📚 Rational and Irrational Numbers | 有理数与无理数
In the IGCSE Edexcel Mathematics syllabus, understanding the difference between rational and irrational numbers is a foundational skill. This topic appears across both Paper 1 and Paper 2, often in questions about number classification, surd manipulation, estimation, and decimal reasoning. In this article, we will explore the formal definitions, decimal characteristics, surds, rationalising the denominator, and the most common examination pitfalls, all supported by worked examples in the style of Edexcel questions.
在 IGCSE Edexcel 数学大纲中，理解有理数与无理数的区别是一项基础技能。该知识点在 Paper 1 和 Paper 2 中都会出现，常见于数的分类、根式运算、估算和小数推理相关的题目中。在本文中，我们将探讨严格定义、小数特征、根式、分母有理化以及最常见的考试陷阱，并附上与 Edexcel 出题风格一致的例题。
1. Defining Rational Numbers | 有理数的定义
A rational number is any number that can be written as a fraction p/q, where p and q are integers and q ≠ 0. The set of rational numbers is denoted by ℚ.
有理数是可以写成 p/q 形式的任何数，其中 p 和 q 是整数，且 q ≠ 0。有理数集用符号 ℚ 表示。
For example, 3/4, -7/2, 0.6 and 5 are all rational. The integer 5 is rational because it can be written as 5/1. Every integer is rational, and every terminating decimal is rational.
例如，3/4、-7/2、0.6 和 5 都是有理数。整数 5 是有理数，因为可以写成 5/1。每个整数都是有理数，每个有限小数也都是有理数。
The set ℚ includes three major groups of numbers:
有理数集 ℚ 包括三大类数：
- Integers | 整数：…, -2, -1, 0, 1, 2, …
- Terminating decimals | 有限小数：0.5, 0.125, 3.75
- Recurring decimals | 循环小数：0.333…, 0.181818…
Every number in one of these three forms can always be rewritten as a fraction of two integers. For instance, 0.181818… = 2/11, which shows that recurring decimals are rational.
这三种形式中的每一种数都可以改写为两个整数之比。例如，0.181818… = 2/11，这表明循环小数是有理数。
| Number | Decimal form | Rational? | Reason |
| 0.625 | terminating | Yes | = 5/8 |
| 0.777… | recurring | Yes | = 7/9 |
| -4 | integer | Yes | = -4/1 |
2. Defining Irrational Numbers | 无理数的定义
An irrational number is a number that cannot be written in the form p/q, where p and q are integers and q ≠ 0. In decimal form, an irrational number is non-terminating and non-recurring — its digits continue forever without a fixed repeating pattern.
无理数是不能写成 p/q 形式的数，其中 p 和 q 是整数且 q ≠ 0。在十进制形式中，无理数是无限不循环小数——其数字永远延续，且不出现固定的循环模式。
Well-known examples are the square roots of non-square integers, such as √2, √3 and √5, and the special constant π:
著名的例子包括非平方整数的平方根，如 √2、√3 和 √5，以及特殊常数 π：
√2 = 1.41421356…
π = 3.14159265…
Other irrational numbers include √2 + 1, 2√3, and numbers like 0.101001000100001…, where the pattern of zeros grows each time and never repeats.
其他无理数包括 √2 + 1、2√3 以及像 0.101001000100001… 这样的数，其中的零的个数每次递增，永不循环。
Key point: negative irrational numbers also exist, for example -√2, which is the opposite of √2 on the number line.
关键点：负无理数也存在，例如 -√2，它是 √2 在数轴上的相反数。
3. Decimal Expansions of Numbers | 数的小数展开
The decimal expansion of a rational number always either terminates or eventually recurs. For example, 1/8 = 0.125 terminates, while 2/3 = 0.666… recurs.
有理数的小数展开要么有限终止，要么最终循环。例如，1/8 = 0.125 有限终止，而 2/3 = 0.666… 循环。
In contrast, an irrational number has a decimal expansion that never terminates and never settles into a repeating pattern. No finite amount of digits can fully describe it — we always need to use a symbol or approximation instead.
相比之下，无理数的小数展开永远不会终止，也永远不会进入循环模式。无论写出多少位数字，都无法完全描述它——我们只能借助根号或 π 等符号来表示，或使用近似值。
A useful exam test is this: if you can write the number as a fraction of two integers, it is rational; if you cannot, it is irrational. Recurring decimals can always be converted into fractions using algebra, so they are always rational.
一个有用的考试检验方法是：如果你能把一个数写成两个整数之比，它就是有理数；如果不能，就是无理数。循环小数总可以用代数方法化成分数，因此它们始终是有理数。
For example, to convert 0.777… to a fraction, let x = 0.777…, then 10x = 7.777…, and subtracting gives 9x = 7, so x = 7/9.
例如，要将 0.777… 化为分数，设 x = 0.777…，则 10x = 7.777…，相减得 9x = 7，所以 x = 7/9。
| Number | Decimal form | Classification |
| √2 | 1.41421356… | Irrational |
| π | 3.14159265… | Irrational |
| 0.123123123… | recurring | Rational (= 41/333) |
| 22/7 | 3.142857142857… | Rational |
Notice that 22/7 is only an approximation of π; it is rational because its decimal recurs, whereas π is genuinely irrational. This is a classic Edexcel trap.
请注意，22/7 仅仅是 π 的近似值；它是有理数，因为其小数循环，而 π 是真正的无理数。这是 Edexcel 的经典陷阱。
4. Surds: Recognising Irrational Roots | 根式：识别无理根
A surd is an irrational number expressed using a root symbol, such as √2, √3, or ³√5. A surd arises when we take a square root (or cube root, etc.) of a number that is not a perfect square (or perfect cube).
