📚 Cambridge AS Further Maths Terminology Cheat Sheet | 剑桥AS进阶数学术语速记指南
Mastering the technical language of Cambridge AS Further Mathematics is essential for exam success. This guide decodes the key pure-maths vocabulary – from complex numbers to vectors – and pairs each term with its Chinese equivalent, alongside memory-friendly tips. Use it as a daily warm‑up or a last‑minute revision buddy.
掌握剑桥 AS 进阶数学的技术语言是考试成功的基础。本指南解析纯数部分的核心词汇——从复数到向量——并为每个术语配上中文对照和助记提示。你可以把它当作每日热身或考前速查伙伴。
1. Imaginary and Complex Numbers | 虚数与复数
The imaginary unit i is defined by i² = –1, making √–1 = i a valid number outside the real line.
虚数单位 i 定义为 i² = –1，因此 √–1 = i 是一个超脱实数轴的合法数。
A complex number is written as z = a + bi, where a = Re(z) is the real part and b = Im(z) is the imaginary part.
复数记作 z = a + bi，其中 a = Re(z) 是实部，b = Im(z) 是虚部。
When b = 0, the number is purely real; when a = 0, it is purely imaginary (e.g. 5i).
当 b = 0 时，该数为纯实数；当 a = 0 时，为纯虚数（如 5i）。
Equality of complex numbers: a + bi = c + di ⟺ a = c and b = d.
复数相等条件：a + bi = c + di 当且仅当 a = c 且 b = d。
2. Modulus, Argument and Conjugates | 模、辐角与共轭
The modulus |z| of z = a + bi is its distance from the origin on the Argand diagram: |z| = √a² + b².
复数 z = a + bi 的模 |z| 是它在阿根图上到原点的距离：|z| = √a² + b²。
The argument arg(z) = θ is the angle measured from the positive real axis, typically in (–π, π] or [0, 2π).
辐角 arg(z) = θ 是从正实轴量起的角度，通常取区间 (–π, π] 或 [0, 2π)。
The complex conjugate changes the sign of the imaginary part: z* = a – bi (also written as z).
复共轭改变虚部的符号：z* = a – bi（也记作 z）。
Useful facts: z z* = |z|² (always real), and |z*| = |z|. This helps rationalise denominators.
实用性质：z z* = |z|² 恒为实数，且 |z*| = |z|。这常用于分母有理化。
3. Polar Form and Euler’s Formula | 极坐标形式与欧拉公式
Any non‑zero complex number can be expressed in polar form: z = r(cos θ + i sin θ), where r = |z| and θ = arg(z).
任意非零复数可表示为极坐标形式：z = r(cos θ + i sin θ)，其中 r = |z|，θ = arg(z)。
Euler’s relation links exponential and trigonometric functions: eiθ = cos θ + i sin θ.
欧拉公式连接指数函数与三角函数：eiθ = cos θ + i sin θ。
This gives the compact exponential form z = r eiθ, making multiplication and division much simpler.
由此得到紧凑的指数形式 z = r eiθ，极大简化了乘除法运算。
Multiplying in polar form: multiply moduli, add arguments. Dividing: divide moduli, subtract arguments.
极坐标形式的乘法：模相乘，辐角相加。除法：模相除，辐角相减。
4. De Moivre’s Theorem and Roots of Unity | 棣莫弗定理与单位根
De Moivre’s theorem states: (cos θ + i sin θ)n = cos(nθ) + i sin(nθ) for any integer n.
棣莫弗定理指出：对任意整数 n，(cos θ + i sin θ)n = cos(nθ) + i sin(nθ)。
It is a powerful tool for finding powers of complex numbers and for deriving multiple‑angle trig identities.
它是求复数乘方和推导多倍角三角恒等式的有力工具。
The nth roots of unity are the solutions of zn = 1, given by zk = e2kπi/n for k = 0,1,…,n−1.
n 次单位根是方程 zn = 1 的解，由 zk = e2kπi/n 给出，k = 0,1,…,n−1。
They are equally spaced on the unit circle, and their sum is zero. If ω is a primitive root, then 1 + ω + ω² + … + ωn-1 = 0.
它们在单位圆上均匀分布，且总和为零。若 ω 是本原根，则 1 + ω + ω² + … + ωn-1 = 0。
5. Matrices: Order, Arithmetic and Transformations | 矩阵：阶、运算与变换
A matrix of order m × n has m rows and n columns. The element in row i, column j is often denoted aij.
一个 m × n 阶矩阵有 m 行 n 列。第 i 行第 j 列的元素常记作 aij。
Addition and subtraction are performed element‑wise only for matrices of the same order.
加法和减法只能对同阶矩阵逐元素进行。
Multiplication AB is defined only when the number of columns in A equals the number of rows in B. The product is not commutative in general.
矩阵乘法 AB 仅在 A 的列数等于 B 的行数时才有定义。乘法通常不满足交换律。
1s on the leading diagonal and 0s elsewhere; it satisfies AI = IA = A for square matrices.
单位矩阵 I 的主对角线全为 1，其余元素为 0；对方阵满足 AI = IA = A。
gements and shears. A 2×2 transformation matrix maps original coordinates to image points.
矩阵可表示几何变换：旋转、反射、缩放和剪切。2×2 变换矩阵把原坐标映射到像点。
6. Determinant and Inverse of 2×2 Matrices | 2×2矩阵的行列式与逆
The determinant of a 2×2 matrix M = [a b; c d] is det M = ad – bc. It indicates area scale factor and invertibility.
2×2 矩阵 M = [a b; c d] 的行列式为 det M = ad – bc。它表示面积缩放因子和可逆性。
If det M = 0, the matrix is singular (no inverse, transformation collapses the plane).
