Rounding Numbers: Decimal Places and Significant Figures | 数字舍入：小数位与有效数字

📚 Rounding Numbers: Decimal Places and Significant Figures | 数字舍入：小数位与有效数字

Rounding is a fundamental skill in mathematics that helps us simplify numbers while keeping them close to the original value. It makes data easier to work with in everyday situations and scientific calculations. In this guide, we will explore two of the most important methods: rounding to decimal places and rounding to significant figures, which are essential topics in the Cambridge Lower Secondary Mathematics curriculum.

舍入是数学中的一项基本技能，它帮助我们在保持数字接近原值的同时将数值简化，使日常情境和科学计算中的数据处理更加方便。本指南将探讨两种最重要的方法：舍入到指定小数位和舍入到有效数字，这是剑桥初中数学课程中的核心内容。

1. Introduction to Rounding | 舍入简介

Rounding means replacing a number with a nearby number that is shorter or simpler, while keeping its value approximately the same. The result is less precise but easier to read, write and calculate.

舍入是指用一个更短或更简单的近似数字替换原数，同时保持其数值大致不变。结果虽然精度降低，但更易于阅读、书写和计算。

We use rounding when exact values are not needed. For example, saying “the journey is 14.6 km” instead of 14.637 km is often completely acceptable.

当不需要精确数值时，我们会使用舍入。例如，说“旅程是 14.6 公里”而不是 14.637 公里，往往是完全可以接受的。

2. Why Round Numbers? | 为什么要舍入数字？

Rounding makes numbers more manageable. In real life, measurements are rarely exact, so we round to a sensible degree of accuracy. It also helps us compare numbers and spot trends quickly.

舍入使数字更易于处理。在现实生活中，测量很少完全精确，因此我们会舍入到合理的精度。这也有助于快速比较数字和发现趋势。

In mathematics, rounding is used in estimation, checking answers for reasonableness and when dealing with irrational numbers like π (pi).

在数学中，舍入用于估算、检查答案的合理性，以及处理像 π (圆周率) 这样的无理数。

3. Rounding to Decimal Places | 舍入到指定小数位

Rounding to a certain number of decimal places (d.p.) means keeping only that many digits after the decimal point. The digit in the chosen decimal place stays the same or increases by 1 depending on the next digit.

舍入到一定的小数位 (d.p.) 意味着只保留小数点后那么多位数字。所选小数位上的数字保持不变或加 1，取决于后一位数字的大小。

For instance, 3.456 rounded to 2 decimal places is 3.46, because the third decimal digit is 6 (5 or more, so we round up).

例如，3.456 舍入到 2 位小数是 3.46，因为第三位小数是 6 (大于等于 5，因此要进位)。

4. Step-by-Step Method for Decimal Places | 小数位舍入的分步方法

Follow these steps to round a number to a given number of decimal places:

按照以下步骤将数字舍入到指定的小数位数：

Identify the last required digit: count the decimal places from the point. Underline or mark that digit.
确定最后保留的数字： 从小数点开始数出所需的小数位。在该位数字下标上横线或做好标记。
Look at the next digit: check the digit immediately to the right of the marked one.
看下一位数字： 查看标记数字紧右侧的那一位。
Decide to round up or down: if the next digit is 5 or more, increase the marked digit by 1. If it is 4 or less, keep the marked digit unchanged.
决定进位还是舍去： 如果下一位数字大于等于 5，则将标记的数字加 1。如果小于等于 4，则保持标记数字不变。
Remove all digits to the right: drop everything after the marked decimal place.
去掉右侧所有数字： 删除标记小数位之后的所有数字。

Remember: if increasing a digit from 9 to 10, you must carry the extra 1 to the next left digit, just like in addition.

