Number Foundations: Integers, Order of Operations and Powers | 数字基础：整数、运算顺序与幂

📚 Number Foundations: Integers, Order of Operations and Powers | 数字基础：整数、运算顺序与幂

Welcome to the Seventh Grade Topic Series. In this article, we will explore the essential number skills that every Year 7 student must master: integers, the order of operations, powers and roots, factors, and multiples. These ideas appear throughout the IGCSE syllabus and in many real-world situations.

欢迎来到七年级主题系列。在这篇文章中，我们将探索每一位七年级学生都必须掌握的核心数字技能：整数、运算顺序、幂与根、因数与倍数。这些概念贯穿整个 IGCSE 课程，也广泛出现在现实生活场景中。

1. What are Integers? | 什么是整数？

Integers are the set of whole numbers that includes positives, negatives, and zero. In symbols, we write integers as {…, −3, −2, −1, 0, 1, 2, 3, …}. There is no fraction or decimal part in an integer.

整数是包含正数、负数和零的全体整数的集合。用符号表示即为 {…, −3, −2, −1, 0, 1, 2, 3, …}。整数中不包含分数部分或小数部分。

The number line helps us visualise integers. Zero sits in the middle; positive integers go to the right, and negative integers go to the left. The distance from zero to a number is called its absolute value. For example, the absolute value of −5 is 5 because −5 is exactly 5 units away from zero.

数轴帮助我们直观地理解整数。零位于正中间；正整数向右侧延伸，负整数向左侧延伸。从零到某个数的距离被称为该数的绝对值。例如，−5 的绝对值是 5，因为 −5 距离零恰好有 5 个单位。

Negative integers appear in real life when we measure temperatures below zero, bank balances that are overdrawn, or floors below ground level in a building.

负整数在现实生活中无处不在，例如零度以下的温度、银行透支的余额，或者建筑物中地下的楼层。

2. Adding and Subtracting Integers | 整数的加减法

When adding integers with the same sign, add their absolute values and keep the same sign. For example, 4 + 3 = 7 and (−4) + (−3) = −7.

当两个整数同号相加时，将它们的绝对值相加并保留相同的符号。例如，4 + 3 = 7，而 (−4) + (−3) = −7。

When adding integers with different signs, subtract the smaller absolute value from the larger one, and keep the sign of the number with the larger absolute value. For example, 9 + (−4) = 5 because 9 − 4 = 5, and the positive sign is kept. Also (−9) + 4 = −5 because 9 − 4 = 5, and the negative sign is kept.

当两个整数异号相加时，用较大的绝对值减去较小的绝对值，并保留绝对值较大的那个数的符号。例如，9 + (−4) = 5，因为 9 − 4 = 5，保留正号；而 (−9) + 4 = −5，因为 9 − 4 = 5，保留负号。

Subtracting an integer is the same as adding its opposite. Therefore a − b = a + (−b), and a − (−b) = a + b. Let us look at two examples:

减去一个整数等于加上它的相反数。因此 a − b = a + (−b)，而 a − (−b) = a + b。我们来看两个例子：

7 − (−3) = 7 + 3 = 10

−2 − (−6) = −2 + 6 = 4

A useful phrase is “two negatives make a positive” for subtraction. However, be careful: this rule applies only in certain operations, as we will see in the next section.

有一个常用的口诀：“负负得正”。但请注意：这个规律只适用于特定的运算情形，下一节我们就会看到区别。

3. Multiplying and Dividing Integers | 整数的乘除法

Multiplication and division of integers follow a simple sign rule. When the two numbers have the same sign, the answer is positive. When they have different signs, the answer is negative.

整数的乘法和除法遵循一个简单的符号法则。当两个数同号时，结果为正；当两个数异号时，结果为负。

Operation	Example	Result
positive × positive	6 × 3	+18
positive × negative	6 × (−3)	−18
negative × positive	(−6) × 3	−18
negative × negative	(−6) × (−3)	+18

The same rule applies to division. For instance, 12 ÷ (−4) = −3, while (−12) ÷ (−4) = 3. Notice that division by zero is never allowed and has no meaning in mathematics.

除法同样遵循这条法则。例如，12 ÷ (−4) = −3，而 (−12) ÷ (−4) = 3。请注意，除以零在数学中永远不被允许，也没有任何意义。

A common mistake is to confuse the “two negatives make a positive” rule. In addition, negative signs must be handled according to the operation. Always check the operation before applying the sign rule.

一个常见错误是把“负负得正”法则用错地方。在加法中，负号的处理方式不同。在套用符号法则之前，一定要先确认运算的类型。

4. Order of Operations | 运算顺序

When a calculation contains several operations, we must follow a fixed order. The acronym BODMAS stands for Brackets, Orders (powers and roots), Division and Multiplication, Addition and Subtraction.

