Mathematics: Applications and Interpretation HL – Complex Numbers | 复数知识点精讲

📚 Mathematics: Applications and Interpretation HL – Complex Numbers | 复数知识点精讲

Complex numbers often top the list of most searched topics in IB Math AI HL, yet students frequently mistype ‘complex’ as ‘compress’ due to phonetic similarity. This article provides a thorough revision of complex numbers, covering everything from the imaginary unit to De Moivre’s theorem and polynomial equations, ensuring you master this essential area.

复数常常是IB数学应用与解释HL课程中被搜索最多的主题，然而学生们经常因发音相似而误将“复数”的英文“complex”打成“compress”。本文提供复数知识的全面复习，从虚数单位到棣莫弗定理和多项式方程，帮助你彻底掌握这一重要领域。

1. Introduction to Complex Numbers | 复数引言

The real numbers are insufficient to solve equations such as x² + 1 = 0. This led to the introduction of the imaginary unit i, defined by i² = -1. A complex number z can be expressed in Cartesian form as z = a + b i, where a and b are real numbers; a is called the real part (Re(z)) and b the imaginary part (Im(z)).

实数无法解决诸如x² + 1 = 0的方程。这促使引入了虚数单位 i，定义为 i² = -1。复数 z 可以表示为代数形式 z = a + b i，其中 a 和 b 为实数；a 称为实部 (Re(z))，b 称为虚部 (Im(z))。

2. Cartesian Form and the Complex Plane | 代数形式与复平面

Every complex number a + b i corresponds to a unique point (a, b) on the complex plane, also known as the Argand diagram. The horizontal axis represents the real part, and the vertical axis represents the imaginary part. This geometric representation allows us to visualise addition as vector addition.

每个复数 a + b i 对应复平面（也称阿尔冈图）上的唯一点 (a, b)。横轴表示实部，纵轴表示虚部。这种几何表示使我们能够将复数加法可视化为向量加法。

3. Conjugate and Modulus | 共轭与模

The complex conjugate of z = a + b i is denoted by z̄ = a – b i. It reflects the point across the real axis. The modulus |z| = √(a² + b²) gives the distance from the origin to the point. Important properties include z·z̄ = |z|², and |z₁z₂| = |z₁||z₂|.

复数 z = a + b i 的共轭记为 z̄ = a – b i，它是点关于实轴的镜像。模 |z| = √(a² + b²) 表示该点到原点的距离。重要性质包括 z·z̄ = |z|² 以及 |z₁z₂| = |z₁||z₂|。

4. Argument and Polar Form | 辐角与极坐标式

The argument of a non-zero complex number z, denoted arg(z), is the angle θ measured from the positive real axis to the line segment joining the origin to z. It is usually taken in the interval (-π, π] or [0, 2π). Using r = |z| and θ = arg(z), the polar form is z = r (cos θ + i sin θ), also written as r cis θ.

非零复数 z 的辐角，记作 arg(z)，是从正实轴到原点与 z 连线的夹角 θ。通常取区间 (-π, π] 或 [0, 2π)。利用 r = |z| 和 θ = arg(z)，极坐标式为 z = r (cos θ + i sin θ)，也常写作 r cis θ。

5. Multiplication & Division in Polar Form | 极坐标式的乘除

Polar form greatly simplifies multiplication and division. For z₁ = r₁ cis θ₁ and z₂ = r₂ cis θ₂:

z₁z₂ = r₁r₂ cis (θ₁ + θ₂)

z₁ / z₂ = (r₁/r₂) cis (θ₁ – θ₂), r₂ ≠ 0

Thus, multiplication scales the moduli and adds the arguments, while division scales and subtracts arguments.

极坐标式极大简化了乘法和除法。对于 z₁ = r₁ cis θ₁ 和 z₂ = r₂ cis θ₂：
乘法为 z₁z₂ = r₁r₂ cis (θ₁ + θ₂)，除法为 z₁ / z₂ = (r₁/r₂) cis (θ₁ – θ₂)（r₂ ≠ 0）。因此，乘法将模相乘并将辐角相加，而除法则将模相除并将辐角相减。

6. Euler’s Formula and Exponential Form | 欧拉公式与指数形式

Euler’s formula establishes a profound link between exponential and trigonometric functions: e^(iθ) = cos θ + i sin θ. This leads to the exponential form z = r e^(iθ), which is compact and widely used in

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）