根式是用根号表示的无理数，例如 √2、√3 或 ³√5。当我们对一个不是完全平方数（或完全立方数）的数开平方（或开立方等）时，就得到了根式。
For example, √4 = 2 is not a surd, because 4 is a perfect square. But √5 cannot be simplified to an integer, so it is a surd and therefore irrational.
例如，√4 = 2 不是根式，因为 4 是完全平方数。但 √5 不能化简为整数，所以它是根式，因此是无理数。
To place a surd on the number line, compare it with nearby perfect squares. For instance, √7 lies between √4 = 2 and √9 = 3. More precisely, since 2.6² = 6.76 and 2.7² = 7.29, we know √7 lies between 2.6 and 2.7.
要在数轴上定位一个根式，可以将其与附近的完全平方数比较。例如，√7 位于 √4 = 2 和 √9 = 3 之间。更精确地说，因为 2.6² = 6.76，2.7² = 7.29，所以 √7 在 2.6 和 2.7 之间。
5. Simplifying Surds | 化简根式
When simplifying surds, the most important rule is that for positive numbers a and b:
化简根式时，最重要的规则是，对于正数 a 和 b：
√(a × b) = √a × √b
This rule allows us to extract square factors from under the root sign. To simplify a surd, look for the largest perfect square that divides the number inside the root.
这条规则使我们能够将平方因子从根号下提取出来。要化简根式，我们需要找到根号内数的最大完全平方因数。
Example: Simplify √72. Since 72 = 36 × 2, and 36 is a perfect square, we write:
示例：化简 √72。因为 72 = 36 × 2，而 36 是完全平方数，所以：
√72 = √36 × √2 = 6√2
More examples:
更多示例：
- √50 = √25 × √2 = 5√2
- √18 = √9 × √2 = 3√2
- √48 = √16 × √3 = 4√3
There is also a second rule for division: √(a/b) = √a / √b, for positive a and b. For example, √(9/4) = 3/2, a rational result.
还有第二条除法规则：√(a/b) = √a / √b，其中 a、b 为正数。例如，√(9/4) = 3/2，这是一个有理数结果。
6. Rationalising the Denominator | 分母有理化
Edexcel examiners expect you to present fractions with a rational denominator. If a fraction has a surd in its denominator, we rationalise it by multiplying the numerator and denominator by a carefully chosen factor.
Edexcel 考官期望分数的分母为有理数。如果分数的分母含有根式，我们通过将分子和分母同时乘以一个精心选择的因子来将其有理化。
Case 1 — single surd: for 1/√2, multiply top and bottom by √2:
情况一——单个根式：对于 1/√2，分子分母同乘 √2：
1/√2 = (1 × √2)/(√2 × √2) = √2/2
Case 2 — surd plus/minus a rational number: for 1/(√5 + 2), we multiply by the conjugate (√5 – 2). Using the difference of two squares:
情况二——根式加/减有理数：对于 1/(√5 + 2)，我们乘以共轭式 (√5 – 2)。利用平方差公式：
(√a + √b)(√a – √b) = a – b
Thus:
因此：
1/(√5 + 2) = (√5 – 2)/[(√5)² – 2²] = (√5 – 2)/(5 – 4) = √5 – 2
Note that the conjugate has the same two terms but with the opposite sign in the middle. This technique eliminates every surd from the denominator in one step.
注意，共轭式包含相同的两项，但中间符号相反。这种方法可以一步消除分母中的所有根式。
7. Operations with Surds | 根式的运算
Surds can be added and subtracted only when they contain the same irrational part, just like collecting like terms in algebra:
根式只有在含有相同无理数部分时才能相加或相减，就像代数中合并同类项一样：
3√2 + 5√2 = 8√2
However, √2 + √3 cannot be simplified further, because √2 and √3 are different surds. A common error is to write √2 + √3 = √5, which is completely incorrect.
然而，√2 + √3 不能再化简，因为 √2 和 √3 是不同的根式。一个常见错误是写成 √2 + √3 = √5，这完全错误。
Multiplication and division are straightforward:
乘法和除法很简单：
- √2 × √3 = √6
- √2 × √2 = 2
- √6 ÷ √2 = √3
Expanding brackets with surds follows normal algebra rules. For example:
带根式的括号展开遵循普通代数规则。例如：
(1 + √2)² = 1² + 2 × 1 × √2 + (√2)² = 1 + 2√2 + 2 = 3 + 2√2
Remember that (√a)² = a for any positive a, and this fact is frequently tested in Paper 1.
记住，对于任何正数 a，(√a)² = a，这一知识点在 Paper 1 中经常考查。
8. Estimating Irrational Values | 估算无理数的值
Exam questions often ask you to estimate or bracket an irrational number between two integers or between two decimal values. The strategy is to compare squares.
考试题目常要求你估计无理数的大小，或在两个整数之间、两个小数之间确定无理数的范围。策略是比较平方。
To estimate √17, note that 4² = 16 and 5² = 25, so √17 lies between 4 and 5. Since 17 is much closer to 16 than to 25, a good estimate is √17 ≈ 4.1 or 4.2.
要估算 √17，注意 4² = 16，5² = 25，所以 √17 在 4 和 5 之间。由于 17 更接近 16 而不是 25，较好的估计是 √17 ≈ 4.1 或 4.2。
You should also be confident with the approximate values of common irrational constants:
你还应熟悉常见无理常数的近似值：
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