Negative Numbers — Year 7 KS3 Mathematics Guide 负数完全指南

📚 Negative Numbers: A Complete Year 7 KS3 Guide | 负数完全指南（Year 7 KS3）

1. What Are Negative Numbers? The World Left of Zero | 什么是负数：数轴零点的左侧世界

在 Year 7 数学课上，我们第一次认识了一种比零还小的数，它们叫作负数。负数就是小于零的数，例如 -1、-3.5、-100 都是负数。数学家发明负数，是为了描述”缺少”、”低于”或”反向”的数量。想象一条水平数轴：0 在正中间，正数在 0 的右边，负数在 0 的左边。数轴向左延伸得越远，数就越小；向右延伸得越远，数就越大。因此 -10 在 -3 的左边，-10 比 -3 小。

In Year 7 mathematics, we meet a new kind of number for the first time: numbers smaller than zero, called negative numbers. A negative number is any number less than zero, such as -1, -3.5 or -100. Mathematicians invented negative numbers to describe quantities that are “missing”, “below” or “going in the opposite direction”. Imagine a horizontal number line: zero sits in the middle, positive numbers lie to the right of zero, and negative numbers lie to the left. The further left the number line extends, the smaller the numbers become; the further right, the larger. So -10 lies to the left of -3, which means -10 is smaller than -3.

负号 “-” 放在一个数前面，就表示这个数在零以下。注意，负号与减号长得一样，但含义不同：减号是运算符号，表示”减去”；负号是性质符号，表示”这个数是负的”。例如在算式 5 – 3 中，减号表示运算；而在 -5 中，负号说明 5 是负的。到了后面我们会看到，这种区别在运算中非常重要。

The minus sign “-” placed in front of a number shows that the number lies below zero. Note that the minus sign and the subtraction sign look identical, but they mean different things: subtraction is an operation meaning “take away”, while a negative sign is a property of the number itself, meaning “this number is negative”. For example, in the calculation 5 – 3 the minus is an operation, but in -5 the minus tells us that 5 is negative. Later we will see that this distinction matters greatly in calculations.

2. Negative Numbers in Real Life: Temperature, Altitude and Bank Balances | 负数的现实意义：温度、海拔与银行账户

负数并不是数学家的空想，它们在日常生活中随处可见。最典型的例子是温度：北京冬天的最低气温可以达到 -10°C，而莫斯科的冬天甚至可以降到 -30°C。天气预报说”零下五度”，写出来就是 -5°C。温度计上的刻度就是一条竖起来的数轴，0°C 是冰点，冰点以下就是负数温度。

Negative numbers are not a figment of mathematicians’ imagination; they appear everywhere in daily life. The most classic example is temperature: the lowest winter temperature in Beijing can reach -10°C, and winters in Moscow can drop below -30°C. When the weather forecast says “five degrees below zero”, it is written as -5°C. The scale on a thermometer is a vertical number line: 0°C is the freezing point, and everything below freezing is a negative temperature.

第二个常见场景是海拔。海平面的高度记作 0 米，陆地上的高山海拔为正数，例如珠穆朗玛峰约 8848 米；而低于海平面的地方，例如死海沿岸，海拔约为 -430 米。第三，银行账户也可能出现负数：如果你透支了 200 元，账户余额就显示为 -200 元，意思是”你欠银行 200 元”。此外，足球联赛的净胜球、电梯里地下车库的楼层（-1 层、-2 层）都在使用负数。

The second common setting is altitude. Sea level is recorded as 0 metres; mountains above the sea have positive altitudes, such as Mount Everest at about 8848 metres, while places below sea level, such as the shores of the Dead Sea, sit at about -430 metres. Third, bank accounts can go negative too: if you overdraw your account by 200 yuan, the balance reads -200 yuan, meaning “you owe the bank 200 yuan”. In addition, goal difference in football leagues, and the underground car-park floors in lifts (-1, -2), all make use of negative numbers.

