📚 Complex Numbers | 复数
Complex numbers form a cornerstone of advanced mathematics, extending the real number system to include solutions to equations like x2 mplex numbers appear in both Analysis & Approaches (AA) and Applications & Interpretation (AI) at Higher Level, equipping students with the tools to explore polynomial roots, trigonometric identities, and geometric transformations in the complex plane.
复数是高等数学的一块基石，它将实数系扩展到包含诸如 x² + 1 = 0 这类方程的解。在 IB 文凭课程中，复数同时出现在分析与方法 (AA) 以及应用与解释 (AI) 的高水平内容中，为学生提供探索多项式根、三角恒等式以及复平面几何变换的工具。
1. Introduction to Imaginary Numbers | 虚数简介
Historically, the need for complex numbers arose from cubic equations, but the simplest motivation is the equation x2 + 1 = 0. No real number satisfies this equation, so we define the imaginary unit i by the property i2 = −1. Pure imaginary numbers are of the form bi where b is a real number.
历史上，对复数的需求起源于三次方程，但最简单的动机是方程 x² + 1 = 0。没有实数满足这个方程，因此我们定义虚数单位 i，其性质为 i² = −1。纯虚数的形式为 bi，其中 b 是实数。
Using the imaginary unit, we can solve any quadratic equation with a negative discriminant. For instance, the roots of x2 ; 3i. This shows that the real numbers are embedded within a larger, algebraically closed system.
利用虚数单位，我们可以解任何判别式为负的二次方程。例如，方程 x² − 4x + 13 = 0 的根可以通过求根公式得到 x = 2 ± 3i。这表明实数嵌入在一个更大的、代数封闭的系统中。
2. Definition and Standard Form | 定义与标准形式
A complex number z is written in standard form as z = a + bi, where a and b are real numbers. Here, a is the real part, Re(z), and b is the imaginary part, Im(z). If b = 0, z is purely real; if a = 0, z is purely imaginary.
复数 z 的标准形式写作 z = a + bi，其中 a 和 b 是实数。这里，a 为实部，记作 Re(z)；b 为虚部，记作 Im(z)。若 b = 0，则 z 为纯实数；若 a = 0，则 z 为纯虚数。
Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. The set of all complex numbers is denoted by ℂ, and it forms a field under the usual addition and multiplication operations, which we will explore next.
两个复数相等当且仅当它们的实部相等且虚部相等。全体复数构成的集合记作 ℂ，它在我们接下来要探讨的通常加法和乘法运算下构成一个域。
3. Addition, Subtraction, and Multiplication | 加减乘运算
mn; c) + (b ± d)i. Multiplication uses the distributive property together with the rule i2 = −1: (a + bi)(c + di) = ac + adi + bci + bdi2 = (ac − bd) + (ad + bc)i.
复数的加减运算按分量进行：(a + bi) ± (c + di) = (a ± c) + (b ± d)i。乘法利用分配律并结合规则 i² = −1：(a + bi)(c + di) = ac + adi + bci + bdi² = (ac − bd) + (ad + bc)i。
These operations inherit commutativity, associativity, and distributivity from real numbers, making complex arithmetic a natural extension. In IB exam questions, you are often asked to simplify expressions like (2 + 3i)(1 − i) or to find real numbers p and q such that (p + qi)2 = 5 + 12i.
这些运算从实数继承了交换律、结合律和分配律，使得复数算术成为自然的扩展。在 IB 考题中，你经常需要化简诸如 (2 + 3i)(1 − i) 的表达式，或者求实数 p 和 q 使得 (p + qi)² = 5 + 12i。
4. Complex Conjugate and Division | 共轭复数与除法
The complex conjugate of z = a + bi is denoted by z̄ and defined as z̄ phically, conjugation reflects a point across the real axis. Key properties include: z + z̄ = 2a (twice the real part), z − z̄ = 2bi (twice the imaginary part times i), and z z̄ = a2 + b2, a non‑negative real number.
