Standard Form | 科学计数法

📚 Standard Form | 科学计数法

Standard form (also called scientific notation) is a way of writing very large or very small numbers in a compact and consistent format. In your IGCSE Edexcel Mathematics exam, you will often need to convert numbers into standard form, perform calculations with them, and interpret them in real-world contexts.

科学计数法（也称标准形式）是一种用紧凑且一致的方式书写极大或极小数字的方法。在你的 IGCSE Edexcel 数学考试中，你经常需要将数字转换为科学计数法，用其进行计算，并在实际情境中解释它们。

1. The Definition | 定义

A number is written in standard form if it is expressed as:

一个数字以标准形式书写，意味着它被表示为：

A × 10ⁿ

where 1 ≤ A < 10, and n is an integer. The coefficient A must be at least 1 and less than 10. The exponent n tells you how many places the decimal point moves.

其中 1 ≤ A < 10，n 是整数。系数 A 必须大于或等于 1 且小于 10。指数 n 告诉你小数点需要移动多少位。

2. Writing Large Numbers in Standard Form | 将大数写成科学计数法

For numbers greater than 10, n is positive. For example, 5,300,000 can be written as:

对于大于 10 的数，n 为正数。例如，5,300,000 可以写成：

5.3 × 10⁶

Count how many places the decimal point must jump to the left to get a number between 1 and 10. Here, 5,300,000 → 5.3 requires 6 jumps, so the exponent is 6.

数一下小数点必须向左跳多少位才能得到 1 到 10 之间的数。这里，5,300,000 → 5.3 需要跳 6 次，因此指数为 6。

3. Writing Small Numbers in Standard Form | 将小数写成科学计数法

For numbers less than 1, n is negative. For example, 0.00042 can be written as:

对于小于 1 的数，n 为负数。例如，0.00042 可以写成：

4.2 × 10⁻⁴

Move the decimal point to the right until you get a number between 1 and 10. Here, 0.00042 → 4.2 requires 4 jumps, so n = −4.

将小数点向右移动，直到得到一个 1 到 10 之间的数。这里，0.00042 → 4.2 需要移动 4 次，因此 n = −4。

4. Converting Standard Form to Ordinary Numbers | 将科学计数法转换为普通数字

To convert a number in standard form back to ordinary form, multiply A by 10ⁿ by moving the decimal point n places. If n is positive, move right; if n is negative, move left.

将科学计数法还原为普通数字时，通过将小数点移动 n 位来乘以 10ⁿ。如果 n 为正，向右移动；如果 n 为负，向左移动。

2.8 × 10³ = 2800

2.8 × 10³ = 2800
6.01 × 10⁻² = 0.0601

6.01 × 10⁻² = 0.0601

Notice that every shift of the decimal place corresponds to a power of ten.

注意，小数点每移动一位就对应一个 10 的幂。

5. Multiplying Numbers in Standard Form | 科学计数法的乘法

When multiplying two numbers in standard form, multiply the coefficients together and add the exponents:

将两个科学计数法数字相乘时，将系数相乘，并将指数相加：

(A × 10ᵐ) × (B × 10ⁿ) = (A × B) × 10ᵐ⁺ⁿ

Example: (3.0 × 10⁴) × (2.0 × 10²) = (3.0 × 2.0) × 10⁴⁺² = 6.0 × 10⁶. If the new coefficient is 10 or greater, adjust it: e.g. (5.0 × 10³) × (4.0 × 10²) = 20.0 × 10⁵ = 2.0 × 10⁶.

例如：(3.0 × 10⁴) × (2.0 × 10²) = (3.0 × 2.0) × 10⁴⁺² = 6.0 × 10⁶。如果新的系数大于或等于 10，需要调整：例如 (5.0 × 10³) × (4.0 × 10²) = 20.0 × 10⁵ = 2.0 × 10⁶。

6. Dividing Numbers in Standard Form | 科学计数法的除法

When dividing, divide the coefficients and subtract the exponents:

做除法时，将系数相除，并将指数相减：

(A × 10ᵐ) ÷ (B × 10ⁿ) = (A ÷ B) × 10ᵐ⁻ⁿ

Example: (8.0 × 10⁷) ÷ (2.0 × 10³) = 4.0 × 10⁷⁻³ = 4.0 × 10⁴. If the new coefficient is less than 1, adjust: e.g. (2.0 × 10⁵) ÷ (8.0 × 10²) = 0.25 × 10³ = 2.5 × 10².

