📚 Complex Numbers and Functions | 复数与复变函数考点精讲
This revision guide covers the core ideas of complex numbers and simple complex functions required for the IGCSE Edexcel Further Pure Mathematics specification. You will learn how to work with the imaginary unit i, perform arithmetic operations, convert between Cartesian and polar forms, use De Moivre’s theorem, find roots, and explore elementary functions of a complex variable. Every section pairs an English explanation with its Chinese translation to strengthen your understanding and exam readiness.
本复习指南涵盖IGCSE Edexcel Further Pure数学大纲中复数和简单复变函数的核心内容。你将学习虚数单位i的运用、复数算术运算、直角坐标与极坐标形式的转换、德莫弗定理、求复数的根，以及探究基本的复变函数。每一节都把英文解释与中文翻译配对，以加深理解并提高应试能力。
1. Introduction to Complex Numbers | 复数概念
A complex number has the form z = a + bi, where a and b are real numbers and i is the imaginary unit defined by i² = –1. The real part is Re(z) = a, and the imaginary part is Im(z) = b. If b = 0, z is purely real; if a = 0 and b ≠ 0, z is purely imaginary.
复数具有形式 z = a + bi，其中a和b是实数，i是虚数单位，满足 i² = –1。实部 Re(z) = a，虚部 Im(z) = b。若b=0，z为纯实数；若a=0且b≠0，z为纯虚数。
A complex number can be visualised as an ordered pair (a, b) on the Argand plane. This representation helps us interpret addition and multiplication geometrically.
复数可在阿尔冈平面上表示为有序对 (a, b)。这种表示法有助于从几何上理解加法和乘法。
2. Operations with Complex Numbers | 复数运算
Addition and subtraction are performed by combining real parts and imaginary parts separately. For z₁ = a + bi and z₂ = c + di:
加法和减法分别对实部和虚部进行。对于 z₁ = a + bi 和 z₂ = c + di：
z₁ + z₂ = (a + c) + (b + d)i
z₁ – z₂ = (a – c) + (b – d)i
Multiplication uses the distributive law and the fact that i² = –1. The product is (a + bi)(c + di) = ac + adi + bci + bdi² = (ac – bd) + (ad + bc)i.
乘法使用分配律并利用 i² = –1。乘积为 (a + bi)(c + di) = ac + adi + bci + bdi² = (ac – bd) + (ad + bc)i。
Division is carried out by multiplying numerator and denominator by the complex conjugate of the denominator. For (a + bi)/(c + di), multiply by (c – di)/(c – di), giving ((a + bi)(c – di))/(c² + d²). Simplify the numerator to obtain the result in standard form.
除法通过将分子分母同乘分母的共轭复数来实现。对于 (a + bi)/(c + di)，乘以 (c – di)/(c – di)，得到 ((a + bi)(c – di))/(c² + d²)。化简分子即可得到标准形式。
3. Complex Conjugate | 共轭复数
The complex conjugate of z = a + bi is denoted by z* or z̄ and is defined as z̄ = a – bi. Geometrically, conjugation reflects a point across the real axis on the Argand diagram.
复数 z = a + bi 的共轭记为 z* 或 z̄，定义为 z̄ = a – bi。几何上，取共轭将阿尔冈图上的点关于实轴反射。
Key properties: z + z̄ = 2a (a real number), z – z̄ = 2bi (a purely imaginary number), and z z̄ = a² + b² = |z|². Also, the conjugate of a sum is the sum of the conjugates, and the conjugate of a product is the product of the conjugates.
重要性质：z + z̄ = 2a（实数），z – z̄ = 2bi（纯虚数），并且 z z̄ = a² + b² = |z|²。此外，和的共轭等于共轭的和，积的共轭等于共轭的积。
4. Modulus and Argument | 模与辐角
The modulus of a complex number z = a + bi is its distance from the origin on the Argand diagram, denoted |z| = √(a² + b²). It is always a non‑negative real number.