场景 Situation	负数含义 Meaning of the negative
温度 Temperature	零下，低于冰点 Below freezing
海拔 Altitude	低于海平面 Below sea level
银行余额 Bank balance	透支，欠款 Overdraft, money owed
净胜球 Goal difference	失球多于进球 Conceded more than scored
楼层 Floors	地面以下 Below ground level

3. Comparing and Ordering Negative Numbers: Left Means Smaller | 比较与排序负数：数轴上越左越小

比较负数大小最容易犯的错误是”直觉反了”：-5 看起来比 -2 大，因为它有更大的数字 5。但别忘了数轴规则：越靠左的数越小。-5 在 -2 的左边，所以 -5 小于 -2，写作 -5 < -2。反过来，-2 大于 -5，写作 -2 > -5。一个实用的记忆法是”温度法”：-5°C 比 -2°C 更冷，更冷就是更小。

Comparing negative numbers is where intuition most easily goes wrong: -5 looks bigger than -2 because it contains the larger digit 5. But remember the number line rule: the further left, the smaller. Since -5 lies to the left of -2, -5 is less than -2, written -5 < -2. Conversely, -2 is greater than -5, written -2 > -5. A handy memory aid is the “temperature test”: -5°C is colder than -2°C, and colder means smaller.

排序时，可以先把所有数画在数轴上，再从左到右写出，就是从最小到最大。例如把 -3、2、-1、0、4 排序：它们在数轴上的顺序是 -3、-1、0、2、4，所以 -3 < -1 < 0 < 2 < 4。注意 0 比所有负数大，但比所有正数小；任何正数都大于任何负数。这一条规则请务必记牢。

To order a set of numbers, plot them all on the number line first, then read from left to right: that gives the order from smallest to largest. For example, to order -3, 2, -1, 0, 4: their positions on the line are -3, -1, 0, 2, 4, so -3 < -1 < 0 < 2 < 4. Notice that 0 is larger than every negative number but smaller than every positive number, and any positive number is greater than any negative number. Remember this rule firmly.

4. Adding and Subtracting Negative Numbers: The Rules of Sign Combination | 负数加法与减法：符号的”合并”规则

做负数加减法，可以把每个数看作”数轴上的移动”：加正数向右走，加负数向左走；减正数向左走，减负数向右走。例如 3 + (-5)：从 3 出发向左走 5 步，到达 -2，所以 3 + (-5) = -2。再如 -2 + 6：从 -2 出发向右走 6 步，到达 4，所以 -2 + 6 = 4。

For adding and subtracting negative numbers, think of each number as a movement on the number line: adding a positive moves right, adding a negative moves left, subtracting a positive moves left, and subtracting a negative moves right. For example, 3 + (-5): start at 3 and move 5 steps left, arriving at -2, so 3 + (-5) = -2. For -2 + 6: start at -2 and move 6 steps right, arriving at 4, so -2 + 6 = 4.

更快捷的符号合并规则如下：两个符号相同，就合并成一个加号，即正加正得正，负加负得负，并把绝对值相加；两个符号不同，就合并成一个减号，即大绝对值减小绝对值，符号取绝对值较大者的符号。例如 7 + (-3)：符号不同，7 – 3 = 4，符号取正的，答案是 4。又如 -6 + (-4)：符号相同（都是负），6 + 4 = 10，符号取负，答案是 -10。

The faster rule is sign combination: two identical signs merge into a plus, so positive plus positive stays positive and negative plus negative stays negative, and you add the absolute values; two different signs merge into a minus, so you subtract the smaller absolute value from the larger and take the sign of the larger absolute value. For example, 7 + (-3): the signs differ, 7 – 3 = 4, the sign is positive, so the answer is 4. For -6 + (-4): the signs are the same (both negative), 6 + 4 = 10, the sign is negative, so the answer is -10.

