📚 Glossary & Mnemonic Guide for CCEA Year 12 Further Mathematics | CCEA 进阶数学词汇术语速记指南
This article serves as a quick-reference glossary and mnemonic companion for Year 12 students studying CCEA Further Mathematics. It distils essential terminology across Pure Mathematics, Mechanics and Statistics into bilingual pairs, helping you consolidate definitions, recall formulas and avoid common pitfalls.
本文是为学习 CCEA 进阶数学的 12 年级学生准备的双语速记词汇表。它提炼了纯数、力学和统计中的核心术语，通过中英对照的定义帮助大家巩固概念、快速回忆公式并避开常见误区。
1. Proof and Logic | 证明与逻辑
Conjecture – a statement that is believed to be true but has not yet been proved. In mathematical reasoning, a conjecture becomes a theorem once a valid proof is established.
猜想 – 被认为正确但尚未证明的陈述。在数学推理中，一旦提供了有效证明，猜想就变成了定理。
Theorem – a statement that has been proven to be true using logical reasoning and previously established results. Examples include Pythagoras' theorem and the Fundamental Theorem of Calculus.
定理 – 已通过逻辑推理和先前建立的结果证明为真的陈述。例如毕达哥拉斯定理和微积分基本定理。
Implication – a conditional statement of the form 'If P, then Q' hesis and Q is the conclusion. An implication is only false when P is true and Q is false.
蕴涵 – 形如“若 P 则 Q”的条件语句，记作 P → Q。P 是前提，Q 是结论。仅当 P 为真且 Q 为假时蕴涵才为假。
Contrapositive – statement formed by negating and swapping the hypothesis and conclusion of an implication: ¬Q → ¬P. It is logically equivalent to the original implication and is often used in proofs.
逆否命题 – 将一个蕴涵的前提和结论否定并交换位置得到的命题：¬Q → ¬P。它与原蕴涵逻辑等价，常用于证明中。
Proof by contradiction – ion leads to a contradiction (e.g. 1 = 0). Therefore, the original statement must be true.
反证法 – 先假设要证明的命题的否定成立，然后推导出一个矛盾（如 1 = 0），从而说明原命题必定为真。
2. Complex Numbers | 复数
Imaginary unit i – defined such that i2 the fundamental building block of complex numbers and allows solutions to equations like x2 + 1 = 0.
虚数单位 i – 定义为满足 i2 = -1 的数。它是复数的基础，使得 x2 + 1 = 0 这样的方程能够求解。
Complex number z – a and b are real numbers. a is called the real part Re(z) and b the imaginary part Im(z).
复数 z – 形如 z = a + bi 的数，其中 a 和 b 是实数。a 称为实部 Re(z)，b 称为虚部 Im(z)。
Modulus – the distance of a complex number from the origin in the Argand diagram, given by |z| = √a2 + b2. It is always non‑negative.
模 – 复数在阿干特图上到原点的距离，由 |z| = √a2 + b2 给出，总是非负。
Argument – the angle θ (usually in radians) that the line representing z makes with the positive real axis, denoted arg(z). Mnemonic: think of the 'angle of attack' of the vector.
辐角 – 表示复数 z 的向量与正实轴之间的夹角 θ（通常以弧度为单位），记作 arg(z)。记忆法：想象一下向量的“攻击角度”。
Complex conjugate – e conjugate is z* = a - bi. Conjugates are useful for division and have the property zz* = |z|2.
共轭复数 – 对于 z = a + bi，其共轭为 z* = a - bi。共轭在除法中很有用，并且满足 zz* = |z|2。
Polar form and Euler's formula – z = r(cos θ + i sin θ) = reiθ. This links trigonometry and exponentials; de Moivre's theorem states that (cos θ + i sin θ)n = cos nθ + i sin nθ.
极坐标形式与欧拉公式 – z = r(cos θ + i sin θ) = reiθ。它将三角与指数联系起来；棣莫弗定理指出 (cos θ + i sin θ)n = cos nθ + i sin nθ。
3. Matrices & Linear Transformations | 矩阵与线性变换
Matrix – The order of a matrix is m × n, where m is the number of rows and n the number of columns.