复数 z = a + bi 的模是它在阿尔冈平面上到原点的距离，记为 |z| = √(a² + b²)。模永远是一个非负实数。
The argument of z, arg(z), is the angle θ measured from the positive real axis to the line segment joining the origin to the point, typically given in radians. It satisfies tan θ = b/a, taking care to use the correct quadrant. The principal argument usually lies in (–π, π] or [0, 2π).
复数 z 的辐角 arg(z) 是从正实轴到连接原点与点的线段所形成的角 θ，通常以弧度表示。它满足 tan θ = b/a，并需注意确定正确的象限。主辐角通常位于 (–π, π] 或 [0, 2π)。
5. Argand Diagram | 阿尔冈图
The Argand diagram displays complex numbers as points in a plane, with the real part on the horizontal axis and the imaginary part on the vertical axis. Every point (a, b) represents z = a + bi.
阿尔冈图将复数显示为平面上的点，水平轴表示实部，垂直轴表示虚部。每个点 (a, b) 代表 z = a + bi。
Addition of complex numbers corresponds to vector addition: the parallelogram rule. Multiplication by a real number scales the vector; multiplication by i rotates the vector by 90° anticlockwise.
复数的加法对应向量加法，即平行四边形法则。乘以一个实数会缩放向量；乘以 i 则将向量逆时针旋转90度。
6. Polar Form of Complex Numbers | 极坐标形式
Using the modulus r = |z| and argument θ = arg(z), we can write z in polar (or trigonometric) form: z = r(cos θ + i sin θ). This form is particularly useful for multiplication, division, and exponentiation.
利用模 r = |z| 和辐角 θ = arg(z)，我们可以将 z 写成极坐标（或三角）形式：z = r(cos θ + i sin θ)。这种形式在乘法、除法和乘方运算中尤为有用。
The conversion from Cartesian a + bi to polar form requires r = √(a² + b²) and θ = arctan(b/a) adjusted for quadrant. Conversely, a = r cos θ and b = r sin θ.
从直角坐标 a + bi 转换为极坐标形式需要 r = √(a² + b²) 以及经过象限调整后的 θ = arctan(b/a)。反之，a = r cos θ，b = r sin θ。
7. Multiplication and Division in Polar Form | 极坐标下的乘除法
If z₁ = r₁(cos θ₁ + i sin θ₁) and z₂ = r₂(cos θ₂ + i sin θ₂), then multiplication gives z₁z₂ = r₁r₂[cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)]. The moduli multiply and the arguments add.
若 z₁ = r₁(cos θ₁ + i sin θ₁) 且 z₂ = r₂(cos θ₂ + i sin θ₂)，则乘积为 z₁z₂ = r₁r₂[cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)]。模相乘，辐角相加。
Division yields z₁/z₂ = (r₁/r₂)[cos(θ₁ – θ₂) + i sin(θ₁ – θ₂)], provided z₂ ≠ 0. The moduli divide and the arguments subtract.
除法得出 z₁/z₂ = (r₁/r₂)[cos(θ₁ – θ₂) + i sin(θ₁ – θ₂)]，前提是 z₂ ≠ 0。模相除，辐角相减。
8. De Moivre’s Theorem | 德莫弗定理
De Moivre’s theorem extends the polar multiplication rule to powers: for any integer n, [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ). The modulus is raised to the nth power and the argument is multiplied by n.
德莫弗定理将极坐标乘法法则推广到乘方：对任意整数 n，[r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ)。模的 n 次幂，辐角乘以 n。
The theorem can be proved by induction and is a cornerstone for finding powers and roots of complex numbers. It also links to Euler’s formula e^(iθ) = cos θ + i sin θ, though Euler’s form is not essential at IGCSE level.
该定理可以通过数学归纳法证明，是求复数乘方和根的基础。它也关联到欧拉公式 e^(iθ) = cos θ + i sin θ，不过欧拉形式在IGCSE层面不作要求。
9. Roots of Complex Numbers | 复数的根
To find the nth roots of a complex number w, we write w in polar form w = R(cos φ + i sin φ). The nth roots are given by zₖ = R^(1/n) [cos((φ + 2kπ)/n) + i sin((φ + 2kπ)/n)], for k = 0, 1, 2, …, n – 1.
