Standard Form | 科学计数法与标准形式

📚 Standard Form | 科学计数法与标准形式

Standard form, also known as scientific notation, is a compact way of writing extremely large or extremely small numbers. It is written in the form a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. This topic is fundamental for the Edexcel IGCSE Mathematics examination and appears in both calculator and non‑calculator papers.

标准形式，又称科学计数法，是一种简洁地表示极大或极小数值的方法。它写作 a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。这是 Edexcel IGCSE 数学考试的基础内容，在计算器与非计算器试卷中都会出现。

1. What is Standard Form? | 什么是标准形式？

Standard form is used to express numbers that are either very large or very small without writing long strings of zeros. For instance, the distance between Earth and the Sun is about 149,600,000 km, which can be written as 1.496 × 10⁸ km.

标准形式用于表达非常大或非常小的数，避免了书写一长串零。例如，地球到太阳的距离约为 149,600,000 千米，可写成 1.496 × 10⁸ 千米。

Similarly, the charge of an electron is about 0.0000000000000000001602 coulombs, which is expressed as 1.602 × 10⁻¹⁹ C.

同样，电子的电荷量约为 0.0000000000000000001602 库仑，可表示为 1.602 × 10⁻¹⁹ 库仑。

In standard form, the first part a must be at least 1 but less than 10. The exponent n tells us how many places the decimal point has been moved.

在标准形式中，前一部分 a 必须大于等于 1 且小于 10。指数 n 表示小数点移动的位数。

2. Writing Large Numbers in Standard Form | 用标准形式书写大数

To convert a large number into standard form, move the decimal point to the left until only one non‑zero digit remains to its left. Count the number of places moved; this becomes the positive exponent n.

将一个大的普通数字转换为标准形式时，小数点向左移动，直到左边只剩下一位非零数字。移动的位数即为正整数指数 n。

For example, write 345,000 in standard form.

例如，将 345,000 写成标准形式。

345,000 = 3.45 × 10⁵

Here, we moved the decimal point 5 places to the left, and the leading digit is 3, followed by .45.

这里，小数点向左移动了 5 位，首位数字是 3，后面是 .45。

Another example: 80,000,000 = 8 × 10⁷. Note that if there are trailing zeros after a whole number, they are not included in the a part unless they are significant.

另一个例子：80,000,000 = 8 × 10⁷。注意，如果整数末尾有零，这些零不写入 a 部分，除非它们是有效数字。

Write the non‑zero digits with the decimal point after the first digit.
Multiply by 10 raised to the number of places the decimal point moved.

写出非零数字，并将小数点放在第一位之后。
乘以 10 的「小数点移动位数」次方。

3. Writing Small Numbers in Standard Form | 用标准形式书写小数

For numbers less than 1, move the decimal point to the right until one non‑zero digit remains to its left. The number of places moved gives a negative exponent n.

对于小于 1 的数，将小数点向右移动，直到左边只剩一位非零数字。移动的位数给出负指数 n。

For example, write 0.00072 in standard form.

例如，将 0.00072 写成标准形式。

0.00072 = 7.2 × 10⁻⁴

We moved the decimal point 4 places to the right, so n = −4.

小数点向右移动了 4 位，因此 n = −4。

Another example: 0.000009 = 9 × 10⁻⁶. Be careful to count the zeros after the decimal point, including the one before the first non‑zero digit.

另一个例子：0.000009 = 9 × 10⁻⁶。注意小数后面的零的个数，包括第一个非零数字前的零。

4. Multiplying and Dividing in Standard Form | 标准形式的乘除法

When multiplying numbers in standard form, multiply the a parts and add the exponents. For example:

当标准形式的两数相乘时，将 a 部分相乘，并将指数相加。例如：

(2 × 10³) × (3 × 10⁴) = 6 × 10⁷

Because 2 × 3 = 6 and 3 + 4 = 7.

因为 2 × 3 = 6，且 3 + 4 = 7。

When dividing, divide the a parts and subtract the exponents.

相除时，将 a 部分相除，并将指数相减。

(8 × 10⁶) ÷ (2 × 10²) = 4 × 10⁴

Here 8 ÷ 2 = 4 and 6 − 2 = 4.

