AS Edexcel Further Mathematics: Formula & Theorem Quick Reference Handbook | AS Edexcel 进阶数学：公式定理速查手册

📚 AS Edexcel Further Mathematics: Formula & Theorem Quick Reference Handbook | AS Edexcel 进阶数学：公式定理速查手册

This quick reference handbook covers the essential formulas, theorems, and key results required for the AS Edexcel Further Mathematics course. It is designed to help you quickly recall the core content from Further Pure Mathematics 1 (FP1) and related applied modules, ensuring you have a solid foundation for problem‑solving and examinations.

本速查手册涵盖了 AS Edexcel 进阶数学课程所需的基本公式、定理和关键结论。它旨在帮助你快速回顾 Further Pure Mathematics 1 （FP1） 及相关应用模块的核心内容，为解决问题和应对考试打下坚实基础。

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1. Complex Numbers – Basic Operations | 复数——基本运算

A complex number is written as z = a + bi wo complex numbers are equal if and only if both their real and imaginary parts are equal.

复数写作 z = a + bi，其中 a、b 为实数且 i² = −1。实部为 Re(z) = a，虚部为 Im(z) = b。两个复数相等当且仅当它们的实部和虚部分别相等。

Addition and subtraction are performed component‑wise:
(a + bi) ± (c + di) = (a ± c) + (b ± d)i

加减法按分量进行：
(a + bi) ± (c + di) = (a ± c) + (b ± d)i

Multiplication uses i² = −1:
(a + bi)(c + di) = (ac − bd) + (ad + bc)i

乘法利用 i² = −1：
(a + bi)(c + di) = (ac − bd) + (ad + bc)i

Division is achieved by multiplying numerator and denominator by the complex conjugate:
(a + bi)/(c + di) = [(a + bi)(c − di)] / (c² + d²)

除法通过分子分母同乘共轭复数实现：
(a + bi)/(c + di) = [(a + bi)(c − di)] / (c² + d²)

The complex conjugate of z = a + bi is z* = a − bi. Note that z z* = a² + b² is always real and non‑negative.

复数 z = a + bi 的共轭复数为 z* = a − bi。注意 z z* = a² + b² 总是非负实数。

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2. Complex Numbers – Argand Diagram and Modulus‑Argument Form | 复数——Argand 图与模‑辐角形式

yi is represented by the point (x, y) or by a vector from the origin. The modulus of z is |z| = √x² + y² and measures the distance from the origin. The argument of z, denoted arg(z), is the angle θ measured from the positive real axis, usually taken in the range (−π, π] or [0, 2π).

在 Argand 图上，复数 z = x + yi 用点 (x, y) 或从原点出发的向量表示。z 的模为 |z| = √x² + y²，表示到原点的距离。幅角 arg(z) 是从正实轴测量的角度 θ，通常取范围为 (−π, π] 或 [0, 2π)。

The modulus‑argument form is z = r(cos θ + i sin θ), where r = |z| and θ = arg(z). To convert from Cartesian form, use r = √x² + y² and θ = arctan(y/x), adjusting for the correct quadrant.

模‑辐角形式为 z = r(cos θ + i sin θ)，其中 r = |z|，θ = arg(z)。从直角坐标形式转换时，使用 r = √x² + y² 以及 θ = arctan(y/x)，并根据象限调整。

Multiplication and division in modulus‑ m are straightforward. If z₁ = r₁(cos θ₁ + i sin θ₁) and z₂ = r₂(cos θ₂ + i sin θ₂), then:
z₁z₂ = r₁r₂[cos(θ₁+θ₂) + i sin(θ₁+θ₂)]
z₁/z₂ = (r₁/r₂)[cos(θ₁−θ₂) + i sin(θ₁−θ₂)]

模‑辐角形式的乘法和除法十分简洁。若 z₁ = r₁(cos θ₁ + i sin θ₁) 且 z₂ = r₂(cos θ₂ + i sin θ₂)，则有：
z₁z₂ = r₁r₂[cos(θ₁+θ₂) + i sin(θ₁+θ₂)]
z₁/z₂ = (r₁/r₂)[cos(θ₁−θ₂) + i sin(θ₁−θ₂)]

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3. Roots of Polynomials – Sum and Product Relations | 多项式根——和与积的关系

For a quadratic equation ax² + bx + c = 0 with roots α and β, the sum and product of the roots are given by:
α + β = −b/a
αβ = c/a

对于有根 α 和 β 的二次方程 ax² + bx + c = 0，根的和与积为：
α + β = −b/a
αβ = c/a

ven sums and products, and to evaluate symmetric expressions such as α²+β² = (α+β)² − 2αβ.