矩阵 – 按行和列排列的数字矩形阵列。矩阵的阶为 m × n，其中 m 为行数，n 为列数。
Determinant – for a 2×2 matrix A = [[a, b], [c, d]], the determinant is det(A) = ad - bc. A non‑zero determinant means the matrix is invertible (non‑singular).
行列式 – 对于 2×2 矩阵 A = [[a, b], [c, d]]，行列式为 det(A) = ad - bc。非零行列式表示矩阵可逆（非奇异）。
Inverse matrix – A-1 satisfies AA-1 = A-1A = I. For a 2×2 matrix, A-1 = (1/det(A)) [[d, -b], [-c, a]]. Only non‑singular matrices have an inverse.
逆矩阵 – A-1 满足 AA-1 = A-1A = I。对于 2×2 矩阵，A-1 = (1/det(A)) [[d, -b], [-c, a]]。只有非奇异矩阵才有逆。
Identity matrix – iagonal and 0s elsewhere. It acts like the number 1 in matrix multiplication: AI = IA = A.
单位矩阵 – 主对角线为 1、其余位置为 0 的矩阵 I。它在矩阵乘法中相当于数字 1：AI = IA = A。
Linear transformation – a mapping from vectors to vectors that can be represented by a matrix. Examples include rotation by an angle α: [[cos α, -sin α], [sin α, cos α]].
线性变换 – 由矩阵表示的向量到向量的映射。例如旋转 α 角：[[cos α, -sin α], [sin α, cos α]]。
4. Advanced Algebra & Functions | 进阶代数与函数
Discriminant – for the quadratic ax2 + bx + c = 0, the discriminant is Δ = b2 es the nature of the roots: Δ > 0 (two real), Δ = 0 (one repeated), Δ < 0 (no real roots).
判别式 – 对于二次方程 ax2 + bx + c = 0，判别式为 Δ = b2 - 4ac。它决定根的性质：Δ > 0（两不等实根），Δ = 0（一个重根），Δ < 0（无实根）。
Factor theorem – x - a) is a factor if and only if f(a) = 0. It is a special case of the remainder theorem.
因式定理 – 对于多项式 f(x)，(x - a) 是其因式当且仅当 f(a) = 0。它是余数定理的特例。
Partial fractions – a technique to express a rational function as a sum of simpler fractions. It is essential for integrating complicated rational expressions and for series expansions.
部分分式 – 将有理函数表示为较简单分式之和的技巧。对于复杂有理式的积分和级数展开至关重要。
Domain and range – x), while the range is the set of all possible output values (f(x)). Mnemonic: domain = do‑main (input), range = resultant.
定义域与值域 – 函数的定义域是所有可能输入值 (x) 的集合，值域是所有可能输出值 (f(x)) 的集合。记忆法：domain 联想“进去的门”，range 联想“跑出来的范围”。
Inverse function – f-1 verses the effect of f(x). Graphically it is a reflection of y = f(x) in the line y = x. f-1(x) exists only if f is one‑to‑one.
反函数 – f-1(x) 逆转 f(x) 的效果。图像上是 y = f(x) 关于直线 y = x 的反射。仅当 f 是一对一时反函数才存在。
5. Sequences & Series | 数列与级数
Arithmetic sequence – a list of numbers where the difference d between consecutive terms is constant. The n‑th term is a + (n‑1)d; sum to n terms is Sn = n/2 (2a + (n‑1)d).
等差数列 – 相邻项差 d 为常数的数列。第 n 项为 a + (n‑1)d；前 n 项和为 Sn = n/2 (2a + (n‑1)d)。
Geometric sequence – each term is obtained by multiplying the previous term by a constant ratio r. The n‑th term is arn‑1; sum to n terms (r ≠ 1) is Sn = a(1 - rn)/(1 - r).
