Further Maths Core Pure 1 Key Concepts | 进阶数学核心纯数1 知识点精讲

📚 Further Maths Core Pure 1 Key Concepts | 进阶数学核心纯数1 知识点精讲

This comprehensive revision guide covers all the essential topics in the Further Mathematics Core Pure 1 module. From complex numbers and matrices to proof by induction and vectors, each concept is explained with clear definitions, key formulas, and worked examples to help you master the syllabus.

本综合复习指南涵盖了进阶数学核心纯数1模块的所有重要知识点。从复数、矩阵到归纳法证明和向量，每个概念都配有清晰的定义、关键公式和解题示例，助你全面掌握考纲内容。

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1. Complex Numbers – Cartesian Form | 复数的代数形式

A complex number is written in Cartesian form as z = x + iy, where x and y are real numbers and i is the imaginary unit satisfying i² = –1.

复数以代数形式写作 z = x + iy，其中 x 和 y 是实数，i 是虚数单位，满足 i² = –1。

The real part is Re(z) = x and the imaginary part is Im(z) = y. Two complex numbers are equal if and only if their real and imaginary parts are respectively equal.

实部记作 Re(z) = x，虚部记作 Im(z) = y。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: (x₁ + iy₁) ± (x₂ + iy₂) = (x₁ ± x₂) + i(y₁ ± y₂). Multiplication uses distribution together with i² = –1 to combine like terms.

加法和减法按分量进行：(x₁ + iy₁) ± (x₂ + iy₂) = (x₁ ± x₂) + i(y₁ ± y₂)。乘法利用分配律并结合 i² = –1 合并同类项。

The complex conjugate of z is z* = x – iy. Note that z z* = x² + y², a real number. Division is carried out by multiplying numerator and denominator by the conjugate of the denominator.

复数 z 的共轭为 z* = x – iy。注意 z z* = x² + y² 是一个实数。除法通过将分子分母同乘分母的共轭来完成。

z = x + iy, i² = –1, z* = x – iy

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2. Modulus, Argument and Polar Form | 模、辐角与极坐标形式

The modulus of z = x + iy is |z| = √(x² + y²), which represents the distance from the origin in the complex plane.

复数 z = x + iy 的模为 |z| = √(x² + y²)，表示复平面中该点到原点的距离。

The argument of z, denoted arg(z), is the angle θ made with the positive real axis, usually taken in the interval (–π, π]. It satisfies tan θ = y/x, with the quadrant determined by the signs of x and y.

复数 z 的辐角记作 arg(z)，是与正实轴所成的角 θ，通常取区间 (–π, π]。满足 tan θ = y/x，且由 x 和 y 的符号确定象限。

The polar form is z = r(cos θ + i sin θ), where r = |z| and θ = arg(z). Using Euler’s formula it can be written as z = r e^(iθ).

极坐标形式为 z = r(cos θ + i sin θ)，其中 r = |z|，θ = arg(z)。利用欧拉公式可写为 z = r e^(iθ)。

Multiplication and division in polar form become straightforward: |z₁z₂| = |z₁||z₂| and arg(z₁z₂) = arg z₁ + arg z₂; similarly, |z₁/z₂| = |z₁|/|z₂| and arg(z₁/z₂) = arg z₁ – arg z₂.

极坐标形式下乘法和除法变得简便：|z₁z₂| = |z₁||z₂|，arg(z₁z₂) = arg z₁ + arg z₂；类似地，|z₁/z₂| = |z₁|/|z₂|，arg(z₁/z₂) = arg z₁ – arg z₂。

z = r(cos θ + i sin θ) = r e^(iθ)

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3. Solving Equations with Complex Roots | 含复数根的方程求解

For a quadratic equation ax² + bx + c = 0 with real coefficients, if the discriminant b² – 4ac < 0, the roots are a conjugate pair: α and α*.

对于实系数二次方程 ax² + bx + c = 0，若判别式 b² – 4ac < 0，则其根为一对共轭复数 α 和 α*。

Given one complex root, you can reconstruct the quadratic by expanding (z – α)(z – α*). This principle extends to higher-degree polynomials with real coefficients: complex roots always occur in conjugate pairs.

已知一个复数根，可通过展开 (z – α)(z – α*) 重新构造二次式。这一原则可推广至更高次实系数多项式：复数根总是成对出现共轭。

When solving cubic or quartic equations, if one root is complex, the conjugate is also a root. You can then find the remaining real root by comparing coefficients or polynomial division.

求解三次或四次方程时，若有一个复数根，其共轭也必为根。然后可通过比较系数或多多项式除法求出剩余的实根。

z = [ –b ± √(b² – 4ac) ] / 2a

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4. Argand Diagrams and Loci | Argand 图与轨迹

An Argand diagram represents complex numbers as points on a plane with real and imaginary axes. The locus of points satisfying a complex condition can be described geometrically.

Argand 图将复数表示为实轴和虚轴构成的平面上的点。满足某个复数条件的点的轨迹可用几何方式描述。

The equation |z – a| = r describes a circle with centre a and radius r. The inequality |z – a| < r represents the interior of that circle.

方程 |z – a| = r 表示以 a 为圆心、r 为半径的圆。不等式 |z – a| < r 表示该圆的内部区域。

The condition |z – a| = |z – b| gives the perpendicular bisector of the line segment joining points a and b. Meanwhile, arg(z – a) = θ produces a half‑line starting at a, making angle θ with the positive real axis.

条件 |z – a| = |z – b| 给出连接点 a 和点 b 的线段的垂直平分线。而 arg(z – a) = θ 生成起于 a、与正实轴成角 θ 的射线。

Combining loci with inequalities allows you to shade regions such as intersections of a circle and a half-line.

将轨迹与不等式结合，可以标出圆形区域与射线区域的交集等区域。

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5. Matrix Operations and Algebra | 矩阵运算与代数

A matrix is a rectangular array of numbers. Addition and subtraction of matrices of the same dimensions are done element-wise. Scalar multiplication multiplies each entry.

矩阵是一个数字矩形阵列。同阶矩阵的加减法按元素进行，标量乘法则是每个元素乘以该标量。

Matrix multiplication AB is defined when

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