这些关系可通过给定的和与积构造新的二次方程，并可计算对称表达式，例如 α²+β² = (α+β)² − 2αβ。

For a cubic equation ax³ + bx² + cx + d = 0 with roots α, β, γ:
∑α = α+β+γ = −b/a
∑αβ = αβ+βγ+γα = c/a
αβγ = −d/a

对于有根 α, β, γ 的三次方程 ax³ + bx² + cx + d = 0：
∑α = α+β+γ = −b/a
∑αβ = αβ+βγ+γα = c/a
αβγ = −d/a

For a quartic ax⁴ + bx³ + cx² + dx + e = 0 with roots α, β, γ, δ, the pattern extends, with alternating signs and denominator a. The sum of roots is −b/a, the sum of products of roots taken two at a time is c/a, taken three at a time is −d/a, and the product of all roots is e/a (or (−1)⁴e/a).

对于有根 α, β, γ, δ 的四次方程 ax⁴ + bx³ + cx² + dx + e = 0，规律延续，符号交替且分母皆为 a。根的和为 −b/a，两根积之和为 c/a，三根积之和为 −d/a，所有根的积为 e/a（即 (−1)⁴e/a）。

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4. Numerical Solution of Equations | 方程的数值解法

To locate a root, a sign change of f(x) over an interval [a, b] indicates at least one root if f is continuous. The Newton‑Raphson method uses the iteration:
xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ)
starting from a suitable initial approximation x₀. This method converges rapidly if x₀ is close enough to the root and f'(x) does not become zero near the root.

定位根时，如果 f 连续，区间 [a, b] 上 f(x) 的符号改变表明至少存在一个根。牛顿‑拉夫逊方法使用迭代：
xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ)
从一个合适的初始近似值 x₀ 开始。若 x₀ 足够接近根且 f'(x) 在根附近不为零，该方法收敛很快。

The iteration xₙ₊₁ = g(xₙ) converges to a fixed point α if |g'(α)| < 1. You can sometimes rearrange f(x)=0 into the form x = g(x) to use this fixed‑point iteration.

迭代 xₙ₊₁ = g(xₙ) 在 |g'(α)| < 1 时收敛到不动点 α。有时可以把方程 f(x)=0 重写成 x = g(x) 以使用不动点迭代。

When applying iterative methods, always check for convergence by testing the derivative at the suspected root or by observing the values approaching a limit.

应用迭代法时，务必通过检验推测根处的导数或观察数值趋近于某一极限来判断收敛性。

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5. Coordinate Systems – Parametric Equations | 坐标系——参数方程

Parametric equations express the coordinates x and y in terms of a third variable, usually t or θ. To find the Cartesian equation, eliminate the parameter. For the parabola y² = 4ax, a standard parametrisation is x = at², y = 2at.

参数方程用第三个变量（通常是 t 或 θ）表示坐标 x 和 y。要得到直角坐标方程，需消去参数。对于抛物线 y² = 4ax，标准参数形式为 x = at², y = 2at。

The gradient of a curve given parametrically is found by dy/dx = (dy/dt) / (dx/dt). The equation of the tangent at a point with parameter t can then be written using the point‑slope form.

参数给定曲线的梯度通过 dy/dx = (dy/dt) / (dx/dt) 求得。切点参数为 t 处的切线方程可用点斜式写出。

For the rectangular hyperbola xy = c², a common parametrisation is x = ct, y = c/t. Again, differentiation is performed parametrically to obtain tangents and normals.

对于等轴双曲线 xy = c²，常用参数形式为 x = ct, y = c/t。同样，用参数微分法可求切线和法线。

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6. Matrix Algebra – Definitions and Operations | 矩阵代数——定义与运算

matrices can be added or subtracted if they have the same order, by adding corresponding elements.

