Negative Numbers and Directed Numbers: A Complete KS3 CIE Guide — 负数与有向数：KS3 CIE 数学完整指南

1. What Is a Negative Number? The Number Line Extended Left of Zero | 什么是负数？数轴向零的左侧延伸

在小学阶段，我们熟悉的数字几乎都是从 0 开始向右延伸的正数：1、2、3……用来数苹果、量身高、记录温度。但现实生活里有很多数量会”小于零”，例如气温降到冰点以下、银行账户出现透支、电梯下降到地下层。这时我们就需要一套新的数字，把它们放在数轴零点的左侧，叫做负数（negative numbers）。

In primary school, nearly all the numbers we meet stretch to the right of zero on a number line: 1, 2, 3 and so on. We use them to count apples, measure height, and record temperature. But in real life many quantities are “less than zero”: a temperature below freezing, a bank account that is overdrawn, or a lift descending to a basement floor. For these situations we need a new set of numbers, placed to the left of zero on the number line, called negative numbers.

在数学中，负数用数字前面的减号表示，例如 −5 读作”负五”。零既不是正数也不是负数，它是正数与负数之间的分界点。把正数、负数和零放在一起，我们就得到了一条完整的数轴：−4, −3, −2, −1, 0, 1, 2, 3, 4。数轴上越靠右的数字越大，越靠左的数字越小。

In mathematics, a negative number is written with a minus sign in front of the digit, for example −5 is read “negative five”. Zero is neither positive nor negative; it is the dividing point between the two. When we put positives, negatives and zero together, we get a complete number line: −4, −3, −2, −1, 0, 1, 2, 3, 4. On the number line, the further right a number sits, the larger it is, and the further left, the smaller it is.

一个关键点：负数的大小比较和我们直觉相反。−1 其实比 −5 大，因为 −1 在数轴上更靠右。很多学生在排序时容易出错，记住口诀”越靠左越小，越靠右越大”就能避免。

A key point: comparing negative numbers works against our intuition. −1 is actually larger than −5, because −1 sits further to the right on the number line. Many students slip up when ordering negatives; remember the rule “further left is smaller, further right is larger” and you will not go wrong.

2. Reading the Number Line: Ordering and Comparing Negative Integers | 读懂数轴：负数整数的排序与比较

学会读数是掌握负数运算的第一步。以温度计为例，摄氏温度计上 0°C 是冰点，−3°C 表示零下三度，比 0°C 低，比 −10°C 高。把温度计横过来看，它其实就是一条数轴。

Learning to read the number line is the first step to mastering negative arithmetic. Take a thermometer: on a Celsius thermometer, 0°C is freezing point, −3°C means three degrees below zero, lower than 0°C but higher than −10°C. Turn a thermometer on its side and you are looking at a number line.

排序时先把所有数字标在数轴上，然后从左到右依次读出，就是从小到大的顺序。例如把 −7, 3, −1, 0, −4 从小到大排列：标在数轴上后从最左边开始，得到 −7, −4, −1, 0, 3。

To order numbers, first plot them all on a number line, then read them off from left to right, and that is your order from smallest to largest. For example, to arrange −7, 3, −1, 0 and −4 from smallest to largest: plot them, then read from the far left, giving −7, −4, −1, 0, 3.

练习比较大小：−2 和 −6 哪个大？答案 −2 更大，因为它在数轴上更靠右。−8 和 −8 相等（同一个数）。记住，负数永远比正数小，零夹在中间。

Try comparing: which is larger, −2 or −6? The answer is −2, because it sits further right on the number line. −8 and −8 are equal (the same number). Remember, any negative number is smaller than any positive number, and zero sits in between.

3. Adding and Subtracting Negatives: Walk Along the Number Line | 负数的加法与减法：沿着数轴行走

负数的加减法可以想象成在数轴上”行走”。加法表示向右走（如果加的是正数）或向左走（如果加的是负数）。例如 4 + (−3)：从 4 出发，因为加的是负数，向左走 3 步，停在 1。所以 4 + (−3) = 1。

Adding and subtracting negatives can be imagined as “walking” along the number line. Addition means step right (if you add a positive) or step left (if you add a negative). For example, 4 + (−3): start at 4, and because you are adding a negative, step 3 to the left, landing on 1. So 4 + (−3) = 1.

