📚 Integers and Powers | 整数与幂
In Key Stage 3 Cambridge Mathematics, mastering integers (positive and negative whole numbers) and powers forms the foundation for algebra, geometry and data handling. This article systematically explains integer operations, the concept of powers and roots, index notation, and the order of operations. Through clear examples and bilingual explanations, you will build a solid understanding of these essential topics.
在剑桥中学低年级（KS3）数学中，掌握整数（正负整数）和幂是学好代数、几何和数据处理的基础。本文系统地讲解整数运算、幂与根的概念、指数记法以及运算顺序。通过清晰的实例和中英双语讲解，你将打下这些核心知识点的扎实基础。
1. Understanding Integers | 认识整数
Integers include zero, all positive whole numbers and all negative whole numbers. On a number line, positive integers lie to the right of zero and negative integers to the left. The further right a number is, the larger its value; for example, 3 is greater than 2, but -2 is greater than -3 because -2 is to the right of -3.
整数包括零、所有正整数和所有负整数。在数轴上，正整数位于零的右侧，负整数位于左侧。越靠右的数字越大；例如，3大于2，但-2大于-3，因为-2在-3的右边。
Key terms to remember: opposite numbers are pairs like +5 and -5 that are the same distance from zero. The absolute value of an integer is its distance from zero, always non-negative. For instance, | -7 | = 7 and | 4 | = 4.
需要记住的关键术语：相反数是像+5和-5这样到零的距离相等的一对数。整数的绝对值是它到零的距离，总是非负的。例如，| -7 | = 7，| 4 | = 4。
2. Adding and Subtracting Integers | 整数的加法与减法
m the larger and use the sign of the number with the larger absolute value: (-9) + 4 = -5.
同号整数相加时，将它们的绝对值相加并保留共同的符号：(-3) + (-5) = -8。异号整数相加时，用较大的绝对值减去较小的绝对值，并采用绝对值较大的那个数的符号：(-9) + 4 = -5。
nd (-3) - 4 = (-3) + (-4) = -7. Always remember: minus a negative becomes plus a positive.
减去一个整数等于加上它的相反数。例如，2 - (-6) = 2 + 6 = 8，而(-3) - 4 = (-3) + (-4) = -7。始终牢记：减负等于加正。
- (-2) + 7 = 5
- 5 - (-3) = 8
- (-10) - 2 = -12
实例：(-2) + 7 = 5；5 - (-3) = 8；(-10) - 2 = -12。
3. Multiplying and Dividing Integers | 整数的乘法与除法
t 6 × (-3) = -18. The same rule applies to division: (-20) ÷ 4 = -5, and (-36) ÷ (-6) = 6.
整数的乘法和除法遵循简单的符号法则：同号得正，异号得负。因此，(-4) × (-5) = 20，但6 × (-3) = -18。除法适用同样的规则：(-20) ÷ 4 = -5，而(-36) ÷ (-6) = 6。
yields a negative product. For example, (-2) × 3 × (-4) = 24, but (-2) × (-3) × (-1) = -6.
当两个以上整数相乘时，从左到右两两相乘。负因数个数为偶数时积为正，奇数个时积为负。例如，(-2) × 3 × (-4) = 24，而(-2) × (-3) × (-1) = -6。
With division, always perform any sign simplifications first, then divide the absolute values as usual. Avoid dividing by zero — it is undefined.
做除法时，总是先处理符号，再照常对绝对值做除法。避免除以零——这是未定义的。
4. Introduction to Powers | 幂的初步介绍
means 3 × 3 × 3 × 3, where 3 is the base and 4 is the exponent (or index). We read this as 'three to the power of four' or simply 'three to the fourth'. The value is 81.
幂表示同一个底数的重复相乘。表达式3⁴表示3 × 3 × 3 × 3，其中3是底数，4是指数（或幂指数）。我们读作“三的四次方”或简称“三的四次幂”。其值为81。
s 1. Thus, 7⁰ = 1, and (-2)⁰ = 1. The zero power is defined so patterns remain consistent.
