Standard Form (Scientific Notation) | 标准形式（科学计数法）

📚 Standard Form (Scientific Notation) | 标准形式（科学计数法）

Standard form, also called scientific notation, is a compact way of writing very large or very small numbers. It is essential in IGCSE Mathematics because examiners expect you to convert, calculate and apply standard form accurately.

标准形式，也称科学计数法，是一种简洁地表示非常大或非常小的数的方法。在 IGCSE 数学中至关重要，因为考官要求你准确地进行转换、计算和应用标准形式。

1. The Definition of Standard Form | 标准形式的定义

A number is written in standard form if it is expressed as a product of a number between 1 and 10 (including 1 but excluding 10) and a power of 10.

如果一个数被表示为一个介于 1 和 10 之间（包括 1，不包括 10）的数与 10 的幂的乘积，那么这个数就是用标准形式书写的。

N = a × 10ⁿ, where 1 ≤ a < 10 and n is an integer

N = a × 10ⁿ，其中 1 ≤ a < 10，n 为整数

For example, 158 can be written as 1.58 × 10². The number 158 is a useful example because it is neither too large nor too small, yet it demonstrates the key idea clearly.

例如，158 可以写成 1.58 × 10²。数字 158 是一个很有用的例子，因为它既不太大也不太小，却能清晰地展示核心概念。

2. Writing 158 in Standard Form | 将 158 写成标准形式

To write 158 in standard form, follow these steps:

要将 158 写成标准形式，请遵循以下步骤：

Place the decimal point after the first non-zero digit: 1.58.

将小数点放在第一个非零数字之后：1.58。
Count how many places the decimal point has moved from the original number. In 158, the decimal point is understood to be at the end: 158. → 1.58. This is a move of 2 places to the left.

数一数小数点从原数移动了多少位。在 158 中，小数点原本在最后：158. → 1.58。这是向左移动了 2 位。
Since the decimal point moved left, the power of 10 is positive 2.

因为小数点向左移动，所以 10 的指数为正 2。

158 = 1.58 × 10²

This representation tells us that multiplying 1.58 by 100 gives 158. The exponent 2 indicates the number of places the decimal point must move to the right when expanding back.

这种表示告诉我们，将 1.58 乘以 100 得到 158。指数 2 表示展开时小数点需要向右移动的位数。

3. Writing Other Numbers in Standard Form | 将其他数字写成标准形式

The same method applies to very large and very small numbers. Let us adapt 158 to show both cases.

同样的方法适用于非常大和非常小的数字。让我们用 158 的变体来展示这两种情况。

For 158 000, move the decimal point 5 places left to get 1.58:

对于 158 000，将小数点向左移动 5 位得到 1.58：

158 000 = 1.58 × 10⁵

For 0.000158, move the decimal point 4 places right to get 1.58. Because the original number is less than 1, the exponent is negative:

对于 0.000158，将小数点向右移动 4 位得到 1.58。因为原数小于 1，所以指数为负：

0.000158 = 1.58 × 10⁻⁴

Notice that the leading digit 1 is the same in all cases; only the exponent changes according to the magnitude of the number.

请注意，在所有情况下，首位数字 1 都相同；只有指数根据数的大小而变化。

4. Converting from Standard Form to an Ordinary Number | 将标准形式转换回普通数字

To convert 1.58 × 10² back to an ordinary number, multiply 1.58 by 10² (which is 100). Moving the decimal point 2 places to the right gives 158.

要将 1.58 × 10² 转换回普通数字，需将 1.58 乘以 10²（即 100）。将小数点向右移动 2 位得到 158。

If the exponent is negative, move the decimal point to the left. For example:

如果指数为负，则向左移动小数点。例如：

1.58 × 10⁻³ = 0.00158

Always check: a positive exponent means a number greater than or equal to 1, while a negative exponent means a number between 0 and 1.

务必检查：正指数表示大于或等于 1 的数，负指数表示介于 0 和 1 之间的数。

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

When multiplying two numbers in standard form, multiply the a-parts together and add the powers of 10.

当两个标准形式的数相乘时，将前面的 a 部分相乘，并将 10 的指数相加。

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

Example using 158 in standard form:

使用标准形式 158 的示例：

(1.58 × 10²) × (2 × 10³) = (1.58 × 2) × 10²⁺³ = 3.16 × 10⁵

Because 3.16 is already between 1 and 10, no further adjustment is needed.

因为 3.16 已经在 1 到 10 之间，所以无需进一步调整。

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

When dividing, divide the a-parts and subtract the powers of 10.

