📚 De Moivre’s Theorem and Its Applications | 棣莫弗定理及其应用
De Moivre’s Theorem is one of the most elegant and powerful results in complex number theory. It provides a direct connection between complex numbers in polar form and their powers and roots, making it an essential tool in IB Mathematics Analysis and Approaches HL and many other advanced curricula.
棣莫弗定理是复变函数理论中最优美且最有力的结论之一。它将极坐标形式下的复数与其幂和根直接联系起来，是 IB 数学分析与方法 HL 及许多高级课程中的核心工具。
1. Statement of De Moivre’s Theorem | 棣莫弗定理的表述
For any real number θ and any integer n, De Moivre’s Theorem states that:
对于任意实数 θ 和任意整数 n，棣莫弗定理表明：
(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)
This beautiful identity tells us that when a complex number written in polar form is raised to a power, we simply multiply the angle θ by that power while keeping the modulus (radius) raised to the corresponding power.
这个优美的恒等式告诉我们，当极坐标形式下的复数被提升到某个幂时，我们只需将角度 θ 乘以该幂，同时将模长（半径）提升到相应的幂。
More generally, if z = r(cos θ + i sin θ), then for any integer n:
更一般地，若 z = r(cos θ + i sin θ)，则对任意整数 n：
zⁿ = rⁿ[cos(nθ) + i sin(nθ)]
For the special case r = 1, the formula reduces to the simple rotation in the complex plane: each multiplication by cos θ + i sin θ rotates the vector by θ radians counterclockwise.
对于特殊情况 r = 1，该公式简化为复平面上的纯旋转：每次乘以 cos θ + i sin θ 都将向量逆时针旋转 θ 弧度。
2. Proof of De Moivre’s Theorem | 棣莫弗定理的证明
There are several ways to prove De Moivre’s Theorem. The most common method for positive integer powers uses mathematical induction.
证明棣莫弗定理的方法有多种。对于正整数幂，最常用的方法是数学归纳法。
Base case: For n = 1, the statement is trivially true: (cos θ + i sin θ)¹ = cos(1·θ) + i sin(1·θ).
基础情形：当 n = 1 时，命题显然成立：(cos θ + i sin θ)¹ = cos(1·θ) + i sin(1·θ)。
Inductive step: Assume the theorem holds for some positive integer k. Then:
归纳步骤：假设定理对某个正整数 k 成立。则：
(cos θ + i sin θ)ᵏ⁺¹ = (cos θ + i sin θ)ᵏ · (cos θ + i sin θ)
= [cos(kθ) + i sin(kθ)] · (cos θ + i sin θ)
Expanding the product using the distributive law and collecting real and imaginary parts:
使用分配律展开乘积并合并实部和虚部：
= [cos(kθ)cos θ − sin(kθ)sin θ] + i[cos(kθ)sin θ + sin(kθ)cos θ]
Using the addition formulas for sine and cosine, this simplifies to:
利用正弦和余弦的加法公式，这简化为：
= cos[(k+1)θ] + i sin[(k+1)θ]
Thus the theorem holds for k+1, and by induction it holds for all positive integers. The result extends naturally to negative integers and zero using the reciprocal of the polar form.
因此定理对 k+1 成立，由归纳法可知对所有正整数成立。通过取极坐标形式的倒数，结果可自然推广到负整数和零的情况。
3. Powers of Complex Numbers | 复数的幂
One of the most direct applications of De Moivre’s Theorem is computing high powers of complex numbers. Without this theorem, raising a complex number like (1 + i√3) to the power 10 would involve tedious binomial expansion.
棣莫弗定理最直接的应用之一是计算复数的较高次幂。没有这个定理，计算类似 (1 + i√3)¹⁰ 这样的复数幂将需要进行冗长的二项式展开。
Let us work through an example. Consider z = 1 + i√3.