矩阵是一个矩形数字阵列。阶数为 行 × 列。两个矩阵只有在阶数相同时才能进行加减运算，按对应元素相加减。

Multiplication of a matrix by a scalar multiplies every element by that scalar.
k × [a, b; c, d] = [ka, kb; kc, kd]

数与矩阵相乘，用该数乘以矩阵的每一个元素。
k × [a, b; c, d] = [ka, kb; kc, kd]

ry being the sum of the products of corresponding elements from row i of A and column j of B. Note that matrix multiplication is not commutative: generally AB ≠ BA.

矩阵乘法在 A 的列数等于 B 的行数时有定义。若 A 为 m×n，B 为 n×p，则 AB 为 m×p，其 (i, j) 元素为 A 的第 i 行与 B 的第 j 列对应元素乘积之和。注意矩阵乘法不满足交换律，通常 AB ≠ BA。

The identity matrix I has 1s on the main diagonal and 0s elsewhere, satisfying AI = IA = A. A zero matrix contains only zeros.

单位矩阵 I 的主对角线为 1，其余为 0，满足 AI = IA = A。零矩阵的所有元素均为 0。

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7. Matrix Algebra – Determinant and Inverse | 矩阵代数——行列式与逆矩阵

The determinant of a 2×2 matrix
M = [a, b; c, d]
is det(M) = ad − bc.

2×2 矩阵
M = [a, b; c, d]
的行列式为 det(M) = ad − bc。

The inverse of a 2×2 matrix exists if and only if det(M) ≠ 0, and is given by
M⁻¹ = 1/(ad−bc) × [d, −b; −c, a].
The inverse satisfies M M⁻¹ = M⁻¹ M = I.

2×2 矩阵的逆矩阵存在当且仅当 det(M) ≠ 0，并由下式给出：
M⁻¹ = 1/(ad−bc) × [d, −b; −c, a]。
逆矩阵满足 M M⁻¹ = M⁻¹ M = I。

be calculated by expansion along a row or column. The inverse can be found using the adjugate method or by row reduction.

对于 3×3 矩阵，行列式可通过按某一行或某一列展开来计算。逆矩阵可用伴随矩阵法或行变换求得。

Key properties:
det(AB) = det(A)det(B);
det(A⁻¹) = 1/det(A) if A is invertible;
det(kA) = kⁿdet(A) for an n×n matrix A and scalar k.

关键性质：
det(AB) = det(A)det(B)
det(A⁻¹) = 1/det(A)（若 A 可逆）
对于 n×n 矩阵 A 与标量 k，det(kA) = kⁿdet(A)。

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8. Matrix Transformations in the Plane | 平面上的矩阵变换

of the plane can be represented by 2×2 matrices. The image of a point with position vector v is Mv. Common transformation matrices include:

平面的线性变换可用 2×2 矩阵表示。位置向量为 v 的点的像为 Mv。常见的变换矩阵有：

Transformation / 变换	Matrix / 矩阵
Reflection in the x‑axis / 关于 x 轴反射	[1, 0; 0, −1]
Reflection in the y‑axis / 关于 y 轴反射	[−1, 0; 0, 1]
Reflection in the line y = x	[0, 1; 1, 0]
Rotation by angle θ anticlockwise / 逆时针旋转角 θ	[cosθ, −sinθ; sinθ, cosθ]
Enlargement (scale factor k, centre origin) / 以原点为中心放大（缩放因子 k）	[k, 0; 0, k]
Stretch parallel to x‑axis, factor k / 平行于 x 轴拉伸，因子 k	[k, 0; 0, 1]
Stretch parallel to y‑axis, factor k / 平行于 y 轴拉伸，因子 k	[1, 0; 0, k]

A composite transformation corresponds to the product of their matrices, applied from right to left: transformation B followed by A is represented by AB.

复合变换对应于矩阵的乘积，从右向左应用：先 B 后 A 的变换表示为 AB。

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9. Series – Summation of Finite Series | 级数——有限级数的求和

Standard results used in FP1 include:
∑_r=1ⁿ r = n(n+1)/2
∑_r=1ⁿ r² = n(n+1)(2n+1)/6
∑_r=1ⁿ r³ = [n(n+1)/2]²

FP1 中使用的标准结果有：
∑_r=1ⁿ r = n(n+1)/2
∑_r=1ⁿ r² = n(n+1)(2n+1)/6
∑_r=1ⁿ r³ = [n(n+1)/2]²

d), split it into separate sums and use the standard formulae. Constants factor out: ∑ k u_r = k ∑ u_r.

