📚 Standard Form and Significant Figures | 标准形式与有效数字
Standard form, also called scientific notation, is a compact way to write very large or very small numbers. It appears frequently in IGCSE Mathematics papers, as well as in physics and chemistry, because it makes calculations with extreme magnitudes much easier. This guide covers every rule you need to master, with worked examples and common exam pitfalls.
标准形式，也称为科学记数法，是一种以紧凑方式书写非常大或非常小数字的方法。它频繁出现在 IGCSE 数学试卷中，也出现在物理和化学中，因为它使极端数量级的计算变得容易得多。本指南涵盖你需要掌握的每一条规则，并配有例题和常见考试陷阱。
1. What Is Standard Form? | 什么是标准形式？
A number is written in standard form when it is expressed as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer (positive, negative, or zero). The value A is called the coefficient or mantissa, and n is called the exponent or power of 10.
当一个数字写成 A × 10ⁿ 的形式时，它就是标准形式，其中 1 ≤ A < 10，且 n 是整数（正数、负数或零）。值 A 称为系数或尾数，n 称为指数或10的幂。
A × 10ⁿ, where 1 ≤ A < 10 and n ∈ ℤ
For example, 3.2 × 10⁵ is in standard form because 3.2 lies between 1 and 10, and 5 is an integer. However, 32 × 10⁴ is not in standard form because 32 is greater than 10.
例如，3.2 × 10⁵ 是标准形式，因为 3.2 在1和10之间，且5是整数。然而，32 × 10⁴ 不是标准形式，因为32大于10。
2. Writing Large Numbers in Standard Form | 将大数写成标准形式
To convert a large number into standard form, move the decimal point to the left until exactly one non-zero digit remains on its left. Count the number of places you moved the decimal point — this becomes the positive exponent n.
要将一个大数转换为标准形式，将小数点向左移动，直到左边只剩下一个非零数字。数一数小数点移动了多少位——这个位数就是正指数 n。
Example: Write 73 000 in standard form.
例：将 73 000 写成标准形式。
73 000 → 7.3 × 10⁴
The decimal point moved 4 places left, so n = 4. Notice that trailing zeros in the original number are not written in the coefficient.
小数点向左移动了4位，所以 n = 4。注意原数中的末尾零不会写在系数中。
Example: Write 1 250 000 in standard form.
例：将 1 250 000 写成标准形式。
1 250 000 → 1.25 × 10⁶
The decimal point moved 6 places left, and the trailing zeros are dropped.
小数点向左移动了6位，末尾零被省略。
3. Writing Small Numbers in Standard Form | 将小数写成标准形式
For numbers less than 1, move the decimal point to the right until one non-zero digit remains on its left. The number of places moved gives a negative exponent.
对于小于1的数，将小数点向右移动，直到左边剩下一个非零数字。移动的位数对应负指数。
Example: Write 0.000 042 in standard form.
例：将 0.000 042 写成标准形式。
0.000 042 → 4.2 × 10⁻⁵
The decimal point moved 5 places right, so n = −5. The leading zeros are not part of the coefficient.
小数点向右移动了5位，所以 n = −5。前导零不属于系数。
Example: Write 0.003 07 in standard form.
例：将 0.003 07 写成标准形式。
0.003 07 → 3.07 × 10⁻³
The zero between 3 and 7 is significant and must be kept in the coefficient.
3和7之间的零是有效数字，必须保留在系数中。
4. Converting from Standard Form | 从标准形式转换
To convert a number in standard form back into an ordinary number, move the decimal point n places. If n is positive, move the decimal point to the right; if n is negative, move it to the left. Add zeros as placeholders when needed.
要将标准形式的数字转换回普通数字，将小数点移动 n 位。如果 n 为正，向右移动；如果 n 为负，向左移动。需要时用零占位。
Example: Convert 2.6 × 10³ to an ordinary number.
例：将 2.6 × 10³ 转换为普通数字。
2.6 × 10³ = 2600
The exponent 3 moves the decimal point 3 places right: 2.6 → 26 → 260 → 2600.
指数3将小数点向右移动3位：2.6 → 26 → 260 → 2600。
Example: Convert 7.9 × 10⁻³ to an ordinary number.
例：将 7.9 × 10⁻³ 转换为普通数字。
7.9 × 10⁻³ = 0.0079
The exponent −3 moves the decimal point 3 places left with zeros as placeholders.
指数 −3 将小数点向左移动3位，并用零占位。
5. Multiplying and Dividing in Standard Form | 标准形式的乘法与除法
When multiplying two numbers in standard form, multiply the coefficients and add the exponents. When dividing, divide the coefficients and subtract the exponents.
