📚 Standard Form (Scientific Notation) | 标准式（科学记数法）
Standard form, also called scientific notation, is a powerful way to write extremely large or extremely small numbers without writing long strings of zeros. In your IGCSE Mathematics exam, standard form appears in the Number topic and is frequently tested alongside calculations and problem solving.
标准式（又称科学记数法）是一种简洁书写极大或极小数字的强力工具，无需写出一长串零。在 IGCSE 数学考试中，标准式属于“数”这一板块，常与计算和问题解决结合考查。
1. What Is Standard Form? | 什么是标准式？
A < 10 and n is an integer. The value A is called the coefficient or mantissa, and n is called the exponent or power of ten.
当一个数写成 A × 10ⁿ 的形式，并且满足 1 ≤ A < 10、n 为整数时，这个数就是用标准式书写的。A 称为系数（或尾数），n 称为指数（或十的幂次）。
nteger. On the other hand, 32 × 10⁴ is not in standard form because 32 is greater than 10.
例如，3.2 × 10⁵ 是标准式，因为 3.2 在 1 到 10 之间，且 5 是整数；而 32 × 10⁴ 不是标准式，因为 32 大于 10。
2. Writing Large Numbers in Standard Form | 将大数写成标准式
To convert a large number into standard form, move the decimal point to the left until only one non-zero digit remains on its left. Count how many places you moved the decimal point; this count becomes the positive exponent n.
要把一个大数转换成标准式，将小数点向左移动，直到它左边只剩下一个非零数字。你移动的小数位数就是正指数 n。
For example, write 425000 in standard form.
例如，把 425000 写成标准式。
ast zero: 425000. Moving it five places left gives 4.25, so the number becomes 4.25 × 10⁵.
小数点目前位于最后一个零的后面：425000。将它向左移动五位得到 4.25，因此这个数写成 4.25 × 10⁵。
- 2460000 = 2.46 × 10⁶
- 90000 = 9 × 10⁴
- 7000000000 (seven billion) = 7 × 10⁹
- 2460000 = 2.46 × 10⁶
- 90000 = 9 × 10⁴
- 7000000000（七十亿）= 7 × 10⁹
3. Writing Small Numbers in Standard Form | 将小数写成标准式
For numbers smaller than 1, move the decimal point to the right until there is exactly one non-zero digit on its left. The number of places moved becomes a negative exponent n.
对于小于 1 的数，将小数点向右移动，直到它左边恰好只有一个非零数字。移动的位数就是负指数 n。
For example, write 0.00063 in standard form.
例如，把 0.00063 写成标准式。
Moving the decimal point four places to the right gives 6.3, so 0.00063 = 6.3 × 10⁻⁴.
将小数点向右移动四位得到 6.3，所以 0.00063 = 6.3 × 10⁻⁴。
| Decimal | Standard Form | Explanation |
| 0.051 | 5.1 × 10⁻² | Move point 2 places right |
| 0.0004 | 4 × 10⁻⁴ | Move point 4 places right |
| 0.00000091 | 9.1 × 10⁻⁷ | Move point 7 places right |
| 小数 | 标准式 | 说明 |
| 0.051 | 5.1 × 10⁻² | 小数点右移 2 位 |
| 0.0004 | 4 × 10⁻⁴ | 小数点右移 4 位 |
| 0.00000091 | 9.1 × 10⁻⁷ | 小数点右移 7 位 |
4. Converting Standard Form Back to Ordinary Numbers | 将标准式转换回普通数
To convert from standard form back to an ordinary number, move the decimal point in A to the right if n is positive, or to the left if n is negative. The absolute value |n| tells you how many places to move.
要将标准式转换回普通数，当 n 为正时把 A 的小数点向右移动，当 n 为负时向左移动。|n| 表示移动的位数。
If n is positive, add zeros on the right as needed; if n is negative, add zeros on the left as needed.
若 n 为正，必要时在右边补零；若 n 为负，必要时在左边补零。
7.82 × 10⁴ = 78200
7.82 × 10⁻³ = 0.00782
- 3.5 × 10² = 350
- 1.2 × 10⁶ = 1200000
- 6.4 × 10⁻¹ = 0.64
- 8.05 × 10⁻⁵ = 0.0000805
- 3.5 × 10² = 350
- 1.2 × 10⁶ = 1200000
- 6.4 × 10⁻¹ = 0.64
- 8.05 × 10⁻⁵ = 0.0000805
5. Multiplying and Dividing in Standard Form | 标准式的乘除运算
When multiplying two numbers in standard form, multiply the coefficients together and add the exponents.