这里 8 ÷ 2 = 4，且 6 − 2 = 4。

Sometimes the result does not have the a part between 1 and 10. You must convert it. For example:

有时结果的 a 部分不在 1 到 10 之间，此时需要转换。例如：

(5 × 10³) × (4 × 10²) = 20 × 10⁵ = 2.0 × 10⁶

Because 20 × 10⁵ is not standard form; move the decimal one place left and increase the exponent by 1.

因为 20 × 10⁵ 不是标准形式；将小数点左移一位，指数增加 1。

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

To add or subtract numbers in standard form, they must have the same exponent. If they do not, rewrite one of them so that the exponents match.

要在标准形式下进行加减，必须先使两数的指数相同。若指数不同，则改写其中一个数，使指数一致。

Example: (3 × 10⁴) + (5 × 10³). Rewrite 5 × 10³ as 0.5 × 10⁴. Then add:

例如：(3 × 10⁴) + (5 × 10³)。将 5 × 10³ 改写为 0.5 × 10⁴，然后相加：

3 × 10⁴ + 0.5 × 10⁴ = 3.5 × 10⁴

You can also convert both numbers to ordinary form, add, then convert back if necessary.

也可以将两个数都转换成普通数字，相加后再转换回标准形式（如有必要）。

For subtraction, the same rule applies. For example:

减法同理。例如：

(7 × 10⁵) − (2 × 10⁵) = 5 × 10⁵

If the exponents differ, adjust first. For example, (6 × 10⁶) − (4 × 10⁵) becomes 6 × 10⁶ − 0.4 × 10⁶ = 5.6 × 10⁶.

如果指数不同，先调整。例如，(6 × 10⁶) − (4 × 10⁵) 变成 6 × 10⁶ − 0.4 × 10⁶ = 5.6 × 10⁶。

6. Converting between Standard Form and Ordinary Numbers | 标准形式与普通数的互转

To convert from standard form to an ordinary number, move the decimal point according to the exponent. A positive exponent means the number becomes larger; a negative exponent means it becomes smaller.

将标准形式转换为普通数字时，按指数移动小数点。正指数表示数字变大；负指数表示数字变小。

Examples of positive exponents:

正指数的例子：

2.5 × 10³ = 2500
1.004 × 10⁶ = 1,004,000
9 × 10² = 900

2.5 × 10³ = 2500
1.004 × 10⁶ = 1,004,000
9 × 10² = 900

Examples of negative exponents:

负指数的例子：

7.8 × 10⁻² = 0.078
3 × 10⁻⁵ = 0.00003
1.2 × 10⁻¹ = 0.12

7.8 × 10⁻² = 0.078
3 × 10⁻⁵ = 0.00003
1.2 × 10⁻¹ = 0.12

Always check that the ordinary number has the correct place value. Use place holders if needed.

务必检查普通数字的数位是否正确。必要时使用零补位。

7. Ordering and Comparing in Standard Form | 标准形式的大小比较与排序

When comparing two numbers in standard form, first compare the exponents n. A larger n means a larger number, regardless of the value of a. If the exponents are equal, compare the a parts.

比较两个标准形式的大小时，先比较指数 n。指数越大，数值越大，与 a 的值无关。若指数相同，再比较 a 部分。

For example, 2.3 × 10⁵ is greater than 9.9 × 10⁴ because 5 > 4. And 4.1 × 10³ is less than 4.9 × 10³ because 4.1 < 4.9.

例如，2.3 × 10⁵ 大于 9.9 × 10⁴，因为 5 > 4。而 4.1 × 10³ 小于 4.9 × 10³，因为 4.1 < 4.9。

To order a list of standard form numbers, sort by exponent first, then by a part.

给一组标准形式的数排序时，先按指数排序，再按 a 部分排序。

Be careful with negative exponents: the more negative the exponent, the smaller the number. For example, 3 × 10⁻² is larger than 2 × 10⁻³ because −2 > −3.

注意负指数：指数越负，数值越小。例如，3 × 10⁻² 大于 2 × 10⁻³，因为 −2 > −3。

8. Practical Applications and Exam Tips | 实际应用与考试技巧

Standard form is used in science for astronomical distances, atomic sizes, population statistics, and financial calculations. In exam problems, you may be asked to estimate answers, compare quantities, or use a calculator with standard form notation.

