Negative Numbers: A Complete Guide for Year 7 — 负数运算完全指南：七年级数学

📚 Negative Numbers: A Complete Guide for Year 7 | 负数运算完全指南：七年级数学

负数（negative numbers）是七年级数学中最重要、也最容易出错的章节之一。很多同学在小学阶段只接触过正数和零，进入中学后第一次遇到”比零还小的数”，往往会感到困惑：负负为什么得正？减去一个负数为什么要变成加法？本篇文章将用中英双语、结合数轴和生活实例，把负数的概念、四则运算法则、常见错误和应用题完整讲透。

Negative numbers are among the most important and most error-prone topics in Year 7 mathematics. Many students only meet positive numbers and zero in primary school, so the first time they encounter numbers “smaller than zero” in secondary school, they often feel confused: why does a negative times a negative give a positive? Why does subtracting a negative turn into addition? This article explains the concept of negative numbers, the four operation rules, common mistakes and applied problems thoroughly, in both Chinese and English, using the number line and real-life examples.

1. 什么是负数：数轴上的位置与顺序 | What Are Negative Numbers: Position and Order on the Number Line

在小学里，我们认识的自然数 0、1、2、3…… 都表示”有多少个物体”。但世界上有一些量天生就”比零还少”：零下五度的气温、海平面以下三米、银行卡里欠款两百元。为了表示这些量，数学家引入了负数。负数就是在数（positive numbers）前面加上负号（minus sign）的数，例如 -1、-3.5、-100。

In primary school, we learned the natural numbers 0, 1, 2, 3 … which describe “how many objects there are”. But some quantities in the world are naturally “less than zero”: a temperature of five degrees below zero, a point three metres below sea level, a bank account overdrawn by two hundred yuan. To describe these quantities, mathematicians introduced negative numbers. A negative number is a positive number with a minus sign in front of it, for example -1, -3.5 and -100.

理解负数最好的工具是数轴（number line）。数轴是一条水平直线，向右为正方向，向左为负方向，0 是正数和负数的分界点。在数轴上，越靠右的数越大，越靠左的数越小。因此 -2 比 -1 小，-5 比 -3 小；任何负数都小于 0，而任何正数都大于 0。例如：-5 < -2 < 0 < 1 < 3。

The best tool for understanding negative numbers is the number line. A number line is a horizontal straight line: to the right is the positive direction, to the left is the negative direction, and 0 is the boundary between positive and negative numbers. On the number line, numbers further to the right are larger, and numbers further to the left are smaller. Therefore -2 is smaller than -1, and -5 is smaller than -3; every negative number is less than 0, while every positive number is greater than 0. For example: -5 < -2 < 0 < 1 < 3.

请记住一个容易混淆的点：负数的大小比较与它们的”绝对值”大小相反。绝对值（absolute value）表示一个数到 0 的距离，用两条竖线表示，例如 |−5| = 5。虽然 5 比 3 大，但 -5 却比 -3 小，因为 -5 在数轴上更靠左。比较负数时，可以先看绝对值，绝对值大的那个负数反而更小。

Remember one easily confused point: comparing the sizes of negative numbers is the opposite of comparing their absolute values. The absolute value of a number is its distance from 0, written with two vertical bars, for example |−5| = 5. Although 5 is bigger than 3, -5 is smaller than -3, because -5 lies further to the left on the number line. When comparing negative numbers, look at their absolute values first: the negative number with the larger absolute value is actually the smaller one.

规则 Rule	例子 Example
任何正数 > 0 > 任何负数 Any positive > 0 > any negative	-7 < 0 < 0.5
两个负数比较：绝对值大的更小 Of two negatives, the one with the larger absolute value is smaller	|-9|=9, |-4|=4, 所以 -9 < -4
数轴上越靠左越小 Further left on the number line means smaller	-6 < -1 < 2

2. 负数的实际含义：温度、海拔与银行余额 | Real-World Meanings: Temperature, Sea Level and Bank Balances

负数不是数学家凭空发明的抽象符号，它在日常生活中无处不在。最典型的例子是温度。摄氏温度（degrees Celsius）以水的冰点 0°C 为基准：北京冬天可能到 -10°C，这意味着比冰点还低 10 度。天气预报里说的”最低气温零下三度”，用数学符号写出来就是 -3°C。

Negative numbers are not abstract symbols invented out of thin air by mathematicians; they appear everywhere in daily life. The most typical example is temperature. The Celsius scale uses the freezing point of water, 0°C, as its reference: Beijing can reach -10°C in winter, which means 10 degrees lower than freezing. When a weather forecast says “the lowest temperature is three below zero”, the mathematical notation is -3°C.

