Euler's identity | 欧拉恒等式

📚 Euler’s identity | 欧拉恒等式

Euler’s identity is often described as the most beautiful equation in mathematics. It connects five fundamental constants — 0, 1, e, i, and π — in a single, remarkably simple expression: e^(iπ) + 1 = 0. For A-Level students, this identity is not just a curiosity; it emerges naturally from the study of complex numbers, exponential functions, and infinite series.

欧拉恒等式常被誉为数学中最美的公式。它用一个极其简洁的表达式将五个基本常数——0、1、e、i 和 π——联系在一起：e^(iπ) + 1 = 0。对 A-Level 学生而言，这一恒等式不仅是奇妙的结论，更是复数、指数函数与无穷级数学习中的自然产物。

1. What Is Euler’s Identity? | 什么是欧拉恒等式？

Euler’s identity is the special case of Euler’s formula, e^(ix) = cos x + i sin x, when x = π. Substituting π gives e^(iπ) = cos π + i sin π = -1 + 0i = -1. Rearranging yields e^(iπ) + 1 = 0. The equation is celebrated because it unifies arithmetic, analysis, and geometry.

欧拉恒等式是欧拉公式 e^(ix) = cos x + i sin x 在 x = π 时的特例。代入 π 得到 e^(iπ) = cos π + i sin π = -1 + 0i = -1，整理后即为 e^(iπ) + 1 = 0。这个等式之所以被称颂，是因为它将算术、分析和几何统一在了一起。

2. The Five Constants | 五个常数

Each number in the identity has a profound meaning in mathematics:

恒等式中的每个数字在数学中都有深远的意义：

0 — The additive identity, representing nothing and the neutral element for addition.
0 — 加法单位元，代表“无”，是加法的中性元素。
1 — The multiplicative identity, the foundation of all counting and measurement.
1 — 乘法单位元，是所有计数与测量的基础。
e — Euler’s number (≈ 2.71828), the base of natural logarithms, arising naturally in growth and decay.
e — 自然常数（≈2.71828），自然对数的底数，在增长与衰减中自然出现。
i — The imaginary unit, defined by i² = -1, extending the real number system.
i — 虚数单位，定义为 i² = -1，将实数系扩展为复数系。
π — The ratio of a circle’s circumference to its diameter (≈ 3.14159), central to trigonometry and geometry.
π — 圆周率，圆的周长与直径之比（≈3.14159），是三角学与几何学的核心。

3. Complex Numbers and the Exponential Function | 复数与指数函数

A complex number z can be written as z = x + iy, where x and y are real numbers. Its modulus is |z| = √(x² + y²), and its argument is θ = arctan(y/x). Euler’s formula demonstrates that the exponential function, when extended to complex arguments, encodes both magnitude and rotation.

复数 z 可以写成 z = x + iy，其中 x、y 为实数。其模为 |z| = √(x² + y²)，辐角为 θ = arctan(y/x)。欧拉公式表明，指数函数在扩展到复数参数时，同时包含了大小与旋转的信息。

For any real x, the complex exponential is defined by the series:

对任意实数 x，复指数函数由级数定义为：

e^(ix) = cos x + i sin x

This is Euler’s formula, the foundation of the identity.

这就是欧拉公式，也是该恒等式的基础。

4. Maclaurin Series for e^x, sin x, and cos x | e^x、sin x 与 cos x 的麦克劳林级数

The derivation of Euler’s formula relies on the Maclaurin series expansions. These are Taylor series centered at 0:

欧拉公式的推导依赖于麦克劳林级数展开，这些是以 0 为中心的泰勒级数：

e^x = 1 + x/1! + x²/2! + x³/3! + x⁴/4! + …

sin x = x – x³/3! + x⁵/5! – x⁷/7! + …

cos x = 1 – x²/2! + x⁴/4! – x⁶/6! + …

These series converge for all real x, and actually for all complex x as well.