将两个标准形式的数字相乘时，将系数相乘，并将指数相加。相除时，将系数相除，并将指数相减。
(A × 10ᵐ) × (B × 10ⁿ) = (A × B) × 10ᵐ⁺ⁿ
(A × 10ᵐ) ÷ (B × 10ⁿ) = (A ÷ B) × 10ᵐ⁻ⁿ
Example: Calculate (4 × 10⁶) × (3 × 10⁻²).
例：计算 (4 × 10⁶) × (3 × 10⁻²)。
4 × 3 = 12, and 10⁶ × 10⁻² = 10⁴, so the answer is 12 × 10⁴
Since 12 is not between 1 and 10, we must adjust the answer.
由于12不在1和10之间，我们必须调整答案。
12 × 10⁴ = 1.2 × 10⁵
We divide 12 by 10 and multiply the exponent by 10¹, which increases n from 4 to 5.
我们将12除以10，并将指数乘以10¹，使 n 从4增加到5。
Example: Calculate (8 × 10⁷) ÷ (2 × 10³).
例：计算 (8 × 10⁷) ÷ (2 × 10³)。
8 ÷ 2 = 4, and 10⁷ ÷ 10³ = 10⁴, so the answer is 4 × 10⁴
Here the coefficient 4 already lies between 1 and 10, so no adjustment is needed.
这里系数4已经在1和10之间，因此无需调整。
6. Adding and Subtracting in Standard Form | 标准形式的加法与减法
To add or subtract numbers in standard form, you must first rewrite both numbers so that they have the same exponent. Then add or subtract the coefficients and keep the common exponent.
要对标准形式的数字进行加法或减法，必须先将两个数字改写为相同的指数。然后对系数进行加法或减法，并保留共同的指数。
Example: Calculate 3 × 10⁴ + 5 × 10³.
例：计算 3 × 10⁴ + 5 × 10³。
Rewrite 5 × 10³ as 0.5 × 10⁴
3 × 10⁴ + 0.5 × 10⁴ = 3.5 × 10⁴
Alternatively, rewrite both as 10³: 30 × 10³ + 5 × 10³ = 35 × 10³, then adjust to 3.5 × 10⁴. Both methods give the same result.
或者，将两者都改写为10³的形式：30 × 10³ + 5 × 10³ = 35 × 10³，然后调整为 3.5 × 10⁴。两种方法得到相同的结果。
Example: Calculate 6.2 × 10⁶ − 4 × 10⁵.
例：计算 6.2 × 10⁶ − 4 × 10⁵。
Rewrite 4 × 10⁵ as 0.4 × 10⁶
6.2 × 10⁶ − 0.4 × 10⁶ = 5.8 × 10⁶
Remember: you can only add or subtract the coefficients when the powers of 10 match exactly.
记住：只有当10的幂完全相同时，你才能对系数进行加法或减法。
7. Significant Figures — The Rules | 有效数字——规则
Significant figures (s.f.) are the digits in a number that carry meaning and contribute to its precision. Knowing which zeros count as significant is essential for rounding correctly.
有效数字是数字中承载意义并决定其精度的位数。知道哪些零算作有效数字，对于正确四舍五入至关重要。
- All non-zero digits are significant. | 所有非零数字都是有效的。
- Zeros between non-zero digits are significant (e.g. 306 has 3 s.f.). | 非零数字之间的零是有效的（例如 306 有3位有效数字）。
- Leading zeros are NOT significant (e.g. 0.0045 has 2 s.f.). | 前导零不是有效的（例如 0.0045 有2位有效数字）。
- Trailing zeros after a decimal point are significant (e.g. 2.50 has 3 s.f.). | 小数点后的末尾零是有效的（例如 2.50 有3位有效数字）。
- Trailing zeros in a whole number without a decimal point are ambiguous — avoid relying on them. | 没有小数点的整数中的末尾零含义不明确——不要依赖它们。
| Number | Significant Figures | 数值 | 有效数字位数 |
| 405 | 3 | 405 | 3 |
| 0.025 | 2 | 0.025 | 2 |
| 7.00 | 3 | 7.00 | 3 |
| 1000 | ambiguous (1, 2, 3, or 4) | 1000 | 不明确（1、2、3或4） |
| 1.30 × 10³ | 3 | 1.30 × 10³ | 3 |
8. Rounding to Significant Figures | 四舍五入到有效数字
To round a number to a given number of significant figures, count that many digits from the first non-zero digit on the left. Look at the next digit: if it is 5 or more, round the last retained digit up; otherwise, leave it unchanged.
要将一个数字四舍五入到指定位数的有效数字，从左边第一个非零数字开始数那么多位。查看下一位数字：如果为5或更大，则将最后保留的那位进位；否则保持不变。
Example: Round 3478 to 1 s.f. and to 2 s.f.