5. Subtracting a Negative Means Adding: The Double Negative Mystery | 减去负数等于加上正数：双重负号之谜

减法中有一条让很多同学困惑的规则：减去一个负数，等于加上它的相反数，也就是加上一个正数。用算式表达就是 5 – (-3) = 5 + 3 = 8。为什么？回到数轴：减去一个数就是向相反方向移动，减正数向左，那么减负数就向右，向右移动 3 步，结果当然和加 3 一样。

Subtraction contains a rule that confuses many students: subtracting a negative number is the same as adding its opposite, that is, adding a positive. In symbols, 5 – (-3) = 5 + 3 = 8. Why? Back to the number line: subtracting a number means moving in the opposite direction, so subtracting a positive moves left, and subtracting a negative therefore moves right; moving right by 3 gives exactly the same result as adding 3.

于是我们有了”负负得正”的双重负号规则：两个负号并排出现时，它们互相抵消变成加号。-4 – (-6) = -4 + 6 = 2；-10 – (-2) = -10 + 2 = -8。注意后一题：减去 -2 变成加 2，-10 加 2 仍然向左，结果是 -8，不是 -12。常见错误就是把 -10 – (-2) 算成 -12，那其实是 -10 + (-2) 的结果。

This gives us the double negative rule: when two minus signs appear side by side, they cancel each other out and become a plus. -4 – (-6) = -4 + 6 = 2, and -10 – (-2) = -10 + 2 = -8. Note the second example: subtracting -2 becomes adding 2, and -10 plus 2 is still to the left, giving -8, not -12. A common error is to compute -10 – (-2) as -12, which is actually the result of -10 + (-2).

6. Multiplying and Dividing Negative Numbers: The Sign Rules | 负数乘法与除法：正负得负，负负得正

乘法和除法只有两条规则，全部记住就不怕：同号相乘（除）得正，异号相乘（除）得负。也就是说，正正得正，负负得正，正负得负，负正得负。例如 3 x (-4) = -12，(-3) x 4 = -12，(-3) x (-4) = 12。除法同理：(-20) / 5 = -4，20 / (-5) = -4，(-20) / (-5) = 4。

Multiplication and division have only two rules, and once you remember them you are safe: same signs give a positive result, different signs give a negative result. In other words, positive times positive is positive, negative times negative is positive, positive times negative is negative, and negative times positive is negative. For example, 3 x (-4) = -12, (-3) x 4 = -12, and (-3) x (-4) = 12. Division works the same way: (-20) / 5 = -4, 20 / (-5) = -4, and (-20) / (-5) = 4.

符号组合 Sign pair	结果 Result	例子 Example
正 x 正 Positive x positive	正 Positive	2 x 3 = 6
正 x 负 Positive x negative	负 Negative	2 x (-3) = -6
负 x 正 Negative x positive	负 Negative	(-2) x 3 = -6
负 x 负 Negative x negative	正 Positive	(-2) x (-3) = 6

当算式里有多于两个负数相乘时，数一数负号的个数：负号个数为偶数，结果为正；负号个数为奇数，结果为负。例如 (-2) x (-3) x (-4)：三个负号，奇数个，结果必为负，2 x 3 x 4 = 24，所以答案是 -24。而 (-2) x (-3) x (-4) x (-5) 有四个负号，偶数个，答案是正的 120。

When more than two negative numbers are multiplied, count the negative signs: an even number of negatives gives a positive result, and an odd number gives a negative result. For example, (-2) x (-3) x (-4) has three negative signs, an odd number, so the result must be negative; 2 x 3 x 4 = 24, hence the answer is -24. Meanwhile (-2) x (-3) x (-4) x (-5) has four negative signs, an even number, so the answer is positive 120.

