Standard Form | 标准形式

📚 Standard Form | 标准形式

In Edexcel IGCSE Mathematics, standard form (also called scientific notation) is a way of writing very large or very small numbers compactly. It is a common exam topic because it combines powers of 10 with place value and estimation.

在 Edexcel IGCSE 数学中，标准形式（又称科学记数法）是一种简洁地表示非常大或非常小的数字的方法。它是一个常见考点，因为它将10的幂与位值和估算结合起来。

1. What Is Standard Form? | 什么是标准形式？

A number is written in standard form if it is expressed as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer. A must be a decimal number with only one non-zero digit before the decimal point.

如果一个数表示为 A × 10ⁿ，其中 1 ≤ A < 10 且 n 是整数，则这个数就是以标准形式书写的。A 必须是只有一位非零数字在小数点前的十进制数。

For example, 4500 in standard form is 4.5 × 10³. The decimal point is moved three places left to create a number between 1 and 10.

例如，4500 的标准形式是 4.5 × 10³。小数点向左移动三位，得到一个介于1和10之间的数。

2. Converting Large Numbers | 大数的转换

To convert a large number into standard form, move the decimal point left until only one digit remains before the decimal point. Count the number of places moved; this becomes the positive power of 10.

要将一个大数转换为标准形式，需要将小数点向左移动，直到小数点前只剩一位数字。移动的位数即为10的正指数。

86300 = 8.63 × 10⁴

Here the decimal point moves 4 places left, so the power is 4.

这里小数点向左移动了4位，因此指数为4。

3. Converting Small Numbers | 小数的转换

For numbers less than 1, you move the decimal point right until the first non-zero digit is the only digit before the decimal point. The number of moves becomes a negative power of 10.

对于小于1的数，需要将小数点向右移动，直到第一个非零数字是小数点前唯一的数字。移动的位数即为10的负指数。

0.000742 = 7.42 × 10⁻⁴

Notice the original number is smaller than 1, so the exponent is negative.

注意原始数小于1，因此指数为负数。

4. Rules of Powers of 10 | 10的幂的规律

Understanding powers of 10 is essential. Each positive power means “multiply by 10 that many times”; each negative power means “divide by 10 that many times”.

理解10的幂至关重要。每个正指数表示“乘以10多少次”；每个负指数表示“除以10多少次”。

10³ = 1000, so 2.5 × 10³ = 2500.
10⁻² = 0.01, so 6.8 × 10⁻² = 0.068.
10⁰ = 1, so any number multiplied by 10⁰ is unchanged.

10³ = 1000，所以 2.5 × 10³ = 2500。
10⁻² = 0.01，所以 6.8 × 10⁻² = 0.068。
10⁰ = 1，所以任何数乘以 10⁰ 不变。

5. Multiplying in Standard Form | 标准形式的乘法

When multiplying two numbers in standard form, multiply the A parts together, multiply the powers of 10 together using index laws, then adjust the answer into standard form if A is outside the range 1 ≤ A < 10.

将两个标准形式数相乘时，先将 A 部分相乘，再利用指数法则将10的幂相乘，最后如果 A 不在 1 ≤ A < 10 范围内，则需要调整答案为标准形式。

(3 × 10⁴) × (2 × 10⁵) = 6 × 10⁹

If the product of A values is 10 or more, split it off: (4 × 10³) × (5 × 10²) = 20 × 10⁵ = 2 × 10⁶.

如果 A 部分的乘积大于等于10，则拆出10的因子：(4 × 10³) × (5 × 10²) = 20 × 10⁵ = 2 × 10⁶。

6. Dividing in Standard Form | 标准形式的除法

For division, divide the A parts, subtract the exponents, and then normalise the result.

对于除法，先除 A 部分，再指数相减，最后将结果化为标准形式。

(9 × 10⁷) ÷ (3 × 10²) = 3 × 10⁵

If the result of A division is less than 1, adjust by borrowing a power of 10: (6 × 10⁴) ÷ (8 × 10²) = 0.75 × 10² = 7.5 × 10¹.

如果 A 相除的结果小于1，则需借一个10的幂进行修正：(6 × 10⁴) ÷ (8 × 10²) = 0.75 × 10² = 7.5 × 10¹。

7. Adding and Subtracting in Standard Form | 标准形式的加减法

To add or subtract numbers in standard form, convert both numbers to the same power of 10, add/subtract the A parts, and then write the result in standard form.

要对标准形式的数进行加减，需要先将两数化为相同的10的幂，再对 A 部分进行加减，最后将结果写成标准形式。

(2 × 10⁵) + (3 × 10⁵) = 5 × 10⁵

If powers differ, adjust first: (4 × 10⁶) + (7 × 10⁵) = (4 × 10⁶) + (0.7 × 10⁶) = 4.7 × 10⁶.

如果指数不同，先统一指数：(4 × 10⁶) + (7 × 10⁵) = (4 × 10⁶) + (0.7 × 10⁶) = 4.7 × 10⁶。

8. Using Your Calculator | 计算器的使用

Scientific calculators display standard form using a button such as “×10ˣ” or an “EXP” key. To enter 5.2 × 10⁷, press 5.2, then the standard form key, then 7.

科学计算器使用“×10ˣ”或“EXP”键来显示标准形式。要输入 5.2 × 10⁷，按 5.2，再按标准形式键，然后按 7。

When your calculator shows 3.45 08 or 3.45ᴇ8, it means 3.45 × 10⁸. In exams, write your final answer using the form A × 10ⁿ, not calculator notation.

当计算器显示 3.45 08 或 3.45ᴇ8 时，它表示 3.45 × 10⁸。在考试中，请用 A × 10ⁿ 的形式写出最终答案，而不是计算器符号。

9. Common Exam Mistakes | 常见考试错误

Students often lose marks by making these errors:

学生常常因为以下错误而失分：

Writing A with a value outside 1 ≤ A < 10, e.g. 12 × 10⁴.
Using a positive power for a small number, e.g. writing 0.07 as 7 × 10⁺².
Forgetting that 10⁰ = 1, so 8.2 × 10⁰ = 8.2.

A 的取值超出 1 ≤ A < 10，例如写成 12 × 10⁴。
对小数使用正指数，例如将 0.07 写成 7 × 10⁺²。
忘记 10⁰ = 1，所以 8.2 × 10⁰ = 8.2。

10. Practice Questions | 练习问题

Try these Edexcel-style questions before checking your answers.

请在查看答案前尝试以下 Edexcel 风格的练习。

Question	Answer
Write 6 900 000 in standard form.	6.9 × 10⁶
Write 4.2 × 10⁻³ as an ordinary number.	0.0042
Calculate (5 × 10³) × (4 × 10⁵), giving the answer in standard form.	2 × 10⁹
Calculate (1.2 × 10⁴) + (8 × 10³), giving the answer in standard form.	2 × 10⁴

Check that every final answer has A between 1 and 10, and that the power of ten matches the size of the original number.

检查每个最终答案的 A 是否在1到10之间，且10的幂与原始数的大小一致。

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Standard Form | 标准形式