Numbers in Science: Significant Figures and Standard Form | 科学中的数字：有效数字与标准式

📚 Numbers in Science: Significant Figures and Standard Form | 科学中的数字：有效数字与标准式

In any scientific measurement, the numbers we record carry meaning. The number 363, for example, contains three digits, each telling us something about the precision of a measurement. In this article, you will learn how to use significant figures and standard form correctly—a core skill for Edexcel IGCSE Science.

在任何科学测量中，我们记录的数字都带有意义。例如数字 363，包含三个数位，每个数位都告诉我们测量精度的高低。在本文中，你将学习如何正确使用有效数字和标准式，这是 Edexcel IGCSE 科学考试的核心技能。

1. Why Numbers Matter in Science | 为什么数字在科学中重要

Scientists do not just write down any number. A measurement such as 363 cm is different from 363.0 cm: the first has three significant figures, while the second has four. This difference signals how precisely the measurement was made.

科学家不会随意写下数字。像 363 cm 这样的测量与 363.0 cm 是不同的：前者有三位有效数字，而后者有四位。这种差异表明了测量的精确程度。

In Edexcel IGCSE Science, you must be able to read, write and round measurements to the correct number of significant figures. Marks are often lost by careless rounding or by leaving out zeros that are actually meaningful.

在 Edexcel IGCSE 科学考试中，你必须能够按照正确的有效数字位数来读取、书写和舍入测量值。因粗心舍入或因遗漏真正有意义的零而失分的情况非常常见。

2. Understanding Significant Figures | 理解有效数字

Significant figures (sometimes written as “s.f.”) are the digits in a number that carry real, meaningful information about its precision. Non-zero digits are always significant. In the number 363, all three digits are non-zero, so it has 3 significant figures.

有效数字（有时写作 “s.f.”）是数字中承载真实、有意义精度信息的数位。非零数字始终是有效数字。在数字 363 中，三个数位都是非零，因此它有 3 位有效数字。

Zeros can be significant or not, depending on their position. For example, in 403, the zero is between two non-zero digits, so it is significant. In 0.036, the leading zeros are not significant.

零可能是有效数字，也可能不是，取决于它们的位置。例如，在 403 中，零位于两个非零数字之间，因此它是有效的。在 0.036 中，前面的零不是有效数字。

3. Rules for Counting Significant Figures | 数有效数字的规则

Use these rules to count significant figures quickly and accurately.

使用以下规则可以快速准确地数出有效数字位数。

Non-zero digits are always significant: 363 → 3 s.f.
Zero between two non-zero digits is significant: 303 → 3 s.f.
Leading zeros (zeros at the beginning) are not significant: 0.036 → 2 s.f.
Trailing zeros in a number without a decimal point are not significant if they are placeholders: 3600 → 2 s.f. (unless stated otherwise)
Trailing zeros after a decimal point are significant: 36.300 → 5 s.f.

非零数字始终是有效数字：363 → 3 位有效数字
两个非零数字之间的零是有效数字：303 → 3 位有效数字
前导零（开头的零）不是有效数字：0.036 → 2 位有效数字
没有小数点的数字中，末尾的零如果只是占位则不是有效数字：3600 → 2 位有效数字（除非另有说明）
小数点之后的末尾零是有效数字：36.300 → 5 位有效数字

Number	Significant Figures	Reason
363	3	All non-zero digits
0.036	2	Leading zeros not counted
30.6	3	Zero between non-zero digits
3600	2	Trailing zeros without decimal point

4. Rounding to Significant Figures | 四舍五入到有效数字

When you round to a certain number of significant figures, count from the first non-zero digit. For example, round 3634 to 2 significant figures.

当你将数字四舍五入到一定位数的有效数字时，从第一个非零数字开始数。例如，将 3634 四舍五入到 2 位有效数字。

3634 → 3600 (2 s.f.)

The third digit is 3, which is less than 5, so the second digit stays as 6. The rest become place-holding zeros.

第三位数字是 3，小于 5，所以第二位数字保持为 6。其余数字变成占位零。

Another example: round 0.0365 to 2 significant figures.

