📚 Understanding Integers: Operations and Order of Operations | 理解整数：运算与运算顺序
Mastering integers and the order of operations is a cornerstone of Key Stage 3 mathematics. This article explores what integers are, how to add, subtract, multiply and divide them, and why BIDMAS (or PEMDAS) is essential for solving multi‑step problems correctly. We will break down each operation step by step, clarify common pitfalls, and provide worked examples to build your confidence.
掌握整数和运算顺序是 KS3 数学的基石。本文将探讨什么是整数，如何对它们进行加、减、乘、除，以及为什么 BIDMAS（或 PEMDAS）对于正确解决多步问题至关重要。我们将逐步分解每一类运算，澄清常见的陷阱，并通过实例帮助你建立信心。
1. What is an Integer? | 什么是整数？
An integer is any whole number that can be positive, negative or zero. Integers do not include fractions or decimals. The set of integers is written as {…, −3, −2, −1, 0, 1, 2, 3, …}. On a number line, integers are equally spaced points extending infinitely in both directions.
整数是任何正整数、负整数或零，不包括分数或小数。整数的集合写作 {…, −3, −2, −1, 0, 1, 2, 3, …}。在数轴上，整数是等间距的点，向两个方向无限延伸。
A positive integer like +7 can be written simply as 7. A negative integer such as −5 always has a minus sign in front. Zero is neither positive nor negative. Understanding this number family is the first step before performing operations with them.
像 +7 这样的正整数可以直接写作 7。像 −5 这样的负整数前面总是有一个减号。零既不是正数也不是负数。在对整数进行运算之前，理解这个数族是第一步。
2. Adding and Subtracting Integers | 整数的加法与减法
When adding integers with the same sign, add their absolute values and keep the common sign. For example: (−4) + (−9) = −13. When the signs differ, subtract the smaller absolute value from the larger one and take the sign of the number with the larger absolute value: 7 + (−10) = −3.
当同号整数相加时，将它们的绝对值相加并保留共同的符号。例如：(−4) + (−9) = −13。当符号不同时，用较大的绝对值减去较小的绝对值，并取绝对值较大的数的符号：7 + (−10) = −3。
Subtraction can be transformed into addition by changing the sign of the number being subtracted. For instance, 5 − (−3) becomes 5 + (+3) = 8. This ‘add the opposite’ rule makes all subtractions easier to handle: 2 − 7 = 2 + (−7) = −5.
减法可以通过改变被减数的符号转化为加法。例如，5 − (−3) 变为 5 + (+3) = 8。这条 “加上相反数” 的规则使所有减法都变得更容易处理：2 − 7 = 2 + (−7) = −5。
Visualising a number line can help: moving right for addition and left for subtraction, or reversing direction when subtracting a negative.
用数轴想象可以帮助理解：加法向右移动，减法向左移动；当减去一个负数时，方向会反转。
3. Multiplying and Dividing Integers | 整数的乘法与除法
The sign rules for multiplication and division are simple: a positive times a positive gives a positive; a negative times a negative also gives a positive. A negative times a positive (or vice versa) gives a negative. The same applies to division.
乘法和除法的符号规则很简单：正数乘以正数得正数；负数乘以负数也得正数。负数乘以正数（或反过来）得负数。同样的规则也适用于除法。
Examples: (−6) × 4 = −24; (−3) × (−8) = 24; 20 ÷ (−5) = −4; (−42) ÷ (−7) = 6. When multiplying more than two integers, count the number of negative signs: an even number gives a positive product, an odd number gives a negative product.
示例：(−6) × 4 = −24；(−3) × (−8) = 24；20 ÷ (−5) = −4；(−42) ÷ (−7) = 6。当两个以上的整数相乘时，数一数负号的个数：偶数个负号得正积，奇数个负号得负积。
This sign awareness is crucial for later topics like algebra and brackets expansion.
这种符号意识对后续的代数与去括号等内容至关重要。
4. Understanding Negative Numbers | 理解负数
Negative numbers can feel abstract, but they represent real-world ideas such as temperature below zero, debts, or levels below sea level. On a number line, numbers to the left are smaller: −10 < −2. The absolute value of a number is its distance from zero, so |−7| = 7.
负数可能感觉抽象，但它们代表现实世界中的概念，例如零下的温度、债务或海平面以下的高度。在数轴上，越靠左的数字越小：−10 < −2。一个数的绝对值是它到零的距离，因此 |−7| = 7。
When dealing with negative bases and powers, note that (−2)² means −2 × −2 = 4, while −2² means −(2 × 2) = −4. The bracket makes all the difference.
在处理负底数和幂时，注意 (−2)² 表示 −2 × −2 = 4，而 −2² 表示 −(2 × 2) = −4。括号起着决定性的作用。
Always keep the negative sign attached to the number until an operation separates them. Practising with counters or number‑line jumps builds strong intuition.