为了求复数 w 的 n 次根，先将 w 写成极坐标形式 w = R(cos φ + i sin φ)。n 次根由 zₖ = R^(1/n) [cos((φ + 2kπ)/n) + i sin((φ + 2kπ)/n)] 给出，其中 k = 0, 1, 2, …, n–1。
These roots lie equally spaced on a circle of radius R^(1/n) in the Argand diagram. For example, the cube roots of unity (1) are 1, cos(2π/3) + i sin(2π/3), and cos(4π/3) + i sin(4π/3).
这些根均匀分布在阿尔冈图上半径为 R^(1/n) 的圆上。例如，单位1的立方根为 1、cos(2π/3) + i sin(2π/3) 和 cos(4π/3) + i sin(4π/3)。
10. Complex Roots of Quadratic Equations | 二次方程的复数根
If a quadratic equation ax² + bx + c = 0 with real coefficients has discriminant Δ = b² – 4ac < 0, the roots are complex conjugates. They can be expressed using the quadratic formula: x = (–b ± √(4ac – b²) i)/(2a).
若实系数二次方程 ax² + bx + c = 0 的判别式 Δ = b² – 4ac < 0，则根是一对共轭复数。它们可用求根公式表示为：x = (–b ± √(4ac – b²) i)/(2a)。
For instance, x² + 4 = 0 gives x = ±2i. Always simplify the root to the standard a ± bi form and state the conjugate pair explicitly.
例如，x² + 4 = 0 得到 x = ±2i。务必把根化简为标准的 a ± bi 形式，并明确指出这是一对共轭复数。
11. Simple Complex Functions | 简单的复变函数
A complex function takes a complex number as its input and produces a complex output. Common examples at IGCSE Further Pure include f(z) = z² + c or f(z) = az + b, where coefficients may be complex or real.
复变函数以复数为输入并产生复数输出。IGCSE Further Pure中常见的例子有 f(z) = z² + c 或 f(z) = az + b，系数可以是复数或实数。
Evaluating such functions is straightforward: substitute the complex value and simplify using the standard operations. For example, if f(z) = z² + 1 + i and z = i, then f(i) = i² + 1 + i = (–1) + 1 + i = i. This is a simple mapping on the Argand plane.
求这类函数的值很简单：代入复数值，用标准运算化简。例如，若 f(z) = z² + 1 + i 且 z = i，则 f(i) = i² + 1 + i = (–1) + 1 + i = i。这是阿尔冈平面上的一个简单映射。
While iterations of complex functions generate fractals like the Mandelbrot set, these are beyond the IGCSE syllabus. For your exam, focus on substitution, simplification, and illustrating how a function transforms a given set of points.
虽然复变函数的迭代产生了曼德尔布罗特集等分形，但这些超出了IGCSE考纲。考试时需要关注代入、化简以及说明函数如何变换给定的点集。
12. Exam Tips | 考试技巧
Always check your calculator’s complex number mode to verify arithmetic and to convert between Cartesian and polar forms. Set the angle unit to radians unless the question specifies degrees.
务必检查计算器的复数模式，用来验证算术运算以及直角坐标与极坐标形式的转换。除非题目指定使用度数，否则将角度单位设为弧度。
When finding arguments, sketch the point on a quick Argand diagram to confirm the quadrant. Write the principal argument in (–π, π] for Edexcel papers unless instructed otherwise.
求辐角时，快速画出阿尔冈示意图来确定象限。除非另有说明，Edexcel试卷中主辐角一般写成 (–π, π] 范围。
In question sections about roots, list all n roots explicitly and show them symmetrically placed around the circle. Clearly separate real and imaginary parts in final answers.
在关于根的考题部分，要明确列出所有 n 个根，并展示它们在圆上对称分布。最终答案中要把实部和虚部清晰分开。
Finally, practice converting complex numbers to polar form fluently—many problems become easier once you recognise that multiplication adds angles and roots divide the circle.
最后，要熟练练习将复数转换为极坐标形式——一旦掌握乘法角度相加、求根等分圆周的特性，许多问题便会迎刃而解。
Published by TutorHao | Further Pure Mathematics Revision Series | aleveler.com
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