相除时，将前面的 a 部分相除，并减去 10 的指数。

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

Example:

示例：

(1.58 × 10²) ÷ (2 × 10³) = (1.58 ÷ 2) × 10²⁻³ = 0.79 × 10⁻¹

The result 0.79 is less than 1, so we must rewrite it: 0.79 × 10⁻¹ = 7.9 × 10⁻². This final form satisfies the rule 1 ≤ a < 10.

结果 0.79 小于 1，所以我们必须重写：0.79 × 10⁻¹ = 7.9 × 10⁻²。最终形式满足 1 ≤ a < 10。

7. Adding and Subtracting in Standard Form | 标准形式的加法与减法

Before adding or subtracting, both numbers must have the same power of 10. Sometimes this means converting one number into a different index form.

进行加法或减法前，两个数必须具有相同的 10 的指数。有时这意味着要将一个数转换为另一种指数形式。

Example:

示例：

(1.58 × 10²) + (2.3 × 10³) = (0.158 × 10³) + (2.3 × 10³) = 2.458 × 10³

Here 1.58 × 10² becomes 0.158 × 10³, which is equivalent but not yet in standard form. The final answer 2.458 × 10³ satisfies the definition.

这里 1.58 × 10² 变成了 0.158 × 10³，两者等价，但还不是标准形式。最终答案 2.458 × 10³ 符合定义。

8. Common Mistakes to Avoid | 需要避免的常见错误

Writing 158 as 15.8 × 10¹: the a-part must be between 1 and 10, so 15.8 is not allowed.

将 158 写成 15.8 × 10¹：a 部分必须在 1 到 10 之间，所以 15.8 是不允许的。
Using the wrong sign for the exponent: moving the decimal left gives a positive exponent, moving right gives a negative exponent.

指数符号错误：小数点向左移动得到正指数，向右移动得到负指数。
Forgetting to adjust after multiplication or division: if a is 10 or greater, divide it by 10 and increase the exponent by 1; if a is less than 1, multiply it by 10 and decrease the exponent by 1.

乘除后忘记调整：如果 a 大于等于 10，则将其除以 10 并使指数加 1；如果 a 小于 1，则将其乘以 10 并使指数减 1。
Adding or subtracting without matching exponents first.

相加或相减前没有先使指数相同。

9. Real-World Applications | 实际应用

Standard form is used in science and engineering to handle extreme quantities. For example, the distance from the Earth to the Sun is about 1.496 × 10⁸ km, and the mass of a proton is about 1.67 × 10⁻²⁷ kg.

标准形式在科学和工程中用于处理极端的数量。例如，地球到太阳的距离约为 1.496 × 10⁸ km，质子的质量约为 1.67 × 10⁻²⁷ kg。

Our example 158 can also be linked to everyday life: if a bacterium population doubles itself 10 times from 158, the result may be expressed in standard form as 158 × 2¹⁰ = 161 792 ≈ 1.62 × 10⁵.

我们使用的例子 158 也可以联系实际生活：如果一个细菌种群从 158 个开始，每繁殖一次数量翻倍，繁殖 10 次后，结果可用标准形式表示为 158 × 2¹⁰ = 161 792 ≈ 1.62 × 10⁵。

10. Exam Tips and Summary | 考试技巧与总结

In the Edexcel IGCSE examination, standard form questions may appear as short calculations, conversions, or within problem-solving contexts. Always show the original expression before your final answer.

在 Edexcel IGCSE 考试中，标准形式问题可能以简答计算、转换或应用题的形式出现。始终在最终答案前写出原始表达式。

Check that 1 ≤ a < 10 in every final answer.

在每个最终答案中检查 1 ≤ a < 10。
Use your calculator carefully; scientific notation buttons may display as E notation, e.g. 1.58E2 means 1.58 × 10².

小心使用计算器；科学计数法按钮可能显示为 E 记法，例如 1.58E2 表示 1.58 × 10²。
Read the question: sometimes you are asked to give your answer in ordinary numbers, not in standard form.

仔细审题：有时题目要求用普通数字作答，而不是标准形式。

To revise, remember the single anchor example: 158 = 1.58 × 10². From this, you can derive all the rules of standard form by moving the decimal point and counting the exponent.

复习时，请记住这个关键示例：158 = 1.58 × 10²。由此，你可以通过移动小数点和数指数来推导出标准形式的所有规则。

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Standard Form (Scientific Notation) | 标准形式（科学计数法）