof by induction.

用英文和中文向朋友解释一个概念，这会迫使你真正理解它。不妨在链式法则、共轭复数或数学归纳法这类主题上试试。

Finally, trust that every minute spent internalising mathematical language pays back with faster, more accurate exam responses — because once you understand the question perfectly, the maths itself becomes much clearer.

最后，请相信，花在消化数学语言上的每一分钟都会以更快速、更准确的考试作答作为回报——因为一旦你完全理解了题目，数学本身就会变得清晰许多。

Published by TutorHao | SQA Advanced Higher Mathematics Revision Series | aleveler.com

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Year 13 SQA Mathematics: Quick Vocabulary and Terminology Memorisation Guide | SQA 数学：词汇术语速记指南