Number Systems & Surds | 数字系统与根式

📚 Number Systems & Surds | 数字系统与根式

This revision guide covers the key ideas of number classification and surds for the Edexcel IGCSE Mathematics syllabus. You will learn how to place numbers in the correct set, simplify and manipulate surds, rationalise denominators, and apply these skills confidently in exam questions.

本复习指南涵盖 Edexcel IGCSE 数学大纲中数字分类与根式的核心概念。你将学会将数字归入正确的集合、化简和运算根式、有理化分母，并自信地在考试题目中运用这些技能。

1. The Number Family | 数字家族

Numbers are classified into a hierarchy of sets. Knowing the correct name for each type of number is often the very first question in an IGCSE paper.

数字按层级划分为若干集合。要知道每种数字的正确名称，往往是 IGCSE 试卷里的第一道题目。

Natural numbers (ℕ) are positive whole numbers: 1, 2, 3, 4, … Some definitions also include 0.

自然数 (ℕ) 是正整數：1, 2, 3, 4, … 有些定义也包括 0。
Integers (ℤ) include all whole numbers, positive, negative and zero: …, -3, -2, -1, 0, 1, 2, 3, …

整数 (ℤ) 包括所有正负整数和零：…, -3, -2, -1, 0, 1, 2, 3, …
Rational numbers (ℚ) can be written as a fraction p/q where p and q are integers and q ≠ 0. Examples: ½, -0.75, 3, 0.333… = 1/3.

有理数 (ℚ) 可以写成 p/q 的形式，其中 p 和 q 是整数且 q ≠ 0。例如：½, -0.75, 3, 0.333… = 1/3。
Irrational numbers cannot be written as a simple fraction. Their decimal expansions go on forever without repeating: π, √2, √3, √5.

无理数 不能写成简单分数。它们的小数展开无限且不循环：π, √2, √3, √5。
Real numbers (ℝ) include every rational and irrational number.

实数 (ℝ) 包含所有有理数和无理数。

Set	Definition	Examples
Natural ℕ	Counting numbers	1, 2, 3
Integer ℤ	Whole numbers (±)	-2, 0, 5
Rational ℚ	p/q form	½, 0.75
Irrational	Non-repeating, non-terminating	√2, π
Real ℝ	All of the above	Every number you meet

2. What Are Surds? | 什么是根式？

A surd is an irrational number expressed as a root that cannot be simplified to a whole number. In IGCSE you usually work with square roots.

根式是用根号形式表示的无理数，无法化简为整数。在 IGCSE 中通常接触的是平方根。

Only roots that are irrational are called surds. For example:

只有无理数的根才叫根式。例如：

√2, √3, √5, √7 are surds because they cannot be simplified.

√2, √3, √5, √7 是根式，因为它们无法化简。
√4 = 2 and √9 = 3 are not surds because they are integers.

√4 = 2 和 √9 = 3 不是根式，因为它们是整数。

3. Simplifying Surds | 化简根式

To simplify a surd, look for the largest square number that divides the number under the root sign.

化简根式时，要找到能整除根号下数字的最大平方数。

√(a × b) = √a × √b

Example: Simplify √72.

例：化简 √72。

√72 = √(36 × 2) = √36 × √2 = 6√2

Check that the number outside the root is the largest possible integer. Here 36 is the highest square factor of 72.

检查根号外的整数是否已取到最大。这里 36 是 72 的最大平方因子。

Another example: Simplify √50.

另一个例子：化简 √50。

√50 = √(25 × 2) = 5√2

If you use a smaller square factor, you will need to simplify again. Using √(10 × 5) would not help because neither factor is a square.

如果使用较小的平方因子，你可能需要再化简。使用 √(10 × 5) 没有帮助，因为两个因子都不是平方数。

4. Adding and Subtracting Surds | 根式的加减

You can only add or subtract surds when the number under the root sign is identical.

