Exercise 26D: Complex Numbers – Euler and De Moivre | 练习26D：复数——欧拉公式与棣莫弗定理

📚 Exercise 26D: Complex Numbers – Euler and De Moivre | 练习26D：复数——欧拉公式与棣莫弗定理

Mastering Exercise 26D is essential for IB HL Mathematics students who want to confidently handle complex number operations using Euler's formula and De Moivre's theorem. This article provides a comprehensive review of the key concepts, techniques, and common pitfalls, ensuring you are well-prepared for both classroom tests and the final examination.

掌握练习26D对于想自信地运用欧拉公式和棣莫弗定理处理复数运算的IB高级数学学生来说至关重要。本文全面回顾关键概念、技巧和常见错误，确保你为课堂测验和期末考试做好充分准备。

1. Polar Form Recap | 极坐标形式回顾

omplex number z = x + iy can be expressed in polar form as z = r(cosθ + i sinθ), where r = √x² + y² is the modulus and θ = arg(z) is the argument, usually chosen in (-π, π] or [0, 2π).

复数 z = x + iy 可表示为极坐标形式 z = r(cosθ + i sinθ)，其中 r = √x² + y² 是模，θ = arg(z) 是辐角，通常取区间 (-π, π] 或 [0, 2π)。

sian and polar form, use x = r cosθ, y = r sinθ and tanθ = y/x (with quadrant adjustment).

在直角坐标与极坐标之间转换时，使用 x = r cosθ, y = r sinθ 以及 tanθ = y/x（需根据象限调整）。

2. Euler's Formula | 欧拉公式

Euler's formula is one of the most elegant results in mathematics: e^iθ θ. It provides a direct link between the exponential function and trigonometric functions.

欧拉公式是数学中最优美的结果之一：e^iθ = cosθ + i sinθ。它为指数函数与三角函数之间建立了直接联系。

Thus, the polar form can be written in exponential form: z = r e^iθ.

因此，极坐标形式可以写成指数形式：z = r e^iθ。

3. Multiplication and Division in Exponential Form | 指数形式下的乘法与除法

Multiplying two complex numbers z₁ = r₁ e^iθ₁ and z₂ = r₂ e^iθ₂ gives: z₁z₂ = r₁r₂ e^(i(θ₁+θ₂)). The moduli multiply and the arguments add.

两个复数 z₁ = r₁ e^iθ₁ 与 z₂ = r₂ e^iθ₂ 相乘可得：z₁z₂ = r₁r₂ e^(i(θ₁+θ₂))。模相乘，辐角相加。

Division z₁/z₂ = (r₁/r₂) e^(i(θ₁−θ₂)), with moduli dividing and arguments subtracting.

除法 z₁/z₂ = (r₁/r₂) e^(i(θ₁−θ₂))，模相除，辐角相减。

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4. De Moivre's Theorem | 棣莫弗定理

De Moivre' + i sinθ)ⁿ = cos(nθ) + i sin(nθ) for any integer n. In exponential form, this is simply (e^iθ)ⁿ = e^{i nθ}.

棣莫弗定理指出：对于任意整数 n，(cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ)。在指数形式下，即 (e^iθ)ⁿ = e^{i nθ}。

This theorem is used to compute powers of complex numbers efficiently and to derive trigonometric identities.

该定理可用于高效计算复数的幂以及推导三角恒等式。

5. Integer Powers of Complex Numbers | 复数的整数次幂

To find (r e^iθ)ⁿ, apply De Moivre: rⁿ e^{i nθ}, then convert back to Cartesian if needed. For example, (1 + i)⁶: first express 1+i = √2 e^iπ/4, then raise to power: (√2)⁶ e^{i 6π/4} = 8 e^{i 3π/2} = 8(cos 3π/2 + i sin 3π/2) = -8i.

要求 (r e^iθ)ⁿ 时，运用棣莫弗：rⁿ e^{i nθ}，如有需要再转换回直角坐标。例如，(1 + i)⁶：先将 1+i 写成 √2 e^iπ/4，然后求幂：(√2)⁶ e^{i 6π/4} = 8 e^{i 3π/2} = 8(cos 3π/2 + i sin 3π/2) = -8i。

6. Roots of Complex Numbers | 复数的根

To find the n-th roots of z = r e^iθ, use z^1/n = r^1/n 0, 1, ..., n-1. There are exactly n distinct roots, equally spaced on a circle of radius r^1/n.

要求 z = r e^iθ 的 n 次方根，使用 z^1/n = r^1/n e^(i(θ + 2kπ)/n)，其中 k = 0, 1, ..., n-1。恰有 n 个不同的根，它们均匀分布在半径为 r^1/n 的圆上。

Always add 2kπ before dividing by n to capture all roots.

务必在除以 n 前加上 2kπ，以获取所有根。

7. Roots of Unity | 单位根

The n-th roots of unity solve zⁿ = 1. They are given by ω_k = e^{i 2kπ/n} ..., n-1. These points form the vertices of a regular n-gon inscribed in the unit circle.

n 次单位根是方程 zⁿ = 1 的解，由 ω_k = e^{i 2kπ/n} 给出，k = 0, 1, ..., n-1。这些点构成内接于单位圆的正 n 边形的顶点。

sum of all roots = 0, and they are used in polynomial factorisation and symmetry problems.

性质：所有根之和为 0，并用于多项式因式分解和对称性问题。

8. Solving Equations with Euler and De Moivre | 利用欧拉与棣莫弗解方程

Write the constant in polar/exponential form, then extract roots using the formula above.

许多IB题目要求解如 z³ = -8 或 z⁴

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