AS WJEC Further Maths: Essential Formulas & Theorems Quick Reference | AS WJEC 进阶数学：公式定理速查手册

📚 AS WJEC Further Maths: Essential Formulas & Theorems Quick Reference | AS WJEC 进阶数学：公式定理速查手册

This quick reference guide covers the essential formulas and theorems for the AS WJEC Further Mathematics specification, focusing on the core pure topics: complex numbers, matrices, series, proof by induction, roots of polynomials, inequalities, hyperbolic functions, and polar coordinates. Keep it handy for revision and exam practice.

本速查手册涵盖 AS WJEC 进阶数学考试大纲中的核心公式与定理，聚焦纯数学部分：复数、矩阵、级数、数学归纳法、多项式根、不等式、双曲函数和极坐标。方便你随时查阅与备考。

1. Complex Numbers | 复数

A complex number is of the form z = x + iy with x, y ∈ ℝ and i² = −1. The real part is Re(z) = x, the imaginary part is Im(z) = y.

复数形式为 z = x + iy，其中 x, y ∈ ℝ 且 i² = −1。实部为 Re(z) = x，虚部为 Im(z) = y。

plex conjugate is z* = x − iy. It satisfies z + z* = 2x, z − z* = 2iy, and zz* = x² + y² = |z|².

共轭复数为 z* = x − iy。满足 z + z* = 2x、z − z* = 2iy，且 zz* = x² + y² = |z|²。

The modulus of z is |z| = √x² + y² y/x, with the quadrant determined by the signs of x and y. The principal argument is usually in (−π, π].

z 的模为 |z| = √x² + y²。辐角 arg(z) = θ，其中 tan θ = y/x，需根据 x, y 的符号确定象限。主辐角通常取 (−π, π]。

Polar form: z = r(cos θ + i sin θ), where r = |z| ₂ = r₁r₂[cos(θ₁+θ₂) + i sin(θ₁+θ₂)]. Division: z₁/z₂ = (r₁/r₂)[cos(θ₁−θ₂) + i sin(θ₁−θ₂)].

极坐标形式：z = r(cos θ + i sin θ)，其中 r = |z|，θ = arg(z)。乘法：z₁z₂ = r₁r₂[cos(θ₁+θ₂) + i sin(θ₁+θ₂)]。除法：z₁/z₂ = (r₁/r₂)[cos(θ₁−θ₂) + i sin(θ₁−θ₂)]。

De Moivre's theorem: For any integer n, [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ).

棣莫弗定理：对于任意整数 n，[r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ)。

h roots of a complex number w = r(cos θ + i sin θ), solve zⁿ = w. The roots are given by r^1/n i sin((θ + 2kπ)/n)] for k = 0, 1, ..., n−1. They lie symmetrically on a circle of radius r^1/n.

求复数 w = r(cos θ + i sin θ) 的 n 次方根，解 zⁿ = w。根为 r^1/n[cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)]，k = 0, 1, ..., n−1。它们对称分布在半径为 r^1/n 的圆上。

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2. Matrices & Linear Transformations | 矩阵与线性变换

det(M) ≠ 0, then M is invertible and its inverse is M⁻¹ = (1/(ad−bc)) [[d, −b], [−c, a]].

2×2 矩阵 M = [[a, b], [c, d]] 的行列式为 det(M) = ad − bc。若 det(M) ≠ 0，则 M 可逆，逆矩阵为 M⁻¹ = (1/(ad−bc)) [[d, −b], [−c, a]]。

ty matrix I = [[1, 0], [0, 1]] satisfies MI = IM = M. For any invertible matrix, MM⁻¹ = I.

单位矩阵 I = [[1, 0], [0, 1]] 满足 MI = IM = M。对任意可逆矩阵，有 MM⁻¹ = I。

...], ...] (m×n) and B = [[b₁₁, ...], ...] (n×p), the product C = AB is m×p with entries c_ij = Σ_k=1ⁿ a_ik b_kj. Multiplication is not commutative in general.