另一个例子：将 0.0365 四舍五入到 2 位有效数字。

0.0365 → 0.037 (2 s.f.)

Here, the first significant digit is 3, the second is 6, and the third is 5. Since 5 rounds up, 6 becomes 7.

这里，第一个有效数字是 3，第二个是 6，第三个是 5。因为 5 进位，所以 6 变成 7。

5. Multiplying and Dividing with Significant Figures | 乘除中的有效数字

When you multiply or divide, your final answer must have the same number of significant figures as the value with the fewest significant figures in the calculation.

在乘除运算中，最终答案的有效数字位数必须与计算中有效数字位数最少的值相同。

For example, calculate 363 × 24. Here, 363 has 3 s.f. and 24 has 2 s.f. The answer should have 2 s.f.

例如，计算 363 × 24。这里，363 有 3 位有效数字，24 有 2 位有效数字。答案应保留 2 位有效数字。

363 × 24 = 8712 → 8700 (2 s.f.)

If you are using a calculator, do not round the intermediate values. Only round the final answer.

如果使用计算器，不要对中间值舍入，只对最终答案舍入。

6. Adding and Subtracting with Significant Figures | 加减中的有效数字

For addition and subtraction, the rule is different. Your answer must have the same number of decimal places as the value with the fewest decimal places.

加减法的规则不同。你的答案的小数位数必须与加数或减数中小数位数最少的值相同。

For example, 3.6 + 363.00. The first number has 1 decimal place, the second has 2 decimal places. So the answer must have 1 decimal place.

例如，3.6 + 363.00。第一个数有 1 位小数，第二个有 2 位小数，因此答案必须保留 1 位小数。

3.6 + 363.00 = 366.60 → 366.6 (1 d.p.)

Remember: decimal places and significant figures are counted separately.

记住：小数位数和有效数字位数是分别计算的。

7. Standard Form (Scientific Notation) | 标准式（科学计数法）

Standard form is a way of writing very large or very small numbers. A number in standard form is written as:

标准式是一种书写非常大或非常小数字的方法。标准式可以写作：

A × 10ⁿ

where A is a number between 1 and 10, and n is an integer (positive or negative).

其中 A 是大于等于 1 且小于 10 的数，n 是整数（正或负）。

For example, 36300 in standard form is 3.63 × 10⁴. The number 0.0363 is 3.63 × 10⁻².

例如，36300 的标准式是 3.63 × 10⁴。而 0.0363 是 3.63 × 10⁻²。

To convert to standard form, move the decimal point until there is one non-zero digit to its left. Count the moves: left moves give a positive exponent, right moves give a negative exponent.

转换为标准式时，移动小数点直到其左边只有一个非零数字。数一数移动的位数：向左移动得到正指数，向右移动得到负指数。

8. Calculating with Standard Form | 用标准式进行计算

When multiplying or dividing numbers in standard form, multiply or divide the A parts separately, and combine the powers of 10 using the laws of indices.

对标准式数进行乘除时，先分别对 A 部分进行乘除，再用指数法则合并 10 的幂。

For example:

例如：

(3.0 × 10⁴) × (2.0 × 10³) = (3.0 × 2.0) × 10⁴⁺³ = 6.0 × 10⁷

For division:

对于除法：

(9.0 × 10⁶) ÷ (3.0 × 10²) = (9.0 ÷ 3.0) × 10⁶⁻² = 3.0 × 10⁴

Make sure the final answer is in correct standard form, with A between 1 and 10.

确保最终答案符合标准式，即 A 在 1 到 10 之间。

9. Using Numbers in Equations | 在方程中使用数字

In IGCSE Science, you will put numbers into equations such as density, speed, concentration and the ideal gas equation. Always write the equation first, then substitute the values, then calculate, then round.