当一个算式中包含多种运算时，我们必须遵循固定的顺序。缩写 BODMAS 代表：括号（Brackets）、阶（Orders，即幂与根）、除法与乘法（Division and Multiplication）、加法与减法（Addition and Subtraction）。

Division and multiplication are carried out from left to right, and so are addition and subtraction. For example, in 9 − 2 + 3, we work from left to right: 9 − 2 = 7, then 7 + 3 = 10.

除法和乘法按从左到右的顺序进行，加法和减法同样按从左到右的顺序进行。例如，在 9 − 2 + 3 中，我们从左到右计算：9 − 2 = 7，然后 7 + 3 = 10。

Consider the expression 3 + 4 × 2. Without rules, we might add first and get 14. But BODMAS tells us to multiply first, so 4 × 2 = 8 and then 3 + 8 = 11.

来看表达式 3 + 4 × 2。如果没有规则，我们可能会先加得到 14。但 BODMAS 要求先乘，所以 4 × 2 = 8，然后 3 + 8 = 11。

Brackets always come first. In the expression (3 + 4) × 2, we evaluate 3 + 4 = 7 first, then 7 × 2 = 14. Changing the position of brackets changes the result completely.

括号永远优先。在表达式 (3 + 4) × 2 中，我们先算 3 + 4 = 7，再算 7 × 2 = 14。改变括号的位置会完全改变结果。

2 × (5 − 1)² + 6 ÷ 3 = 2 × 4² + 2 = 2 × 16 + 2 = 34

Here we handled the bracket first, then the power, then multiplication and division, and finally addition. Writing down each step prevents careless errors in longer expressions.

在这个例子中，我们先处理括号，再计算幂，然后做乘除法，最后做加法。在较长的算式中逐步写出过程，可以避免粗心导致的错误。

5. Powers and Exponents | 幂与指数

A power is a shorthand way to write repeated multiplication. In the expression 2³, the number 2 is the base and 3 is the exponent. It means 2 × 2 × 2 = 8.

幂是表示重复乘法的简洁写法。在表达式 2³ 中，数字 2 是底数，3 是指数。它表示 2 × 2 × 2 = 8。

We read 2³ as “two to the power of three” or “two cubed”. Similarly, 5² is read as “five squared” and equals 5 × 5 = 25. The square of a number means multiplying it by itself once.

我们把 2³ 读作“2 的 3 次方”或“2 的立方”。类似地，5² 读作“5 的平方”，等于 5 × 5 = 25。一个数的平方表示这个数自乘一次。

Any number raised to the power of 1 equals itself, for example 7¹ = 7. Any non-zero number raised to the power of 0 equals 1, so 9⁰ = 1 and 250⁰ = 1. However, 0⁰ is not defined in this course.

任何数的 1 次方都等于它本身，例如 7¹ = 7。任何非零数的 0 次方都等于 1，所以 9⁰ = 1，250⁰ = 1。但在这个课程中，0⁰ 不作定义。

When multiplying powers with the same base, we add the exponents. For example, 4² × 4³ = 4⁽²⁺³⁾ = 4⁵ = 1024. When dividing powers with the same base, we subtract the exponents, so 5⁶ ÷ 5² = 5⁽⁶⁻²⁾ = 5⁴ = 625.

当同底数的幂相乘时，我们把指数相加。例如，4² × 4³ = 4⁽²⁺³⁾ = 4⁵ = 1024。当同底数的幂相除时，我们把指数相减，因此 5⁶ ÷ 5² = 5⁽⁶⁻²⁾ = 5⁴ = 625。

Powers of ten are especially important in science. Notice the pattern: 10¹ = 10, 10² = 100, 10³ = 1000. The exponent tells us how many zeros appear after the 1.

10 的幂在科学中尤其重要。请注意这个规律：10¹ = 10，10² = 100，10³ = 1000。指数告诉我们数字 1 后面要写多少个零。

6. Square Roots and Cube Roots | 平方根与立方根

The square root of a number is the value that, when multiplied by itself, gives the original number. We write the square root symbol as √. For example, √49 = 7 because 7 × 7 = 49, and √121 = 11 because 11 × 11 = 121.

一个数的平方根是指自乘后得到原数的那个值。我们用符号 √ 表示平方根。例如，√49 = 7，因为 7 × 7 = 49；√121 = 11，因为 11 × 11 = 121。

The cube root of a number is the value that, when used in a product of three equal factors, gives the original number. We write the cube root as ∛. For example, ∛64 = 4 because 4 × 4 × 4 = 64, and ∛1000 = 10 because 10³ = 1000.