矩阵乘法：若 A 为 m×n 矩阵，B 为 n×p 矩阵，则乘积 C = AB 为 m×p 矩阵，元素 c_ij = Σ_k=1ⁿ a_ik b_kj。通常矩阵乘法不满足交换律。

Common linear transformation matrices (acting on column vectors [[x], [y]]): Rotation anticlockwise by angle θ: [[cos θ, −sin θ], [sin θ, cos θ]]. Reflection in the line through origin making angle θ with the positive x-axis: [[cos 2θ, sin 2θ], [sin 2θ, −cos 2θ]]. Enlargement scale factor k: [[k, 0], [0, k]]. Shear parallel to x-axis with factor λ: [[1, λ], [0, 1]].

常见线性变换矩阵（作用于列向量 [[x], [y]]）：逆时针旋转 θ 角：[[cos θ, −sin θ], [sin θ, cos θ]]。关于过原点与正 x 轴夹角为 θ 的直线的反射：[[cos 2θ, sin 2θ], [sin 2θ, −cos 2θ]]。比例因子为 k 的缩放：[[k, 0], [0, k]]。平行于 x 轴且剪切因子为 λ 的剪切：[[1, λ], [0, 1]]。

When two transformations are combined, the matrix for 'M₁ followed by M₂' is M₂M₁ (right to left). The determinant gives the area scale factor of the transformation; negative determinant indicates a reflection orientation.

组合两个变换时，“先 M₁ 后 M₂”的矩阵为 M₂M₁（从右到左）。行列式给出变换的面积尺度因子；负行列式表示镜像翻转。

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3. Roots of Polynomials | 多项式根

ax² + bx + c = 0 (a ≠ 0) with roots α and β: Sum of roots α + β = −b/a. Product αβ = c/a.

对于二次方程 ax² + bx + c = 0 (a ≠ 0)，根为 α 和 β：根之和 α + β = −b/a；根之积 αβ = c/a。

0 (a ≠ 0) with roots α, β, γ: Σα = α + β + γ = −b/a. Σαβ = αβ + βγ + γα = c/a. αβγ = −d/a.

对于三次方程 ax³ + bx² + cx + d = 0 (a ≠ 0)，根为 α, β, γ：α + β + γ = −b/a；αβ + βγ + γα = c/a；αβγ = −d/a。

the original ones. Useful expansions: α² + β² = (α+β)² − 2αβ; α³ + β³ = (α+β)³ − 3αβ(α+β).

你可以构造具有变换后的根的新多项式。例如，若新根为 α²，则利用原对称式推导。常用展开：α² + β² = (α+β)² − 2αβ；α³ + β³ = (α+β)³ − 3αβ(α+β)。

: Σα = −b/a, Σαβ = c/a, Σαβγ = −d/a, αβγδ = e/a. Similar patterns hold for higher degrees.

对于四次方程，根为 α, β, γ, δ：Σα = −b/a，Σαβ = c/a，Σαβγ = −d/a，αβγδ = e/a。更高次方程也类似。

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4. Summation of Series | 级数求和

Standard finite sums (from r = 1 to n): Σ r = ½ n(n+1). Σ r² = ⅙ n(n+1)(2n+1). Σ r³ = ¼ n²(n+1)² = [½ n(n+1)]².

标准有限求和公式（r 从 1 到 n）：∑ r = n(n+1)/2；∑ r² = n(n+1)(2n+1)/6；∑ r³ = n²(n+1)²/4 = [n(n+1)/2]²。

Σ r + Σ b. Σ (r² ± r) = Σ r² ± Σ r. These allow you to break down more complicated series.

线性性质：∑(a r + b) = a∑r + ∑b；∑(r² ± r) = ∑r² ± ∑r。利用这些性质可将复杂级数拆解。

For series given by a polynomial in r, express as combination of r³, r², r, constant, then apply standard formulas. Also, use method of differences for telescoping series.

对于多项式形式的级数，将其表示为 r³, r², r 和常数的组合，再套用标准公式。此外，可对裂项级数使用差分法。

When proving summation formulae by induction, the key step is showing S_k+1 = S_k + a_k+1 simplifies to the desired expression for n = k+1.

