Place Value and Number Systems — 位值与数制系统 (KS3 Cambridge Mathematics)

一、What is Place Value? | 什么是位值？

Place value is one of the most fundamental concepts in mathematics. It is the idea that the position of a digit within a number determines its actual value. For example, in the number 347, the digit ‘3’ is in the hundreds place, so it represents 300, not just 3. The same digit ‘3’ in the number 83 represents only 3. This concept allows us to represent any quantity using only ten digits (0-9), simply by changing where they appear. Without place value, we would need a unique symbol for every possible number – an impossible task. The ancient Romans used a non-place-value system with letters (I, V, X, L, C, D, M), which made arithmetic extremely cumbersome. The Hindu-Arabic place value system, introduced to Europe in the Middle Ages, revolutionised mathematics by making complex calculations straightforward and systematic.

位值是数学中最基本的概念之一。它指的是数字在数中的位置决定了它的实际值。例如，在数字 347 中，数字 “3” 位于百位，因此它代表 300，而不仅仅是 3。同样的数字 “3” 在 83 中只代表 3。这个概念使我们仅用十个数字（0-9）就能表示任何数量，只需改变它们出现的位置即可。如果没有位值系统，我们需要为每一个可能的数字创建一个独特的符号 – 这是不可能完成的任务。古罗马人使用的就是一个没有位值的系统，用字母（I、V、X、L、C、D、M）来表示数字，这使得算术运算极其繁琐。中世纪传入欧洲的印度-阿拉伯位值系统通过使复杂计算变得直接和系统化，彻底革新了数学。

二、The Decimal (Base-10) System | 十进制系统

The number system we use every day is called the decimal system, or base-10. This means each place value is ten times larger than the place to its right. Starting from the rightmost digit, we have the ones (units) place, then tens (10), hundreds (100), thousands (1000), and so on. Each step to the left multiplies the value by 10. This pattern continues indefinitely in both directions – we can also go smaller than 1 with decimal places like tenths, hundredths, and thousandths. The word “decimal” comes from the Latin word “decimus,” meaning “tenth,” which perfectly captures the essence of the system. Why base-10? Most historians believe it is because humans have ten fingers, making base-10 the most natural counting system for our species. However, other bases exist in mathematics and computing: binary (base-2) is the language of computers, using only 0 and 1; hexadecimal (base-16) is used in programming and colour codes; and the ancient Babylonians used base-60, which survives today in our measurement of time (60 seconds, 60 minutes) and angles (360 degrees).

我们日常使用的数制叫做十进制，即基数为 10。这意味着每个位值比它右边的位大十倍。从最右边的数字开始，我们有个位，然后是十位（10）、百位（100）、千位（1000），以此类推。每向左移动一位，数值就乘以 10。这种模式在两个方向上都可以无限延伸 – 我们还可以用小数位（十分位、百分位、千分位）来表示小于 1 的数。”decimal” 一词源自拉丁语 “decimus”，意为”第十”，完美地概括了这个系统的本质。为什么是十进制？大多数历史学家认为这是因为人类有十根手指，使得十进制成为我们物种最自然的计数系统。然而，数学和计算中还存在其他进制：二进制（基数为 2）是计算机的语言，只使用 0 和 1；十六进制（基数为 16）用于编程和颜色代码；古巴比伦人使用六十进制，至今仍存在于我们对时间（60 秒、60 分钟）和角度（360 度）的测量中。

三、Understanding Hundreds, Tens, and Units (HTU) | 理解百位、十位和个位

For KS3 students, the first step in mastering place value is understanding the three basic columns: hundreds (H), tens (T), and units (U). Take the number 256: the ‘2’ is in the hundreds column (2 × 100 = 200), the ‘5’ is in the tens column (5 × 10 = 50), and the ‘6’ is in the units column (6 × 1 = 6). Adding these together gives us 200 + 50 + 6 = 256. This decomposition is the foundation of all arithmetic operations – addition, subtraction, multiplication, and division all depend on understanding these column values. The Cambridge KS3 curriculum emphasises partitioning numbers into their constituent parts as a core skill. Students who can fluently partition 847 into 800 + 40 + 7 will find it much easier to perform mental arithmetic, understand the column method for addition and subtraction, and later grasp algebraic concepts like expanding brackets. A useful exercise is to practice reading three-digit numbers aloud while pointing to each column, reinforcing the connection between the written digit and its place value.