一个数的立方根是指三个相同因数相乘后得到原数的那个值。我们用符号 ∛ 表示立方根。例如，∛64 = 4，因为 4 × 4 × 4 = 64；∛1000 = 10，因为 10³ = 1000。

Perfect squares are numbers whose square roots are integers. The first twelve perfect squares are:

完全平方数是指平方根仍为整数的数。前十二个完全平方数是：

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144

Knowing these values by heart speeds up many calculations. For example, √144 = 12 and √81 = 9. Similarly, the first few perfect cubes are 1, 8, 27, 64, 125 and 216.

熟记这些数值可以大大加快计算速度。例如，√144 = 12，√81 = 9。同样，前几个完全立方数是 1、8、27、64、125 和 216。

Note that the square root of a negative integer is not a real number at this level. For instance, √−9 has no real value, because no real number multiplied by itself equals a negative number.

请注意，在目前这个学习阶段，负整数的平方根不是实数。例如，√−9 没有实数值，因为没有任何实数的自乘会等于负数。

7. Prime Numbers and Factors | 质数与因数

A prime number is a positive integer with exactly two different factors: 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17 and 19. Notice that 2 is the only even prime number.

质数是指恰好只有两个不同因数的正整数：1 和它本身。前几个质数是 2、3、5、7、11、13、17 和 19。请注意，2 是唯一的偶数质数。

Numbers with more than two factors are called composite numbers. For example, 12 has factors 1, 2, 3, 4, 6 and 12, so 12 is composite. The number 1 is neither prime nor composite; it has only one factor.

具有两个以上因数的数称为合数。例如，12 的因数有 1、2、3、4、6 和 12，因此 12 是合数。数字 1 既不是质数也不是合数，因为它只有一个因数。

Prime factorisation means writing a number as a product of its prime factors. A factor tree is a clear method for this. For example, we can break down 36 as follows:

质因数分解是指把一个数写成它的质因数乘积的形式。因数树是一种清晰的方法。例如，我们可以像下面这样分解 36：

36 = 2 × 18 = 2 × 2 × 9 = 2 × 2 × 3 × 3 = 2² × 3²

Prime factorisation is the foundation for finding the LCM and HCF in the next two sections. Always start with the smallest prime that divides the number, then continue until all factors are prime.

质因数分解是接下来求最小公倍数和最大公因数的基础。始终从能整除这个数的最小质数开始，然后继续分解，直到所有因数都是质数为止。

8. Multiples and the Lowest Common Multiple | 倍数与最小公倍数

A multiple of a number is the result of multiplying that number by a positive integer. The multiples of 4 are 4, 8, 12, 16, 20, 24, and so on. Every number has infinitely many multiples.

一个数的倍数是指该数与某个正整数相乘得到的积。4 的倍数是 4、8、12、16、20、24 等等。每个数都有无穷多个倍数。

The lowest common multiple (LCM) of two or more numbers is the smallest positive number that is a multiple of each of them. We can find the LCM by listing multiples:

两个或多个数的最小公倍数（LCM）是同时是每一个数的倍数的最小正整数。我们可以通过列举倍数来求 LCM：

Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, …

4 的倍数：4、8、12、16、20、24、28、32、36……
Multiples of 6: 6, 12, 18, 24, 30, 36, …

6 的倍数：6、12、18、24、30、36……
The smallest common multiple is 12, so LCM(4, 6) = 12.

最小的公共倍数是 12，因此 LCM(4, 6) = 12。

Another method uses prime factorisation. Write each number as a product of primes, then take the highest power of every prime that appears. For 8 = 2³ and 12 = 2² × 3, the highest powers are 2³ and 3, so LCM = 2³ × 3 = 24.

另一种方法使用质因数分解。把每个数写成质因数的乘积，然后取每个出现的质数的最高次幂。对于 8 = 2³ 和 12 = 2² × 3，最高次幂是 2³ 和 3，因此 LCM = 2³ × 3 = 24。

The LCM is useful when adding or subtracting fractions with different denominators. It also helps solve problems about repeating events, such as two buses arriving at the same stop again.

LCM 在异分母分数加减法时非常有用。它也帮助我们解决重复性事件的问题，例如两辆公交车再次同时到达同一个站点。

9. The Highest Common Factor | 最大公因数

The highest common factor (HCF) of two or more numbers is the largest positive integer that divides each of them exactly. It is also called the greatest common divisor (GCD).

两个或多个数的最大公因数（HCF）是能同时整除每一个数的最大正整数。它也被称为最大公约数（GCD）。

For example, the factors of 18 are 1, 2, 3, 6, 9 and 18. The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. The common factors are 1, 2, 3 and 6, so the HCF is 6. We write HCF(18, 24) = 6.