复数 z = a + bi 的共轭记作 z̄，定义为 z̄ = a − bi。从图形上看，共轭相当于将点关于实轴反射。重要性质包括：z + z̄ = 2a（实部的两倍），z − z̄ = 2bi（虚部乘 i 的两倍），以及 z z̄ = a² + b²，这是一个非负实数。
Division by a non‑ r by the conjugate of the denominator: (a + bi) / (c + di) = ((a + bi)(c − di)) / (c2 + d2). This process expresses the result in standard form a' + b'i.
除以一个非零复数 c + di 可通过将分子与分母同乘以分母的共轭来完成：(a + bi) / (c + di) = ((a + bi)(c − di)) / (c² + d²)。这一过程将结果表示为标准形式 a' + b'i。
5. Modulus and Argument | 模与辐角
The modulus of a complex number z = a + bi, written |z|, is the distance from the origin in the complex plane (Argand diagram): |z| = √(a2 + b2). It generalises the absolute value of a real number and satisfies |z1z2| = |z1||z2| and |z1/z2| = |z1|/|z2|.
复数 z = a + bi 的模，记作 |z|，是复平面（阿根图）上该点到原点的距离：|z| = √(a² + b²)。它推广了实数的绝对值，且满足 |z₁z₂| = |z₁||z₂| 以及 |z₁/z₂| = |z₁|/|z₂|。
+ bi, then tan θ = b/a, but the quadrant must be determined by the signs of a and b.
复数 z 的辐角，记作 arg(z)，是正实轴与连接原点和 z 的线段之间的夹角 θ，通常以弧度度量。主辐角 Arg(z) 的取值范围为 (−π, π] 或 [0, 2π)。若 z = a + bi，则 tan θ = b/a，但必须由 a 和 b 的符号确定象限。
6. Polar Form | 极坐标形式
Using modulus and argument, any non‑zero complex number can be expressed in polar form: z = r(cos θ + i sin θ), where r = |z| is representation is particularly useful for multiplication, division, and exponentiation.
利用模和辐角，任何非零复数都可以表示为极坐标形式：z = r(cos θ + i sin θ)，其中 r = |z|，θ = arg(z)。这种表示法在乘法、除法和乘方运算中尤为有用。
In polar form, multiplying two complex numbers multiplies their moduli and adds their arguments: z1z2 = r1r2[cos(θ1 + θ2) + i sin(θ1 + θ2)]. Division divides the moduli and subtracts the arguments. This geometric interpretation reveals that multiplication by i corresponds to a rotation of π/2 about the origin.
在极坐标形式下，两个复数相乘等于模相乘、辐角相加：z₁z₂ = r₁r₂[cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)]。除法则是模相除、辐角相减。这一几何解释表明，乘以 i 相当于绕原点旋转 π/2。
7. Exponential Form and Euler's Formula | 指数形式与欧拉公式
Euler's formula, eiθ = cos θ + i sin θ, links exponential and trigonometric functions and provides a compact exponential form: z = r eiθ. Here r = |z| ta; = arg(z). This form dramatically simplifies complex multiplication and exponentiation.
欧拉公式 eiθ = cos θ + i sin θ 将指数函数与三角函数联系起来，并给出了紧凑的指数形式：z = r eiθ。这里 r = |z|，θ = arg(z)。这种形式极大地简化了复数乘法和乘方运算。
Using exponential form, multiplication becomes z1z2 = r1r2 e^(i(θ1 + θ2)), and division becomes z1/z2 = (r1/r2) e^(i(θ1 − θ2)). The famous identity eiπ ental constants. In IB HL, you will often switch between standard, polar, and exponential forms to suit the operation.
利用指数形式，乘法变为 z₁z₂ = r₁r₂ e^(i(θ₁ + θ₂))，除法变为 z₁/z₂ = (r₁/r₂) e^(i(θ₁ − θ₂))。著名的恒等式 eiπ + 1 = 0 优雅地结合了五个基本常数。在 IB 高水平中，你将经常根据运算需要在标准形式、极坐标形式和指数形式之间切换。
8. De Moivre's Theorem | 棣莫弗定理
De Moivre's theorem states that for any real number θ and integer n, (cos θ + i sin θ)n = cos(nθ) + i sin(nθ). Combined with the binomial theorem, this allows us to express cos(nθ) and sin(nθ) as polynomials in cos θ and sin θ.