对于 KS3 学生来说，掌握位值的第一步是理解三个基本列：百位（H）、十位（T）和个位（U）。以数字 256 为例：”2″ 在百位列（2 × 100 = 200），”5″ 在十位列（5 × 10 = 50），”6″ 在个位列（6 × 1 = 6）。将这些相加得到 200 + 50 + 6 = 256。这种分解是所有算术运算的基础 – 加法、减法、乘法和除法都依赖于对这些列值的理解。Cambridge KS3 课程强调将数字分解为组成部分是一项核心技能。能够熟练地将 847 分解为 800 + 40 + 7 的学生会发现，他们更容易进行心算，理解加法和减法的列式方法，以及日后掌握代数概念如展开括号。一个有用的练习是大声读出三位数，同时指向每一列，加强书面数字与其位值之间的联系。

四、Extending to Thousands, Millions, and Beyond | 扩展到千位、百万位及以上

Once students are comfortable with three-digit numbers, the place value system extends naturally to larger numbers. After the hundreds comes the thousands (1000), then ten thousands (10,000), hundred thousands (100,000), and millions (1,000,000). The pattern is consistent: every three digits form a new group, and we use commas or spaces to separate these groups for readability. For instance, 4,528,361 is read as “four million, five hundred twenty-eight thousand, three hundred sixty-one.” Each group of three digits follows exactly the same hundreds-tens-units pattern, just at a different scale. In KS3 Cambridge Mathematics, students need to be comfortable with numbers up to at least one million, and they should also be introduced to billions (1,000,000,000) in context – for example, the population of the Earth (approximately 8.2 billion) or the distance to the Sun (approximately 150 million kilometres). Understanding place value at this scale helps students make sense of large numbers they encounter in science, geography, and everyday news.

当学生熟悉三位数后，位值系统自然地扩展到更大的数字。百位之后是千位（1000），然后是万位（10,000）、十万位（100,000）和百万位（1,000,000）。规律是一致的：每三个数字形成一个新的组，我们使用逗号或空格将这些组分开以便于阅读。例如，4,528,361 读作”four million, five hundred twenty-eight thousand, three hundred sixty-one”。每组三个数字遵循完全相同的百位-十位-个位模式，只是在不同的规模上。在 KS3 Cambridge Mathematics 中，学生需要自如地处理至少到百万位的数字，并且还应该了解十亿（1,000,000,000）的概念 – 例如，地球人口（约 82 亿）或到太阳的距离（约 1.5 亿公里）。在这个规模上理解位值有助于学生理解在科学、地理和日常新闻中遇到的大数字。

五、Reading and Writing Large Numbers in Words | 大数的英文读写规则

In the Cambridge KS3 curriculum, students are expected to read and write large numbers both in figures and in words. When writing numbers in words, remember these key rules: use hyphens for numbers from twenty-one to ninety-nine (e.g., “thirty-four”, “seventy-eight”), and use “and” before the tens and units when they follow hundreds (e.g., “one hundred and twenty-five”). For numbers in the millions, group by thousands and apply the same pattern. Example: 6,042,519 is written as “six million, forty-two thousand, five hundred and nineteen.” Note that we do not say “and” between the millions and thousands – only before the final tens and units. A common error among KS3 students is inserting extra “ands” (e.g., “six million and forty-two thousand”) which is grammatically incorrect in standard British English number conventions. Cambridge examiners will deduct marks for incorrectly written number words, so precision matters.

在 Cambridge KS3 课程中，学生应该能够用数字和文字两种方式读写大数。用文字书写数字时，请记住这些关键规则：从 21 到 99 的数字使用连字符（例如 “thirty-four”、”seventy-eight”），当十位和个位跟在百位后面时使用 “and”（例如 “one hundred and twenty-five”）。对于百万级的数字，按千分组并应用相同的模式。例如：6,042,519 写作 “six million, forty-two thousand, five hundred and nineteen”。注意我们在百万和千之间不说 “and” – 只在最后的十位和个位之前使用。KS3 学生常见的错误是插入多余的 “and”（例如 “six million and forty-two thousand”），这在标准英式英语数字惯例中是不合语法的。Cambridge 考官会因错误书写数字单词而扣分，因此精确性很重要。

