A-Level AQA Further Maths: Formula & Theorem Quick Reference | A-Level AQA 进阶数学：公式定理速查手册

📚 A-Level AQA Further Maths: Formula & Theorem Quick Reference | A-Level AQA 进阶数学：公式定理速查手册

This handy reference collects the most important formulas and theorems for AQA A-Level Further Mathematics. Use it for quick revision before exams. All key results from pure core topics are presented with paired English and Chinese explanations, so you can review efficiently and deepen understanding.

这份速查手册汇总了 AQA A-Level 进阶数学最重要的公式和定理，供考前快速回顾。纯数核心主题的所有关键结论均以英中对照的形式呈现，帮助你高效复习并加深理解。

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1. Complex Numbers | 复数

Complex numbers extend the real numbers by introducing the imaginary unit i. They are fundamental in further pure mathematics, linking algebra with trigonometry and enabling powerful techniques such as de Moivre’s theorem.

复数通过引入虚数单位 i 来扩展实数。它们是进阶纯数的基础，将代数与三角联系起来，并支持 de Moivre 定理等强大方法。

i² = -1

The imaginary unit i is defined by i² = -1. A general complex number is written z = a + bi where a and b are real.

虚数单位 i 满足 i² = -1。复数的一般形式为 z = a + bi，其中 a 与 b 为实数。

z* = a – bi

The complex conjugate is z* = a – bi. It satisfies z z* = a² + b² = |z|² and is useful in division and polynomial equations.

复共轭为 z* = a – bi，满足 z z* = a² + b² = |z|²，在除法和多项式方程中非常有用。

|z| = √(a² + b²)

The modulus of z is |z| = √(a² + b²), representing the distance from the origin in the Argand diagram.

复数 z 的模为 |z| = √(a² + b²)，表示 Argand 图中到原点的距离。

arg(z) = θ

The argument arg(z) is the angle the vector makes with the positive real axis, usually given in the range (-π, π].

辐角 arg(z) 是向量与正实轴的夹角，通常取值范围为 (-π, π]。

eⁱθ = cos θ + i sin θ

Euler’s formula connects the exponential function with trigonometric functions, giving the compact polar form z = r eⁱθ.

欧拉公式将指数函数与三角函数联系起来，得到简洁的极形式 z = r eⁱθ。

(r(cos θ + i sin θ))ⁿ = rⁿ (cos nθ + i sin nθ)

De Moivre’s theorem is used for raising complex numbers to integer powers and for deriving multiple-angle identities.

De Moivre 定理用于将复数提升到整数次幂以及推导倍角恒等式。

z^(1/n) = r^(1/n) [cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)], k = 0,1,…,n-1

The n-th roots of a complex number form a regular n-gon on the Argand diagram with centre at the origin.

复数的 n 次方根在 Argand 图上构成以原点为中心的正 n 边形。

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

Matrices represent linear transformations in 2D and 3D. AQA Further Maths requires fluency with matrix operations, determinants, inverses, eigenvalues and eigenvectors.

矩阵表示二维和三维的线性变换。AQA 进阶数学要求熟练进行矩阵运算、行列式、逆矩阵、特征值与特征向量的计算。

Matrix multiplication: (AB)ᵢⱼ = Σₖ Aᵢₖ Bₖⱼ

The product of two matrices is defined by multiplying rows of the first by columns of the second. Order matters: AB ≠ BA in general.

两个矩阵的乘积由第一个矩阵的行与第二个矩阵的列相乘定义。顺序很重要：一般 AB ≠ BA。

det(A) = ad – bc (for 2×2 matrix [[a,b],[c,d]])

The determinant gives the area scale factor of the transformation. A matrix is singular if its determinant is zero.

行列式给出变换的面积缩放因子。如果行列式为零，则矩阵是奇异的。

A⁻¹ = (1/det(A)) [[d, -b],[-c, a]]

The inverse of a 2×2 matrix reverses the transformation; it exists only when det(A) ≠ 0.

2×2 矩阵的逆可以逆转变换；仅当 det(A) ≠ 0 时存在。

det(A – λI) = 0

The characteristic equation yields eigenvalues λ. For each λ, eigenvectors v satisfy (A – λI)v = 0.

特征方程给出特征值 λ。对于每个 λ，特征向量 v 满足 (A – λI)v = 0。

A = PDP⁻¹

Diagonalisation expresses a matrix as the product of eigenvector matrix P, diagonal matrix D of eigenvalues, and P⁻¹. This simplifies powers, Aⁿ = PDⁿP⁻¹.

对角化将矩阵表示为特征向量矩阵 P、特征值对角矩阵 D 和 P⁻¹ 的乘积，便于求幂 Aⁿ = PDⁿP⁻¹。

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3. Summation of Series | 级数求和

Standard summations of powers of natural numbers are essential for evaluating series and applying the method of differences.

自然数幂的标准求和公式对于求级数以及差分法至关重要。

∑ r (r = 1 to n) = ½ n(n+1)

The sum of the first n natural numbers is n(n+1)/2.

前 n 个自然数的和为 n(n+1)/2。

∑ r² (r = 1 to n) = ⅙ n(n+1)(2n+1)

The sum of squares formula is n(n+1)(2n+1)/6.

平方和公式为 n(n+1)(2n+1)/6。

∑ r³ (r = 1 to n) = ¼ n²(n+1)²

The sum of cubes equals the square of the sum of natural numbers.

立方和等于自然数之和的平方。

∑ (uₙ₊₁ – uₙ) = uₙ₊₁ – u₁

Telescoping sums using the method of differences collapse to a simple expression involving only the first and last terms.

使用差分法的裂项求和可化简为仅涉及首末项的简单表达式。

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4. Roots of Polynomials | 多项式的根

Relationships between roots and coefficients speed up solving polynomial equations without finding individual roots explicitly.

根与系数的关系可以在不显式求出每个根的情况下加速求解多项式方程。

α + β = -b/a, αβ = c/a

For a quadratic ax² + bx + c = 0, the sum and product of roots α, β are given by these formulas.

对于二次方程 ax² + bx + c = 0，根 α, β 的和与积由上述公式给出。

∑α = -b/a, ∑αβ = c/a, αβγ = -d/a

For a cubic ax³ + bx² + cx + d = 0, three symmetric sums link the roots without solving them.

对于三次方程 ax³ + bx² + cx + d = 0，三个对称和将根联系起来而无须求解。

Root transformations: replace x by (y – k) or ky etc.

Use substitution to find new equations whose roots are related to those of the original polynomial.

通过代换找到新方程，其根与原多项式的根相关联。

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5. Hyperbolic Functions | 双曲函数

Hyperbolic functions are analogues of trigonometric functions and appear in integration, differential equations and coordinate geometry.

双曲函数是三角函数的对应物，出现在积分、微分方程和坐标几何中。

sinh x = (eˣ – e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2

The definitions in terms of exponentials make derivatives easy: d/dx(sinh x) = cosh x, d/dx(cosh x) = sinh x.

用指数定义使得求导简单：d/dx(sinh x) = cosh x，d/dx(cosh x) = sinh x。

cosh² x – sinh² x = 1

This fundamental identity mirrors the trigonometric identity but with a minus sign contrast.

这个基本恒等式与三角恒等式类似，但符号不同。

arsinh x = ln(x + √(x²+1)), arcosh x = ln(x + √(x²-1)), artanh x = ½ ln((1+x)/(1-x))Published by TutorHao | A-Level 进阶数学 Revision Series | aleveler.com

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