Standard Form | 标准式（科学计数法）

📚 Standard Form | 标准式（科学计数法）

Standard form, also known as scientific notation, is a way of writing very large or very small numbers in a compact and consistent format. It is a key topic in the Edexcel IGCSE Mathematics syllabus and appears in many problem-solving and calculator-based questions.

标准式，又称科学计数法，是一种将非常大或非常小的数字以紧凑且一致的形式书写的方法。它是Edexcel IGCSE数学课程中的一个关键考点，也常常出现在许多解答题和计算器相关题目中。

1. What is Standard Form? | 什么是标准式？

A number is written in standard form if it is expressed in the form A × 10ⁿ, where 1 ≤ A < 10 and n is an integer. The value A is called the mantissa (or coefficient), and n is the exponent.

一个数符合以下形式即被称为标准式：A × 10ⁿ，其中1 ≤ A < 10，n为整数。A称为尾数（或系数），n称为指数。

A × 10ⁿ, where 1 ≤ A < 10, n ∈ ℤ

For example, 3.2 × 10⁴ is in standard form, but 32 × 10³ is not, because the coefficient 32 is not less than 10.

例如，3.2 × 10⁴是标准式，但32 × 10³不是，因为系数32不小于10。

2. Writing Large Numbers | 大数的标准式写法

To write a large number in standard form, move the decimal point so that only one non-zero digit remains to its left. Count how many places you move the decimal point: this becomes the positive exponent of 10.

要将一个大数写成标准式，移动小数点使得其左边只剩下一位非零数字。小数点移动的位数即为10的正指数。

Example: Write 240,000 in standard form.

例子：将240,000写成标准式。

Move the decimal point from the end of the number to after the 2: 2.40000. This is 5 places to the left, so the exponent is 5.

将小数点从数字末尾移到2后面：2.40000。这需要向左移动5位，因此指数为5。

240,000 = 2.4 × 10⁵

3. Writing Small Numbers | 小数的标准式写法

For small numbers (less than 1), move the decimal point to the right until there is exactly one non-zero digit to its left. The number of places moved becomes the negative exponent of 10.

对于小于1的小数，向右移动小数点，直到其左边恰好只有一位非零数字。移动的位数即为10的负指数。

Example: Write 0.00056 in standard form.

例子：将0.00056写成标准式。

Move the decimal point to the right until it sits after the 5: 5.6. This is 4 places, so the exponent is −4.

将小数点向右移动，直到它在5后面：5.6。一共移动4位，因此指数为−4。

0.00056 = 5.6 × 10⁻⁴

4. Converting to and from Standard Form | 与普通数互转

To convert a number from standard form to ordinary form, multiply the coefficient by 10ⁿ. If n is positive, move the decimal point n places to the right; if n is negative, move it n places to the left.

要将标准式转换为普通数，将系数乘以10ⁿ。若n为正，小数点向右移动n位；若n为负，小数点向左移动n位。

7.8 × 10³ = 7800
4.2 × 10⁻² = 0.042
9.01 × 10⁶ = 9,010,000

Always insert placeholder zeros when the decimal point moves beyond the available digits.

当小数点移动超出原有数字范围时，务必补上占位零。

5. Multiplying and Dividing in Standard Form | 标准式的乘除

When multiplying or dividing numbers in standard form, deal with the coefficients and the powers of 10 separately. Then adjust the result so that it is again in standard form.

对标准式进行乘除时，分别处理系数和10的幂。然后将结果调整回标准式。

Multiplication: multiply the coefficients, add the exponents.

乘法：系数相乘，指数相加。

(a × 10ᵐ) × (b × 10ⁿ) = (a × b) × 10ᵐ⁺ⁿ

Example: (3 × 10⁵) × (4 × 10²) = 12 × 10⁷ = 1.2 × 10⁸.

例子：(3 × 10⁵) × (4 × 10²) = 12 × 10⁷ = 1.2 × 10⁸。

Division: divide the coefficients, subtract the exponents.

除法：系数相除，指数相减。

(a × 10ᵐ) ÷ (b × 10ⁿ) = (a ÷ b) × 10ᵐ⁻ⁿ

Example: (8 × 10⁶) ÷ (2 × 10³) = 4 × 10³.

例子：(8 × 10⁶) ÷ (2 × 10³) = 4 × 10³。

6. Adding and Subtracting in Standard Form | 标准式的加减

To add or subtract numbers in standard form, the exponents must be the same. Rewrite one number so that its power of 10 matches the other, then add or subtract the coefficients.

标准式的加减要求指数相同。改写其中一个数，使其10的幂与另一个一致，然后对系数进行加减。

Example: (4.2 × 10³) + (7.8 × 10²)

例子：(4.2 × 10³) + (7.8 × 10²)

Rewrite 7.8 × 10² as 0.78 × 10³. Then (4.2 + 0.78) × 10³ = 4.98 × 10³.