计算两个标准式相乘时，将系数相乘，并将指数相加。
(A × 10ᵐ) × (B × 10ⁿ) = (A × B) × 10ᵐ⁺ⁿ
For division, divide the coefficients and subtract the exponents.
计算除法时，将系数相除，并将指数相减。
(A × 10ᵐ) ÷ (B × 10ⁿ) = (A ÷ B) × 10ᵐ⁻ⁿ
Example 1: (3 × 10⁴) × (2 × 10⁵) = 6 × 10⁹.
例 1：(3 × 10⁴) × (2 × 10⁵) = 6 × 10⁹。
Example 2: (8 × 10⁷) ÷ (2 × 10²) = 4 × 10⁵.
例 2：(8 × 10⁷) ÷ (2 × 10²) = 4 × 10⁵。
10³) = 20 × 10⁷. Since 20 is not between 1 and 10, rewrite it as 2 × 10¹ × 10⁷ = 2 × 10⁸.
有时系数会变成 10 或更大，例如 (5 × 10⁴) × (4 × 10³) = 20 × 10⁷。由于 20 不在 1 到 10 之间，需要改写：20 × 10⁷ = 2 × 10¹ × 10⁷ = 2 × 10⁸。
6. Adding and Subtracting in Standard Form | 标准式的加减运算
Standard form is not ideal for addition and subtraction. The safest method is to convert both numbers to the same power of ten, or convert them to ordinary numbers first, then perform the operation and write the answer back in standard form.
标准式并不适合直接进行加减运算。最稳妥的方法是将两个数化成相同的十的幂次，或者先将它们转换成普通数，完成运算后再把答案写回标准式。
Example: calculate 3 × 10⁴ + 2 × 10³.
例如：计算 3 × 10⁴ + 2 × 10³。
Method 1: rewrite 2 × 10³ = 0.2 × 10⁴, then 3 × 10⁴ + 0.2 × 10⁴ = 3.2 × 10⁴.
方法一：把 2 × 10³ 改写为 0.2 × 10⁴，则 3 × 10⁴ + 0.2 × 10⁴ = 3.2 × 10⁴。
Method 2: calculate in ordinary numbers: 30000 + 2000 = 32000 = 3.2 × 10⁴.
方法二：用普通数计算：30000 + 2000 = 32000 = 3.2 × 10⁴。
5 × 10⁵ − 3 × 10⁴ = 500000 − 30000 = 470000 = 4.7 × 10⁵
7. Using Your Calculator | 使用计算器
Most scientific calculators have a key labelled ×10ˣ or EXP. To enter 4.2 × 10⁷, press 4.2 then the EXP (or ×10ˣ) key, then 7. On the screen you will usually see 4.2 × 10⁷ or 4.2E7.
大多数科学计算器都有一个标有 ×10ˣ 或 EXP 的按键。要输入 4.2 × 10⁷，先按 4.2，再按 EXP（或 ×10ˣ）键，然后按 7。屏幕上通常会显示 4.2 × 10⁷ 或 4.2E7。
If the answer in your calculator window is given in standard form, you should copy it down using the same notation. Do not write the letter E in your final answer.
如果计算器窗口中的答案以标准式显示，你应当用相同的记法抄写下来。注意不要在最终答案中写字母 E。
negative powers, enter 4.2, press EXP, then press (−) followed by 5 to obtain 4.2 × 10⁻⁵.
对于负指数，输入 4.2，按 EXP，再按 (−) 键后输入 5，即可得到 4.2 × 10⁻⁵。
8. Ordering Numbers in Standard Form | 比较标准式的大小
To compare numbers in standard form, first compare the powers of ten. A number with a greater exponent is always larger.
比较标准式大小时，先比较十的幂次。指数较大的数一定更大。
e equal, compare the coefficients A. For example, 6.1 × 10⁴ > 5.9 × 10⁴ because 6.1 > 5.9.