第二个常见场景是海拔（height above sea level）。地理学以海平面为 0 米基准，珠穆朗玛峰的海拔约 8848 米，而吐鲁番盆地的艾丁湖湖面低于海平面约 154 米，记为 -154 米。飞机飞行的高度、潜水员下潜的深度，也都用正负数来区分”海平面之上”与”海平面之下”。

The second common context is height above sea level. Geography uses sea level as the 0-metre reference: Mount Everest is about 8848 metres above sea level, while Aydingkol Lake in the Turpan Basin lies about 154 metres below sea level, written as -154 metres. Aircraft altitudes and diver depths also use positive and negative numbers to distinguish “above sea level” from “below sea level”.

第三个场景是银行账户与财务。存入 500 元记作 +500（或直接写 500），透支 200 元记作 -200。余额为 -200 表示”欠银行 200 元”。温度计、电梯楼层（地下车库 B1、B2）、比赛净胜球数（goal difference）、游戏得分，全都是负数在日常中的用武之地。

The third context is bank accounts and finance. Depositing 500 yuan is recorded as +500 (or simply 500), while an overdraft of 200 yuan is recorded as -200. A balance of -200 means “you owe the bank 200 yuan”. Thermometers, lift floors (basements B1, B2), goal differences in football, and game scores are all places where negative numbers do real work in daily life.

理解负数的现实意义非常重要：它能帮助你把抽象的运算规则”翻译”成可以想象的情景。例如”温度从 5°C 下降到 -3°C，一共降了多少度”，这个问题本质上就是计算 5 – (-3)，答案是 8 度。有了生活背景，负数的加减就不再是死记硬背的符号游戏。

Understanding the real meaning of negative numbers is very important: it helps you “translate” abstract operation rules into situations you can imagine. For example, “the temperature falls from 5°C to -3°C; how many degrees does it drop in total?” is essentially calculating 5 – (-3), and the answer is 8 degrees. With a real-life background, adding and subtracting negative numbers is no longer a game of memorising symbols by rote.

3. 同号相加：正正得正，负负得负 | Adding Numbers with the Same Sign: Positive plus Positive, Negative plus Negative

加法法则的第一条：同号（same sign）的两个数相加，结果的符号不变，绝对值相加。也就是说，两个正数相加得正数，两个负数相加得负数，数值部分就是两个绝对值的和。例如 3 + 5 = 8，(-3) + (-5) = -8。

The first rule of addition: when two numbers with the same sign are added, the sign of the result stays the same, and the absolute values are added. In other words, two positive numbers give a positive result, two negative numbers give a negative result, and the numerical part is the sum of the two absolute values. For example, 3 + 5 = 8 and (-3) + (-5) = -8.

为什么两个负数相加还是负数？回到数轴上看：从 0 出发，先向左走 3 步到达 -3，再向左走 5 步，就到达 -8。两次都向左，方向没有改变，只是距离越走越远。用温度来理解：零下 3 度再降温 5 度，当然变成零下 8 度。

Why does adding two negative numbers still give a negative? Go back to the number line: starting from 0, walk 3 steps to the left to reach -3, then walk 5 more steps to the left, and you arrive at -8. Both walks are to the left, so the direction never changes; you simply travel further and further away. Think in terms of temperature: if it is -3 degrees and it gets 5 degrees colder, of course it becomes -8 degrees.

在书写时要注意括号的使用。习惯上，当负数和运算符号连在一起时，我们加上括号避免混淆，例如 (-3) + (-5)，而不是写成 -3 + -5（虽然两种写法数学上等价，但考试中请按教材规范书写）。如果题目没有括号，例如 -3 – 5，它表示的是 (-3) – (+5)，结果仍然是 -8。

Be careful with brackets when writing. By convention, when a negative number sits next to an operation sign, we add brackets to avoid confusion, for example (-3) + (-5), rather than writing -3 + -5 (although both forms are mathematically equivalent, follow your textbook convention in exams). If a question has no brackets, for example -3 – 5, it means (-3) – (+5), and the result is still -8.