7. Order of Operations with Negative Numbers: The Power of Brackets | 运算顺序：括号与负数平方的陷阱

Year 7 已经学过运算顺序 BIDMAS：先算括号（Brackets），再算指数（Indices），然后是除法与乘法（Division and Multiplication），最后是加法与减法（Addition and Subtraction）。引入负数后，最经典的陷阱是指数与负号的配合：-3² 与 (-3)² 结果完全不同。

By Year 7 you already know the order of operations BIDMAS: Brackets first, then Indices, then Division and Multiplication, and finally Addition and Subtraction. Once negative numbers enter the picture, the classic trap is how indices interact with the minus sign: -3² and (-3)² give completely different results.

在 -3² 中，没有括号，指数 2 只作用于 3，不作用于负号，所以 -3² = -(3 x 3) = -9。而在 (-3)² 中，括号把 -3 整个括起来，指数作用于整个负数，所以 (-3)² = (-3) x (-3) = 9。一句话：负号的平方，必须先加括号才得正；不加括号，负号留在外面。考试中这是高频考点，务必看清括号。

In -3² there is no bracket, so the index 2 applies only to the 3, not to the minus sign, giving -3² = -(3 x 3) = -9. In (-3)², however, the bracket encloses the whole of -3, so the index applies to the entire negative number, giving (-3)² = (-3) x (-3) = 9. In one sentence: to square a negative number you must bracket it first to get a positive; without brackets, the minus sign stays outside. This is a high-frequency exam point, so always look carefully for brackets.

再看一个综合例子：2 + 3 x (-4) – (-5)。按 BIDMAS：先算乘法 3 x (-4) = -12，算式变成 2 + (-12) – (-5)；接着从左到右，2 + (-12) = -10，-10 – (-5) = -10 + 5 = -5。所以整道题的结果是 -5。注意不能先算 2 + 3，因为加法在乘法之后。

Now consider a combined example: 2 + 3 x (-4) – (-5). By BIDMAS: first the multiplication 3 x (-4) = -12, so the expression becomes 2 + (-12) – (-5); then working left to right, 2 + (-12) = -10, and -10 – (-5) = -10 + 5 = -5. So the whole expression evaluates to -5. Note that you must not add 2 + 3 first, because addition comes after multiplication.

8. Common Mistakes and Traps: Why Negative Numbers Go Wrong | 常见错误与陷阱：为什么总是算错

几乎所有 Year 7 学生都在负数上栽过跟头。第一个高频错误是”比较大小时直觉颠倒”，把 -5 当成比 -2 大。对策：永远回到数轴或温度去验证，-5°C 更冷，所以 -5 更小。第二个高频错误是漏写负号：例如 6 – 9 算成 3，正确答案是 -3。记住：小的正数减大的正数，结果必为负。

Almost every Year 7 student has tripped over negative numbers. The first high-frequency error is reversing intuition when comparing, treating -5 as larger than -2. The remedy: always go back to the number line or to temperature, -5°C is colder, so -5 is smaller. The second common error is dropping the minus sign: for example computing 6 – 9 as 3, when the correct answer is -3. Remember: a smaller positive minus a larger positive always gives a negative result.

错误错误 Wrong	正确 Correct	原因 Reason
-5 > -2	-5 < -2	数轴上越左越小 Further left is smaller
-10 – (-2) = -12	-10 – (-2) = -8	减负等于加正 Subtracting a negative adds
-3² = 9	-3² = -9	指数不作用于负号 Index applies to 3 only
(-3) x (-4) = -12	(-3) x (-4) = 12	负负得正 Negative x negative is positive
6 – 9 = 3	6 – 9 = -3	小减大必为负 Smaller minus larger is negative

第三个陷阱是忘记”减负得正”而把负号直接丢掉。第四个陷阱是依赖计算器却不理解原理：计算器能给你答案，但考试时你必须在纸上独立完成。每次算完，养成”验号”的习惯：先定符号，再算数值，两步分开做，错误率会大幅下降。

The third trap is dropping the minus sign without applying “subtracting a negative adds”. The fourth trap is relying on a calculator without understanding the principles: a calculator gives you the answer, but in the exam you must work independently on paper. After every calculation, form the habit of “checking the sign first”: decide the sign, then compute the value, keeping the two steps separate; this dramatically reduces your error rate.