例如：(8.0 × 10⁷) ÷ (2.0 × 10³) = 4.0 × 10⁷⁻³ = 4.0 × 10⁴。如果新的系数小于 1，需要调整：例如 (2.0 × 10⁵) ÷ (8.0 × 10²) = 0.25 × 10³ = 2.5 × 10²。

7. Adding and Subtracting in Standard Form | 科学计数法的加减法

Before adding or subtracting, rewrite both numbers so that they have the same exponent. Then add or subtract the coefficients.

在进行加减法之前，先将两个数字重新写成相同的指数形式。然后将系数相加或相减。

3.2 × 10⁴ + 4.5 × 10³ = 3.2 × 10⁴ + 0.45 × 10⁴ = 3.65 × 10⁴

Similarly for subtraction. Never add or subtract the exponents in this case.

减法同理。在这种情况下，绝不能将指数相加或相减。

8. Standard Form with Negative Numbers | 负数与科学计数法

Standard form can also represent negative numbers. The negative sign simply attaches to the coefficient A. For example, −250,000 = −2.5 × 10⁵.

科学计数法也可以表示负数。负号简单地附在系数 A 上。例如，−250,000 = −2.5 × 10⁵。

In calculations, the usual rules of signs apply. Pay attention to the sign of the coefficient and the exponent independently.

在计算中，通常的正负号规则依然适用。注意不要混淆系数的符号和指数的正负。

9. Solving Word Problems with Standard Form | 用科学计数法解决应用题

Many real-world quantities, such as the distance from Earth to the Sun (1.5 × 10⁸ km) or the mass of a proton (1.67 × 10⁻²⁷ kg), are written in standard form.

许多现实世界中的量，例如地球到太阳的距离（1.5 × 10⁸ 公里）或质子的质量（1.67 × 10⁻²⁷ 公斤），都以科学计数法书写。

When a question gives values in standard form, keep them in standard form throughout the calculation to avoid errors with place value.

当题目给出的数值采用科学计数法时，在计算过程中应始终保持科学计数法形式，以避免位数对位错误。

10. Common Mistakes | 常见错误

Students often forget that A must be less than 10. For example, 12.3 × 10⁵ is not correct standard form; it should be adjusted to 1.23 × 10⁶.

学生经常忘记 A 必须小于 10。例如，12.3 × 10⁵ 不是正确的科学计数法；它应调整为 1.23 × 10⁶。

Another common mistake is confusing negative exponents with negative numbers. 3.2 × 10⁻² is positive, equal to 0.032, not −32.

另一个常见错误是将负指数与负数混淆。3.2 × 10⁻² 是一个正数，等于 0.032，而不是 −32。

Always check that a small number has a negative exponent and a large number has a positive exponent.

始终检查：小数对应负指数，大数对应正指数。

11. Calculator Skills | 计算器技巧

Your scientific calculator has a key labelled ×10ˣ or EXP. Use it to enter numbers in standard form directly. For example, to enter 4.2 × 10⁷, press 4.2, then EXP, then 7.

你的科学计算器上有一个标有 ×10ˣ 或 EXP 的按键。使用它可以直接输入科学计数法数字。例如，要输入 4.2 × 10⁷，先按 4.2，再按 EXP，然后按 7。

When the calculator displays a result like 5.32E8, this means 5.32 × 10⁸. Make sure you can interpret this notation confidently.

当计算器显示类似 5.32E8 的结果时，它表示 5.32 × 10⁸。请确保你能熟练理解这种显示形式。

12. Exam-Style Practice | 考试型练习

Let us solve a typical exam question. The speed of light is 3.0 × 10⁸ m/s. How far does light travel in 2.5 × 10² seconds?

让我们解决一道典型考试题。光速为 3.0 × 10⁸ 米/秒。光在 2.5 × 10² 秒内传播多远？

Distance = speed × time = (3.0 × 10⁸) × (2.5 × 10²) = 7.5 × 10¹⁰ m.

距离 = 速度 × 时间 = (3.0 × 10⁸) × (2.5 × 10²) = 7.5 × 10¹⁰ 米。

Always write your final answer in standard form if the original values are in standard form, and show all working clearly.

如果原始值采用科学计数法，那么最终答案通常也应用科学计数法书写，并清晰展示所有计算步骤。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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Standard Form | 科学计数法