在 IGCSE 科学中，你需要将数字代入方程，例如密度、速度、浓度和理想气体方程。一定要先写出方程，再代入数值，然后计算，最后舍入。

For example, density = mass ÷ volume. If mass = 363 g and volume = 150 cm³, then density = 363 ÷ 150 = 2.42 g/cm³. Since 363 has 3 s.f. and 150 has 2 s.f., the answer should be written as 2.4 g/cm³ (2 s.f.).

例如，密度 = 质量 ÷ 体积。如果质量 = 363 g，体积 = 150 cm³，则密度 = 363 ÷ 150 = 2.42 g/cm³。由于 363 有 3 位有效数字，150 有 2 位有效数字，答案应写成 2.4 g/cm³（2 位有效数字）。

Always include the correct unit in your final answer. Even a correct number with the wrong unit will lose marks.

始终在最终答案中包含正确的单位。即使数字正确，单位错误也会失分。

10. Common Mistakes to Avoid | 常见错误避免

Many students lose marks because of small mistakes. Here are the most common ones.

许多学生因为小错误而失分。以下是最常见的几种。

Counting leading zeros as significant: 0.036 is NOT 3 s.f. It is 2 s.f.
Keeping too many digits after a calculation: always round to the required s.f.
Forgetting to convert to standard form when the exponent changes sign: 0.000363 = 3.63 × 10⁻⁴, not 10⁴.
Writing units incorrectly: for example, writing grams instead of kilograms after a conversion.
Using a rounded value in the next step of a multi-step calculation. Keep the full value in your calculator until the final step.

把前导零算作有效数字：0.036 不是 3 位有效数字，而是 2 位有效数字。
计算后保留过多数字：始终舍入到所需的有效数字位数。
忘记符号变化的指数转换：0.000363 = 3.63 × 10⁻⁴，而不是 10⁴。
单位写错：例如，换算后把克写成千克。
在多步计算中使用了已舍入的中间值。在最终步骤之前，应在计算器中保留完整数值。

11. Exam Tips for Edexcel IGCSE | Edexcel IGCSE 考试提示

In the Edexcel IGCSE exam, you may be asked to “give your answer to an appropriate number of significant figures”. You must decide whether to use significant figures or decimal places based on the operation.

在 Edexcel IGCSE 考试中，你可能会被要求 “将答案保留到适当的有效数字位数”。你必须根据运算类型来决定使用有效数字还是小数位数。

Read the question carefully. If it gives data such as 36.0 cm³, the zero after the decimal point is significant. That tells you how precise the answer should be.

仔细阅读题目。如果题目给出像 36.0 cm³ 这样的数据，小数点后的零是有效数字。这告诉你答案应保留的精度。

Always show your working. Even if the final number is slightly wrong, you can still earn method marks.

始终写出计算过程。即使最终数字稍有错误，你仍然可以获得方法分。

12. Practice Questions | 练习题

Try these questions to test your understanding.

尝试以下题目来测试你的理解。

How many significant figures are in 4.063 × 10⁻³?
Round 6.783 to 2 significant figures.
Calculate 50.2 × 3.5 and give your answer to the correct number of significant figures.
Express 0.000460 in standard form.
Add 12.43 and 5.6 and give your answer to the correct number of decimal places.

4.063 × 10⁻³ 有多少位有效数字？
将 6.783 四舍五入到 2 位有效数字。
计算 50.2 × 3.5，并将答案保留到正确的有效数字位数。
将 0.000460 表示为标准式。
计算 12.43 + 5.6，并将答案保留到正确的小数位数。

Answers: 4 s.f.; 6.8; 50.2 has 3 s.f., 3.5 has 2 s.f., product = 175.7 → 180 (2 s.f.); 4.60 × 10⁻⁴; 12.43 + 5.6 = 18.03 → 18.0 (1 d.p.).

答案：4 位有效数字；6.8；50.2 有 3 位有效数字，3.5 有 2 位有效数字，乘积 = 175.7 → 180（2 位有效数字）；4.60 × 10⁻⁴；12.43 + 5.6 = 18.03 → 18.0（1 位小数）。

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Numbers in Science: Significant Figures and Standard Form | 科学中的数字：有效数字与标准式