减法则表示方向翻转。减去一个负数，等于加上它的相反数。−2 − (−5) 可以写成 −2 + 5 = 3。口诀：”负负得正”在减法里同样适用：两个负号相遇，变成加号。

Subtraction means the direction flips. Subtracting a negative number is the same as adding its opposite. −2 − (−5) can be rewritten as −2 + 5 = 3. The rule “negative and negative make positive” applies to subtraction too: two minus signs meeting become a plus.

再看一例：−3 − 2。从 −3 出发，减去正数 2，向左走 2 步，停在 −5。所以 −3 − 2 = −5。练习时最好真的画出数轴，用手指或铅笔”走”一遍，比死记硬背可靠得多。

Another example: −3 − 2. Start at −3, subtract positive 2, step 2 to the left, landing on −5. So −3 − 2 = −5. When practising, it helps to actually draw the number line and “walk” it with a finger or pencil; this is far more reliable than memorising.

4. The Sign Rules for Multiplication and Division: Why Two Negatives Make a Positive | 乘除法的符号法则：为什么负负得正

乘法和除法比加减法更依赖符号规则。核心只有两条：同号相乘除得正，异号相乘除得负。具体来说：正 × 正 = 正，负 × 负 = 正，正 × 负 = 负，负 × 正 = 负。除法完全一样。

Multiplication and division depend more heavily on sign rules than addition and subtraction. There are really only two rules: same signs give a positive, different signs give a negative. In detail: positive × positive = positive, negative × negative = positive, positive × negative = negative, negative × positive = negative. Division works exactly the same way.

例如 (−4) × 6 = −24（异号得负），(−4) × (−6) = 24（同号得正），(−24) ÷ 6 = −4（异号得负），(−24) ÷ (−6) = 4（同号得正）。

For example, (−4) × 6 = −24 (different signs, negative result), (−4) × (−6) = 24 (same signs, positive result), (−24) ÷ 6 = −4 (different signs, negative), and (−24) ÷ (−6) = 4 (same signs, positive).

为什么负负得正？可以从”乘法的意义”理解。3 × 2 表示”2 的三倍”，即 2 + 2 + 2 = 6。那么 (−3) × 2 表示”正 2 的负三倍”，等于三次减去 2，即 0 − 2 − 2 − 2 = −6。而 (−3) × (−2) 表示”负 2 的负三倍”，等于三次减去负 2（即三次加上 2），得到 +6。这个推理能真正解释规则，而不是死记。

Why do two negatives make a positive? We can understand it through the meaning of multiplication. 3 × 2 means “three times 2”, that is 2 + 2 + 2 = 6. Then (−3) × 2 means “negative three times positive 2”, which is subtracting 2 three times: 0 − 2 − 2 − 2 = −6. And (−3) × (−2) means “negative three times negative 2”, which is subtracting negative 2 three times (that is, adding 2 three times), giving +6. This reasoning truly explains the rule instead of asking you to memorise it.

5. Order of Operations with Negatives: Brackets, Powers and BIDMAS | 含负数的运算顺序：括号、乘方与 BIDMAS

当负数与乘方、括号混在一起时，最容易出错。记住运算顺序 BIDMAS（括号、指数、除法、乘法、加法、减法）。特别注意两个陷阱：(−3)² 和 −3² 是不同的！(−3)² = 9，因为括号把负号一起平方了；而 −3² = −9，因为没有括号时，指数只作用于 3，负号最后才加上。

When negatives mix with powers and brackets, mistakes are easiest to make. Remember the order of operations BIDMAS (Brackets, Indices, Division, Multiplication, Addition, Subtraction). Watch two traps in particular: (−3)² and −3² are different! (−3)² = 9, because the bracket squares the sign together with the number; but −3² = −9, because without brackets the index only applies to the 3, and the minus sign is applied last.

