Equality in A-Level Mathematics | A-Level 数学中的等式与恒等关系

📚 Equality in A-Level Mathematics | A-Level 数学中的等式与恒等关系

In A-Level Mathematics, equality is the foundation of equations, identities and proof. Understanding what ‘=’ really means helps you avoid errors when manipulating expressions, solving equations and proving statements. This article covers the key ideas Edexcel students need for the Pure Mathematics specification.

在 A-Level 数学中，等式是方程、恒等式和证明的基础。理解 ‘=’ 的确切含义有助于避免在化简表达式、解方程和证明陈述时出错。本文涵盖 Edexcel 纯数学考试大纲中学生需要掌握的关键概念。

1. The Meaning of Equality | 等式的含义

In mathematics, the equals sign = states that two expressions have the same value. For example, 2 + 3 = 5 is a statement of equality. Equality can also be used to declare that two different-looking expressions represent the same quantity for all allowed inputs.

在数学中，等号 = 表示两个表达式具有相同的值。例如 2 + 3 = 5 就是一个等式陈述。等式还可以用来说明两个形式不同的表达式在所有允许的输入下表示相同的量。

A statement of equality can be true, false or conditional. 4 + 1 = 6 is false; 3x = 12 is true only when x = 4; and (x + 1)² = x² + 2x + 1 is true for every real x.

一个等式陈述可以是真、假或有条件的。4 + 1 = 6 是假等式；3x = 12 仅在 x = 4 时为真；而 (x + 1)² = x² + 2x + 1 对所有实数 x 都为真。

2. Equations Versus Identities | 方程与恒等式的区别

An equation is a condition that may be satisfied by specific values only. For instance, 2x + 1 = 7 is an equation because it holds only for x = 3. In Edexcel wording, solving an equation means finding all values that make the equality true.

方程是可能只由特定值满足的条件。例如 2x + 1 = 7 是一个方程，因为它仅在 x = 3 时成立。在 Edexcel 的表述中，解方程意味着找出使等式成立的所有值。

An identity is a special type of equality that is true for all permitted values. It is often written with the triple bar ≡ rather than =. For example, x² − 1 ≡ (x − 1)(x + 1). This identity holds for every real number x.

恒等式是一种特殊等式，对所有允许的值都成立。它常用三横线符号 ≡ 而非 = 表示。例如 x² − 1 ≡ (x − 1)(x + 1)。这个恒等式对所有实数 x 都成立。