4. 异号相加：数轴上的”走格子” | Adding Numbers with Different Signs: Walking Steps on the Number Line

加法法则的第二条：异号（different signs）的两个数相加，结果的符号由绝对值较大的那个数决定，数值部分是大绝对值减去小绝对值。例如 7 + (-4)：7 的绝对值大，所以结果为正，数值为 7 – 4 = 3，即 7 + (-4) = 3。又如 (-9) + 4：9 的绝对值大，结果为负，数值为 9 – 4 = 5，所以 (-9) + 4 = -5。

The second rule of addition: when two numbers with different signs are added, the sign of the result is decided by the number with the larger absolute value, and the numerical part is the larger absolute value minus the smaller one. For example 7 + (-4): 7 has the larger absolute value, so the result is positive, and the numerical part is 7 – 4 = 3, so 7 + (-4) = 3. Another example: (-9) + 4: 9 has the larger absolute value, the result is negative, and 9 – 4 = 5, so (-9) + 4 = -5.

用数轴理解异号相加最直观。计算 3 + (-7)：从 0 出发先向右走 3 步到 3，再向左走 7 步，最后停在 -4。你也可以换个顺序理解：向左走的 7 步先”抵消”掉向右的 3 步，还剩下向左的 4 步，所以答案是 -4。异号相加的本质就是”抵消”（cancelling out）。

The number line makes different-sign addition most intuitive. To calculate 3 + (-7): start from 0, walk 3 steps to the right to reach 3, then walk 7 steps to the left, finally stopping at -4. You can also think in a different order: the 7 leftward steps first “cancel” the 3 rightward steps, leaving 4 leftward steps, so the answer is -4. The essence of different-sign addition is cancelling out.

类比”正负数相抵”：你可以把正数想象成赚到的钱，负数想象成花掉的钱。今天赚了 7 元又花了 4 元，净赚 3 元，即 7 + (-4) = 3；如果赚了 4 元却花了 9 元，净亏 5 元，即 4 + (-9) = -5。赚钱花钱的直觉和数轴的方向完全一致。

Here is an analogy for positive and negative numbers cancelling: imagine positive numbers as money earned and negative numbers as money spent. Today you earn 7 yuan and spend 4 yuan, a net gain of 3 yuan, so 7 + (-4) = 3; if you earn 4 yuan but spend 9 yuan, you have a net loss of 5 yuan, so 4 + (-9) = -5. The earning-and-spending intuition matches the direction of the number line perfectly.

算式 Calculation	口诀 Shortcut	答案 Answer
5 + (-2)	正大，结果正 Positive wins	3
(-5) + 2	负大，结果负 Negative wins	-3
(-4) + 9	正大，结果正 Positive wins	5
(-8) + (-1)	同号，相加 Same sign, add	-9

5. 减法与负号：减去一个负数等于加上它的相反数 | Subtraction and the Minus Sign: Subtracting a Negative Is Adding Its Opposite

减法法则是最让学生头疼的一条：减去一个数，等于加上这个数的相反数（opposite）。也就是说，减法可以统一变成加法来处理：a – b = a + (-b)，而 a – (-b) = a + b。关键结论：减去一个负数，等于加上一个正数，负负得正！

The subtraction rule is the one that troubles students most: subtracting a number is the same as adding its opposite. In other words, subtraction can always be converted into addition: a – b = a + (-b), and a – (-b) = a + b. The key conclusion: subtracting a negative number is the same as adding a positive number, and two negatives make a positive!

用数轴验证一下：计算 4 – (-3)。”减”在数轴上表示”向左走”，但 -3 本身又表示”向左 3 步”，连续两个向左的指令互相抵消，就变成了向右 3 步，于是 4 – (-3) = 4 + 3 = 7。这就是为什么”减负等于加正”。

Verify this on the number line: calculate 4 – (-3). “Subtract” on the number line means “walk left”, but -3 itself also means “walk 3 steps left”; two consecutive leftward instructions cancel each other out, becoming 3 steps to the right, so 4 – (-3) = 4 + 3 = 7. This is why “subtracting a negative equals adding a positive”.