再看含括号的例子：计算 10 − 3 × (−2)。按 BIDMAS，先算乘法 3 × (−2) = −6，再用 10 减去 −6，即 10 + 6 = 16。很多人误算成 10 − 3 = 7，再 × (−2) = −14，这就错了。

Now a bracketed example: work out 10 − 3 × (−2). Following BIDMAS, do the multiplication first: 3 × (−2) = −6, then subtract −6 from 10, that is 10 + 6 = 16. Many students wrongly compute 10 − 3 = 7 first, then × (−2) = −14, which is incorrect.

含乘方的混合题：(−2)³ ÷ (−4)。先算 (−2)³ = −8（负数的奇数次方仍是负数），再除以 −4，同号相除得正，结果为 2。

A mixed question with powers: (−2)³ ÷ (−4). First compute (−2)³ = −8 (an odd power of a negative stays negative), then divide by −4; same signs give a positive, so the answer is 2.

6. Negative Numbers in Real Life: Temperature, Money and Elevation | 现实生活中的负数：温度、金钱与海拔

负数的真正价值在于描述现实世界。温度是最直观的例子：北京冬天 −5°C，哈尔滨可能 −25°C。两地温差 = 较高温度 − 较低温度，例如 3 − (−5) = 8，即相差 8 度。这种”温差”问题在 CIE 考试中非常常见。

The real value of negative numbers is describing the real world. Temperature is the most intuitive example: a Beijing winter day at −5°C, or Harbin at −25°C. The temperature difference between two places equals the higher temperature minus the lower, for example 3 − (−5) = 8, an 8-degree difference. These “temperature difference” questions appear very often in CIE papers.

金钱方面，负数表示欠债或透支。如果账户余额是 −£40，表示你欠银行 40 英镑；再存入 60 英镑，余额变成 −40 + 60 = 20 英镑。海拔高度也用正负数：海平面为 0 米，珠穆朗玛峰约 +8848 米，死海约 −430 米。

With money, negatives mean debt or an overdraft. If a balance is −£40, you owe the bank 40 pounds; deposit 60 pounds and the balance becomes −40 + 60 = 20 pounds. Elevation also uses positive and negative: sea level is 0 metres, Mount Everest is about +8848 m, and the Dead Sea about −430 m.

7. Directed Numbers on a Vertical Scale: Above and Below Sea Level | 竖直刻度上的有向数：海平面之上与之下

有向数（directed numbers）强调数字带有方向：正数向上/向右，负数向下/向左。竖直数轴在测量问题里特别有用。假设一艘潜艇从海平面下潜 120 米，记作 −120；随后上浮 45 米，当前位置是 −120 + 45 = −75 米，仍在水下 75 米。

Directed numbers emphasise that numbers carry direction: positives go up or right, negatives go down or left. A vertical number line is especially useful in measurement problems. Suppose a submarine dives 120 metres from sea level, recorded as −120; then it rises 45 metres, so its new position is −120 + 45 = −75 metres, still 75 metres underwater.

这种”起点 + 变化量 = 终点”的模型适用于所有有向数问题。变化量向上为正、向下为负。练习：电梯从地下二层（−2）上升 5 层，到达 +3 层。−2 + 5 = 3。

This “start + change = end” model works for every directed-number problem. A change upwards is positive, downwards is negative. Practise: a lift rises 5 floors from the second basement floor (−2) and reaches +3. Indeed, −2 + 5 = 3.

8. Finding the Difference: Subtraction as the Gap Between Two Numbers | 求差值：减法就是两个数之间的间隔

“求差”是负数应用题的另一种常见形式。两个数的差 = 大数 − 小数，结果永远是正数。但更稳健的方法是直接用”数轴上两点的距离”，它等于两数之差的绝对值。

“Finding the difference” is another common type of negative-number problem. The difference between two numbers equals the larger minus the smaller, and the result is always positive. But a more robust method is to think of “the distance between two points on the number line”, which equals the absolute value of their difference.