9. Problem Solving with Negative Numbers: Strategies for Word Problems | 负数应用题：实际场景中的解题策略

应用题的关键是把生活语言翻译成数学语言。经典题型一：温度变化。某地早晨气温 -3°C，中午上升了 8°C，问中午气温。上升 8°C 就是加 8：-3 + 8 = 5，中午 5°C。如果傍晚又下降 12°C，那么 5 – 12 = -7，傍晚 -7°C。注意”上升”对应加，”下降”对应减。

The key to word problems is translating everyday language into mathematical language. Classic type one: temperature change. The morning temperature is -3°C and it rises by 8°C by noon; what is the noon temperature? A rise of 8°C means add 8: -3 + 8 = 5, so noon is 5°C. If it then falls by 12°C by evening, then 5 – 12 = -7, so the evening temperature is -7°C. Note that “rises” maps to addition and “falls” maps to subtraction.

经典题型二：海拔差。山顶海拔 1200 米，谷底海拔 -150 米，问山顶比谷底高多少米。求”相差多少”用减法：1200 – (-150) = 1200 + 150 = 1350 米。这一步最容易错，因为”高多少”被想成”1200 – 150″；但实际上谷底在海平面以下，要跨过 0 米，所以必须处理负号。

Classic type two: altitude difference. A mountain top is at 1200 metres and a valley floor is at -150 metres; how much higher is the top than the valley? “How much higher” means subtraction: 1200 – (-150) = 1200 + 150 = 1350 metres. This step is the easiest to get wrong, because “higher by how much” tempts you into 1200 – 150; but the valley is below sea level, so you must cross 0 metres and therefore handle the negative sign.

经典题型三：比分与净胜球。一支球队第一轮净胜球为 -3，第二轮又丢了 2 个球（净胜球再减 2），问两轮合计。计算：-3 + (-2) = -5。如果第三轮进了 9 球丢了 1 球（净胜 +8），合计 -5 + 8 = 3，最终净胜球为正的 3。解题步骤建议：第一步找关键词定运算（上升加、下降减、相差减）；第二步写算式；第三步先定符号再算数值；第四步回代检查合理性。

Classic type three: scores and goal difference. A team finishes round one with a goal difference of -3, then concedes 2 more goals in round two (goal difference falls by 2); what is the total after two rounds? Calculate: -3 + (-2) = -5. If in round three they score 9 and concede 1 (a gain of +8), the total becomes -5 + 8 = 3, a positive goal difference of 3. Suggested problem-solving steps: first, find the key words to decide the operation (rises means add, falls means subtract, difference means subtract); second, write the calculation; third, fix the sign before computing the value; fourth, check that the answer makes sense in the story.

10. Mixed Practice and Challenge Questions | 综合练习与提高题

下面的练习题覆盖了本章所有知识点，请先在纸上独立完成，再对照答案。第 1 题：计算 8 + (-3)。第 2 题：计算 -7 + (-2)。第 3 题：计算 5 – (-9)。第 4 题：计算 -4 – 6。第 5 题：计算 (-6) x 4。第 6 题：计算 (-8) x (-5)。第 7 题：计算 (-36) / (-9)。第 8 题：计算 -2² 与 (-2)²。第 9 题：把 -7、3、-1、0、-4 从小到大排列。第 10 题：某地温度从 -6°C 上升 10°C，再下降 4°C，最终温度是多少？

The practice questions below cover every knowledge point in this chapter. Please work through them independently on paper before checking the answers. Question 1: calculate 8 + (-3). Question 2: calculate -7 + (-2). Question 3: calculate 5 – (-9). Question 4: calculate -4 – 6. Question 5: calculate (-6) x 4. Question 6: calculate (-8) x (-5). Question 7: calculate (-36) / (-9). Question 8: calculate -2² and (-2)². Question 9: arrange -7, 3, -1, 0, -4 from smallest to largest. Question 10: a place starts at -6°C, rises 10°C, then falls 4°C; what is the final temperature?