切记：如果某位数字从 9 变为 10，则需将额外的 1 进位到左侧相邻位，就像加法那样。

5. Examples of Rounding to Decimal Places | 小数位舍入示例

Let’s practise with some examples:

我们用一些例子来练习：

Round 7.3486 to 1 d.p.: The first decimal is 3. The next digit is 4 (less than 5), so keep 3 → 7.3
将 7.3486 舍入到 1 位小数： 第一位小数是 3。下一位是 4 (小于 5)，保留 3 → 7.3
Round 7.3486 to 2 d.p.: The second decimal is 4. Next digit is 8 (5 or more), so round up 4 to 5 → 7.35
将 7.3486 舍入到 2 位小数： 第二位小数是 4。下一位是 8 (大于等于 5)，将 4 进位为 5 → 7.35
Round 7.3486 to 3 d.p.: Third decimal is 8. Next digit is 6 (5 or more), so 8 becomes 9 → 7.349
将 7.3486 舍入到 3 位小数： 第三位小数是 8。下一位是 6 (大于等于 5)，8 变为 9 → 7.349

Notice how trailing zeros must be kept to show the required accuracy: 2.10 rounded to 2 d.p. stays 2.10, not 2.1.

注意，末尾的零必须保留以表示所需的精度：2.10 舍入到 2 位小数后仍为 2.10，而不是 2.1。

6. Rounding to Significant Figures | 舍入到有效数字

A significant figure (s.f.) is any digit in a number that contributes to its precision. For numbers without a decimal point, we start counting from the first non-zero digit. This method pays attention to the overall size of the number, not the position of the decimal point.

有效数字 (s.f.) 是指数字中任何有助于提高精度的数字。对于没有小数点的数，我们从第一个非零数字开始计数。这种方法关注的是数字的整体大小，而不是小数点的位置。

For example, 0.004205 has four significant figures (4, 2, 0, 5) because the leading zeros are not counted.

例如，0.004205 有四位有效数字 (4, 2, 0, 5)，因为前导零不计为有效数字。

7. Step-by-Step Method for Significant Figures | 有效数字舍入的分步方法

To round to n significant figures:

要舍入到 n 位有效数字：

Locate the first significant digit: start from the left, find the first non-zero digit. This is the 1st s.f.
找到第一个有效数字： 从左侧开始，找到第一个非零数字。这是第 1 位有效数字。
Count n significant digits: count digits to the right, including zeros that are between or after non-zero digits, until you have n significant digits. Mark the nth one.
数出 n 位有效数字： 向右继续计数，包括非零数字之间或之后的零，直到获得 n 位有效数字。标记第 n 位。
Look at the next digit: examine the digit immediately to the right of the marked one.
看下一位数字： 查看标记数字紧右侧的那一位。
Apply the rounding rule: if the next digit is 5 or above, round up the marked digit; otherwise leave it unchanged.
应用舍入规则： 如果下一位数字大于等于 5，则将标记数字进位；否则保持不变。
Replace remaining digits: for numbers greater than 1, fill the places after the marked digit with zeros to keep the place value. For decimals, you usually drop digits, but keep necessary trailing zeros to show the number of significant figures.
替换剩余数字： 对于大于 1 的数，在标记数字之后用零填充以保持其位值。对于小数，通常删去多余数字，但保留必要的末尾零以表明有效数字的位数。

8. Examples of Rounding to Significant Figures | 有效数字舍入示例

Observe how the position of the last significant digit changes:

观察最后一位有效数字的位置如何变化：

654.98 to 3 s.f.: The first 3 s.f. are 6,5,4. Next digit is 9 → round 4 up to 5 → fill with zeros → 655
654.98 舍入到 3 位有效数字： 前三位有效数字是 6,5,4。下一位是 9 → 4 进位为 5 → 后面补零 → 655
0.007248 to 2 s.f.: First non-zero is 7 (1st s.f.), then 2 (2nd s.f.). Next digit is 4 → keep 2 → result: 0.0072
0.007248 舍入到 2 位有效数字： 第一个非零是 7 (第 1 位有效数字)，然后是 2 (第 2 位)。下一位是 4 → 保留 2 → 结果: 0.0072
5096 to 2 s.f.: First two s.f. are 5 and 0. Next digit is 9 → 0 becomes 1 → fill with zeros → 5100
5096 舍入到 2 位有效数字： 前两位有效数字是 5 和 0。下一位是 9 → 0 变为 1 → 补零 → 5100

In the last example, 5100 clearly shows rounding, even though it looks like a round hundred.

