📚 A-Level Maths: Core Methods of Rationalising the Denominator | A-Level 数学：分母有理化的核心方法
A fraction whose denominator contains a square root is often not in its simplest mathematical form. For instance, 1/√2 is exact, but it is awkward to use in calculations. By multiplying the numerator and denominator by a suitable factor, we can move the root into the numerator and leave a rational denominator. This process is known as rationalising the denominator and it is a standard requirement across many A-Level pure mathematics questions.
当分数的分母含有平方根时，这个分数通常还不算是最简洁的数学形式。例如 1/√2 虽然精确，但计算起来并不方便。通过分子分母同时乘以一个合适的因式，我们可以把根号移到分子上，使分母变为有理数。这个过程称为分母有理化，是 A-Level 纯数学中许多题目要求掌握的标准技巧。
1. What Does Rationalising the Denominator Mean? | 什么是有理化分母？
In algebra, a rational number is any number that can be written as p/q, where p and q are integers and q ≠ 0. A surd is an irrational number expressed with a root, such as √2, √3 or √5. The denominator of a fraction is the number below the fraction line. Rationalising the denominator means changing a fraction so that its denominator contains no surd, while keeping the value of the fraction exactly the same.
在代数中，有理数是可以写成 p/q 的数，其中 p、q 都是整数且 q ≠ 0。surd（无理根式）是用根号表示的无理数，例如 √2、√3 或 √5。分数的分母就是分数线下面的数。分母有理化就是在保持分数值完全不变的前提下，把分母中的根号消去。
For example, 1/√2 is equal to √2/2. The first expression has an irrational denominator, while the second has the rational denominator 2. Both expressions are exactly the same number, but the second form is usually considered simpler and more convenient.
例如，1/√2 等于 √2/2。第一个表达式分母是无理数，而第二个表达式的分母是整数 2。这两个表达式完全相等，但第二种形式通常被认为更简洁、更方便。
2. Why Do We Need This Method? | 为什么需要分母有理化？
There are three main reasons to rationalise a denominator in A-Level mathematics:
在 A-Level 数学中，分母有理化主要有三个原因：
Standard form and mark schemes: past exam questions often require the final answer to have a rational denominator. If you leave 1/√2 as your answer, you may lose a method or accuracy mark.
标准形式与评分要求：历年考题往往要求最终答案的分母为有理数。如果你把 1/√2 直接作为答案，可能会失去方法分或精确分。
Easier computation: it is much easier to estimate √2/2 than to divide 1 by 1.4142135.... Rationalising produces a clearer numerical value.
便于计算：估算 √2/2 比用 1 ÷ 1.4142135... 容易得多。有理化之后数值更清晰。
Algebraic clarity: rational denominators make it easier to add, subtract, compare and simplify fractions in larger expressions.
代数表达更清晰：有理分母便于在更大的表达式中进行加减、比较和化简。
In short, rationalising is not just a decorative step; it is a practical tool that keeps algebra tidy and accurate.
简而言之，有理化不只是为了让式子好看，而是一个实用的代数工具，它能让运算更加整洁和准确。
3. Basic Rule: a/√b | 基本法则：a/√b
The simplest case occurs when the denominator is a single square root, such as √b. The basic rule is to multiply the fraction by 1 in the form √b/√b. Because √b × √b = b, the denominator becomes the rational number b.
最简单的情况是分母为单独的平方根，例如 √b。基本法则就是把分数乘以形如 √b/√b 的 1。由于 √b × √b = b，分母就会变成有理数 b。
a/√b = (a√b)/b, where b > 0
Worked example 1: rationalise 3/√5.
例题 1：化简 3/√5 的分母。