例如求 −6 和 4 的差。用数轴距离：从 −6 走到 4，先走 6 步到 0，再走 4 步到 4，共 10 步，所以差是 10。算式表达：4 − (−6) = 4 + 6 = 10。

For example, find the difference between −6 and 4. Using number-line distance: to get from −6 to 4, walk 6 steps to 0, then 4 steps to 4, for 10 steps in total, so the difference is 10. In symbols: 4 − (−6) = 4 + 6 = 10.

温差、海拔差、比分差（例如高尔夫计分中低于标准杆用负数表示）都可用同一思路解决。核心始终是：把两个数放到同一条数轴上，数一数它们之间隔了多少个单位。

Temperature differences, elevation gaps, and score differences (for example, in golf, below par is recorded as negative) all use the same idea. The core idea is always: put the two numbers on the same number line and count how many units separate them.

9. Common Mistakes and How to Avoid Them | 常见错误与避坑方法

负数学习中有几个高频错误，值得专门警惕。第一，忽略符号只看数字大小：误以为 −8 > −3。纠正：在数轴上定位，−8 更靠左，所以 −8 < −3。

Several high-frequency mistakes crop up when learning negatives, and they deserve special attention. First, ignoring the sign and comparing only the digits, wrongly thinking −8 > −3. Fix: locate them on the number line; −8 is further left, so −8 < −3.

第二，−3² 与 (−3)² 混淆。第三，减法中”负负得正”用错位置：−5 − 3 不等于 −5 + 3。记住只有”减号后面跟着负数”时才变加，−5 − (−3) = −5 + 3 = −2，而 −5 − 3 = −8。

Second, confusing −3² with (−3)². Third, misapplying “two negatives make a positive” in subtraction: −5 − 3 does not equal −5 + 3. Remember that only when a minus sign is followed by a negative number does it turn into plus: −5 − (−3) = −5 + 3 = −2, whereas −5 − 3 = −8.

第四，乘法口诀背反：负 × 负得正，很多人误记成得负。可以把”负负得正”类比成语言里的双重否定：”我不是不饿” = “我饿”，两个否定抵消，变成肯定。

Fourth, memorising the multiplication rule backwards: negative × negative is positive, but many misremember it as negative. You can relate “two negatives make a positive” to double negatives in language: “I am not not hungry” means “I am hungry”; two negations cancel into an affirmation.

10. Worked Examples: Step-by-Step Solutions | 例题精讲：分步解答

例题 1：计算 −8 + 12 − 5。从左到右：−8 + 12 = 4，再 4 − 5 = −1。答案 −1。

Example 1: Work out −8 + 12 − 5. Left to right: −8 + 12 = 4, then 4 − 5 = −1. Answer: −1.

例题 2：计算 6 − (−9)。减负数变加：6 + 9 = 15。答案 15。

Example 2: Work out 6 − (−9). Subtracting a negative becomes addition: 6 + 9 = 15. Answer: 15.

例题 3：计算 (−5) × 4 ÷ (−2)。先乘：(−5) × 4 = −20；再除：−20 ÷ (−2) = 10。答案 10。

Example 3: Work out (−5) × 4 ÷ (−2). Multiply first: (−5) × 4 = −20; then divide: −20 ÷ (−2) = 10. Answer: 10.

例题 4：某城市早晨气温 −4°C，中午上升 9°C，夜间又下降 12°C。求夜间气温。−4 + 9 = 5，5 − 12 = −7。答案 −7°C。

Example 4: A city is −4°C in the morning, rises 9°C by noon, then falls 12°C overnight. Find the overnight temperature. −4 + 9 = 5, then 5 − 12 = −7. Answer: −7°C.

11. Practice Questions to Test Yourself | 自测练习题

试着独立完成以下题目，全部围绕负数运算。

Try these questions on your own; they all revolve around negative-number arithmetic.