让我们看一个例子。考虑 z = 1 + i√3。
First, we find the modulus and argument of z:
首先，我们求 z 的模和辐角：
r = √(1² + (√3)²) = √4 = 2
θ = arctan(√3/1) = π/3
Thus z = 2(cos π/3 + i sin π/3). Applying De Moivre’s Theorem:
因此 z = 2(cos π/3 + i sin π/3)。应用棣莫弗定理：
z¹⁰ = 2¹⁰ [cos(10 · π/3) + i sin(10 · π/3)]
= 1024 [cos(10π/3) + i sin(10π/3)]
Since 10π/3 = 2π + 4π/3, we can reduce the angle to its principal value:
由于 10π/3 = 2π + 4π/3，我们可以将角度化简到主值范围：
z¹⁰ = 1024 [cos(4π/3) + i sin(4π/3)] = 1024 (−1/2 − i√3/2) = −512 − 512i√3
Notice how the calculation reduces to trigonometric evaluation rather than algebraic expansion. This is the power of De Moivre’s Theorem.
注意计算如何简化为三角函数求值而非代数展开。这正是棣莫弗定理的力量所在。
4. Roots of Complex Numbers | 复数的根
De Moivre’s Theorem is also essential for finding all n-th roots of a complex number. If z = r(cos θ + i sin θ), then the n distinct n-th roots are given by:
棣莫弗定理也是求解复数 n 次方根的关键工具。若 z = r(cos θ + i sin θ)，则其 n 个不同的 n 次方根由下式给出：
z^(1/n) = r^(1/n) [cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)]
for k = 0, 1, 2, …, n−1.
其中 k = 0, 1, 2, …, n−1。
The addition of 2kπ reflects the periodic nature of sine and cosine: since adding a full rotation of 2π does not change the value of cos θ or sin θ, the powers of z produce multiple distinct roots as k varies.
加入 2kπ 反映了正弦和余弦的周期性：由于增加完整的 2π 旋转不会改变 cos θ 或 sin θ 的值，随着 k 的变化，z 的幂会产生多个不同的根。
For example, to find the cube roots of unity (solutions to z³ = 1), we note that 1 = cos 0 + i sin 0. Applying the formula with n = 3:
例如，要求单位根的立方根（即方程 z³ = 1 的解），我们注意 1 = cos 0 + i sin 0。令 n = 3 代入公式：
z_k = cos(2kπ/3) + i sin(2kπ/3), k = 0, 1, 2
This gives:
得到：
| k | Root | Value |
| 0 | cos 0 + i sin 0 | 1 |
| 1 | cos(2π/3) + i sin(2π/3) | −1/2 + i√3/2 |
| 2 | cos(4π/3) + i sin(4π/3) | −1/2 − i√3/2 |
Geometrically, these three roots lie on the unit circle and form the vertices of an equilateral triangle. This geometric interpretation is a beautiful illustration of the connection between algebra and geometry.
在几何上，这三个根位于单位圆上，并构成一个等边三角形的三个顶点。这种几何解释优美地展示了代数与几何之间的联系。
5. Trigonometric Identities from De Moivre’s Theorem | 用棣莫弗定理推导三角恒等式
De Moivre’s Theorem provides a systematic method for deriving multiple-angle formulas in trigonometry. By expanding (cos θ + i sin θ)ⁿ using the binomial theorem and comparing real and imaginary parts, we obtain expressions for cos(nθ) and sin(nθ) in terms of powers of cos θ and sin θ.
棣莫弗定理为推导三角学中的多倍角公式提供了一种系统方法。通过使用二项式定理展开 (cos θ + i sin θ)ⁿ，并比较实部和虚部，我们就能获得用 cos θ 和 sin θ 的幂表示的 cos(nθ) 和 sin(nθ) 的表达式。
For example, take n = 3:
例如，取 n = 3：
(cos θ + i sin θ)³ = cos 3θ + i sin 3θ
Expanding the left side using the binomial theorem:
使用二项式定理展开左边：
= cos³θ + 3i cos²θ sin θ − 3cos θ sin²θ − i sin³θ
Equating real parts:
比较实部：
cos 3θ = cos³θ − 3cos θ sin²θ = 4cos³θ − 3cos θ
Equating imaginary parts:
比较虚部：
sin 3θ = 3cos²θ sin θ − sin³θ = 3sin θ − 4sin³θ
These well-known triple-angle formulas emerge naturally from the complex exponential form. In general, for any positive integer n, the binomial expansion of the left side yields a host of trigonometric identities.