六、Place Value in Decimal Numbers | 小数中的位值

The place value system does not stop at the units column – it extends to the right of the decimal point to represent fractions. The first place after the decimal point is the tenths (1/10), followed by hundredths (1/100), thousandths (1/1000), and so on. For example, in 0.375, the ‘3’ represents 3/10, the ‘7’ represents 7/100, and the ‘5’ represents 5/1000. Together, 0.375 = 375/1000 = 3/8. The value of each digit is still determined by its position relative to the decimal point, following the same logical pattern but in the opposite direction – each step to the right divides the value by 10. A crucial concept for KS3 students is that adding zeros to the right of a decimal does not change its value: 0.5, 0.50, and 0.500 are all equal. This is because each extra zero simply confirms that there are zero hundredths, zero thousandths, etc. However, adding zeros between the decimal point and a non-zero digit DOES change the value: 0.5 is not the same as 0.05, because the ‘5’ has moved from the tenths place to the hundredths place.

位值系统并不止于个位列 – 它向右延伸过小数点来表示分数。小数点后的第一位是十分位（1/10），然后是百分位（1/100）、千分位（1/1000），以此类推。例如，在 0.375 中，”3″ 代表 3/10，”7″ 代表 7/100，”5″ 代表 5/1000。合在一起，0.375 = 375/1000 = 3/8。每个数字的值仍然由其相对于小数点的位置决定，遵循相同的逻辑模式但方向相反 – 每向右移动一位，值就除以 10。KS3 学生需要理解的一个关键概念是，在小数末尾加零不会改变其值：0.5、0.50 和 0.500 都相等。这是因为每个额外的零只是确认了百分位、千分位为零。然而，在小数点和非零数字之间加零确实会改变值：0.5 与 0.05 不同，因为 “5” 从十分位移到了百分位。

七、Comparing and Ordering Numbers Using Place Value | 用位值比较和排列数字

Place value provides a systematic method for comparing numbers of any size. The rule is simple: start from the leftmost digit and compare each column in turn. The first column where the digits differ determines which number is larger. For example, to compare 45,672 and 45,627, we see that the ten-thousands, thousands, and hundreds digits are the same (4, 5, and 6). At the tens column, 7 > 2, so 45,672 > 45,627. This method works equally well for decimal numbers – just align the decimal points and compare digit by digit from left to right. When comparing decimals like 0.425 and 0.43, many students mistakenly think 0.425 is larger because 425 > 43. The correct approach is to compare digit by digit after the decimal point: the tenths digit is 4 in both, but in the hundredths place, 2 < 3, so 0.425 < 0.43. Adding a trailing zero to make both numbers have the same number of decimal places (0.425 vs 0.430) can help students visualise the comparison correctly.

位值为比较任何大小的数字提供了一种系统方法。规则很简单：从最左边的数字开始，依次比较每一列。第一个出现不同数字的列决定了哪个数字更大。例如，比较 45,672 和 45,627，我们看到万位、千位和百位的数字相同（4、5、6）。在十位列，7 > 2，所以 45,672 > 45,627。这种方法同样适用于小数 – 只需对齐小数点，然后从左到右逐位比较。比较像 0.425 和 0.43 这样的小数时，许多学生错误地认为 0.425 更大，因为 425 > 43。正确的方法是在小数点后逐位比较：十分位数字都是 4，但在百分位上，2 < 3，所以 0.425 < 0.43。在末尾加零使两个数字具有相同的小数位数（0.425 vs 0.430），可以帮助学生正确地可视化比较。

八、Rounding to Significant Places and Decimal Places | 有效位数和小数位数的舍入

Rounding is a direct application of place value knowledge. When rounding to the nearest ten, we look at the units digit: if it is 5 or more, we round up; if it is 4 or less, we round down. For example, 347 rounded to the nearest 10 is 350 (because the units digit is 7, which is 5 or more). For the nearest 100, we look at the tens digit: 347 rounded to the nearest 100 is 300 (tens digit is 4, which is less than 5). For rounding to significant figures (s.f.), we identify the first non-zero digit as the most significant, then apply the same rounding rule to the digit that follows. For example, 0.004738 to 2 s.f. is 0.0047 (the first two significant digits are 4 and 7, and the third digit, 3, is less than 5, so we do not round up). KS3 Cambridge exams often ask students to round the same number to different levels: nearest 10, nearest 100, 1 decimal place (d.p.), and 2 significant figures – all in the same question, testing whether students truly understand place value rather than just memorising rules.

