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&# ecause 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。
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。
s;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。
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）。
ifferent 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&# mes: 0 − 2 − 2 − 2 = −6. And (−3) × (−2) means “negative three times negative 2&# ), 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。
DMAS (Brackets, Indices, Division, Multiplication, Addition, Subtraction). Watch two traps in particular: (−3)² and −3² are different! nus;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。
y 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。
us;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。
ure 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。
ion 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。
es, 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&# 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。
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&# nus;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。
minus; 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。
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。
nus;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。
night 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。
−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。
+ 3 = 4 and a new y = 1 − 4 = −3, so the new position is (4, −3). Here “down&# d 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。
s;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。
: 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。
olds. 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。
ng arithmetic sequences. For example, the sequence 11, 7, 3, −1, −5, −9&# 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。
− 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? 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。
s −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”。
ets 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&# ives appear everywhere in temperature, money and elevation problems; the core models are “start + change = end” and “distance on the number line equals the difference”.
更多咨询请联系16621398022（同微信）
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图