标准形式在科学中用于天文距离、原子尺寸、人口统计和金融计算。考试中，你可能需要估算答案、比较数量，或使用计算器中的标准形式符号。

Calculator tip: To enter a number like 1.5 × 10⁸, press the ×10ˣ or EXP key on your calculator. Do not type “1.5×10^8” as a multiplication of numbers.

计算器技巧：输入 1.5 × 10⁸ 这类数时，使用计算器上的 ×10ˣ 或 EXP 键。不要打成 “1.5×10^8” 这种普通乘法。

When solving word problems, always check whether the final answer should be in standard form or ordinary form. Read the question carefully.

解决应用题时，始终检查最终答案要求标准形式还是普通形式。请仔细读题。

Also, remember to give your answer to an appropriate degree of accuracy, often 3 significant figures, unless told otherwise.

同时，请按要求的精度给出答案，通常是 3 位有效数字，除非另有说明。

9. Common Mistakes and How to Avoid Them | 常见错误及避免方法

One common mistake is writing a ≥ 10. For example, writing 12 × 10⁴ instead of 1.2 × 10⁵. Always check that a is between 1 and 10.

常见错误之一是写成 a ≥ 10。例如，将 1.2 × 10⁵ 误写成 12 × 10⁴。始终检查 a 是否在 1 与 10 之间。

Another mistake is miscounting the exponents when moving the decimal point. Practice counting with the zeros. For example, 0.00023: the decimal moves 4 places? No, 0.00023 = 2.3 × 10⁻⁴. Count: from the original decimal to after the leading digit is 4 places right.

另一个错误是移动小数点时数错指数。请练习数零。例如，0.00023：小数点移动 4 位？不对，0.00023 = 2.3 × 10⁻⁴。数：从原来小数点向右移动到首位数字后，是 4 位。

When adding or subtracting, failing to match exponents is a frequent error. Always make exponents equal first.

加减时不去统一指数是常见错误。务必先使指数相同。

Forgetting that a negative exponent means a smaller number can also cause ordering mistakes. Use a number line if needed.

忘记负指数表示更小的数也会导致排序错误。必要时可借助数轴。

10. Practice Questions | 练习题

Test your understanding with these questions. Answers are provided below.

用以下问题测试你的理解。答案在下方。

Write 6,300,000 in standard form.
Write 0.000045 in standard form.
Calculate (4 × 10⁵) × (2 × 10³), giving your answer in standard form.
Calculate (9 × 10⁷) ÷ (3 × 10²), giving your answer in standard form.
Add 2.5 × 10⁴ and 1.8 × 10³, giving your answer in standard form.
Which is larger: 7.2 × 10⁻³ or 8.9 × 10⁻⁴?

将 6,300,000 写成标准形式。
将 0.000045 写成标准形式。
计算 (4 × 10⁵) × (2 × 10³)，结果用标准形式表示。
计算 (9 × 10⁷) ÷ (3 × 10²)，结果用标准形式表示。
计算 2.5 × 10⁴ + 1.8 × 10³，结果用标准形式表示。
比较 7.2 × 10⁻³ 和 8.9 × 10⁻⁴ 的大小。

Answers: 1) 6.3 × 10⁶ 2) 4.5 × 10⁻⁵ 3) 8 × 10⁸ 4) 3 × 10⁵ 5) 2.68 × 10⁴ 6) 7.2 × 10⁻³ is larger because −3 > −4.

答案：1) 6.3 × 10⁶ 2) 4.5 × 10⁻⁵ 3) 8 × 10⁸ 4) 3 × 10⁵ 5) 2.68 × 10⁴ 6) 7.2 × 10⁻³ 更大，因为 −3 > −4。

By mastering standard form, you gain a powerful tool for handling extreme numbers in mathematics and science. Always practise converting, calculating, and comparing to build confidence for your IGCSE exam.

掌握标准形式，你就拥有了在数学和科学中处理极端数值的有力工具。不断练习转换、计算和比较，为你的 IGCSE 考试建立信心。

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Standard Form | 科学计数法与标准形式