舍入是位值知识的直接应用。当舍入到最接近的十位时，我们看个位数字：如果是 5 或以上，则向上舍入；如果是 4 或以下，则向下舍入。例如，347 舍入到最接近的 10 是 350（因为个位数字是 7，大于等于 5）。对于最接近的 100，我们看十位数字：347 舍入到最接近的 100 是 300（十位数字是 4，小于 5）。对于有效数字舍入，我们将第一个非零数字确定为最有效数字，然后对后面的数字应用相同的舍入规则。例如，0.004738 舍入到 2 位有效数字是 0.0047（前两个有效数字是 4 和 7，第三个数字 3 小于 5，所以不向上舍入）。KS3 Cambridge 考试经常会要求学生对同一个数字进行不同级别的舍入：最接近的 10、最接近的 100、1 位小数和 2 位有效数字 – 全部在同一道题中，测试学生是否真正理解位值，而不仅仅是记忆规则。

九、Multiplying and Dividing by Powers of 10 | 乘以和除以 10 的幂

One of the most elegant applications of place value is multiplying and dividing by 10, 100, 1000, and other powers of 10. When multiplying a whole number by 10, each digit moves one place to the left – the units become tens, the tens become hundreds, and a zero fills the empty units place. For example, 47 × 10 = 470. The ‘4’ moves from tens to hundreds (40 becomes 400), and the ‘7’ moves from units to tens (7 becomes 70). When dividing by 10, each digit moves one place to the right: 470 / 10 = 47. For decimal numbers, the key insight is that the decimal point itself does not move; instead, all the digits shift relative to it. So 3.25 × 100 = 325 because each digit moves two places left. Understanding this conceptually – rather than just memorising “add a zero” or “move the decimal point” – prevents common errors when multiplying decimals: 0.4 × 10 = 4, not 0.40. The “add a zero” shortcut fails for decimals and leads to the widespread misconception that 0.4 × 10 = 0.40.

位值最优雅的应用之一是乘以和除以 10、100、1000 以及其他 10 的幂。当一个整数乘以 10 时，每个数字向左移动一位 – 个位变成十位，十位变成百位，一个零填充空出的个位。例如，47 × 10 = 470。”4″ 从十位移到百位（40 变成 400），”7″ 从个位移到十位（7 变成 70）。除以 10 时每个数字向右移动一位：470 / 10 = 47。对于小数来说，关键的洞察是小数点本身并不移动；而是所有数字相对于小数点移动。所以 3.25 × 100 = 325，因为每个数字向左移动两位。从概念上理解这一点 – 而不仅仅是记住”加个零”或”移动小数点” – 可以防止在乘以小数时出现常见错误：0.4 × 10 = 4，而不是 0.40。”加个零”的快捷方式对小数是无效的，会导致 0.4 × 10 = 0.40 这种普遍的错误认知。

十、Solving Word Problems with Place Value | 用位值解决文字应用题

Cambridge KS3 assessments frequently test place value through word problems that require multiple steps of reasoning. A typical question might ask: “Sarah has 2847 pounds in her savings account. She withdraws 500 pounds. How much does she have left? What digit is now in the hundreds place?” Solving this requires: 2847 – 500 = 2347, and the hundreds digit is 3. Another common question type asks: “Using the digits 3, 7, 1, and 9, form the largest possible four-digit number and the smallest possible four-digit number. What is the difference between them?” The largest is 9731, the smallest is 1379, and the difference is 9731 – 1379 = 8352. This type of question tests understanding that the most significant digit contributes most to the number’s size. A more challenging variant asks students to find how many different four-digit numbers can be made from a set of digits – introducing basic combinatorics grounded in place value reasoning. These layered problems prepare students for the problem-solving demands of GCSE and beyond.