要求 ∑ (ar³ + br² + cr + d) 这类级数的和，将其拆分为单独的和并利用标准公式。常数可以提出来：∑ k u_r = k ∑ u_r。

The method of differences is used when the terms of a series can be expressed as f(r) − f(r+1) or a similar telescoping form. Most terms cancel, leaving a simple expression in n.

差分法用于当级数的项可表示为 f(r) − f(r+1) 或类似的裂项形式时。大部分项会抵消，只留下一个关于 n 的简单表达式。

To find ∑_r=1ⁿ +1))], write the partial fraction 1/(r(r+1)) = 1/r − 1/(r+1), then sum to get 1 − 1/(n+1).

要求 ∑_r=1ⁿ [1/(r(r+1))]，先写成分部分式 1/(r(r+1)) = 1/r − 1/(r+1)，再求和得 1 − 1/(n+1)。

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10. Proof by Induction | 归纳法证明

Mathematical induction is used to prove statements that hold for all positive integers n. The process has four clear steps:
1. Basis step: Verify the statement is true for the smallest value, usually n = 1.
2. Inductive hypothesis: Assume the statement is true for n = k.
3. Inductive step: Show that if it is true for n = k, then it is also true for n = k + 1.
4. Conclusion: By mathematical induction, the statement is true for all positive integers n.

数学归纳法用于证明对所有正整数 n 成立的命题。该过程分为四个清晰的步骤：
1. 奠基步骤： 验证命题对最小的 n 成立，通常取 n = 1。
2. 归纳假设： 假设命题对 n = k 成立。
3. 归纳步骤： 证明若对 n = k 成立，则对 n = k + 1 也成立。
4. 结论： 由数学归纳法，命题对所有正整数 n 成立。

ive step often involves writing f(k+1) as an expression that clearly contains the assumed factor, e.g., f(k+1) = a·f(k) + multiple of the divisor.

FP1 中常见的归纳法主题包括证明求和公式、整除性结果以及矩阵的幂。处理整除问题时，归纳步骤通常将 f(k+1) 写为明显包含假设因子的表达式，例如 f(k+1) = a·f(k) + 除数的倍数。

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11. Complex Polynomials and De Moivre – Extension Topics | 复数多项式与棣莫弗定理——延伸专题

While De Moivre’s theorem is formally an FP2 topic, AS students may encounter simple uses of complex conjugates in polynomial equations. If a polynomial has real coefficients, any complex root z occurs with its conjugate z*. This means that complex roots of real polynomials always appear in conjugate pairs.

虽然棣莫弗定理属于 FP2 的内容，但 AS 学生可能会遇到复共轭在多项式方程中的简单应用。若多项式的系数均为实数，任何复数根 z 都与其共轭 z* 成对出现。这意味着实系数多项式的复数根总是成对出现。

For a quadratic with real coefficients, if one root is p + qi, the other root must be p − qi. The sum of the roots is then 2p and the product is p² + q².

对于实系数二次方程，若一根为 p + qi，则另一根必为 p − qi。此时根的和为 2p，积为 p² + q²。

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12. Useful Algebraic Identities and Techniques | 常用代数恒等式与技巧

In Further Mathematics, fluency with algebraic manipulation is vital. Remember the factorisation of the sum and difference of cubes:
a³ + b³ = (a + b)(a² − ab + b²)
a³ − b³ = (a − b)(a² + ab + b²)

在进阶数学中，熟练的代数运算是至关重要的。请记住立方和与立方差的因式分解：
a³ + b³ = (a + b)(a² − ab + b²)
a³ − b³ = (a − b)(a² + ab + b²)

When dealing with inverse matrices, the cancellation law AB = AC does not O does not force either matrix to be zero; these are important pitfalls in matrix algebra.

处理逆矩阵时，由 AB = AC 不能推出 B = C，除非 A 可逆。类似地，AB = O 并不意味着其中一个矩阵是零矩阵；这些是矩阵代数中常见的重要陷阱。

Always check your solutions for extraneous roots when you square both sides of an equation or manipulate identities. For rational functions, partial fraction decomposition is an invaluable tool before summation.

当对方程两边平方或进行恒等变形时，务必检查是否有增根。对于有理函数，部分分式分解在求和前是非常有用的工具。

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