生活类比：今天的气温是 4°C，天气预报说明天比今天”低 -3 度”（也就是高 3 度），明天的气温就是 4 – (-3) = 7°C。”低负三度”这种表达虽然绕口，但在数学题里经常出现。另一个类比是欠债：你欠别人 3 元（-3），如果这笔债被免除（减去 -3），你的财富就增加了 3 元。

A real-life analogy: today’s temperature is 4°C, and the forecast says tomorrow will be “3 degrees lower than the negative” (that is, 3 degrees higher), so tomorrow’s temperature is 4 – (-3) = 7°C. The phrase “lower by negative three degrees” sounds awkward, but it appears often in maths questions. Another analogy is debt: you owe someone 3 yuan (-3), and if that debt is forgiven (subtracting -3), your wealth increases by 3 yuan.

熟练之后，请记住这两条等价变形：见到 “x – (-y)” 直接改写成 “x + y”；见到 “x + (-y)” 改写成 “x – y”。例如 8 – (-2) = 8 + 2 = 10，(-6) – (-1) = -6 + 1 = -5。把减法全部转化为加法后，就可以统一使用”同号相加、异号相抵”的法则了。

Once you are fluent, remember these two equivalent transformations: whenever you see “x – (-y)”, rewrite it as “x + y”; whenever you see “x + (-y)”, rewrite it as “x – y”. For example 8 – (-2) = 8 + 2 = 10, and (-6) – (-1) = -6 + 1 = -5. Once all subtraction is converted to addition, you can uniformly apply the “same sign adds, different signs cancel” rule.

6. 乘法与除法的符号法则：同号为正，异号为负 | Sign Rules for Multiplication and Division: Same Signs Give Positive, Different Signs Give Negative

乘法和除法遵循同一条符号法则：同号相乘（除）得正，异号相乘（除）得负，数值部分照常计算。具体来说：(正) × (正) = 正，(负) × (负) = 正，(正) × (负) = 负，(负) × (正) = 负。例如 3 × 4 = 12，(-3) × (-4) = 12，(-3) × 4 = -12，3 × (-4) = -12。

Multiplication and division follow the same sign rule: same signs give a positive result, different signs give a negative result, and the numerical part is calculated normally. Specifically: positive times positive is positive, negative times negative is positive, positive times negative is negative, and negative times positive is negative. For example 3 × 4 = 12, (-3) × (-4) = 12, (-3) × 4 = -12, and 3 × (-4) = -12.

“负负得正”为什么成立？可以用重复加法来直观理解。3 × (-4) 表示 3 个 -4 相加，即 (-4) + (-4) + (-4) = -12，这很自然。而 (-3) × (-4) 可以理解为”-(3 × (-4))”，也就是 -(-12) = 12。另一种理解：乘以负数相当于”反向”，方向反转两次就回到原方向，正如转身两次回到面对原处。

Why does “negative times negative make positive” hold? You can understand it intuitively through repeated addition. 3 × (-4) means adding -4 three times, that is (-4) + (-4) + (-4) = -12, which is natural. And (-3) × (-4) can be understood as “-(3 × (-4))”, that is -(-12) = 12. Another way to see it: multiplying by a negative means “reversing direction”, and reversing direction twice returns you to the original direction, just as turning around twice leaves you facing the same way.

除法完全同理：(-20) ÷ 5 = -4，20 ÷ (-5) = -4，(-20) ÷ (-5) = 4。你可以随时用乘法来检验除法结果：因为 (-4) × 5 = -20，所以 (-20) ÷ 5 = -4 一定正确。除法的符号法则与乘法完全一致，可以合并记忆为一句口诀：”同号得正，异号得负”（Same signs positive, different signs negative）。

Division works exactly the same way: (-20) ÷ 5 = -4, 20 ÷ (-5) = -4, and (-20) ÷ (-5) = 4. You can always check a division result with multiplication: since (-4) × 5 = -20, (-20) ÷ 5 = -4 must be correct. The sign rule for division is identical to multiplication, so memorise both with one phrase: “same signs positive, different signs negative”.

多个负数连乘时要小心：两个负数相乘得正，三个负数相乘得负，四个负数相乘又得正。规律是：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) × (-3) × (-4) 有三个负号，结果为负数：-24；再加一个 (-1) 变成四个负号，结果为正：24。

Be careful when multiplying several negative numbers together: two negatives give a positive, three negatives give a negative, and four negatives give a positive again. The pattern is: an even number of minus signs gives a positive result, and an odd number of minus signs gives a negative result. For example (-2) × (-3) × (-4) has three minus signs and the result is negative: -24; multiply by another (-1) to make four minus signs and the result becomes positive: 24.