例：将 3478 四舍五入到1位有效数字和2位有效数字。
3478 → 3000 (1 s.f.) | 3478 → 3500 (2 s.f.)
The place value of the first retained digit determines the zeros needed. For 1 s.f., the first digit is in the thousands place, so we write 3000.
第一个保留数字的数位决定了需要补多少个零。对于1位有效数字，第一个数字在千位，所以我们写3000。
Example: Round 0.004 56 to 1 s.f. and to 2 s.f.
例：将 0.004 56 四舍五入到1位有效数字和2位有效数字。
0.004 56 → 0.005 (1 s.f.) | 0.004 56 → 0.0046 (2 s.f.)
Leading zeros never count as significant figures; they only fix the position of the decimal point.
前导零永远不算有效数字；它们只用于确定小数点的位置。
9. Standard Form and Significant Figures Together | 标准形式与有效数字结合
Exam questions often ask you to write a number in standard form rounded to a stated number of significant figures. The exponent stays exactly the same — only the coefficient A is rounded.
考试题经常要求你将一个数字以标准形式写出，并四舍五入到指定位数的有效数字。指数完全保持不变——只对系数 A 进行四舍五入。
Example: Write 2 384 000 in standard form, rounded to 2 s.f.
例：将 2 384 000 写成标准形式，并四舍五入到2位有效数字。
2 384 000 = 2.384 × 10⁶ → 2.4 × 10⁶ (2 s.f.)
Example: Write 0.000 067 89 in standard form, rounded to 1 s.f.
例：将 0.000 067 89 写成标准形式，并四舍五入到1位有效数字。
0.000 067 89 = 6.789 × 10⁻⁵ → 7 × 10⁻⁵ (1 s.f.)
This two-step process — first convert to standard form, then round the coefficient — is the safest way to avoid errors.
这种两步过程——先转换为标准形式，再对系数四舍五入——是最稳妥的避免错误的方法。
10. Common Exam Mistakes | 常见考试错误
- Writing 25 × 10³ instead of 2.5 × 10⁴. The coefficient must always satisfy 1 ≤ A < 10. | 写成 25 × 10³ 而不是 2.5 × 10⁴。系数必须始终满足 1 ≤ A < 10。
- Moving the decimal in the wrong direction for negative exponents. A negative exponent means smaller, so move left. | 对负指数时小数点的移动方向搞反。负指数意味着更小，所以向左移动。
- Forgetting to adjust the final answer after multiplication when A ≥ 10. | 乘法后忘记在 A ≥ 10 时调整最终答案。
- Counting leading zeros as significant figures when rounding decimals. | 四舍五入小数时将前导零计为有效数字。
- Adding exponents when dividing, or subtracting them when multiplying. | 除法时加指数，或乘法时减指数。
- Writing 0.000 042 as 42 × 10⁻⁶ instead of 4.2 × 10⁻⁵. | 将 0.000 042 写成 42 × 10⁻⁶ 而不是 4.2 × 10⁻⁵。
11. Practice Questions | 练习题
Convert each number to standard form: (a) 920 000, (b) 0.000 031, (c) 5 600 000 000.
将每个数字转换为标准形式：(a) 920 000，(b) 0.000 031，(c) 5 600 000 000。
Answers: (a) 9.2 × 10⁵, (b) 3.1 × 10⁻⁵, (c) 5.6 × 10⁹.
答案：(a) 9.2 × 10⁵, (b) 3.1 × 10⁻⁵, (c) 5.6 × 10⁹。
Calculate: (a) (2 × 10⁴) × (3 × 10⁷), (b) (9 × 10⁸) ÷ (3 × 10²), (c) 5 × 10⁶ + 2 × 10⁵.
计算：(a) (2 × 10⁴) × (3 × 10⁷)，(b) (9 × 10⁸) ÷ (3 × 10²)，(c) 5 × 10⁶ + 2 × 10⁵。
Answers: (a) 6 × 10¹¹, (b) 3 × 10⁶, (c) 5.2 × 10⁶.
答案：(a) 6 × 10¹¹，(b) 3 × 10⁶，(c) 5.2 × 10⁶。
Round each number to 3 s.f.: (a) 24 567, (b) 0.003 048, (c) 9.876 × 10⁵.
将每个数字四舍五入到3位有效数字：(a) 24 567，(b) 0.003 048，(c) 9.876 × 10⁵。
Answers: (a) 24 600, (b) 0.003 05, (c) 9.88 × 10⁵.
答案：(a) 24 600，(b) 0.003 05，(c) 9.88 × 10⁵。
A bacteria population doubles every hour. After 24 hours the population is 1.68 ×
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