答案与简解：第 1 题 5，异号相减取正号。第 2 题 -9，同号相加取负号。第 3 题 14，减负得正，5 + 9。第 4 题 -10，相当于 -4 + (-6)。第 5 题 -24，异号得负。第 6 题 40，负负得正。第 7 题 4，同号相除得正。第 8 题 -9 与 9，注意括号区别。第 9 题 -7 < -4 < -1 < 0 < 3。第 10 题：-6 + 10 – 4 = 0，最终 0°C。全部做对的同学已经掌握了 Year 7 负数的核心；做错的同学请对照上面的规则找出错在哪一步。

Answers and brief solutions: Question 1: 5, different signs, subtract and take the positive sign. Question 2: -9, same signs, add and take the negative sign. Question 3: 14, subtracting a negative adds, 5 + 9. Question 4: -10, equivalent to -4 + (-6). Question 5: -24, different signs give negative. Question 6: 40, negative times negative is positive. Question 7: 4, same signs in division give positive. Question 8: -9 and 9, note the difference the brackets make. Question 9: -7 < -4 < -1 < 0 < 3. Question 10: -6 + 10 – 4 = 0, so the final temperature is 0°C. If you answered every question correctly, you have mastered the core of Year 7 negative numbers; if not, go back to the rules above and find exactly which step went wrong.

Summary | 总结

本章我们完整学习了 Year 7 负数的核心知识。首先，负数是小于零的数，在数轴上位于 0 的左侧，越左越小，任何负数都小于 0 小于任何正数。其次，负数广泛存在于温度、海拔、银行余额等现实场景中，把生活语言翻译成加减运算时要抓住”上升加、下降减、相差减”等关键词。第三，加法减法遵循符号合并规则：同号相加、异号相减取绝对值大者的符号；减去一个负数等于加上它的相反数。

In this chapter we have studied the core of Year 7 negative numbers. First, negative numbers are numbers less than zero, located to the left of 0 on the number line; the further left, the smaller, and every negative number is less than 0 and less than every positive number. Second, negative numbers appear widely in real life, in temperature, altitude and bank balances, and when translating everyday language into arithmetic you should catch key words such as “rises means add, falls means subtract, difference means subtract”. Third, addition and subtraction follow the sign-combination rules: same signs add, different signs subtract and take the sign of the larger absolute value; subtracting a negative is the same as adding its opposite.

第四，乘法和除法遵循两条简洁的规则：同号得正，异号得负；多个负数相乘时，负号个数为奇数则结果为负，为偶数则结果为正。第五，运算顺序 BIDMAS 在负数中同样适用，特别要警惕 -3² 与 (-3)² 的区别：没有括号时，指数不作用于负号。最后，任何计算都建议”先定符号，再算数值”，并用数轴或温度等实际场景验证答案是否合理。掌握这些规则并反复练习，负数将不再是 Year 7 数学的拦路虎。

Fourth, multiplication and division follow two simple rules: same signs give positive, different signs give negative; when several negative numbers are multiplied, an odd count of negative signs gives a negative result and an even count gives a positive result. Fifth, the order of operations BIDMAS applies equally with negatives, and you must be especially wary of the difference between -3² and (-3)²: without brackets, the index does not apply to the minus sign. Finally, for any calculation, it is wise to “fix the sign first, then compute the value”, and to use the number line or real-life settings such as temperature to check whether the answer is sensible. Master these rules and practise repeatedly, and negative numbers will no longer be a stumbling block in Year 7 mathematics.

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