7. 负数与括号：BIDMAS 运算顺序 | Negative Numbers and Brackets: The BIDMAS Order of Operations

当负数、括号、乘方和四则运算混在一起时，必须严格遵守运算顺序 BIDMAS：先算括号（Brackets），再算指数（Indices），然后乘除（Division and Multiplication，从左到右），最后加减（Addition and Subtraction，从左到右）。口诀可以记为”先括号、后乘方、再乘除、最后加减”。

When negative numbers, brackets, powers and the four operations are mixed together, you must strictly follow the order of operations BIDMAS: Brackets first, then Indices, then Division and Multiplication (from left to right), and finally Addition and Subtraction (from left to right). You can remember it as “brackets first, then powers, then multiply and divide, then add and subtract”.

看看括号如何改变结果。计算 10 – 3 + 2：按从左到右的顺序，10 – 3 = 7，再加 2 得 9。但如果题目写成 10 – (3 + 2)，先算括号内 3 + 2 = 5，再算 10 – 5 = 5。同一个题目，括号不同，答案完全不同。遇到带负号的括号时尤其要小心，例如 8 – (-3 + 5) = 8 – 2 = 6。

See how brackets change the result. Calculate 10 – 3 + 2: working from left to right, 10 – 3 = 7, then adding 2 gives 9. But if the question is written as 10 – (3 + 2), you first work out the bracket: 3 + 2 = 5, then 10 – 5 = 5. The same numbers with different brackets give completely different answers. Be especially careful with brackets containing negative numbers, for example 8 – (-3 + 5) = 8 – 2 = 6.

去括号法则（removing brackets）也是高频考点：括号前是加号，去掉括号后各项符号不变；括号前是减号，去掉括号后各项都要变号（正变负、负变正）。例如 a + (b – c) = a + b – c，而 a – (b – c) = a – b + c。用数值检验：7 – (3 – 2) = 7 – 3 + 2 = 6，与直接计算 7 – 1 = 6 一致。

The rule for removing brackets is also a frequent exam topic: when a plus sign stands before a bracket, the signs of all terms inside stay unchanged after removing the bracket; when a minus sign stands before a bracket, every term inside must change sign (positive becomes negative, negative becomes positive). For example a + (b – c) = a + b – c, while a – (b – c) = a – b + c. Check with numbers: 7 – (3 – 2) = 7 – 3 + 2 = 6, which agrees with computing 7 – 1 = 6 directly.

题目 Question	正确步骤 Correct Steps	答案 Answer
(-2) × (5 – 8)	先算括号 5 – 8 = -3，再算 (-2) × (-3)	6
-3²	先算乘方 3² = 9，再加负号（无括号！）	-9
(-3)²	括号内 -3 整体平方	9
12 ÷ (-2) × 3	从左到右：12 ÷ (-2) = -6，再乘 3	-18

特别注意 -3² 与 (-3)² 的区别：-3² 表示”3 的平方的相反数”，答案是 -9；而 (-3)² 表示”负三的平方”，(-3) × (-3) = 9。这一字之差是考试中最经典的陷阱题，每年都有大量学生在此失分。记住：负号在括号内才参与乘方，在括号外则最后处理。

Pay special attention to the difference between -3² and (-3)²: -3² means “the opposite of 3 squared” and equals -9, while (-3)² means “negative three squared”, that is (-3) × (-3) = 9. This tiny difference is the classic trap question in exams, and large numbers of students lose marks on it every year. Remember: the minus sign takes part in the power only when it is inside the brackets; outside the brackets it is handled last.

8. 常见误区：学生最容易犯的五个错误 | Common Misconceptions: The Five Mistakes Students Make Most

误区一：把 -5 和 5 当成”一样的数”。它们的绝对值确实都是 5，但在数轴上方向完全相反。-5 表示”零下五度”，5 表示”零上五度”，相差 10 度。任何比较大小的题目，都要先看符号，再看绝对值。

Mistake 1: treating -5 and 5 as “the same number”. Their absolute values are indeed both 5, but on the number line they point in completely opposite directions. -5 means “five below zero” and 5 means “five above zero”, a difference of 10 degrees. In any size-comparison question, look at the sign first, then at the absolute value.