用归纳法证明求和公式时，关键步骤是证明 S_k+1 = S_k + a_k+1 能化简为 n = k+1 的目标表达式。

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5. Proof by Induction | 数学归纳法

1). Induction hypothesis: assume P(k) is true for some arbitrary k ≥ 1. Induction step: use the hypothesis to prove that P(k+1) is true. Then conclude that P(n) holds for all positive integers n.

数学归纳法原理包含三个主要部分：基础步骤——验证 P(n) 对初始值（通常 n=1）成立；归纳假设——假设 P(k) 对某个 k ≥ 1 成立；归纳步骤——利用假设证明 P(k+1) 成立。然后得出结论：P(n) 对所有正整数 n 均成立。

isible by 4), matrix powers (e.g., Mⁿ formula), and inequalities (e.g., 2ⁿ > n for n ≥ 1).

常见应用包括证明：求和公式（如 Σr²）、整除性命题（如 5ⁿ−1 能被 4 整除）、矩阵幂公式（如 Mⁿ 的表达式）以及不等式（如 n ≥ 1 时 2ⁿ > n）。

For matrix induction, compute M^k+1 = M^k M, substitute the assumption for M^k, and simplify using matrix multiplication to match the predicted form.

对于矩阵归纳，计算 M^k+1 = M^k M，代入对 M^k 的假设，通过矩阵乘法化简以匹配预测形式。

Be careful to state the conclusion clearly: 'Hence, by the principle of mathematical induction, P(n) is true for all n ∈ ℕ.'

务必清晰写出结论：“因此，由数学归纳法原理，P(n) 对所有自然数 n 均成立。”

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6. Inequalities | 不等式

Basic rules: if a > b and c > 0 then ac > bc; if a > b and c < 0 then ac < bc. Squaring both sides is safe only when both sides are non-negative.

基本规则：若 a > b 且 c > 0，则 ac > bc；若 a > b 且 c < 0，则 ac < bc。两边平方仅当两边均非负时才成立。

To solve quadratic inequalities, find the roots of the corresponding equation and sketch a sign diagram or graph. For ax² + bx + c > 0, the solution is outside the roots if a > 0.

解二次不等式时，先求对应方程的根，然后画出符号图或图像。对于 ax² + bx + c > 0，若 a > 0，解在两根之外。

The AM–GM inequality for two non-negative numbers a, b states: (a + b)/2 ≥ √ab d only if a = b. This can be used to prove further inequalities, often by rewriting terms.

两个非负数的算术-几何平均不等式：(a + b)/2 ≥ √ab，当且仅当 a = b 时取等。可用于证明更多不等式，常通过重写项来实现。

A useful starting point is the fact that (x − y)² ≥ 0 ⇒ x² + y² ≥ 2xy, and similarly (x + y)² ≥ 0 ⇒ x² + y² ≥ −2xy.

常用的出发点是 (x − y)² ≥ 0 ⇒ x² + y² ≥ 2xy，类似地 (x + y)² ≥ 0 ⇒ x² + y² ≥ −2xy。

When proving inequalities, often work backwards from the desired result to a known true statement, but present the final proof in forward logical order.

证明不等式时，常从目标结论倒推至已知真命题，但最终证明应按前向逻辑顺序呈现。

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7. Hyperbolic Functions | 双曲函数

Definitions: sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, tanh x = sinh x / cosh x = (e²ˣ − 1)/(e²ˣ + 1). Their reciprocals are sech x = 1/cosh x, cosech x = 1/sinh x, coth x = 1/tanh x.

定义：sinh x = (eˣ − e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2，tanh x = sinh x / cosh x = (e²ˣ − 1)/(e²ˣ + 1)。它们的倒数为 sech x = 1/cosh x，cosech x = 1/sinh x，coth x = 1/tanh x。

a product of two sinh terms (or implicit sinh²). For example, cos²θ + sin²θ = 1 becomes cosh²x − sinh²x = 1.

奥斯本法则：将三角恒等式中 sin 换为 i sinh，cos 换为 cosh，并改变含有

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