只有当根号内的数字相同时，才能对根式进行加减运算。

a√x + b√x = (a + b)√x

Example: Simplify 3√5 + 2√5.

例：化简 3√5 + 2√5。

3√5 + 2√5 = 5√5

Before adding or subtracting, simplify each surd first. For example:

在加减之前，先化简每个根式。例如：

√8 + √2 = 2√2 + √2 = 3√2

Without simplifying first, students often wrongly write √8 + √2 = √10. This is incorrect because the roots are different.

如果不先化简，学生常会错误地写成 √8 + √2 = √10。这是不对的，因为根号内的数字不同。

5. Multiplying and Dividing Surds | 根式的乘除

Multiplication and division of surds follow direct rules.

根式的乘除遵循直接的法则。

√a × √b = √(ab) √a ÷ √b = √(a/b)

Example: Multiply √3 × √6.

例：计算 √3 × √6。

√3 × √6 = √18 = √(9 × 2) = 3√2

Always simplify the product at the end. A common exam mark is awarded for the final simplified form.

最后务必化简乘积。考试中常有一分给最终化简形式。

Example: Divide √15 ÷ √5.

例：计算 √15 ÷ √5。

√15 ÷ √5 = √(15/5) = √3

Multiplying a surd by itself gives a rational number.

根式乘以其自身会得到一个有理数。

√a × √a = a

This property is essential for rationalising denominators later.

这个性质对后面的有理化分母至关重要。

6. Expanding Brackets with Surds | 展开含根式的括号

Expanding brackets with surds works exactly like expanding algebraic brackets. Use FOIL or the distributive law.

含根式的括号展开与代数括号展开完全相同。使用 FOIL 或分配律。

Example: Expand (3 + √2)(1 + √2).

例：展开 (3 + √2)(1 + √2)。

(3 + √2)(1 + √2) = 3 + 3√2 + √2 + 2 = 5 + 4√2

Notice that √2 × √2 = 2, producing an integer term.

注意 √2 × √2 = 2，产生了一个整数项。

Example: Expand (√5 + 2)².

例：展开 (√5 + 2)²。

(√5 + 2)² = (√5)² + 2(√5)(2) + 2² = 5 + 4√5 + 4 = 9 + 4√5

Be careful with the middle term: it always appears in a squared binomial.

注意中间项：在二项式平方中一定会出现。

The difference of two squares is very useful:

平方差公式非常有用：

(a + √b)(a − √b) = a² − b

This is exactly what we need for rationalising binomial denominators.

这正是有理化含二项式分母时所需要的。

7. Rationalising the Denominator | 有理化分母

An exam answer should never leave a surd in the denominator. Rationalising means rewriting the fraction so the denominator becomes rational.

考试答案不应把根式留在分母中。有理化就是把分数重写为分母为有理数的形式。

Case 1: Denominator is a single surd | 情形1：分母为单个根式

Multiply numerator and denominator by that surd.

将分子和分母同时乘以该根式。

a/√b = a√b/b

Example: Rationalise 4/√3.

例：有理化 4/√3。

4/√3 = (4 × √3)/(√3 × √3) = 4√3/3

Example: Rationalise 3/(2√5).

例：有理化 3/(2√5)。

3/(2√5) = (3 × √5)/(2√5 × √5) = 3√5/10

Case 2: Denominator is a binomial | 情形2：分母为二项式

Multiply numerator and denominator by the conjugate of the denominator.

将分子和分母同时乘以分母的共轭式。

Conjugate of a + √b is a − √b

a + √b 的共轭式是 a − √b

Example: Rationalise 5/(2 + √3).

例：有理化 5/(2 + √3)。

5/(2 + √3) × (2 − √3)/(2 − √3) = 5(2 − √3)/[(2)² − (√3)²]

= (10 − 5√3)/(4 − 3) = 10 − 5√3

The denominator becomes 1, but usually it is just a small integer. Never lose the numerator adjustment when multiplying by the conjugate.