始终将负号与数字视为一体，直到运算将它们分开。借助计数片或数轴跳跃进行练习可以建立牢固的直觉。
5. The Order of Operations (BIDMAS) | 运算顺序 (BIDMAS)
BIDMAS tells us the order in which operations should be performed in a calculation: Brackets, Indices, Division, Multiplication, Addition, Subtraction. Some regions use BODMAS (O for Orders) or PEMDAS. The rules are the same: work from left to right for division and multiplication, and again for addition and subtraction.
BIDMAS 告诉我们运算的顺序：括号、指数、除法、乘法、加法、减法。有些地区使用 BODMAS（O 代表 Order）或 PEMDAS。规则是相同的：对于除法和乘法，从左到右计算；对于加法和减法，同样从左到右。
For example, in 3 + 6 × 2, we multiply first: 3 + 12 = 15. If we added first, we would get 18, which is wrong. This structure prevents ambiguity in multi‑step problems.
例如，在 3 + 6 × 2 中，我们先乘法：3 + 12 = 15。如果先做加法，会得到 18，这是错误的。这种结构避免了多步问题中的歧义。
Memorising BIDMAS is just the beginning – understanding why each step is needed is what makes a confident mathematician.
记住 BIDMAS 只是开始，理解每一步的原因才能成为自信的数学学习者。
6. Brackets First | 括号优先
Brackets are used to group parts of an expression that must be calculated first. Work from the innermost set outwards if there are nested brackets. Common types include round brackets ( ), square brackets [ ] and sometimes curly brackets { }.
括号用于将表达式中必须首先计算的部分组合在一起。如果有嵌套括号，则从最内层向外计算。常见的类型包括圆括号 ( )、方括号 [ ]，有时还有花括号 { }。
Example: 2 × (3 + 5) = 2 × 8 = 16. Without brackets, 2 × 3 + 5 = 11, which is a different result. When a number sits directly next to a bracket, it implies multiplication: 4(7 − 2) = 4 × 5 = 20.
示例：2 × (3 + 5) = 2 × 8 = 16。如果没有括号，2 × 3 + 5 = 11，这是一个不同的结果。当一个数字直接位于括号旁边时，意味着乘法：4(7 − 2) = 4 × 5 = 20。
Always rewrite complex expressions clearly, showing every step to avoid missing hidden brackets.
始终清晰地重写复杂表达式，显示每一步，以避免遗漏隐藏的括号。
7. Indices (Powers) | 指数（幂）
An index (or power) tells you how many times to multiply a base by itself. 5² means 5 × 5 = 25; 2³ means 2 × 2 × 2 = 8. The index can also be 1, which just gives the base number, or 0, which gives 1 (as long as the base is not zero).
指数（或幂）告诉你将底数与自身相乘多少次。5² 表示 5 × 5 = 25；2³ 表示 2 × 2 × 2 = 8。指数也可以是 1，结果就是底数本身；指数为 0 时，结果为 1（只要底数不为零）。
In BIDMAS, indices come straight after brackets. For example, 3 × 2² = 3 × 4 = 12. Be careful when indices are combined with negation: −3² is read as −(3²) = −9.
在 BIDMAS 中，指数紧随括号之后。例如，3 × 2² = 3 × 4 = 12。当指数与负号结合时要小心：−3² 应读作 −(3²) = −9。
Square roots and cube roots are inverse operations of squaring and cubing: √16 = 4 because 4² = 16; ∛27 = 3. These are often treated at the same rank as powers in the order of operations.
平方根和立方根是平方和立方的逆运算：√16 = 4，因为 4² = 16；∛27 = 3。在运算顺序中，它们通常与幂处于同等级别。
8. Division and Multiplication | 除法与乘法
Division and multiplication have equal priority. When a calculation contains both, carry them out in the order they appear, from left to right. For example, 12 ÷ 3 × 2 = (12 ÷ 3) × 2 = 4 × 2 = 8; doing multiplication first would give 12 ÷ 6 = 2, which is incorrect.
除法和乘法具有相同的优先级。当计算中同时包含两者时，按从左到右的顺序执行。例如，12 ÷ 3 × 2 = (12 ÷ 3) × 2 = 4 × 2 = 8；先做乘法会得到 12 ÷ 6 = 2，这是错误的。
Rewrite division as multiplication by a reciprocal to avoid mistakes: 12 ÷ 3 × 2 is the same as 12 × (1/3) × 2 = 8. This approach is especially useful when variables are involved later in algebra.
将除法改写为乘以倒数可以避免错误：12 ÷ 3 × 2 等同于 12 × (1/3) × 2 = 8。这种方法在后续代数中涉及变量时尤其有用。
Always treat a fraction bar as a hidden bracket: the numerator and denominator are computed fully before division.
永远将分数线视为隐藏的括号：分子和分母在除法之前先完全计算。
9. Addition and Subtraction | 加法与减法
Addition and subtraction are the last operations to perform, and they also share equal priority. Work from left to right. For instance, 10 − 3 + 2 = 7 + 2 = 9, not 10 − 5 = 5. Treat subtraction as adding a negative to keep things clear.