棣莫弗定理指出，对于任何实数 θ 和整数 n，有 (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)。结合二项式定理，该定理允许我们将 cos(nθ) 和 sin(nθ) 表示为关于 cos θ 和 sin θ 的多项式。
For example, using n = 3: (cos θ + i sin θ)3 = cos 3θ + i sin 3θ. Expanding the left side by the binomial theorem gives cos3θ + 3i cos2θ sin θ − 3 cos θ sin2θ − i sin3θ. Equating real and imaginary parts yields the triple‑angle formulas: cos 3θ = 4 cos3θ − 3 cos θ, sin 3θ = 3 sin θ − 4 sin3θ.
例如，取 n = 3：(cos θ + i sin θ)³ = cos 3θ + i sin 3θ。用二项式定理展开左边得 heta; + 3i cos²θ sin θ − 3 cos θ sin²θ − i sin³θ。令实部和虚部相等即得到三倍角公式：cos 3θ = 4 cos³θ − 3 cos θ，sin 3θ = 3 sin θ − 4 sin³θ。
9. Roots of Complex Numbers | 复数的根
To solve zn plex number w, we write w in polar form w = R(cos φ + i sin φ) and apply De Moivre's theorem in reverse. The n distinct roots are given by: zk = √[n]{R} [cos(φ/n + 2πk/n) + i sin(φ/n + 2πk/n)], for k = 0, 1, ..., n − 1.
要求解 zⁿ = w，其中 w 是复数，我们将 w 写成极坐标形式 w = R(cos φ + i sin φ) 并逆向应用棣莫弗定理。n 个不同的根由下式给出：zₖ = ⁿ√R [cos(φ/n + 2πk/n) + i sin(φ/n + 2πk/n)]，其中 k = 0, 1, ..., n−1。
Geometrically, these n roots lie on a circle of radius √[n]{R} and are equally spaced by an angle of 2π/n. This explains why the cube roots of unity, solving z3 = 1, are 1, e2πi/3 and e4πi/3, forming an equilateral triangle in the complex plane.
从几何上看，这 n 个根位于半径为 ⁿ√R 的圆上，且彼此间隔相等的角度 2π/n。这就解释了为什么解 z³ = 1 的单位立方根是 1、e2πi/3 和 e4πi/3，它们在复平面上构成一个等边三角形。
In IB examination contexts, you may be asked to find all roots of an equation like z4 quadratic equations with complex coefficients. Always present your answers in standard form a + bi unless polar form is requested.
在 IB 考试情境中，你可能被要求找出诸如 z⁴ = −16 的所有根并作图，或者求解具有复系数的二次方程。除非明确要求极坐标形式，否则始终以标准形式 a + bi 呈现答案。
10. Complex Numbers in Geometry and Applications | 复数在几何中的应用
so translations, rotations, and dilations become simple arithmetic. Multiplication by r eiθ corresponds to a scaling by factor r and a rotation by angle θ about the origin.
复数为平面几何提供了一种强有力的语言。点 (x, y) 对应 z = x + yi，因此平移、旋转和伸缩变为简单的算术运算。乘以 r eiθ 对应于按因子 r 缩放并绕原点旋转角度 θ。
Straight lines and circles can be elegantly described using complex equations. For instance, |z − z0| = R represents a circle of radius R centred at z0, while |z − z1| = |z − z2| describes the perpendicular bisector of the segment joining z1 and z2. Additionally, the fundamental theorem of algebra assures that every non‑constant polynomial with complex coefficients has at least one complex root, underlining the central role of complex numbers in mathematics.
直线和圆可以用复数方程优雅地描述。例如，|z − z₀| = R 表示以 z₀ 为圆心、半径为 R 的圆，而 |z − z₁| = |z − z₂| 描述的是连接 z₁ 和 z₂ 的线段的垂直平分线。此外，代数基本定理保证每个非常数的复系数多项式至少有一个复根，突显了复数在数学中的核心地位。
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