这里分母变为 1，但通常只是一个小整数。乘以共轭式时绝不要遗漏分子上的调整。

8. Surds and Indices | 根式与指数

Surds can be written as fractional indices. This is frequently tested when simplifying algebraic expressions.

根式可以写成分数指数形式。这在化简代数表达式时经常考查。

√a = a¹ᐟ² ⁿ√a = a¹ᐟⁿ (aᵐ)ⁿ = aᵐⁿ

Example: Write √x³ in index form.

例：将 √x³ 写成指数形式。

√x³ = (x³)¹ᐟ² = x³ᐟ²

Example: Evaluate 16⁻¹ᐟ².

例：计算 16⁻¹ᐟ²。

16⁻¹ᐟ² = 1/(16¹ᐟ²) = 1/√16 = ¼

Remember the reciprocal first, then square root.

记住先取倒数，再开平方根。

This link between surds and indices means a question can be solved either by index rules or by surd manipulation. Choose the method you find most reliable.

根式与指数的关联意味着同一个问题既可以用指数法则解决，也可以用根式运算解决。选择你觉得最可靠的方法。

9. Common Mistakes & Exam Tips | 常见错误与考试技巧

Many students lose marks on surd questions due to small but avoidable errors.

许多学生在根式题目上失分，是因为一些细小但可避免的错误。

Mistake: √a + √b = √(a + b). This is always wrong. You can only combine like surds.

错误：√a + √b = √(a + b)。 这永远不正确。只能合并同类根式。
Mistake: √a² = a. For real numbers, √a² = |a|, but in IGCSE questions a is usually positive.

错误：√a² = a。 在实数范围内，√a² = |a|，但在 IGCSE 题目中 a 通常为正。
Mistake: Leaving the denominator with a surd. Always rationalise before writing the final answer.

错误：分母中留下根式。 在写最终答案前一定要有理化。
Tip: Show the intermediate step of simplifying √12 = 2√3. It earns method marks.

提示： 写出化简过程的中间步骤 √12 = 2√3，可以获得方法分。
Tip: Check whether your answer can be simplified further by finding another square factor.

提示： 检查答案是否能继续化简，看看是否还有平方因子。
Tip: In multiple-choice or calculator questions, use a calculator to check by converting surds to decimals.

提示： 在选择题或可用计算器的题目中，可以把根式转换为小数来检查。

10. Practice Questions | 练习

Try these questions before checking the answers. They cover every skill from this guide.

先尝试以下题目，再对照答案。它们涵盖本指南的各项技能。

Question	Answer
1. Simplify √128	8√2
2. Simplify 3√5 − √20 + √45	4√5
3. Expand (2 + √3)(1 − √3)	−1 − √3
4. Rationalise 3/(√7)	3√7/7
5. Rationalise 6/(√5 − 1)	(3√5 + 3)/2
6. Express 27⁻²ᐟ³ in the form a/b	1/9

Worked solution for Question 2:

第2题解答过程：

3√5 − √20 + √45 = 3√5 − 2√5 + 3√5 = 4√5

Because √20 = √(4 × 5) = 2√5 and √45 = √(9 × 5) = 3√5.

因为 √20 = √(4 × 5) = 2√5，√45 = √(9 × 5) = 3√5。

Worked solution for Question 5:

第5题解答过程：

6/(√5 − 1) × (√5 + 1)/(√5 + 1) = 6(√5 + 1)/(5 − 1) = 6(√5 + 1)/4 = (3√5 + 3)/2

Always simplify the final fraction fully.

最后一定要把分数化到最简。

Mastering number systems and surds gives you a strong foundation for algebra, geometry and calculus at A Level. Keep practising until every step feels automatic.

掌握数字系统与根式，会为 A Level 的代数、几何和微积分打下坚实基础。持续练习，直到每一步都变得自然熟练。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Number Systems & Surds | 数字系统与根式