加法和减法是最后执行的运算，它们也具有相同的优先级。从左到右计算。例如，10 − 3 + 2 = 7 + 2 = 9，而不是 10 − 5 = 5。将减法视为加上一个负数可以让思路更清晰。
In long expressions, it helps to underline or circle the part you are computing at each stage. This reduces the chance of skipping operations or mixing up signs.
在较长的表达式中，在每个阶段对你正在计算的部分划线或画圈会有帮助。这可以减少跳过运算或混淆符号的机会。
Remember that the minus sign belongs to the number that follows, so it travels with that number when reordering terms with caution.
记住，减号属于紧跟在后的数字，因此在小心地重新排列各项时它会随该数移动。
10. Working with Multiple Operations | 混合运算
When a problem contains brackets, indices, multiplication, division, addition and subtraction all together, apply BIDMAS step by step. Example: 3 + 4 × (2 + 1)². First, brackets: 2 + 1 = 3. Then indices: 3² = 9. Next multiplication: 4 × 9 = 36. Finally, addition: 3 + 36 = 39.
当一个问题同时包含括号、指数、乘法、除法、加法和减法时，请一步步应用 BIDMAS。示例：3 + 4 × (2 + 1)²。首先，括号：2 + 1 = 3。接着指数：3² = 9。然后乘法：4 × 9 = 36。最后加法：3 + 36 = 39。
Write out each step on a new line, keeping the remaining parts of the expression unchanged. This method prevents errors and makes it easy to check your work.
每一步写在新的一行上，保持表达式的其余部分不变。这种方法可以防止错误，并且便于检查你的作业。
If you encounter division and multiplication in the same step, strictly move left to right. Pause after each major stage to ensure no operation is missed.
如果在同一步骤中遇到除法和乘法，请严格从左到右进行。在每个主要阶段之后暂停一下，确保没有遗漏任何运算。
11. Common Mistakes to Avoid | 常见错误避免
One frequent error is adding before multiplying because the operations were read left to right without applying BIDMAS. Always scan the whole expression first. Another is mishandling negative signs with indices: −5² is often mistakenly calculated as 25 instead of −25.
一个常见错误是由于未应用 BIDMAS 而直接从左到右读取表达式，结果先做了加法再做乘法。务必先浏览整个表达式。另一个错误是在处理指数和负号时出错：−5² 经常被错误地算作 25，而不是 −25。
Forgetting that a number in front of a bracket means multiplication can lead to trouble. 2(3+4) is 2×7, but some pupils write 23 or 27. Always insert the missing multiplication symbol mentally.
忘记括号前的数字表示乘法可能会导致麻烦。2(3+4) 表示 2×7，但有些学生会写成 23 或 27。请在心里补上缺失的乘号。
Division and multiplication being treated with a strict hierarchy rather than left‑to‑right order can also cause mistakes. Practise with mixed examples to break these habits.
在处理除法和乘法时，如果设定严格的等级区分而非从左到右的顺序，也会导致错误。通过混合练习来打破这些习惯。
12. Practice Examples and Tips | 练习示例与提示
Try solving these step by step: (a) 8 − 2 × (5 − 3)²; (b) (−4) × 6 ÷ (−2); (c) 20 ÷ 4 × (1 + 1)². For (a), brackets first: 5 − 3 = 2; indices: 2² = 4; multiplication: 2 × 4 = 8; then 8 − 8 = 0. For (b), (−4) × 6 = −24; −24 ÷ (−2) = 12. For (c), 20 ÷ 4 = 5; brackets: 1+1=2; indices: 2²=4; then 5 × 4 = 20.
请逐步尝试解答以下题目：(a) 8 − 2 × (5 − 3)²；(b) (−4) × 6 ÷ (−2)；(c) 20 ÷ 4 × (1 + 1)²。对于 (a)，先括号：5 − 3 = 2；指数：2² = 4；乘法：2 × 4 = 8；然后 8 − 8 = 0。对于 (b)，(−4) × 6 = −24；−24 ÷ (−2) = 12。对于 (c)，20 ÷ 4 = 5；括号：1+1=2；指数：2²=4；然后 5 × 4 = 20。
Checking your answer by recomputing with a different grouping or using a calculator that respects BIDMAS can give you extra confidence. Consistent practice with small daily sets of problems builds speed and accuracy.
通过不同的分组方式重新计算答案，或使用遵守 BIDMAS 的计算器进行检验，可以为你增添信心。坚持每天做一组小练习题，能够提高速度和准确性。
Keep a list of error patterns you have made, and review them before tests. Understanding integers and order of operations thoroughly now will unlock success in algebra, geometry and beyond.
将你犯过的错误类型列成清单，并在测试前加以复习。此刻彻底掌握整数和运算顺序，将为你在代数、几何及更广领域的学习开启成功之门。
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