Cambridge KS3 评估经常通过需要多步推理的文字题来测试位值。一道典型的题目可能会问：”Sarah 的储蓄账户中有 2847 英镑。她取出了 500 英镑。她还剩多少钱？现在百位的数字是什么？”解决这个问题需要：2847 – 500 = 2347，百位数字是 3。另一种常见问题类型是：”使用数字 3、7、1 和 9，组成最大的四位数和最小的四位数。它们之间的差是多少？”最大的是 9731，最小的是 1379，差是 9731 – 1379 = 8352。这类问题测试的是学生对最有影响力的数字位数对数字大小的贡献最大的理解。一个更具挑战性的变体要求计算从一组数字中可以组成多少个不同的四位数 – 引入了基于位值推理的基本组合数学。这些分层问题为学生在 GCSE 及以后的数学学习中应对更高要求的解题做好准备。

十一、Estimation and Approximation Using Place Value | 使用位值进行估算和近似

Estimation is a practical life skill that depends entirely on place value understanding. To estimate the product of 48 and 312, we round each number to its most significant place: 48 rounds to 50 (nearest ten), and 312 rounds to 300 (nearest hundred). The estimate is 50 × 300 = 15,000, which is close to the actual answer 14,976. This technique is invaluable for checking the reasonableness of calculator answers – if a student calculates 48 × 312 and gets 1,497.6, they can immediately recognise this is an order of magnitude too small because the estimate is 15,000. Cambridge KS3 assessments increasingly test estimation skills alongside exact calculations, reflecting the real-world importance of being able to judge whether an answer “makes sense.” The ability to estimate well comes directly from understanding which digits carry the most weight in a number – the essence of place value.

估算是一项完全依赖于位值理解的实际生活技能。要估算 48 和 312 的乘积，我们将每个数字舍入到其最有影响力的位：48 舍入到 50（最接近的十位），312 舍入到 300（最接近的百位）。估算结果是 50 × 300 = 15,000，接近实际答案 14,976。这种技巧对于检查计算器答案的合理性非常有价值 – 如果学生计算 48 × 312 得到 1,497.6，他们可以立即认识到这个结果太小了一个数量级，因为估算值是 15,000。Cambridge KS3 评估越来越多地将估算技能与精确计算一同考查，反映了能够判断答案是否”合理”这一能力的现实重要性。良好的估算能力直接来自于理解数字中哪些位数权重最大 – 这就是位值的本质。

十二、Place Value on the Number Line | 数轴上的位值表示

The number line is one of the most powerful visual tools for understanding place value. By placing numbers on a line, students can see the relative spacing between values and understand that the distance between 300 and 400 is exactly the same as the distance between 2300 and 2400 – both represent a difference of 100 in the hundreds place. KS3 Cambridge textbooks frequently use number lines to teach ordering, rounding, and the concept of intervals. A typical exercise might ask students to estimate the value of an unlabelled point on a number line marked at 0, 100, 200, and 300 – the student must use place value reasoning to determine whether the point is closer to 200 or 300 and estimate accordingly (perhaps 260 or 270). Number lines also help students visualise decimal place value: a line marked from 3.0 to 4.0 with ten equal divisions lets students see that each division represents one tenth (0.1), building intuition for the continuous nature of the real number system beyond whole numbers.

数轴是理解位值最强大的可视化工具之一。通过将数字放在一条线上，学生可以看到值之间的相对间距，并理解 300 和 400 之间的距离与 2300 和 2400 之间的距离完全相同 – 两者都代表百位上 100 的差异。KS3 Cambridge 教材经常使用数轴来教授排序、舍入和区间的概念。一个典型的练习可能要求学生估算在标记为 0、100、200 和 300 的数轴上一个未标记点的值 – 学生必须使用位值推理来确定该点更接近 200 还是 300，并据此估算（可能是 260 或 270）。数轴还可以帮助学生可视化小数位值：一条从 3.0 标记到 4.0 并分为十个等分的线，让学生看到每个等分代表十分之一（0.1），从而建立起对实数系统中超越整数的连续性的直觉。

十三、Negative Numbers and Place Value | 负数与位值

When KS3 students first encounter negative numbers, place value understanding must be extended carefully. The digits in a negative number like -47 still follow the same place value rules: the ‘4’ represents 4 tens (40) and the ‘7’ represents 7 units – but the entire quantity is negative, so -47 = -(40 + 7). This becomes particularly important when comparing negative numbers. Many students initially believe that -47 is larger than -23 because 47 > 23, but the correct ordering on the number line is -47 < -23 because -47 is further to the left. The place value of the digits is the same regardless of the negative sign; it is the sign that determines the direction on the number line. Cambridge KS3 assessments often combine negative numbers with place value in ordering exercises: "Put these numbers in ascending order: -340, 67, -89, 120, -205." Students must mentally compare place values while also respecting the negative signs - a skill that demands careful attention to both the magnitude and the direction of each number.

