IB Mathematics: Argand Diagrams, Modulus and Conjugate | IB数学：阿尔冈图、模与共轭

📚 IB Mathematics: Argand Diagrams, Modulus and Conjugate | IB数学：阿尔冈图、模与共轭

Complex numbers can be visualised. In the Argand diagram every complex number z = x + yi is represented as a point with coordinates (x, y). The horizontal axis is the real axis and the vertical axis is the imaginary axis. Once this link between algebra and geometry is made, the modulus measures distance and the conjugate produces a mirror image.

复数是可以被“看见”的。在阿尔冈图中，每个复数 z = x + yi 都表示为一个坐标为 (x, y) 的点。横轴称为实轴，纵轴称为虚轴。一旦代数与几何之间建立起这种联系，模就度量距离，共轭则产生镜像。

1. What is an Argand Diagram? | 什么是阿尔冈图？

The Argand diagram is a two-dimensional plane used to display complex numbers. It is often called the complex plane. For a complex number z = x + yi, the real part Re(z) = x is measured along the horizontal axis, while the imaginary part Im(z) = y is measured along the vertical axis.

阿尔冈图是用于展示复数的二维平面，通常也被称为复平面。对于复数 z = x + yi，实部 Re(z) = x 沿水平轴度量，虚部 Im(z) = y 沿垂直轴度量。

• Each complex number corresponds to exactly one point in the plane, and every point represents exactly one complex number.

  每个复数对应平面中唯一一个点，而平面中的每个点也对应唯一一个复数。

• The point (x, y) can also be seen as the endpoint of a position vector from the origin O to that point.

  点 (x, y) 也可以看作从原点 O 指向该点的位置向量的终点。

• The diagram is named after the Swiss mathematician Jean-Robert Argand.

  该图形以瑞士数学家让-罗贝尔·阿尔冈的名字命名。

2. Plotting Complex Numbers | 在阿尔冈图上表示复数

To plot z = 3 + 2i, start at the origin, move 3 units to the right along the real axis, then move 2 units upward along the imaginary axis. The point (3, 2) is the Argand diagram representation of 3 + 2i.

要画出 z = 3 + 2i，从原点出发，沿实轴向右移动 3 个单位，再沿虚轴向上移动 2 个单位。点 (3, 2) 就是 3 + 2i 在阿尔冈图中的表示。

• Negative real parts move left; negative imaginary parts move downward.

  负实部表示向左移动，负虚部表示向下移动。

• For example, -2 + 3i is plotted as (-2, 3), and 4 – i is plotted as (4, -1).

  例如，-2 + 3i 对应点 (-2, 3)，而 4 – i 对应点 (4, -1)。

• Addition of complex numbers follows vector addition: z₁ + z₂ is the diagonal of the parallelogram formed by z₁ and z₂.

  复数加法遵循向量加法：z₁ + z₂ 是由 z₁ 和 z₂ 构成的平行四边形的对角线。

• The difference z₁ – z₂ represents the vector from the point z₂ to the point z₁.

  差 z₁ – z₂ 表示从点 z₂ 指向点 z₁ 的向量。

3. The Modulus | 模

For z = x + yi, the modulus is defined by

|z| = √(x² + y²)

The modulus is the distance from the origin to the point (x, y). Since it is a length, |z| ≥ 0, and |z| = 0 if and only if z = 0.

模定义为 |z| = √(x² + y²)。它表示从原点到点 (x, y) 的距离。由于它是长度，因此 |z| ≥ 0，且 |z| = 0 当且仅当 z = 0。

For example, if z = 3 + 4i, then |z| = √(3² + 4²) = √25 = 5. The point (3, 4) lies exactly 5 units from the origin.

例如，若 z = 3 + 4i，则 |z| = √(3² + 4²) = √25 = 5。点 (3, 4) 到原点的距离正好是 5 个单位。

• |z| = |z̄|, because replacing y by -y does not change x² + y².

  |z| = |z̄|，因为把 y 换成 -y 不会改变 x² + y² 的值。

• The modulus is the r in polar form; if z = r(cos θ + i sin θ), then r = |z|.

  模是极坐标形式中的 r；若 z = r(cos θ + i sin θ)，则 r = |z|。

• The triangle inequality states |z + w| ≤ |z| + |w|.

  三角不等式指出 |z + w| ≤ |z| + |w|。

4. The Complex Conjugate | 共轭复数

The conjugate of z = x + yi is written z̄ and is defined by

z̄ = x – yi

Geometrically, the conjugate z̄ is the reflection of

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