任何非零数的零次方都等于1。因此，7⁰ = 1，(-2)⁰ = 1。零次方这么定义是为了保证模式的一致性。
= -4. The exponent applies only to the number directly before it unless brackets are used.
负底数的幂需要注意括号。(-2)² = (-2) × (-2) = 4，但-2²表示-(2²) = -4。除非使用括号，指数只作用于紧靠在它前面的数。
5. Squares and Square Roots | 平方与平方根
0² = 0. Perfect squares like 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 are useful to memorise.
一个数的平方是该数乘以自身的结果。记作n²。例如，5² = 25，(-4)² = 16，0² = 0。像1, 4, 9, 16, 25, 36, 49, 64, 81, 100这样的完全平方数值得记住。
square root of a number asks: what number, multiplied by itself, gives the original number? The symbol is √. Since 5² = 25, √25 = 5. By definition, the square root is non-negative, so √25 is 5, not -5, even though (-5)² also equals 25. When we solve x² = 25, we write x = ±5.
一个数的平方根问的是：哪个数乘以自身得到原数？符号是√。因为5² = 25，所以√25 = 5。根据定义，平方根是非负的，因此√25为5而不是-5，尽管(-5)²也等于25。在求解x² = 25时，我们写作x = ±5。
You can estimate square roots of non-perfect squares by identifying between which two integers they lie. For instance, √30 lies between 5 and 6 because 5² = 25 and 6² = 36.
对于非完全平方数，你可以通过确定它位于哪两个整数之间来估计它的平方根。例如，√30介于5和6之间，因为5² = 25，6² = 36。
6. Cubes and Cube Roots | 立方与立方根
d (-3)³ = (-3) × (-3) × (-3) = -27. Notice that the cube of a negative number is negative.
一个数的立方是该数乘以自身两次：n³ = n × n × n。例如，2³ = 8，3³ = 27，而(-3)³ = (-3) × (-3) × (-3) = -27。注意负数的立方是负数。
The cube root reverses this operation. The symbol is ∛. Since 2³ = 8, ∛8 = 2. Similarly, ∛ because (-3)³ = -27. Unlike square roots, cube roots of negative numbers are real numbers.
立方根是这个运算的逆运算。符号是∛。因为2³ = 8，所以∛8 = 2。类似地，∛(-27) = -3，因为(-3)³ = -27。与平方根不同，负数的立方根是实数。
Perfect cubes include 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. Knowing these helps in recognising cube roots quickly.
完全立方数包括1, 8, 27, 64, 125, 216, 343, 512, 729, 1000。认识这些数字有助于快速识别立方根。
7. Index Notation | 指数记法
x (or exponent) tells how many times. For example, 2 × 2 × 2 × 2 = 2⁴, and a × a × a = a³.
指数记法是重复相乘的一种简写形式。底数是参与相乘的数，指数表明相乘的次数。例如，2 × 2 × 2 × 2 = 2⁴，而a × a × a = a³。
t the indices: 5⁴ ÷ 5² = 5⁴⁻² = 5² = 25. These laws only apply when the base is identical.
同底数幂相乘时，指数相加：2³ × 2² = 2³⁺² = 2⁵ = 32。相除时，指数相减：5⁴ ÷ 5² = 5⁴⁻² = 5² = 25。这些法则仅适用于底数相同的情况。
the indices: (4²)³ = 4²×³ = 4⁶. Also, any number to the power of 1 equals itself: 9¹ = 9.