当 KS3 学生初次接触负数时，位值理解必须谨慎扩展。像 -47 这样的负数中的数字仍然遵循相同的位值规则：”4″ 代表 4 个十（40），”7″ 代表 7 个一 – 但整个量是负的，所以 -47 = -(40 + 7)。这在比较负数时变得尤为重要。许多学生最初认为 -47 大于 -23，因为 47 > 23，但在数轴上正确的排序是 -47 < -23，因为 -47 更靠左。无论是否有负号，数字的位值都是相同的；是符号决定了数轴上的方向。Cambridge KS3 评估经常在排序练习中将负数与位值结合起来："将这些数字按升序排列：-340，67，-89，120，-205。"学生必须在比较位值的同时还要考虑负号 - 这是一项需要同时关注每个数字的量级和方向的技能。

十四、Common Misconceptions and How to Avoid Them | 常见误解及如何避免

Even capable KS3 students can harbour misconceptions about place value that persist into later years if not addressed. One of the most common is the “zero is nothing” fallacy: students treat zero as meaningless rather than as a placeholder that defines the value of other digits. In the number 507, the zero in the tens place is essential – without it, we would have 57, a completely different number. Another common error is misreading the place value of digits after operations: when adding 199 + 1, some students write 1910 because they incorrectly carry over to create a new column. A third misconception is the belief that longer decimals are always larger: 0.375 is not larger than 0.4, despite having more digits. The surest defence against these errors is consistent practice with place value charts, base-10 blocks or diagrams, and verbalising the reasoning behind each step. Teachers using the Cambridge framework are encouraged to have students explain “why” a digit has a particular value, not just “what” the value is.

即使是有能力的 KS3 学生也可能存在位值方面的误解，如果不加以纠正，这些误解会持续到以后的学习阶段。最常见的一个误解是”零没有意义”：学生将零视为无意义的，而不是定义其他数字值的占位符。在数字 507 中，十位的零是必不可少的 – 没有它，我们得到的是 57，一个完全不同的数字。另一个常见错误是误读运算后数字的位值：在计算 199 + 1 时，一些学生会写成 1910，因为他们错误地向新的一列进位。第三个误解是认为位数更多的小数总是更大：尽管 0.375 有更多位数，但它并不比 0.4 大。防范这些错误的最可靠方法是持续使用位值表练习，使用 base-10 积木或图形，并口头解释每一步的推理。使用 Cambridge 框架的教师被鼓励让学生解释”为什么”某个数字具有特定的值，而不仅仅是”什么”值。

Summary | 总结

Place value and the number system form the backbone of all numerical understanding in mathematics. From reading and writing multi-digit numbers to performing complex calculations, comparing quantities, rounding, estimation, and working with decimals, every numerical skill builds upon a secure grasp of what each digit represents based on its position. KS3 Cambridge Mathematics places strong emphasis on these fundamentals precisely because they are prerequisite knowledge for topics students will encounter throughout their secondary education: fractions, percentages, standard form (scientific notation), algebraic manipulation, and later, logarithms and trigonometry. Students who invest time in mastering place value now will find that higher-level mathematics becomes significantly more accessible later. The patterns learned in the decimal system also provide a foundation for understanding other number bases and the binary system that underpins all modern computing. In short, place value is not just a KS3 topic – it is a lifelong mathematical tool.

位值和数制系统构成了数学中所有数值理解的骨干。从读写多位数到执行复杂计算、比较数量、舍入取整、估算以及处理小数，每一项数值技能都建立在对每个数字根据其位置代表什么的牢固掌握之上。KS3 Cambridge Mathematics 非常重视这些基础知识，正是因为它们是学生在整个中学教育中将要接触的诸多主题的前提知识：分数、百分比、标准形式（科学记数法）、代数运算，以及后来的对数和三角学。现在花时间掌握位值的学生会发现，以后的高等数学会变得更容易理解。在十进制系统中学到的模式也为理解其他进制以及支撑所有现代计算的二进制系统提供了基础。简而言之，位值不仅仅是一个 KS3 主题 – 它是一个终身的数学工具。

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