在最后一个例子中，5100 清楚地显示了舍入结果，尽管它看起来像一个整百数。

9. Comparing Decimal Places and Significant Figures | 比较小数位与有效数字

It is crucial to understand the difference. Rounding to decimal places focuses only on digits after the decimal point, while rounding to significant figures considers the whole number’s magnitude. A large number like 23 456 rounded to 3 s.f. is 23 500, whereas rounding it to 2 d.p. would make no sense (it would be 23 456.00).

理解二者的区别至关重要。小数位舍入只关注小数点后的数字，而有效数字舍入则考虑整个数的大小。像 23 456 这样的大数舍入到 3 位有效数字是 23 500，而将其舍入到 2 位小数则没有意义 (结果为 23 456.00)。

The table below summarises the key contrasts:

下表总结了主要对比：

Decimal Places (d.p.)	Significant Figures (s.f.)
Count from the decimal point	Count from the first non-zero digit
Mainly used for measurements with a unit	Used in science and engineering to express precision
3.7562 to 2 d.p. → 3.76	3.7562 to 3 s.f. → 3.76

Interestingly, for decimals less than 1, the results can look the same, but the meaning is different.

有趣的是，对于小于 1 的小数，结果可能看起来相同，但含义不同。

10. Common Mistakes to Avoid | 常见错误及避免方法

Mistake 1: Forgetting to carry when rounding up a 9. For example, 3.497 to 2 d.p. → look at 7, next digit is 7 (≥5) → round 9 up to 10 → carry 1 to the 4, making it 5 → result 3.50.

错误 1：对 9 进位时忘了向前进一位。例如，3.497 舍入到 2 位小数 → 看 9，下一位是 7 (≥5) → 9 进位为 10 → 向 4 进 1，4 变为 5 → 结果是 3.50。

Mistake 2: Dropping necessary zeros. 1.20 to 2 s.f. is 1.2? No—1.20 has 3 s.f., and rounding to 2 s.f. gives 1.2, but that loses the original precision. As an original number, 1.20 to 2 s.f. is 1.2 indeed. However, if you are asked to round 1.204 to 2 s.f., the answer is 1.2, and you do not need a trailing zero because it is exactly 2 s.f. Just avoid confusion: rounding 1.204 to 2 s.f. = 1.2.

错误 2：丢掉必要的零。1.20 舍入到 2 位有效数字是 1.2 吗？不是——1.20 本身有 3 位有效数字，舍入到 2 位有效数字确实得到 1.2，但会丢失原始精度。如果要将 1.204 舍入到 2 位有效数字，答案是 1.2，不需要末尾零。只需记住不要随意删除表明精度的零即可。

Mistake 3: Starting to count significant figures from the decimal point. Always start from the first non-zero digit.

错误 3：从小数点开始计数有效数字。一定要从第一个非零数字开始。

11. Practice Questions and Tips | 练习题与技巧

Test your understanding with these quick questions:

用这些快速问题测验你的理解：

Round 86.476 to 1 d.p. → 86.5
将 86.476 舍入到 1 位小数 → 86.5
Round 0.005438 to 2 s.f. → 0.0054
将 0.005438 舍入到 2 位有效数字 → 0.0054
Round 4.998 to 2 d.p. → 5.00
将 4.998 舍入到 2 位小数 → 5.00

Tip: Always identify the cut-off digit and the digit to its right. Then apply the rule: 5 or more, let it soar; 4 or less, let it rest.

技巧：始终确定保留位和其右的数字。然后应用规则：五或以上，往上进；四或以下，原地静。

12. Summary | 总结

Rounding to decimal places and significant figures are both methods of approximation. Decimal places depend on the position after the decimal point, while significant figures relate to the precision from the first non-zero digit. Mastering these skills builds a strong foundation for estimation, science experiments and everyday problem-solving. Practise regularly with different numbers, and always double-check the digit you are rounding.

舍入到小数位和有效数字都是近似值的方法。小数位取决于小数点后的位置，而有效数字关系到从第一个非零数字开始的精度。掌握这些技能能为估算、科学实验和日常问题解决打下坚实基础。经常用不同数字进行练习，并始终仔细核对要舍入的那一位数字。

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Rounding Numbers: Decimal Places and Significant Figures | 数字舍入：小数位与有效数字