幂的乘方要把指数相乘：(4²)³ = 4²×³ = 4⁶。此外，任何数的1次方等于它本身：9¹ = 9。
| Law | Example |
|---|---|
| aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ |
| aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 7⁵ ÷ 7² = 7³ |
| (aᵐ)ⁿ = aᵐ×ⁿ | (3²)⁴ = 3⁸ |
指数法则速查表：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐ×ⁿ。
8. Order of Operations with Integers | 整数运算顺序
When an expression contains multiple operations, we follow the order commonly known as BIDMAS or PEMDAS: Brackets, Indices (powers), Division and Multiplication (from left to right), Addition and Subtraction (from left to right). This ensures everyone gets the same answer.
当一个表达式中含有多种运算时，我们遵循通常称为BIDMAS或PEMDAS的顺序：括号、指数（幂）、除法和乘法（从左到右）、加法和减法（从左到右）。这确保每个人得到相同的答案。
3. Then indices: 3² = 9. Next, multiplication: 2 × 9 = 18. Finally addition: 3 + 18 = 21.
例子：计算3 + 2 × (5 - 2)²。首先，括号：(5 - 2) = 3。然后，指数：3² = 9。接着，乘法：2 × 9 = 18。最后，加法：3 + 18 = 21。
t 10 - 5 = 5. The same left-to-right rule applies to multiplication and division combined.
如果只剩下加法和减法，就从左到右计算。对于10 - 3 + 2，先算10 - 3 = 7，然后是7 + 2 = 9，而不是10 - 5 = 5。同样的从左到右规则也适用于乘法和除法混合的情况。
e powers. In -3², the index applies only to 3, so -3² = -9. To square -3, write (-3)² = 9.
注意幂里的负号。在-3²中，指数只作用于3，所以-3² = -9。要对-3进行平方，需写成(-3)² = 9。
9. Real-life Applications | 实际应用
Integers appear in everyday contexts — temperatures below zero, bank overdrafts, floors below ground level, and elevations below sea level. Understanding negative numbers helps us describe such situations mathematically.
整数出现在日常生活中——零下的温度、银行透支、地下楼层和海平面以下的高度。理解负数能帮助我们在数学上描述这些情境。
king with speeds, energies, or when solving right-angled triangle problems with Pythagoras' theorem.
幂和根在科学和技术中至关重要。正方形面积公式用到了平方（A = s²）。立方体体积公式用到了立方（V = s³）。在处理速度、能量或者用毕达哥拉斯定理解直角三角形的时候，会用到平方根。
is about 3 × 10⁸ m/s. Understanding indices makes working with such numbers much simpler.
工程师用指数记法来表示非常大和非常小的数（科学记数法）。例如，光速大约是3 × 10⁸ 米/秒。理解指数使处理这类数字变得简单多了。
10. Common Mistakes to Avoid | 常见错误避免
with -5 - (-3). Rewrite as -5 + 3 = -2. Always convert subtraction to adding the opposite.
一个常见错误是处理负数的减法时出错。避免掉入这样的陷阱：认为-5 - 3等于-8，却在-5 - (-3)上犯糊涂。把它重写为-5 + 3 = -2。始终把减法转换为加上相反数。
Another mistake is forgetting the difference between (-2)² and -2². The brackets make all the difference. Without brackets, the exponent is only applied to the 2, not the negative sign.
另一个错误是忘记(-2)²和-2²的区别。括号起到了决定性的作用。没有括号时，指数只应用于数字2，不包括负号。
In the order of operations, students sometimes do addition before subtraction. Remember they are of equal priority and must be done left to right. Using a calculator incorrectly by trusting it to read your mind also leads to errors: enter brackets explicitly.
在运算顺序中，学生有时会先做加法再做减法。请记住它们的优先级相同，必须从左到右进行。错误地使用计算器，以为它会读懂你的意思，也会导致错误：请明确输入括号。
× 3 = 81. Take time to write out the multiplication at first to build strong foundations.
最后，学习幂的时候，不要把底数和指数相乘：3⁴不是3 × 4 = 12。它是重复相乘：3 × 3 × 3 × 3 = 81。开始时花点时间把乘法展开写，打好扎实的基础。
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