📚 Proof of the Distributive Law | 分配律的证明
d constantly, but at A-Level you need to understand why they are true and how to prove them from definitions.
在 AQA A-Level 数学中，证明是核心考查能力。分配律 a(b+c)=ab+ac 及其向量形式 a·(b+c)=a·b+a·c 被频繁使用，但在 A-Level 阶段你需要理解它们为何成立，以及如何从定义出发进行证明。
1. Statement and notation | 陈述与记号
r real numbers a, b and c, the distributive law states that a(b+c)=ab+ac and (a+b)c=ac+bc.
对于实数 a、b 和 c，分配律指出 a(b+c)=ab+ac 以及 (a+b)c=ac+bc。
In vector algebra, there are related versions. If λ is a scalar and u, v are vectors, then scalar multiplication distributes over vector addition:
在向量代数中，也有相应的版本。若 λ 是标量，u、v 是向量，则数乘对向量加法满足分配律：
λ(u+v) = λu + λv
For the dot product, if a, b, c are vectors in ℝⁿ, then the dot product distributes over vector addition:
对于点积，若 a、b、c 是 ℝⁿ 中的向量，则点积对向量加法满足分配律：
a·(b+c) = a·b + a·c
2. Why proof matters in AQA | 为什么证明在 AQA 考试中重要
AQA assessment objectives require clear, logical proof by deduction. Examiners expect you to justify algebraic manipulations rather than simply state them as facts.
AQA 的考查目标要求清晰、合乎逻辑的演绎证明。考官希望你能说明代数变形的依据，而不是直接把结论当作事实写出来。
The distributive law is used in expanding brackets, simplifying surds, working with complex numbers, and manipulating vector equations. A secure proof helps you avoid sign errors and circular reasoning.
分配律广泛用于去括号、化简根式、处理复数以及向量方程的变形。掌握证明过程有助于避免符号错误和循环论证。
Many A-Level proof questions ask you to 'prove that' or 'show that'. Starting from a known distributive law is often the first valid step.
许多 A-Level 证明题会要求你“证明”或“说明”。从已知的分配律出发，往往是第一个合理的步骤。
3. Proof for positive integers by repeated addition | 正整数情形的重复加法证明
Let a, b and c be positive integers. Multiplication by a positive integer is repeated addition.
设 a、b 和 c 为正整数。正整数乘法就是重复加法。
a(b+c) = (b+c) + (b+c) + ... + (b+c) (a times)
Since addition is associative and commutative, the a copies of b can be grouped together and the a copies of c can be grouped together.
由于加法满足结合律和交换律，可以把 a 个 b 合在一起，把 a 个 c 也合在一起。
a(b+c) = (b+...+b) + (c+...+c) = ab + ac
This gives a rigorous first-principles proof for the natural numbers.
这给出了自然数情形下从第一性原理出发的严格证明。
4. Algebraic status: axiom and extension | 代数地位：公理与推广
At A-Level, for the real number system the distributive law is usually taken as an axiom. This means it is a fundamental rule assumed to be true without proof.
在 A-Level 阶段，实数系的分配律通常被当作一条公理。这意味着它是一条被假定为真、不需证明的基本规则。
However, the law can be proved for integers and rationals by extending the repeated-addition argument and using the rules of signs and fractions.
不过，对于整数和
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