等比数列 – 每一项由前一项乘以常数公比 r 得到。第 n 项为 arn‑1；前 n 项和 (r ≠ 1) 为 Sn = a(1 - rn)/(1 - r)。
Sum to infinity – for a geometric series with |r| < 1, S∞ = a/(1 - r). The series converges; otherwise it diverges.
无穷和 – 对于 |r| < 1 的几何级数，S∞ = a/(1 - r)。该级数收敛；否则发散。
Binomial expansion – for a positive integer n, (1 + x)n = 1 + nC1x + nC2x2 + ... + xn. The coefficients are given by Pascal’s triangle or the formula nCr = n!/(r!(n‑r)!).
二项展开 – 对于正整数 n，(1 + x)n = 1 + nC1x + nC2x2 + ... + xn。系数由帕斯卡三角形或公式 nCr = n!/(r!(n‑r)!) 给出。
Sigma notation – compactly represents a sum: Σ (from k=1 to n) ak means a1 + a2 + ... + an. Useful for writing series concisely.
求和符号 – 紧凑表示求和：Σ (k=1 到 n) ak 表示 a1 + a2 + ... + an。用于简洁地书写级数。
6. Calculus Techniques | 微积分技巧
Derivative from first principles – f'(x) = limh→0 [f(x+h) - f(x)]/h. It defines the gradient of the tangent to a curve at a point.
第一原理求导 – f'(x) = limh→0 [f(x+h) - f(x)]/h。它定义了曲线在某点切线的斜率。
Chain rule – if y = g(f(x)), then dy/dx = g'(f(x)) · f'(x). Mnemonic: 'derivative of the outside times derivative of the inside'.
链式法则 – 若 y = g(f(x))，则 dy/dx = g'(f(x)) · f'(x)。口诀：“外函数的导数乘以内函数的导数”。
Product and quotient rules – product: d(uv)/dx = u'v + uv'. Quotient: d(u/v)/dx = (u'v - uv')/v2. The quotient rule can be sung to the tune of 'Hi‑diddle‑dee‑dee'!
乘法法则与除法法则 – 乘法：d(uv)/dx = u'v + uv'。除法：d(u/v)/dx = (u'v - uv')/v2。除法法则可以套用口诀“上导下不导减去下导上不导，分母平方”。
Integration as the reverse of differentiation – the indefinite integral ∫ f(x) dx = F(x) + C, where F'(x) = f(x). The constant C accounts for the lost constant during differentiation.
积分是微分的逆运算 – 不定积分 ∫ f(x) dx = F(x) + C，其中 F'(x) = f(x)。常数 C 补偿了微分时丢失的常数。
Definite integral and the Fundamental Theorem – ∫ba f(x) dx = F(b) - F(a). It gives the net area under the curve between x = a and x = b.
定积分与基本定理 – ∫ba f(x) dx = F(b) - F(a)。它给出了曲线在 x=a 到 x=b 之间的净面积。
Integration by substitution – a technique where you replace awkward parts of the integrand with a new variable u, effectively reversing the chain rule.
换元积分法 – 用新变量 u 替换被积函数中复杂部分的技巧，本质上是链式法则的逆用。
7. Vectors & 3D Geometry | 向量与三维几何
Vector magnitude – for a vector a = xi + yj + zk, the magnitude is |a| = √(x2 + y2 + z2). It represents the length of the vector.
向量的大小（模） – 对于向量 a = xi + yj + zk，其模为 |a| = √(x2 + y2 + z2)，代表向量的长度。
Unit vector – th magnitude 1, often used to indicate direction. The unit vector in the direction of a is â = a/|a|.
单位向量 – 长度为 1 的向量，常用来表示方向。沿 a 方向的单位向量为 â = a/|a|。
Scalar (dot) product – a · b = |a||b| sults in a scalar and is useful for finding angles and testing perpendicularity (a·b = 0).
标量积（点积） – a · b = |a||b| cos θ，θ 为两向量夹角。结果是一个标量，用于求夹角和判断垂直（a·b = 0）。
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