Edexcel Further Maths Core Pure 1: Key Concepts and Exam Techniques | Edexcel 进阶数学核心纯数 1：核心概念与考试技巧

📚 Edexcel Further Maths Core Pure 1: Key Concepts and Exam Techniques | Edexcel 进阶数学核心纯数 1：核心概念与考试技巧

Edexcel Further Maths Core Pure 1 brings together complex numbers, matrices, series, roots of polynomials, induction, vectors and volumes of revolution. This revision guide walks through the essential methods and common exam traps for each topic.

Edexcel 进阶数学核心纯数 1 汇集了复数、矩阵、级数、多项式根、归纳法、向量和旋转体体积等内容。本复习指南逐题讲解各专题的核心方法与常见考试陷阱。

1. Complex Numbers: the Imaginary Unit and Arithmetic | 复数：虚数单位与运算

In Core Pure 1, complex numbers extend the real number system with i² = -1. A complex number z = a + bi has real part a and imaginary part b. Addition, subtraction and multiplication follow normal algebra; division uses the complex conjugate.

在核心纯数 1 中，复数通过 i² = -1 扩展实数系。复数 z = a + bi 的实部是 a，虚部是 b。加减乘遵循普通代数规则，除法需使用共轭复数。

For example, (3 + 2i)(1 – i) = 3 – 3i + 2i – 2i² = 5 – i, since i² = -1. The conjugate z* = a – bi gives z z* = a² + b², a real number. This is used to divide: (a+bi)/(c+di) = [(a+bi)(c-di)]/(c²+d²).

例如，(3 + 2i)(1 – i) = 3 – 3i + 2i – 2i² = 5 – i，因为 i² = -1。共轭复数 z* = a – bi 满足 z z* = a² + b²，是实数。这用于除法：(a+bi)/(c+di) = [(a+bi)(c-di)]/(c²+d²)。

2. The Argand Diagram and Modulus-Argument Form | 阿尔冈图与模-辐角形式

Complex numbers can be plotted on an Argand diagram with x-axis as real part and y-axis as imaginary part. The modulus |z| = √(a² + b²) is distance from origin, and the argument θ is the angle with the positive real axis.

复数可以在阿尔冈图上表示，x 轴是实部，y 轴是虚部。模 |z| = √(a² + b²) 是到原点的距离，辐角 θ 是与正实轴的夹角。

z = r(cos θ + i sin θ), |z₁z₂| = |z₁||z₂|, arg(z₁z₂) = arg z₁ + arg z₂

Modulus-argument form: z = r(cos θ + i sin θ). Multiplying two complex numbers multiplies moduli and adds arguments. Loci such as |z – (2 + i)| = 3 describe circles; |z – a| = |z – b| describes the perpendicular bisector; arg(z – a) = α describes a half-line. Sketching these is a common exam task.

模-辐角形式：z = r(cos θ + i sin θ)。两个复数相乘时，模相乘，辐角相加。轨迹如 |z – (2 + i)| = 3 表示圆；|z – a| = |z – b| 表示垂直平分线；arg(z – a) = α 表示半直线。绘制这些轨迹是常见考试任务。

3. 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 integer n. It is used to find powers of complex numbers and to express sin nθ or cos nθ in terms of powers of sin θ and cos θ.

棣莫弗定理指出 (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ（n 为整数）。它用于求复数的幂，以及将 sin nθ 或 cos nθ 表示为 sin θ 和 cos θ 的幂。

To find the nth roots of a complex number, write z = r(cos θ + i sin θ). The roots are z_k = r^1/n[cos((θ+2πk)/n) + i sin((θ+2πk)/n)] for k = 0,1,…,n-1.

求复数的 n 次根时，设 z = r(cos θ + i sin θ)。根为 z_k = r^1/n[cos

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