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The Argand Diagram, Modulus, and Conjugate | 阿尔冈图、模与共轭

2026-08-03 02:01 · 本文部分内容由AI辅助创作

📚 The Argand Diagram, Modulus, and Conjugate | 阿尔冈图、模与共轭

The Argand diagram provides a powerful geometric way to visualise complex numbers as points or vectors in a plane. Two fundamental concepts arising from this representation are the modulus (distance from the origin) and the complex conjugate (reflection across the real axis). Mastering these ideas is essential for solving equations, describing loci, and manipulating complex numbers in IB Mathematics. This article offers a thorough revision of the Argand diagram, modulus, and conjugate, together with key properties and worked examples.

阿尔冈图（Argand diagram）提供了一种强有力的几何方法，将复数可视化为平面上的点或向量。从这种表示中衍生出的两个基本概念是模（到原点的距离）和共轭复数（关于实轴的反射）。掌握这些概念对于在IB数学中解方程、描述轨迹以及操作复数至关重要。本文将对阿尔冈图、模和共轭进行全面的复习，并涵盖其关键性质及例题。

1. The Argand Diagram | 阿尔冈图

ts the imaginary part (Im) of a complex number. Every complex number z = x + iy (with x, y ∈ ℝ n to (x, y). This visualisation turns algebraic operations into geometric transformations.

阿尔冈图是一个坐标平面，其中水平轴表示复数的实部（Re），垂直轴表示虚部（Im）。每个复数 z = x + iy（x, y ∈ ℝ）都唯一对应于点 (x, y) 或从原点到 (x, y) 的位置向量。这种可视化将代数运算转化为几何变换。

by the point (3, 2). Moving along the real axis changes the real part, while moving parallel to the imaginary axis changes the imaginary part. The Argand diagram is named after Jean-Robert Argand, who popularised this geometric interpretation.

例如，z = 3 + 2i 由点 (3, 2) 表示。沿实轴移动会改变实部，而平行于虚轴移动则改变虚部。阿尔冈图以推广此几何解释的让-罗贝尔·阿尔冈命名。

2. Definition of Modulus | 模的定义

The modulus (or absolute value) of a complex number z = x + iy, denoted by |z| m the point (x, y) to the origin (0, 0) in the Argand diagram. By the Pythagorean theorem, |z| = √x² + y². The modulus is always a non-negative real number and is zero if and only if z = 0.

复数 z = x + iy 的模（或绝对值），记作 |z|，是阿尔冈图中点 (x, y) 到原点 (0, 0) 的距离。根据勾股定理，|z| = √x² + y²。模总是一个非负实数，当且仅当 z = 0 时模为零。

|z| = √x² + y²

To compute the modulus of 3 − 4i, we substitute x = 3 and y = −4: |3 − 4i| = √(3² + (−4)²) = √9 + 16 = √25 = 5. Geometrically, the point (3, −4) is exactly 5 units away from the origin.

要计算 3 − 4i 的模，代入 x = 3 及 y = −4：|3 − 4i| = √(3² + (−4)²) = √9 + 16 = 5。从几何上看，点 (3, −4) 正好距离原点 5 个单位。

3. Properties of Modulus | 模的性质

The modulus possesses several algebraic properties that are extremely useful in manipulation.

模具有若干在运算中极为有用的代数性质。

sions and solve modulus equations without expanding into x+iy form each time. For example, |(1+i)⁴| = |1+i|⁴ = (√2)⁴ = 4.

这些性质使我们无需每次都展开为 x+iy 的形式便能化简表达式并解模方程。例如，|(1+i)⁴| = |1+i|⁴ = (√2)⁴ = 4。

4. Definition of Complex Conjugate | 共轭复数的定义

tained by changing the sign of the imaginary part. It is commonly denoted by z* or with an overline: z trically, the conjugate corresponds to the reflection of the point z across the real axis.

复数 z = x + iy 的共轭是通过改变虚部的符号得到的，通常记作 z* 或用上划线表示。因此，若 z = x + iy，则其共轭为 z* = x − iy。从几何上看，共轭对应于将点 z 关于实轴反射。

z* = x − iy

= 0), the conjugate is the number itself; for a purely imaginary number (x = 0), z* = −z.

对于 z = 2 + 3i，其共轭为 z* = 2 − 3i，即关于实轴的镜像。对于纯实数（y = 0），共轭即其本身；对于纯虚数（x = 0），z* = −z。

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5. Properties of Complex Conjugate | 共轭的性质

The complex conjugate satisfies a set of identities that simplify work with fractions and proofs.

