Working with 9i: Complex Numbers Prelim Answers | 9i 的复数运算：初试答案精讲

📚 Working with 9i: Complex Numbers Prelim Answers | 9i 的复数运算：初试答案精讲

In A-level Mathematics, a simple expression such as 9i can appear in many different types of exam question. Understanding how to handle pure imaginary numbers is essential for complex number topics, including modulus, argument, conjugates, powers, reciprocals and equation solving. This article walks through the key skills using 9i as a central example and gives clear preliminary answers.

在 A-level 数学中，像 9i 这样简单的表达式可以出现在许多不同类型的考试题中。掌握如何处理纯虚数对于复数主题至关重要，包括模、辐角、共轭、幂、倒数和方程求解。本文以 9i 为核心例子，讲解关键技能并给出清晰的初步答案。

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1. Understanding 9i as a Complex Number | 理解 9i 作为复数

A complex number is written in the form z = a + bi, where a is the real part and b is the imaginary part. For z = 9i, the real part is 0 and the imaginary part is 9. Therefore 9i is a pure imaginary number.

复数写作 z = a + bi，其中 a 是实部，b 是虚部。对于 z = 9i，实部为 0，虚部为 9。因此 9i 是一个纯虚数。

z = 0 + 9i, Re(z) = 0, Im(z) = 9

This means that on the complex plane, the point representing 9i lies on the imaginary axis, not on the real axis.

这意味着在复平面上，表示 9i 的点位于虚轴上，而不是实轴上。

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2. Plotting 9i on an Argand Diagram | 在 Argand 图上表示 9i

An Argand diagram uses the horizontal axis for the real part and the vertical axis for the imaginary part. Since 9i has zero real part and imaginary part 9, it is plotted at the point (0, 9).

Argand 图用横轴表示实部，纵轴表示虚部。因为 9i 的实部为零，虚部为 9，所以它绘制在点 (0, 9) 处。

This point is on the positive imaginary axis. It is directly above the origin, so the line from the origin to the point is vertical.

该点位于正虚轴上。它在原点的正上方，因此从原点到该点的线段是垂直的。

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3. Modulus of 9i | 9i 的模

The modulus of a complex number z = a + bi is given by |z| = √(a² + b²). For z = 9i, we have a = 0 and b = 9.

复数 z = a + bi 的模由 |z| = √(a² + b²) 给出。对于 z = 9i，我们有 a = 0，b = 9。

|9i| = √(0² + 9²) = √81 = 9

The modulus is 9. Geometrically, this is the distance from the origin to the point (0, 9) on the Argand diagram.

模为 9。从几何上看，这是从原点到 Argand 图上点 (0, 9) 的距离。

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4. Argument of 9i | 9i 的辐角

The argument of a complex number is the angle θ measured from the positive real axis to the line joining the origin to the point, usually given in radians or degrees. For z = 9i, the point lies on the positive imaginary axis, so the angle is 90° or π/2 radians.

复数的辐角是从正实轴到连接原点与该点的线段所成的角 θ，通常以弧度或度表示。对于 z = 9i，该点位于正虚轴上，因此角度为 90° 或 π/2 弧度。

arg(9i) = π/2 rad = 90°

When writing the argument, it is important to state the range used, such as -π < θ ≤ π, unless the question specifies otherwise.

书写辐角时，重要的是要说明所使用的范围，例如 -π < θ ≤ π，除非题目另有规定。

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5. Complex Conjugate of 9i | 9i 的共轭复数

The complex conjugate of z = a + bi is z* = a – bi. For z = 9i, we replace the sign of the imaginary part.

复数 z = a + bi 的共轭复数是 z* = a – bi。对于 z = 9i，我们改变虚部的符号。

(9i)* = -9i

On the Argand diagram, conjugation reflects the point across the real axis. Therefore 9i is reflected to the point (0, -9).

在 Argand 图上，共轭将点关于实轴反射。因此 9i 被反射到点 (0, -9)。

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6. Squaring 9i and Powers of i | 9i 的平方与 i 的幂

To square 9i, use the rule i² = -1. Multiply the constants and simplify.

要计算 9i 的平方，使用规则 i² = -1。将常数相乘并化简。

(9i)² = 9² × i² = 81 × (-1) = -81

This is a real negative number. It shows that squaring a pure imaginary number can produce a real result. In general, (bi)² = -b².

这是一个负实数。它表明对纯虚数平方可以产生实数结果。一般来说，(bi)² = -b²。

For higher powers, remember the cycle: i¹ = i, i² = -1, i³ = -i, i⁴ = 1, and then the pattern repeats every four powers.

对于更高次幂，请记住循环：i¹ = i，i² = -1，i³ = -i，i⁴ = 1，然后该模式每四次幂重复一次。

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7. Multiplying 9i by Other Complex Numbers | 9i 与其他复数相乘

Multiplication of complex numbers works like expanding brackets. For example, multiply 9i by 2 + 3i.

复数的乘法就像展开括号一样。例如，将 9i 乘以 2 + 3i。

9i(2 + 3i) = 18i + 27i² = 18i + 27(-1) = -27 + 18i

The result is -27 + 18i. Notice that the product has both real and imaginary parts even though one factor was a pure imaginary number.

结果是 -27 + 18i。请注意，尽管一个因数是纯虚数，但乘积同时具有实部和虚部。

You can also use the general rule: (a + bi)(c + di) = (ac – bd) + (ad + bc)i. Here a = 0, b =

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