误区二：认为”减去一个数”和”减去一个负数”一样。3 – 5 = -2，但 3 – (-5) = 8，结果差得很远。只要见到减号后面跟着负号，立即把”减负”改写成”加正”，再做加法。这个动作要形成肌肉记忆。

Mistake 2: thinking “subtracting a number” and “subtracting a negative number” are the same. 3 – 5 = -2, but 3 – (-5) = 8; the results are far apart. Whenever you see a minus sign followed by a negative number, immediately rewrite “subtracting a negative” as “adding a positive” and then add. This action should become muscle memory.

误区三：混淆”大数减小数”的顺序。很多人习惯用”大数减去小数”，于是把 5 – 8 算成 3。正确的做法是严格从左到右：5 – 8 = -3。数轴上从 5 向左走 8 步，停在 -3。任何时候不要擅自交换被减数和减数。

Mistake 3: confusing the order of “big number minus small number”. Many people are used to “larger minus smaller”, so they compute 5 – 8 as 3. The correct approach is strictly left to right: 5 – 8 = -3. On the number line, walk 8 steps left from 5 and you stop at -3. Never swap the minuend and subtrahend on your own.

误区四：漏掉符号。计算 (-3) × 4 时算出数值 12 却忘记写负号，写成 12。每做完一步，都要回头检查结果的符号是否与法则一致。建议在草稿纸上先写出符号判断（”异号，结果负”），再写数值，最后合并。

Mistake 4: dropping the sign. When calculating (-3) × 4, students work out the value 12 but forget the minus sign and write 12. After every step, check that the sign of the result agrees with the rules. On your rough paper, first write the sign judgement (“different signs, result negative”), then the value, and finally combine them.

误区五：-3² 与 (-3)² 不分。前者是 -9，后者是 9。这个错误在七年级乃至九年级的考试中都反复出现。破解方法很简单：看到乘方，先看负号是否在括号内，在括号内就一起乘方，在括号外就最后加负号。

Mistake 5: confusing -3² with (-3)². The first is -9, the second is 9. This error keeps appearing in exams from Year 7 all the way to Year 9. The solution is simple: when you see a power, check whether the minus sign is inside the brackets; if it is, include it in the power; if it is outside, apply the minus sign last.

9. 综合应用题：温度差、海拔差与账目计算 | Applied Problems: Temperature Differences, Height Differences and Account Calculations

应用题最能检验你对负数运算是否真正理解。第一类经典题目是温度差：某地早晨气温 -4°C，中午升到 9°C，问温度上升了多少度？列式 9 – (-4) = 9 + 4 = 13，上升了 13 度。注意”从 -4 到 9″跨越的格数是 13，而不是 5。

Applied problems best test whether you truly understand negative number operations. The first classic type is temperature difference: one morning the temperature is -4°C and at noon it rises to 9°C; how many degrees did it rise? The calculation is 9 – (-4) = 9 + 4 = 13, so it rose 13 degrees. Note that the number of steps from -4 to 9 is 13, not 5.

第二类经典题目是海拔差：珠穆朗玛峰海拔 8848 米，吐鲁番艾丁湖湖面海拔 -154 米，两者的相对高度是多少？列式 8848 – (-154) = 8848 + 154 = 9002 米。这类题的关键是识别”高差 = 高处海拔 – 低处海拔”，而低处海拔是负数时，就变成了加。

The second classic type is height difference: Mount Everest is 8848 metres above sea level and Aydingkol Lake is -154 metres above sea level; what is the vertical separation between them? The calculation is 8848 – (-154) = 8848 + 154 = 9002 metres. The key to this type is recognising that “difference = higher altitude minus lower altitude”, and when the lower altitude is negative, the subtraction becomes addition.

第三类经典题目是账目计算。小明的账户余额是 -35 元（欠款 35 元），他存入 100 元后又转账支出 20 元，问最终余额？列式：-35 + 100 – 20 = 65 – 20 = 45（先算 -35 + 100 = 65），最终余额 45 元。也可以分步计算：存入后余额 -35 + 100 = 65 元，支出后 65 – 20 = 45 元。

The third classic type is account calculations. Xiaoming’s account balance is -35 yuan (a debt of 35 yuan); he deposits 100 yuan and then transfers out 20 yuan. What is the final balance? Calculation: -35 + 100 – 20 = 65 – 20 = 45 (first -35 + 100 = 65), so the final balance is 45 yuan. You can also work step by step: after the deposit the balance is -35 + 100 = 65 yuan, and after the transfer out it is 65 – 20 = 45 yuan.