第 1 题：把 −3, 5, −9, 0, −1 从小到大排列。第 2 题：计算 −7 + (−6)。第 3 题：计算 10 − (−4)。第 4 题：计算 (−8) × (−3)。第 5 题：计算 (−12) ÷ 4。第 6 题：计算 (−2)² − 3 × (−4)。

Question 1: Arrange −3, 5, −9, 0, −1 from smallest to largest. Question 2: Work out −7 + (−6). Question 3: Work out 10 − (−4). Question 4: Work out (−8) × (−3). Question 5: Work out (−12) ÷ 4. Question 6: Work out (−2)² − 3 × (−4).

参考答案：第 1 题 −9, −3, −1, 0, 5；第 2 题 −13；第 3 题 14；第 4 题 24；第 5 题 −3；第 6 题 4 + 12 = 16。

Answers: Question 1: −9, −3, −1, 0, 5. Question 2: −13. Question 3: 14. Question 4: 24. Question 5: −3. Question 6: 4 + 12 = 16.

12. The Coordinate Grid: Plotting Points with Negative Coordinates | 坐标网格：绘制带负坐标的点

负数也把坐标系从”第一象限”扩展到了整个平面。在七年级，学生开始学习四个象限（quadrants）：右上为第一象限（正、正），左上为第二象限（负、正），左下为第三象限（负、负），右下为第四象限（正、负）。

Negative numbers also extend the coordinate grid beyond the first quadrant to the whole plane. In Year 7, students begin to work with the four quadrants: the top-right is the first quadrant (positive, positive), top-left the second (negative, positive), bottom-left the third (negative, negative), and bottom-right the fourth (positive, negative).

一个点的坐标写作 (x, y)，其中 x 是横向位置，y 是纵向位置。点 (−3, 2) 表示从原点向左 3 个单位、再向上 2 个单位，落在第二象限。点 (−2, −5) 落在第三象限。理解坐标符号与象限的对应关系，是后续学习函数图像、平移与反射的基础。

A point’s coordinates are written (x, y), where x is the horizontal position and y the vertical. The point (−3, 2) means 3 units left from the origin, then 2 units up, landing in the second quadrant. The point (−2, −5) lands in the third quadrant. Understanding how coordinate signs map to quadrants is the foundation for later work on function graphs, translations and reflections.

平移（translation）可以直观地用负数表示方向。把点 (1, 1) 向右 3、向下 4 平移，新的 x = 1 + 3 = 4，新的 y = 1 − 4 = −3，所以新位置是 (4, −3)。这里”向下”用减法（加负数）来表达，与前面数轴行走的思路完全一致。

Translation can be described intuitively with negatives. Translating the point (1, 1) by 3 right and 4 down gives a new x = 1 + 3 = 4 and a new y = 1 − 4 = −3, so the new position is (4, −3). Here “down” is expressed as subtraction (adding a negative), exactly the same number-line walking idea as before.

13. Solving Simple Equations with Negative Solutions | 解含有负数解的简单方程

七年级的方程虽然简单，但解常常是负数。例如解 x + 5 = 2：两边同时减去 5，得到 x = 2 − 5 = −3。很多学生在看到”答案是负数”时会犹豫，其实负数的解完全合法。

Year 7 equations are simple, but their solutions are often negative. For example, solve x + 5 = 2: subtract 5 from both sides to get x = 2 − 5 = −3. Many students hesitate when the answer comes out negative, but a negative solution is perfectly valid.

再如解 3x = −12：两边同时除以 3，x = −12 ÷ 3 = −4。又如解 x − 4 = −7：两边加 4，x = −7 + 4 = −3。解方程的黄金法则”等式两边同时做同一操作”对负数同样适用。

Another example: solve 3x = −12. Divide both sides by 3: x = −12 ÷ 3 = −4. Or solve x − 4 = −7: add 4 to both sides, x = −7 + 4 = −3. The golden rule of equation solving, “do the same operation to both sides”, works just as well with negatives.

检验答案：把解代回原方程。对于 x + 5 = 2，代入 x = −3：−3 + 5 = 2，等式成立。养成”代入检验”的习惯，可以立刻发现自己是否在符号上出了错。

Check your answer by substituting it back. For x + 5 = 2, substitute x = −3: −3 + 5 = 2, which holds. Building the habit of substitution-checking will instantly reveal any sign mistakes.