这些常见的三倍角公式自然而然地从复指数形式中涌现出来。一般地，对任意正整数 n，左边的二项式展开可以产生大量三角恒等式。
Additionally, by setting tan θ = t and using the ratio of imaginary to real parts, one can derive formulas for tan(nθ) in terms of t. This is particularly useful when solving polynomial equations involving tangent functions.
此外，令 tan θ = t，并利用虚部与实部的比值，可以推导出用 t 表示的 tan(nθ) 公式。这在求解涉及正切函数的多项式方程时尤其有用。
6. Euler’s Formula and the Exponential Form | 欧拉公式与指数形式
De Moivre’s Theorem is intimately connected with Euler’s famous formula:
棣莫弗定理与欧拉著名的公式紧密相连：
e^(iθ) = cos θ + i sin θ
If we raise both sides to the power n, we obtain (e^(iθ))ⁿ = e^(inθ) = cos(nθ) + i sin(nθ), which is precisely De Moivre’s Theorem. In this sense, De Moivre’s Theorem is a direct consequence of the laws of exponents when applied to complex exponentials.
如果我们将两边都提升到 n 次幂，得到 (e^(iθ))ⁿ = e^(inθ) = cos(nθ) + i sin(nθ)，这正是棣莫弗定理。在这个意义上，棣莫弗定理是指数法则应用于复指数时的直接结果。
This exponential form simplifies many calculations. For instance, multiplying two complex numbers in exponential form simply requires adding their exponents:
这种指数形式极大地简化了许多计算。例如，两个指数形式的复数相乘只需将它们的指数相加：
r₁e^(iθ₁) · r₂e^(iθ₂) = r₁r₂e^(i(θ₁+θ₂))
This property extends to division and powers with remarkable ease:
这一性质使得除法与幂运算也变得极其简便：
r₁e^(iθ₁) / r₂e^(iθ₂) = (r₁/r₂)e^(i(θ₁−θ₂))
(re^(iθ))ⁿ = rⁿe^(inθ)
In IB examinations, candidates are often expected to recognize and convert between rectangular, polar, and exponential forms fluently.
在 IB 考试中，考生通常需要熟练地在直角坐标形式、极坐标形式和指数形式之间进行转换。
7. Geometric Interpretations | 几何解释
De Moivre’s Theorem has profound geometric meaning. In the complex plane, a complex number z = r(cos θ + i sin θ) corresponds to a point at distance r from the origin, making an angle θ with the positive real axis.
棣莫弗定理具有深远的几何意义。在复平面上，复数 z = r(cos θ + i sin θ) 对应于距原点距离为 r、与正实轴夹角为 θ 的点。
When we multiply z by itself, the new point has distance r² and angle 2θ. Thus, squaring a complex number squares its distance from the origin and doubles its angle. More generally, raising to the n-th power multiplies the distance by rⁿ and the angle by n.
当我们用 z 乘以自身时，新点与原点的距离为 r²，角度为 2θ。因此，复数平方运算将距离平方并将角度加倍。更一般地，n 次幂将距离变为 rⁿ，角度变为 nθ。
This rotation-dilation interpretation is used in computer graphics, signal processing, and physics to describe rotations in two dimensions. The unit circle is invariant under the action of complex numbers with modulus 1: they represent pure rotations.
这种旋转-缩放解释被广泛应用于计算机图形学、信号处理和物理学中，用于描述二维旋转。单位圆在模长为 1 的复数作用下保持不变：它们表示纯旋转。
For roots, the geometric picture is equally rich. The n n-th roots of any complex number are equally spaced on a circle centered at the origin, with angular separation of 2π/n between consecutive roots.
对于根而言，几何图像同样丰富。任意复数的 n 个 n 次方根等间距地分布在一个以原点为圆心的圆上，相邻根之间的角度差为 2π/n。
This fact allows us to construct regular n-gons in the complex plane and to solve geometric problems algebraically.