如果指数相同，就比较系数 A。例如，6.1 × 10⁴ > 5.9 × 10⁴，因为 6.1 > 5.9。
ative exponents: 8 × 10⁻² is greater than 3 × 10⁻², but 3 × 10⁻² is greater than 3 × 10⁻³.
注意负指数的情况：8 × 10⁻² 大于 3 × 10⁻²，但 3 × 10⁻² 大于 3 × 10⁻³。
Correct order: 2.5 × 10⁻⁷ < 4.8 × 10⁻⁵ < 9.1 × 10⁻² < 1.6 × 10³
正确顺序：2.5 × 10⁻⁷ < 4.8 × 10⁻⁵ < 9.1 × 10⁻² < 1.6 × 10³
9. Real-World Applications | 实际应用
light is about 3 × 10⁸ m/s, and the mass of an electron is approximately 9.11 × 10⁻³¹ kg.
科学家们每天都在使用标准式。光速约为 3 × 10⁸ m/s，电子的质量约为 9.11 × 10⁻³¹ kg。
der of 10²² metres. Biologists measure the size of a virus, which can be about 1 × 10⁻⁷ m.
天文学家测量遥远星系的距离可达 10²² 米量级；生物学家测量病毒的大小，约为 1 × 10⁻⁷ m。
ons, and global energy consumption. For example, the world population is close to 8 × 10⁹.
在金融和人口统计中，标准式用于表示国家债务、人口数量和全球能源消耗。例如，世界人口接近 8 × 10⁹。
10. Common Mistakes to Avoid | 常见错误避免
12 × 10⁵ as the final answer. Always adjust so that 1 ≤ A < 10. Correct answer: 1.2 × 10⁶.
错误一：把 12 × 10⁵ 作为最终答案。应始终调整使 1 ≤ A < 10。正确答案：1.2 × 10⁶。
Mistake 2: confusing the direction of the decimal move for small numbers. A negative exponent means the decimal point moves to the left, not the right.
错误二：混淆小数点的移动方向。负指数意味着小数点向左移动，而不是向右。
Mistake 3: when multiplying, accidentally subtracting the exponents. Remember: multiplication means add powers; division means subtract powers.
错误三：在乘法中错误地相减指数。记住：乘法意味着指数相加；除法意味着指数相减。
Mistake 4: losing zeros when converting back. Check that your final ordinary number has roughly the expected size by comparing powers of ten.
错误四：转换回普通数时漏掉零。通过比较十的幂次来检查最终普通数的量级是否合理。
11. Exam-Style Practice Questions | 考试风格练习题
Try these questions without a calculator first, then check with a calculator.
先尝试不用计算器完成以下题目，再用计算器检查。
| Question | Answer |
| Write 83000 in standard form. | 8.3 × 10⁴ |
| Write 0.00042 in standard form. | 4.2 × 10⁻⁴ |
| Calculate (2 × 10³) × (4 × 10⁵). | 8 × 10⁸ |
| Calculate (9 × 10⁶) ÷ (3 × 10²). | 3 × 10⁴ |
| Write 6.5 × 10⁻³ as an ordinary number. | 0.0065 |
| 题目 | 答案 |
| 将 83000 写成标准式。 | 8.3 × 10⁴ |
| 将 0.00042 写成标准式。 | 4.2 × 10⁻⁴ |
| 计算 (2 × 10³) × (4 × 10⁵)。 | 8 × 10⁸ |
| 计算 (9 × 10⁶) ÷ (3 × 10²)。 | 3 × 10⁴ |
| 将 6.5 × 10⁻³ 写成普通数。 | 0.0065 |
12. Summary | 总结
rs give positive exponents; small numbers give negative exponents. Multiply by adding powers and divide by subtracting powers. For addition and subtraction, match the powers first.
标准式写作 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数。大数对应正指数，小数对应负指数。乘法指数相加，除法指数相减。加减运算时先统一幂次。
Always check that your coefficient is between 1 and 10, and remember to use the standard form button on your calculator correctly. With regular practice, standard form becomes a quick and reliable skill in the exam.
始终检查系数是否在 1 到 10 之间，并正确使用计算器上的标准式按键。通过规律练习，标准式将成为考试中快速而可靠的一项技能。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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