做应用题的通用步骤：第一步，把题目中的文字翻译成数学算式，特别注意”下降””欠””低于””减少”等词往往对应负数；第二步，按法则计算，草稿上标明每一步的符号；第三步，把答案翻译回生活语言，检查是否符合常理。例如温度差不可能是负数（除非题目问方向），余额不能答成”欠款”与”存款”混淆。

A universal method for applied problems: step one, translate the words of the question into a mathematical expression, paying special attention to words like “falls”, “owes”, “below” and “decreases” which often correspond to negative numbers; step two, calculate according to the rules, marking the sign of each step on your rough paper; step three, translate the answer back into everyday language and check it makes sense. For example, a temperature difference should not be negative (unless the question asks about direction), and a balance should not confuse “debt” with “savings”.

10. 课堂测验：10 道自测题及答案 | Quick Quiz: 10 Self-Test Questions with Answers

下面 10 道题覆盖本篇文章的所有知识点。建议先独立完成，再对照答案批改，并把做错的题目抄进错题本，写明错误原因。

The 10 questions below cover every knowledge point in this article. Try to complete them independently first, then mark your work against the answers, and copy any wrong questions into your mistake notebook with the reason for the error written down.

题号 No.	题目 Question	答案 Answer
1	比较大小：-7 与 -3	-7 < -3
2	计算：(-6) + (-9)	-15
3	计算：12 + (-7)	5
4	计算：(-5) – (-8)	3
5	计算：(-4) × (-7)	28
6	计算：(-36) ÷ 9	-4
7	计算：(-2) × (-3) × (-5)	-30
8	计算：-4² 与 (-4)²	-16 与 16
9	计算：10 – (6 – 9)	13
10	气温从 -8°C 升到 5°C，上升几度？	13 度

第 8 题的答案常常让同学惊讶：-4² = -16，因为它是”4 的平方的相反数”；(-4)² = 16，因为负号在括号内一起平方。第 9 题先算括号：6 – 9 = -3，再算 10 – (-3) = 13。第 10 题列式 5 – (-8) = 13。如果你全部做对，说明本章掌握得非常好；如果有错，请回到对应小节重新阅读。

The answer to question 8 often surprises students: -4² = -16, because it is “the opposite of 4 squared”; (-4)² = 16, because the minus sign is inside the brackets and is squared together. For question 9, work out the bracket first: 6 – 9 = -3, then 10 – (-3) = 13. For question 10, the calculation is 5 – (-8) = 13. If you got them all right, you have mastered this chapter very well; if you made mistakes, go back and re-read the corresponding section.

Summary | 总结

本篇文章围绕七年级数学的核心难点”负数”展开了系统讲解：我们从数轴出发理解负数的位置与大小比较，用温度、海拔和银行余额理解负数的现实意义，接着逐一掌握同号相加、异号相加、减负变加正、乘除符号法则和 BIDMAS 运算顺序，最后通过五个常见误区和十道自测题巩固所学。

This article gave a systematic explanation of “negative numbers”, the core difficulty of Year 7 mathematics: we started from the number line to understand the position and size comparison of negative numbers, used temperature, altitude and bank balances to understand their real meaning, then mastered same-sign addition, different-sign addition, subtracting a negative becomes adding a positive, the sign rules of multiplication and division, and the BIDMAS order of operations, and finally consolidated everything with five common misconceptions and ten self-test questions.

请记住本章最重要的三句话：第一，数轴是理解负数的万能工具，任何时候想不清楚就画数轴；第二，减法一律化为加法，见到”减负”就写”加正”；第三，乘除的符号看”同号得正、异号得负”，负号个数为偶数结果为正。把这些规则练成习惯，负数章节的题目就不再是失分点，而会成为你的得分项。

Remember the three most important sentences of this chapter: first, the number line is the universal tool for understanding negative numbers, so draw one whenever you are unsure; second, always convert subtraction into addition, and write “add the positive” whenever you see “subtract a negative”; third, the sign of multiplication and division follows “same signs positive, different signs negative”, and an even number of minus signs gives a positive result. Turn these rules into habits, and negative number questions will stop being a place where you lose marks and become a place where you gain them.

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