14. Negative Numbers in Sequences and Patterns | 数列与规律中的负数

数列是七年级数学的重点，负数常常出现在等差递减的数列里。例如一个等差数列：11, 7, 3, −1, −5, −9……每一项都比前一项少 4。要找出下一项，只需继续减 4：−9 − 4 = −13。

Sequences are a major Year 7 topic, and negative numbers often appear in decreasing arithmetic sequences. For example, the sequence 11, 7, 3, −1, −5, −9… decreases by 4 each time. To find the next term, simply subtract 4 again: −9 − 4 = −13.

写出这类数列的通项（第 n 项）公式，需要用到负数的乘法。上面这个数列的第 n 项是 15 − 4n：当 n = 1 时得 11，n = 2 时得 7，n = 5 时得 15 − 20 = −5。当 15 − 4n 的结果为负时，就说明这一项落在了零以下。

Writing the nth-term formula for such a sequence requires negative multiplication. The nth term of the sequence above is 15 − 4n: when n = 1 we get 11, when n = 2 we get 7, and when n = 5 we get 15 − 20 = −5. When 15 − 4n turns negative, that term has fallen below zero.

还可以反过来问：−13 是这个数列的第几项？解方程 15 − 4n = −13，移项得 −4n = −28，两边除以 −4 得 n = 7。所以 −13 是第 7 项。这类题目把”数列”与”解方程”两个技能结合了起来。

You can also ask the reverse: which position in the sequence is −13? Solve 15 − 4n = −13, rearrange to −4n = −28, divide both sides by −4 to get n = 7. So −13 is the 7th term. Questions like this combine the “sequences” and “solving equations” skills together.

15. Rounding and Estimating with Negative Quantities | 负数量的四舍五入与估算

四舍五入的规则对负数同样有效，但方向要小心。一般规则：看保留位后一位数字，大于等于 5 就进位，小于 5 就舍去。例如 −3.7 四舍五入到整数是 −4（因为 0.7 大于 0.5，向”更负”方向进一位），而 −3.2 四舍五入到整数是 −3。

The rounding rules apply to negatives too, but the direction needs care. The general rule: look at the digit after the kept place; round up if it is 5 or more, round down otherwise. For example, −3.7 rounds to −4 to the nearest integer (because 0.7 exceeds 0.5, it rounds further into the negative), while −3.2 rounds to −3.

估算（estimation）在负数情境里同样有用。比如估算 −48.6 ÷ 7.1，先四舍五入为 −49 ÷ 7 = −7。真实值是 −6.845，估算值 −7 相当接近。估算能帮我们在心算时快速检验答案是否合理。

Estimation is just as useful with negatives. To estimate −48.6 ÷ 7.1, first round to −49 ÷ 7 = −7. The true value is −6.845, so the estimate of −7 is quite close. Estimation lets us quickly sanity-check whether an answer is reasonable when calculating mentally.

Summary | 总结

总结本文要点：负数是小于零的数，用数轴可以直观地排序、比较和运算；加减法用”数轴行走”理解，减法中”减负数等于加相反数”；乘除法遵循”同号得正、异号得负”，负负得正可从乘法意义推理得出；运算顺序要遵守 BIDMAS，特别注意 (−3)² 与 −3² 的区别；最后，负数在温度、金钱、海拔等现实问题中无处不在，核心模型是”起点 + 变化量 = 终点”和”数轴上的距离即差值”。

To summarise: negative numbers are numbers less than zero, and the number line lets us order, compare and calculate visually. Addition and subtraction are understood as “walking the number line”, and in subtraction “subtracting a negative equals adding its opposite”. Multiplication and division follow “same signs positive, different signs negative”, and the two-negatives rule can be reasoned out from the meaning of multiplication. The order of operations must follow BIDMAS, with special care for the difference between (−3)² and −3². Finally, negatives appear everywhere in temperature, money and elevation problems; the core models are “start + change = end” and “distance on the number line equals the difference”.

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