若 det M = 0，该矩阵为奇异矩阵（无逆矩阵，变换把平面压扁）。
The inverse is given by M⁻¹ = 1/(ad−bc) [d −b; −c a], provided det M ≠ 0.
逆矩阵公式为 M⁻¹ = 1/(ad−bc) [d −b; −c a]，前提是 det M ≠ 0。
To check, multiply: M M⁻¹ = I. Also, det(M⁻¹) = 1/det(M).
验证方法：M M⁻¹ = I。此外，det(M⁻¹) = 1/det(M)。
7. Roots of Polynomial Equations | 多项式方程的根
For a polynomial P(z) with real coefficients, complex roots occur in conjugate pairs. If z₀ is a root, then z₀* is also a root.
对于实系数多项式 P(z)，复根成共轭对出现。若 z₀ 是根，则 z₀* 也是根。
o sums and products of roots. For cubic z³ + pz² + qz + r = 0: Σα = −p, Σαβ = q, αβγ = −r.
韦达定理将系数与根的和、积联系起来。对于三次方程 z³ + pz² + qz + r = 0：Σα = −p，Σαβ = q，αβγ = −r。
is a multiple root. The derivative P′(z) will share that root if the multiplicity is ≥ 2.
若根重复，则为重根。如果重数 ≥ 2，导数 P′(z) 也将共享该根。
Given one complex root, you can often find others by polynomial division and conjugate pairing, without heavy computation.
已知一个复根时，常可利用多项式除法与共轭配对求其余根，避免繁重计算。
8. Summation of Finite Series | 有限级数求和
Standard results for AS Further Maths: Σr=1n r = ½n(n+1), Σr=1n r² = ⅙n(n+1)(2n+1), Σr=1n r³ = ¼n²(n+1)².
AS 进阶数学的标准求和公式：Σr=1n r = ½n(n+1)，Σr=1n r² = ⅙n(n+1)(2n+1)，Σr=1n r³ = ¼n²(n+1)²。
The sigma notation Σ expands a series compactly. The index r runs from the lower to the upper limit.
求和符号 Σ 紧凑地展开级数。指标 r 从下限遍历到上限。
Linear combinations can be handled by decomposing: e.g. Σ(3r² − 2r) = 3Σr² − 2Σr.
线性组合可拆分处理：如 Σ(3r² − 2r) = 3Σr² − 2Σr。
Method of differences is used when terms telescope, leaving only a few at the start and end.
当项可以裂项相消时，使用差分法，最终只留下首尾少量项。
9. Proof by Induction | 数学归纳法证明
Induction has three steps: base case (prove true for n = 1), inductive hypothesis (assume true for n = k), and inductive step (prove for n = k+1).
数学归纳法分三步：基础情形（验证 n = 1 成立），归纳假设（假设 n = k 成立），归纳递推（证明 n = k+1 成立）。
It is essential to write a concluding sentence: " if true for n = k then true for n = k+1, the statement holds for all positive integers n."
必须写出总结句：“由于 n=1 时命题成立，且若 n=k 成立可推出 n=k+1 成立，因此命题对所有正整数 n 均成立。”
Induction can be used for summation formulae, divisibility proofs (e.g. 5n − 1 is divisible by 4), and matrix powers.
归纳法可用于求和公式、整除性证明（如 5n − 1 能被 4 整除）以及矩阵乘方。
10. Vectors in Three Dimensions | 三维向量
A 3D vector is written as xi + yj + zk or as a column vector [x, y, z]ᵀ. Unit vectors i, j, k point along the coordinate axes.
3D 向量写作 xi + yj + zk 或列向量 [x, y, z]ᵀ。单位向量 i, j, k 分别指向坐标轴方向。
The magnitude (length) of a vector a = xi + yj + zk is |a| = √x² + y² + z².
向量 a = xi + yj + zk 的模（长度）为 |a| = √x² + y² + z²。
The scalar (dot) product: a · b = |a||b| cos θ, also computed as a₁b₁ + a₂b₂ + a₃b₃. It gives a scalar.
数量积（点乘）：a · b = |a||b| cos θ，也可通过坐标 a₁b₁ + a₂b₂ + a₃b₃ 计算。结果为标量。
Two non‑zero vectors are perpendicular if and only if a · b = 0. This is the key test for orthogonality.
两个非零向量垂直当且仅当 a · b = 0。这是判断正交性的核心方法。
The vector product a × b yields a vector perpendicular to both a and b. Its magnitude is |a||b| sin θ, equal to the area of the parallelogram spanned by them.
向量积 a × b 得到与 a、b 均垂直的向量。其模为 |a||b| sin θ，等于 a 和 b 张成的平行四边形面积。
11. Key Terminology Summary Table | 关键术语速览表
| English term / symbol | 中文术语 | Quick explanation |
|---|---|---|
| Complex number z = a + bi | 复数 | a: real part; b: imaginary part |
| Modulus |z| | 模 | Distance from origin in Argand diagram |
| Argument arg(z) | 辐角 | Angle with positive real axis |
| Conjugate z* | 共轭复数 | a – bi; reflects in real axis |
| Euler form r eiθ | 欧拉形式 | r = modulus, θ = argument |
| De Moivre (cosθ + i sinθ)n | 棣莫弗定理 | cos nθ + i sin nθ |
| Matrix order m×n | 矩阵阶数 | Rows × columns |
| Determinant det M | 行列式 | ad – bc for 2×2; area scale factor |
| Inverse M⁻¹ | 逆矩阵 | M M⁻¹ = I; exists only if det ≠ 0 |
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