这个事实使我们能够在复平面上构造正 n 边形，并用代数方法解决几何问题。
8. Sum and Product of Roots | 根的和与积
A useful consequence of De Moivre’s Theorem concerns the sum and product of the n-th roots of a number.
棣莫弗定理的一个有用推论涉及一个数的 n 次方根的和与积。
If z = r(cos θ + i sin θ) and its n-th roots are w₀, w₁, …, wₙ₋₁, then:
若 z = r(cos θ + i sin θ)，其 n 次方根为 w₀, w₁, …, wₙ₋₁，则：
Sum = w₀ + w₁ + … + wₙ₋₁ = 0 (if n ≥ 2)
Because the roots are symmetric around the origin, their vector sum cancels out. This can be proved by observing that the sum of a geometric series of the form:
由于这些根关于原点对称排列，它们的向量和相互抵消。这可以通过观察等比级数的和来证明：
1 + ω + ω² + … + ωⁿ⁻¹ = 0
where ω = e^(2πi/n) is a primitive n-th root of unity.
其中 ω = e^(2πi/n) 是一个本原 n 次单位根。
Meanwhile, the product of the n-th roots of z is given by:
同时，z 的 n 个 n 次方根的乘积为：
w₀ · w₁ · … · wₙ₋₁ = (−1)ⁿ⁻¹ z
This property is useful in polynomial theory when relating roots to coefficients, and it can simplify calculations in contest-style problems.
这一性质在多项式理论中将根与系数联系起来时非常有用，并且可以简化竞赛型问题的计算。
9. Applications in Polynomial Equations | 在多项式方程中的应用
De Moivre’s Theorem allows us to solve certain polynomial equations directly, especially those where the unknown is expressed through trigonometric identities.
棣莫弗定理使我们能够直接求解某些多项式方程，尤其是那些未知量通过三角恒等式表达的方程。
A classic example is solving the polynomial equation x⁵ − 1 = 0. Its roots are:
一个经典例子是求解多项式方程 x⁵ − 1 = 0。它的根是：
x_k = cos(2kπ/5) + i sin(2kπ/5), k = 0, 1, 2, 3, 4
These are the fifth roots of unity. Because these roots are symmetrical on the unit circle, one can derive exact values of cos(2π/5) and cos(4π/5) using algebraic manipulation.
这些是五次单位根。由于这些根在单位圆上对称分布，可以通过代数推导得到 cos(2π/5) 和 cos(4π/5) 的精确值。
Furthermore, equations of the form zⁿ = a + bi can always be solved using the general root formula, as long as we carefully handle the principal argument and the addition of 2kπ.
此外，形如 zⁿ = a + bi 的方程总可以使用一般根公式求解，只要仔细处理主辐角和 2kπ 的加入即可。
In more advanced problems, De Moivre’s Theorem helps convert trigonometric equations into polynomial ones, and vice versa. This duality is a key technique in solving non-standard equations.
在更高级的问题中，棣莫弗定理帮助将三角方程转换为多项式方程，反之亦然。这种对偶性是解决非标准方程的关键技巧。
10. Inequalities and Modulus Arguments | 不等式与模辐角问题
De Moivre’s Theorem also plays a role in problems involving the modulus and argument of complex numbers, particularly in proving inequalities about powers.
棣莫弗定理在涉及复数的模和辐角的问题中也扮演重要角色，特别是在证明关于幂的不等式时。
For example, consider z = r(cos θ + i sin θ). Then |zⁿ| = rⁿ. If 0 < r < 1, then |zⁿ| decreases geometrically as n increases. If r > 1, then |zⁿ| grows without bound.
例如，考虑 z = r(cos θ + i sin θ)。则 |zⁿ| = rⁿ。若 0 < r < 1，则 |zⁿ| 随 n 增大呈几何级数递减；若 r > 1，则 |zⁿ| 无界增长。
This simple observation is foundational in complex analysis, where understanding the behavior of powers of complex numbers leads to convergence criteria for complex series.
