📚 Some Important Mathematical Results | 一些重要的数学结果
Mathematics is built upon a set of fundamental results that provide the language and logic for exploring patterns, modelling the natural world, and solving problems. These theorems and formulas – from Pythagoras to Bayes – are cornerstones of the IB curriculum and essential tools for further study.
数学建立在一系列基本结果之上，这些结果为探索规律、建模自然世界和解决问题提供了语言与逻辑。从勾股定理到贝叶斯定理，这些定理和公式是 IB 课程的基石，也是进一步学习的重要工具。
1. Pythagoras' Theorem | 勾股定理
In any right‑ ² = c² . It is fundamental to Euclidean geometry, distance calculations, and trigonometry.
在任意直角三角形中，斜边的平方等于两直角边的平方和。若直角边为 a、b，斜边为 c，则可表示为 a² + b² = c²。它是欧氏几何、距离计算和三角学的基础。
a² + b² = c²
2. The Quadratic Formula | 二次公式
The solutions to the quadratic equation ax² + bx + c = 0 (where a ≠ 0) are given by x = (−b ± √b² − 4ac criminant and determines whether the roots are real and distinct, real and equal, or complex.
二次方程 ax² + bx + c = 0 (a ≠ 0) 的解由公式 x = (−b ± √b² − 4ac) / (2a) 给出。根号下的式子 Δ = b² − 4ac 称为判别式，它决定了根是相异实数、相等实数还是复数。
x = (−b ± √b² − 4ac) / (2a)
3. The Binomial Theorem | 二项式定理
ion of (a + b)ⁿ is given by ∑ₖ₌₀ⁿ ⁿCₖ aⁿ⁻ᵏ bᵏ, where the binomial coefficients are ⁿCₖ = n! / (k!(n−k)! theorem is essential for approximations, series expansions, and probability distributions.
对于任意实数 n，(a + b)ⁿ 的展开式为 ∑ₖ₌₀ⁿ ⁿCₖ aⁿ⁻ᵏ bᵏ，其中二项式系数为 ⁿCₖ = n! / (k!(n−k)!). 当 n 为正整数时，该和为有限项；对于其他 n，则成为无穷级数。此定理在近似计算、级数展开和概率分布中至关重要。
(a + b)ⁿ = ∑ₖ₌₀ⁿ ⁿCₖ aⁿ⁻ᵏ bᵏ
4. Euler's Formula | 欧拉公式
One of the most elegant results in mathematics, Euler's formula links complex exponentials with trigonometric functions: eiθ = cos θ + i sin θ, where i² = −1. It leads to the celebrated identity eiπ + 1 = 0, which unites five fundamental constants in a single equation.
数学中最优雅的结果之一，欧拉公式将复指数与三角函数联系起来：eiθ = cos θ + i sin θ，其中 i² = −1。它引出了著名的恒等式 eiπ + 1 = 0，将五个基本常数融汇于一个方程。
eiθ = cos θ + i sin θ
5. De Moivre's Theorem | 棣莫弗定理
A direct consequence of Euler's formula, De Moivre' + i sin(nθ). It is a powerful tool for raising complex numbers to powers and for finding the nth roots of unity, which are central to complex analysis and signal processing.
作为欧拉公式的直接推论，棣莫弗定理指出：对于任意实数 θ 和整数 n，(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)。它是计算复数乘方和求 n 次单位根的强有力工具，在复分析和信号处理中具有核心地位。
(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)
6. The Fundamental Theorem of Calculus | 微积分基本定理
iation and integration. If f is continuous on [a, b] and F is an antiderivative of f (so F' = f), then by computing the antiderivative at the limits, forming the backbone of integral calculus.
该定理确立了微分与积分之间的互逆关系。若 f 在 [a, b] 上连续，且 F 是 f 的一个原函数（即 F' = f），则 ∫ₐᵇ f(x) dx = F(b) − F(a)。它使得我们可以通过计算原函数在上下限的值来求出定积分，构成了积分学的主干。
∫ₐᵇ f(x) dx = F(b) − F(a)
7. The Number e and Natural Growth | 自然常数 e 与自然增长
e of natural logarithms and describes continuous growth and decay processes. The exponential function eˣ is notable for being its own derivative, a property unique up to a constant multiple.
无理数 e ≈ 2.71828 源自基本极限 e = lim (n→∞) (1 + 1/n)ⁿ。它是自然对数的底，描述了
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