共轭复数满足一系列恒等式，可简化分式运算和证明过程。

These are easily proved by writing z = x + iy. The product property z z* = |z| ecially important because it expresses the modulus squared in a form free of square roots.

这些性质可通过令 z = x + iy 轻松证明。乘积性质 z z* = |z|² 尤为重要，因为它以不含根号的形式表达了模的平方。

6. Relationship Between Modulus and Conjugate | 模与共轭的关系

One of the most useful identities connecting modulus and conjugate is z z* = |z|². Because |z| is a non‑negative real number, we can express the reciprocal of a non‑zero complex number elegantly:

连接模与共轭最有用的恒等式之一是 z z* = |z|²。由于 |z| 是非负实数，我们可以优雅地表示非零复数的倒数：

1/z = z* / |z|², for z ≠ 0

This formula is the basis for dividing complex numbers: to evaluate (a+ib)/(c+id), multiply numerator and denominator by the conjugate of the denominator, giving ((a+ib)(c−id)) / (c²+d²).

该公式是复数除法的基础：要计算 (a+ib)/(c+id)，将分子和分母同时乘以分母的共轭，得到 ((a+ib)(c−id))/(c²+d²)。

Moreover, |z*| = |z| that a number is real if and only if z = z*, and purely imaginary if and only if z = −z*.

此外，|z*| = |z|，因此共轭不改变点到原点的距离。另一个联系是：一个数为实数当且仅当 z = z*，为纯虚数当且仅当 z = −z*。

7. Distance and Midpoint in the Argand Diagram | 阿尔冈图中的距离与中点

The distance between two points representing complex numbers z₁ and z₂ is given by |z₁ − z₂|. This is a direct consequence of the modulus definition, since the vector from z₂ to z₁ is z₁ − z₂. The midpoint of the segment joining z₁ and z₂ is (z₁ + z₂)/2.

表示复数 z₁ 和 z₂ 的两点之间的距离由 |z₁ − z₂| 给出。这是模定义的直接推论，因为从 z₂ 到 z₁ 的向量为 z₁ − z₂。连接 z₁ 与 z₂ 的线段的中点为 (z₁ + z₂)/2。

Many loci are expressed in terms of distance. The equation |z − z₀| = r describes a circle with centre z₀ and radius r. The equation |z − z₁| = |z − z₂| . Adding an argument condition, such as arg(z − z₀) = θ, defines a ray from z₀ at angle θ.

许多轨迹可用距离来表示。方程 |z − z₀| = r 描述了一个以 z₀ 为圆心、r 为半径的圆。方程 |z − z₁| = |z − z₂| 描述了连接 z₁ 与 z₂ 的线段的垂直平分线。添加辐角条件，如 arg(z − z₀) = θ，则定义了一条从 z₀ 出发、辐角为 θ 的射线。

8. Solving Equations Using Modulus and Conjugate | 利用模与共轭解方程

tuting z = x + iy and equating real and imaginary parts, or exploiting the identity z z* = |z|².

模与共轭的性质使我们能够解决那些通过展开求解会非常繁琐的方程。常用的方法包括代换 z = x + iy 并令实部和虚部分别相等，或者利用恒等式 z z* = |z|²。

Example: Solve |z| − z = 1 + 2i.

例题：解方程 |z| − z = 1 + 2i。

Let z = x + iy. Then |z| = √x² + y². Substituting gives: √x² + y² − (x + iy) = 1 + 2i. Equate real parts: √x² + y² − x = 1. Equate imaginary parts: −y = 2 ⇒ y = −2. Substitute y = −2 into the real equation: √x² + 4 − x = 1 ⇒ √x² + 4 = x + 1. Squaring both sides: x² + 4 = x² + 2x + 1 ⇒ 2x = 3 ⇒ x = 3/2. Check: √( (3/2)² + 4 ) = √9/4+16/4=√25/4=5/2; 5/2 − 3/2 = 1, correct. Thus z = 3/2 − 2i.

设 z = x + iy，则 |z| = √x² + y²。代入得：√x² + y² − (x + iy) = 1 + 2i。实部相等：√x² + y² − x = 1。虚部相等：−y = 2 ⇒ y = −2。将 y = −2 代入实部方程：√x² + 4 − x = 1 ⇒ √x² + 4 = x + 1。两边平方：x² + 4 = x² + 2x + 1 ⇒ 2x = 3 ⇒ x = 3/2。检验成立，因此 z = 3/2 − 2i。

9. Representing Loci and Regions | 轨迹与区域的表示

Inequalities involving modulus describe regions in the Argand diagram. For a fixed complex number z₀ and a real number r > 0:

涉及模的不等式描述了阿尔冈图中的区域。对于固定的复数 z₀ 和正实数 r > 0：