例如，18 的因数是 1、2、3、6、9 和 18。24 的因数是 1、2、3、4、6、8、12 和 24。公共因数有 1、2、3 和 6，因此最大公因数是 6。我们记为 HCF(18, 24) = 6。

Using prime factorisation, we find the HCF by taking the lowest power of each common prime. Since 18 = 2 × 3² and 24 = 2³ × 3, the common primes are 2 and 3. The lowest powers are 2¹ and 3¹, so HCF = 2 × 3 = 6.

使用质因数分解时，我们取每个公共质数的最低次幂来求 HCF。因为 18 = 2 × 3²，24 = 2³ × 3，公共质数是 2 和 3。最低次幂是 2¹ 和 3¹，所以 HCF = 2 × 3 = 6。

There is a beautiful relationship between HCF and LCM for two positive integers a and b:

对于两个正整数 a 和 b，HCF 与 LCM 之间有一个美妙的关系：

HCF(a, b) × LCM(a, b) = a × b

We can check this with 18 and 24: HCF = 6, LCM = 72, and 6 × 72 = 432 = 18 × 24. This relationship is a quick way to check your answers.

我们可以用 18 和 24 来验证：HCF = 6，LCM = 72，而 6 × 72 = 432 = 18 × 24。这个关系可以用来快速检验答案是否正确。

10. Real-World Applications | 实际应用

Negative numbers are used for temperature. On a cold day, the temperature might fall from 3 °C to −5 °C. The change is −5 − 3 = −8, meaning a drop of 8 degrees.

负数常用来表示温度。在寒冷的日子里，气温可能从 3 °C 下降到 −5 °C。温度变化为 −5 − 3 = −8，表示下降了 8 度。

Bank balances also use integers. If you have $20 and spend $30, your balance becomes −$10, meaning you owe the bank $10. Adding and subtracting signed amounts helps us keep track of money.

银行余额也使用整数。如果你有 20 美元并花了 30 美元，你的余额变成 −10 美元，表示你欠银行 10 美元。对带符号的金额做加减法，帮助我们管理财务。

The order of operations appears when we calculate costs. Suppose tickets cost $8 each and there is a one-time booking fee of $12 for a group of 5 people. The total cost is 5 × 8 + 12 = 40 + 12 = $52, not 5 × (8 + 12) = $100.

运算顺序在计算费用时也很常见。假设每张票 8 美元，5 人团体一次性预订费 12 美元。总费用是 5 × 8 + 12 = 40 + 12 = 52 美元，而不是 5 × (8 + 12) = 100 美元。

Powers help us understand area and volume. A square carpet with side length 6 metres has an area of 6² = 36 square metres. A cube-shaped box with edge 3 metres has a volume of 3³ = 27 cubic metres.

幂帮助我们理解面积和体积。边长为 6 米的正方形地毯，面积为 6² = 36 平方米。棱长为 3 米的立方体箱子，体积为 3³ = 27 立方米。

Factors and multiples help us divide objects fairly. If you have 20 apples and 16 oranges, the largest number of identical gift bags you can make is the HCF: 4 bags, each with 5 apples and 4 oranges.

因数与倍数帮助我们公平地分配物品。如果你有 20 个苹果和 16 个橙子，能制作的最大相同礼袋数量就是 HCF：4 袋，每袋装有 5 个苹果和 4 个橙子。

11. Exam Tips and Practice | 考试技巧与练习

Here are the most common mistakes students make in Year 7 number work, followed by our advice.

以下是七年级学生在数字计算中最常见的错误，以及我们给出的建议。

Mistake: forgetting the sign rule when multiplying two negatives.

错误：计算两个负数相乘时忘记符号法则。
Advice: remember that same signs give a positive result, and different signs give a negative result.

建议：记住同号得正，异号得负。
Mistake: calculating from left to right without using BODMAS.

错误：不使用 BODMAS，而只是从左到右直接计算。
Advice: always look for brackets and powers before doing multiplication or addition.

建议：在做乘法或加法之前，先寻找括号和幂。
Mistake: confusing LCM with HCF, or dividing instead of listing multiples.

错误：混淆 LCM 与 HCF，或者用除法代替列举倍数。
Advice: read the question carefully: LCM asks for a common multiple, HCF asks for a common factor.

建议：仔细读题：LCM 求的是公共倍数，HCF 求的是公共因数。

Try these practice questions, then check the answers below.

请尝试

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Number Foundations: Integers, Order of Operations and Powers | 数字基础：整数、运算顺序与幂