这一简单观察是复分析的基础，理解复数幂的行为可以帮助建立复级数的收敛判据。
The triangle inequality in combination with De Moivre’s Theorem can also be used to bound the modulus of sums of complex exponentials, which arises frequently in Fourier analysis and signal processing:
三角不等式与棣莫弗定理结合使用，还可以用来给出复指数之和的模长范围，这在傅里叶分析和信号处理中经常出现：
|e^(iθ₁) + e^(iθ₂) + … + e^(iθₙ)| ≤ n
Equality holds when all the angles θ₁, θ₂, …, θₙ are equal modulo 2π.
当所有角度 θ₁, θ₂, …, θₙ 在模 2π 意义下相等时，等号成立。
11. Common Pitfalls and Exam Tips | 常见易错点与考试建议
Students frequently make mistakes when applying De Moivre’s Theorem. Here are the most common pitfalls and how to avoid them.
学生在应用棣莫弗定理时经常犯错。以下是最常见的误区以及如何避免它们。
- Forgetting to reduce the angle: After computing nθ, always reduce it to a convenient range. Calculators may give a wrong angle if not set to radian mode.
- 忽略角度化简：计算完 nθ 后，务必将其化简到合适的范围。如果计算器没有设置为弧度模式，得到的角度可能是错误的。
- Mixing up modulus and argument: The modulus r must be raised to the power n independently of the angle multiplication. A common error is to raise the whole expression inside parentheses, or to forget the modulus entirely when r ≠ 1.
- 混淆模与辐角：模 r 必须独立地进行 n 次幂运算，与角度乘法分开。常见错误是将整个括号内的表达式一起取幂，或在 r ≠ 1 时完全忽略模。
- Using the wrong branch of the root: When solving zⁿ = w, remember that there are exactly n distinct roots, corresponding to k = 0, 1, …, n−1. Taking fewer values or using the same angle repeatedly loses solutions.
- 使用错误的根分支：求解 zⁿ = w 时，必须记住恰好有 n 个不同的根，对应 k = 0, 1, …, n−1。取值太少或重复使用同一角度都会遗漏解。
Exam tip: In IB papers, always write complex numbers in polar form before applying powers or roots. Show the modulus and argument explicitly, and state the formula you are using. Partial credit is often awarded for clear method even if the final answer is slightly wrong.
考试建议：在 IB 试卷中，在计算幂或根之前务必先将复数写成极坐标形式。明确写出模和辐角，并列出所用公式。即使最终结果略有偏差，清晰的解题过程也常常能获得部分分数。
12. Summary and Practice Problems | 总结与练习题目
De Moivre’s Theorem is a cornerstone of complex number mathematics. Its ability to link algebra, trigonometry, and geometry makes it an invaluable tool for solving a wide variety of problems.
棣莫弗定理是复数数学的基石。它将代数、三角和几何联系起来的能力，使其成为解决各种问题的宝贵工具。
To master the theorem, practice converting between forms, computing powers and roots, and deriving trigonometric identities. The following practice problems will help consolidate your understanding:
要掌握这一定理，需要练习形式转换、计算幂与根以及推导三角恒等式。以下练习有助于巩固理解：
- Compute (1 − i√3)⁶ and express the answer in the form a + bi.
- 计算 (1 − i√3)⁶ 并以 a + bi 的形式表达结果。
- Find all fourth roots of −16 and plot them in the complex plane.
- 求 −16 的所有四次方根并在复平面上标出它们。
- Derive an expression for sin 5θ in terms of sin θ and cos θ using De Moivre’s Theorem.
- 使用棣莫弗定理推导 sin 5θ 关于 sin θ 和 cos θ 的表达式。
- Solve the equation z⁴ = 8 − 8i√3, giving your answers in polar form.
- 求解方程 z⁴ = 8 − 8i√3，并以极坐标形式给出答案。
De Moivre’s Theorem rewards systematic practice. Work through these problems carefully, and you will find that even complex-looking questions become manageable with a clear step-by-step approach.
棣莫弗定理需要系统练习。认真完成这些题目，你会发即使看似复杂的问题，只要循序渐进、步骤清晰，也能迎刃而解。
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