这些级数对所有实数 x 收敛，实际上对所有复数 x 也收敛。

5. Derivation of Euler’s Formula | 欧拉公式的推导

To derive Euler’s formula, replace x in the series for e^x with ix, where i² = -1, i³ = -i, i⁴ = 1, and so on. Then separate the real and imaginary parts:

为了推导欧拉公式，将 e^x 的级数中的 x 替换为 ix，利用 i² = -1，i³ = -i，i⁴ = 1 等关系，然后分离实部和虚部：

e^(ix) = 1 + ix/1! – x²/2! – ix³/3! + x⁴/4! + ix⁵/5! – …

Grouping real terms and imaginary terms gives:

将实部与虚部分组：

e^(ix) = (1 – x²/2! + x⁴/4! – …) + i(x – x³/3! + x⁵/5! – …)

The first bracket is cos x, and the second bracket is sin x. Therefore e^(ix) = cos x + i sin x.

第一个括号正是 cos x，第二个括号正是 sin x。因此 e^(ix) = cos x + i sin x。

6. Substituting x = π | 代入 x = π

Now set x = π in Euler’s formula. Since sin π = 0 and cos π = -1, we obtain:

现在在欧拉公式中令 x = π。因为 sin π = 0，cos π = -1，得到：

e^(iπ) = cos π + i sin π = -1 + 0i = -1

Adding 1 to both sides yields the famous identity:

两边同时加 1，即得著名的恒等式：

e^(iπ) + 1 = 0

This result is astonishing because it shows that a number involving the irrational e, the imaginary unit i, and the irrational π simplifies to a simple negative integer, -1, and then to zero when 1 is added.

这个结果令人惊叹：一个涉及无理数 e、虚数单位 i 和无理数 π 的表达式，竟然化简为一个简单的负整数 -1，再加上 1 就变成了 0。

7. Geometric Interpretation on the Complex Plane | 复平面上的几何解释

Euler’s formula e^(iθ) = cos θ + i sin θ describes a point on the unit circle in the complex plane. The real part is cos θ and the imaginary part is sin θ. Thus e^(iθ) is a rotation by θ radians from the positive real axis.

欧拉公式 e^(iθ) = cos θ + i sin θ 描述的是复平面单位圆上的点。实部为 cos θ，虚部为 sin θ。因此 e^(iθ) 相当于从正实轴旋转 θ 弧度。

When θ = π, the point is at (-1, 0), exactly one half-turn around the circle. This geometric view helps make sense of why e^(iπ) equals -1.

当 θ = π 时，点位于 (-1, 0)，正好是圆上的半圈。这种几何视角有助于理解为什么 e^(iπ) 等于 -1。

Unit circle with e^(iπ) at (-1,0) — Figure: Unit circle showing e^(iπ) as rotation by π radians.

（注：此处示意图仅用于直观理解，考试中不需要绘图。）

8. Implications and Applications | 意义与应用

Euler’s identity is not just a beautiful equation; it has deep implications across mathematics and its applications:

欧拉恒等式不仅是优美的等式，它在数学及其应用中具有深远意义：

It bridges exponential functions and trigonometric functions, allowing complex exponentials to be used in solving differential equations and in circuit analysis.
它将指数函数与三角函数联系起来，使复指数可用于求解微分方程和电路分析。
It forms the basis for Fourier analysis, where signals are decomposed into complex sinusoidal components.
它是傅里叶分析的基础，信号可分解为复数正弦分量。
In quantum mechanics, the wave function is often expressed using complex exponentials, e^(i(p·r – Et)/ħ).
在量子力学中，波函数常用复指数表示，如 e^(i(p·r – Et)/ħ)。
In engineering, it simplifies the analysis of alternating current, control systems, and signal processing.
在工程学中，它简化了交流电、控制系统和信号处理的分析。

9. Exam Relevance for AQA A-Level Mathematics | AQA A-Level 数学考试相关性

In the AQA A-Level Mathematics specification, Euler’s formula itself is primarily studied in the Further Mathematics course. However, the underlying concepts — complex numbers, Maclaurin series, and functions — appear in the core A-Level Mathematics syllabus. For example, students are expected to work with complex numbers (in Further Maths) and with standard series expansions (in Pure Maths).