将7.8 × 10²改写为0.78 × 10³。然后(4.2 + 0.78) × 10³ = 4.98 × 10³。

Ensure the final answer remains in standard form. If the coefficient exceeds 10, realign the decimal point and increase the exponent by 1.

务必保证最终答案仍是标准式。如果系数大于等于10，重新移动小数点并将指数加1。

Example: (6 × 10⁴) + (5 × 10⁴) = 11 × 10⁴ = 1.1 × 10⁵.

例子：(6 × 10⁴) + (5 × 10⁴) = 11 × 10⁴ = 1.1 × 10⁵。

7. Standard Form on Calculators | 计算器上的标准式

Scientific calculators often display very large or very small results in standard form. The display usually shows something like ‘3.2E5’, which means 3.2 × 10⁵. The ‘E’ stands for exponent.

科学计算器通常以标准式显示非常大或非常小的结果。显示屏上可能显示为“3.2E5”，即3.2 × 10⁵。这里的“E”表示指数。

To enter a number in standard form, use the ×10ⁿ key or the EXP key.
Always write the answer with ×10ⁿ in your working, not the ‘E’ notation.
Check whether your calculator is in standard form mode if it shows unusual results.

输入标准式时，使用×10ⁿ键或EXP键。

在答题书写中，始终写成×10ⁿ的形式，不要使用“E”记号。

如果计算器显示异常结果，检查是否处于标准式模式。

8. Common Exam Mistakes and Tips | 常见考试错误与技巧

Many students lose marks through small errors in standard form questions. Here are the most common pitfalls and how to avoid them.

许多学生在标准式题目中因小错误而失分。以下是常见的陷阱以及如何避免它们。

Mistake / 错误	Correction / 纠正
Writing 0.5 × 10⁷ instead of 5 × 10⁶	Coefficient must be between 1 and 10.
Confusing negative exponents with negative numbers	10⁻³ = 0.001, not −1000.
Forgetting to change the exponent when multiplying by 10	Adjust both coefficient and exponent accordingly.

书写标准式时，系数必须在1到10之间。例如，0.5 × 10⁷应改写为5 × 10⁶。

负指数表示的是10的倒数幂，不是负数本身。例如，10⁻³ = 0.001。

当系数因进位变化时，不要忘记同步调整指数。

Always check your answer by converting it back to ordinary form, especially when rounding is involved.

检查答案时，尤其是涉及四舍五入时，建议将其转换回普通数验证。

9. Standard Form in Real-World Problems | 标准式在实际问题中的应用

Standard form is essential in science and business for expressing quantities such as distances between stars, the size of viruses, national debts, and the speed of light.

标准式在科学和商业中至关重要，用于表示恒星距离、病毒尺寸、国家债务、光速等量。

The mass of a proton is approximately 1.67 × 10⁻²⁷ kg.
The population of Earth is roughly 7.9 × 10⁹ people.
The distance from the Earth to the Sun is about 1.5 × 10⁸ km.

质子的质量约为1.67 × 10⁻²⁷ kg。

地球人口大约为7.9 × 10⁹人。

地球到太阳的距离约为1.5 × 10⁸ km。

In exam questions, read the units carefully. Sometimes you must convert units before writing the final answer in standard form.

在考试题目中，请仔细阅读单位。有时需要先进行单位换算，再将最终答案写成标准式。

10. Practice Questions | 练习题目

Try these typical Edexcel IGCSE style questions to test your understanding.

尝试以下典型的Edexcel IGCSE风格题目，以测试你的理解。

Write 0.000045 in standard form.
Write 8.1 × 10⁻⁵ as an ordinary number.
Calculate (2.5 × 10³) × (4 × 10²). Give your answer in standard form.
Work out (9 × 10⁷) ÷ (3 × 10⁻²). Give your answer in standard form.
Express (6 × 10³) + (5 × 10²) in standard form.

1. 将0.000045写成标准式。

2. 将8.1 × 10⁻⁵写成普通数。

3. 计算(2.5 × 10³) × (4 × 10²)，用标准式表示。

4. 计算(9 × 10⁷) ÷ (3 × 10⁻²)，用标准式表示。

5. 将(6 × 10³) + (5 × 10²)表示为标准式。

Answers: 1. 4.5 × 10⁻⁵, 2. 0.000081, 3. 1 × 10⁶, 4. 3 × 10⁹, 5. 6.5 × 10³

答案：1. 4.5 × 10⁻⁵，2. 0.000081，3. 1 × 10⁶，4. 3 × 10⁹，5. 6.5 × 10³

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Standard Form | 标准式（科学计数法）