在 AQA A-Level 数学大纲中，欧拉公式主要在 Further Mathematics（进阶数学）中学习。然而，其基础概念——复数、麦克劳林级数和函数——出现在 A-Level 核心数学大纲中。例如，学生需要掌握复数（进阶数学）和标准级数展开（纯数学）。

Even if Euler’s identity is not asked directly, the ability to manipulate e^(ix) and to connect exponential and trigonometric forms is a valuable skill. Exam questions may ask you to:

即使不直接考查欧拉恒等式，熟练处理 e^(ix) 并将指数形式与三角形式相互转换也是一项重要技能。考试可能会要求你：

Use the Maclaurin series to find approximations to functions.
使用麦克劳林级数求函数的近似值。
Express complex numbers in modulus-argument form, including using e^(iθ).
用模-辐角形式表示复数，包括使用 e^(iθ)。
Derive trigonometric identities using Euler’s formula (e.g., cos 2θ = Re(e^(i2θ))).
利用欧拉公式推导三角恒等式（例如 cos 2θ = Re(e^(i2θ))）。

10. Common Pitfalls and Tips | 常见错误与提示

Students often make mistakes when working with Euler’s identity. Here are some tips to avoid them:

学生在处理欧拉恒等式时常犯错误。以下是一些避免错误的提示：

Pitfall \| 常见错误	Tip \| 提示
Forgetting that i² = -1	Always simplify powers of i cyclically: i, -1, -i, 1.
Using degrees instead of radians in series	All trigonometric series and derivatives assume radians.
Confusing e^(iπ) with e^(πi) (same) or with (e^i)^π	Treat e^(iπ) as a single notation; avoid ambiguous exponent groupings.
Neglecting the imaginary part when adding 1	Since the imaginary part is 0, adding 1 shifts the real part from -1 to 0.

11. Practice Exercise | 练习题

Try the following problem to test your understanding:

请尝试以下问题来检验你的理解：

Question: Using Euler’s formula, derive expressions for cos 2θ and sin 2θ in terms of cos θ and sin θ.

问题：利用欧拉公式推导 cos 2θ 和 sin 2θ 关于 cos θ 与 sin θ 的表达式。

Solution: Start from e^(i2θ) = (e^(iθ))² = (cos θ + i sin θ)². Expanding gives cos²θ – sin²θ + 2i sin θ cos θ. But also e^(i2θ) = cos 2θ + i sin 2θ. Equating real and imaginary parts:

解答：从 e^(i2θ) = (e^(iθ))² = (cos θ + i sin θ)² 出发，展开得 cos²θ – sin²θ + 2i sin θ cos θ。同时 e^(i2θ) = cos 2θ + i sin 2θ。比较实部和虚部：

cos 2θ = cos²θ – sin²θ

sin 2θ = 2 sin θ cos θ

This demonstrates the power of Euler’s formula in deriving standard identities.

这展示了欧拉公式在推导标准恒等式时的强大作用。

12. Conclusion | 结语

Euler’s identity is a masterpiece of mathematical elegance. It encapsulates the relationship between exponential growth and rotation, and it is a testament to the unity of mathematics. For A-Level students, mastering the derivation and underlying principles not only prepares you for exams but also gives you a deeper appreciation of how seemingly separate areas of mathematics are interconnected.

欧拉恒等式是数学优雅性的杰作。它浓缩了指数增长与旋转之间的联系，也是数学统一性的证明。对 A-Level 学生而言，掌握其推导和底层原理，不仅为考试做好准备，更能让你深刻体会到数学中看似不同分支之间的内在联系。

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Euler’s identity | 欧拉恒等式