Standard Form | 标准形式（科学记数法）

📚 Standard Form | 标准形式（科学记数法）

Standard form, also known as scientific notation, is a concise way of writing very large or very small numbers as a product of a decimal number between 1 and 10 and an integer power of 10. It is a fundamental topic in the IGCSE Mathematics syllabus and appears in both Paper 1 (non-calculator) and Paper 2 (calculator) examinations.

标准形式（又称科学记数法）是将非常大或非常小的数字简洁地表示为一个 1 到 10 之间的小数与 10 的整数次幂之积的方法。它是 IGCSE 数学大纲中的一个基础知识点，在卷一（非计算器）和卷二（计算器）考试中都会出现。

1. What Is Standard Form? | 什么是标准形式？

Every number in standard form is written in the form A × 10ⁿ. There are two strict rules: the value of A must be greater than or equal to 1 but less than 10, and n must be an integer. If the original number is large (greater than or equal to 10), then n is positive. If the original number is small (between 0 and 1), then n is negative.

标准形式中的每个数都写作 A × 10ⁿ。这里有两条严格规则：A 的值必须大于或等于 1 且小于 10；n 必须是整数。如果原数是较大的数（大于或等于 10），那么 n 为正；如果原数是较小的数（介于 0 和 1 之间），那么 n 为负。

A × 10ⁿ, where 1 ≤ A < 10 and n is an integer / 其中 1 ≤ A < 10，n 为整数

For example, 3.5 × 10³ is in standard form because 3.5 is between 1 and 10, and 3 is an integer. However, 35 × 10² is not standard form because 35 is greater than 10, and 0.35 × 10⁴ is not standard form because 0.35 is less than 1. The number zero cannot be written in standard form at all.

例如，3.5 × 10³ 是标准形式，因为 3.5 介于 1 和 10 之间，且 3 是整数。然而，35 × 10² 不是标准形式，因为 35 大于 10；0.35 × 10⁴ 也不是标准形式，因为 0.35 小于 1。数字 0 完全不能用标准形式表示。

2. Converting Large Numbers to Standard Form | 将大数转换为标准形式

To convert a large number into standard form, place the decimal point after the first non-zero digit. Then count how many places the decimal point has moved; this number becomes the exponent of 10. Since the original number is large, the exponent is positive.

要把一个较大数转换为标准形式，应将小数点放在第一个非零数字之后。然后数一数小数点移动了多少位，这个位数就是 10 的指数。由于原数较大，指数为正数。

Write down the first non-zero digit followed by the remaining significant digits with a decimal point after the first digit. / 写下第一个非零数字及其后的有效数字，并在第一个数字后加上小数点。
Count the number of places the decimal point has moved from its original position. / 数出小数点从原来位置移动了多少位。
Write the result as A × 10ⁿ, where n is that count. / 将结果写成 A × 10ⁿ 的形式，n 即为移动的位数。

Ordinary Number / 普通数字	Standard Form / 标准形式
5 200	5.2 × 10³
721 000	7.21 × 10⁵
4 000 000 000	4 × 10⁹

Notice that trailing zeros are not written as significant figures; they are captured by the power of 10. For instance, 4 000 000 000 becomes 4 × 10⁹, not 4.000 × 10⁹ unless the zeros are significant in a measurement context.

注意，末尾的零不作为有效数字写出，它们由 10 的幂来体现。例如，4 000 000 000 写成 4 × 10⁹，而不是 4.000 × 10⁹，除非在测量语境下这些零是有效数字。

3. Converting Small Numbers to Standard Form | 将小数转换为标准形式

For numbers between 0 and 1, the decimal point must move to the right to sit after the first non-zero digit. The number of places moved gives a negative exponent of 10. The value of A is still between 1 and 10.

对于介于 0 和 1 之间的数，小数点需要向右移动到第一个非零数字之后。移动的位数对应 10 的负指数。A 的值仍然介于 1 和 10 之间。

Ordinary Number / 普通数字	Standard Form / 标准形式
0.0063	6.3 × 10⁻³
0.000 000 52	5.2 × 10⁻⁷
0.000 000 000 1	1 × 10⁻¹⁰

A quick check: 6.3 × 10⁻³ = 6.3 ÷ 10³ = 0.0063. So a negative exponent simply means repeated division by 10. You can always verify your answer by expanding the power of 10.

快速检验：6.3 × 10⁻³ = 6.3 ÷ 10³ = 0.0063。因此负指数只是表示反复除以 10。你总是可以通过展开 10 的幂来验证答案。

4. Converting Standard Form to Ordinary Numbers | 将标准形式转换为普通数字

To convert a number in standard form back to an ordinary number, move the decimal point n places. If n is positive, move the decimal point to the right; if n is negative, move it to the left. Insert zeros when there are not enough digits.

要将标准形式的数还原为普通数字，需要将小数点移动 n 位。如果 n 为正，小数点向右移动；如果 n 为负，小数点向左移动。位数不足时用零补足。

4.25 × 10⁶ = 4 250 000 (move the decimal point 6 places right / 小数点向右移动 6 位)
7.3 × 10⁻⁴ = 0.00073 (move the decimal point 4 places left / 小数点向左移动 4 位)
1.5 × 10⁰ = 1.5 (any number to the power 0 equals 1 / 任何数的 0 次幂都等于 1)

Be especially careful with numbers like 3.00 × 10². Here A already has two decimal places, so the result is 300, not 3.00. The exponent tells you how far the decimal point travels, not how many zeros to add.

特别要注意像 3.00 × 10² 这样的数。这里 A 本身有两位小数，但结果是 300，而不是 3.00。指数告诉你小数点移动的距离，而不是需要添加多少个零。

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

When multiplying two numbers written in standard form, multiply the A values together, and separately add the exponents of 10. This follows directly from the index law aᵐ × aⁿ = aᵐ⁺ⁿ.

两个标准形式的数相乘时，将 A 部分相乘，同时将 10 的指数相加。这直接来源于指数法则 aᵐ × aⁿ = aᵐ⁺ⁿ。

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

Worked example: (2 × 10⁴) × (3 × 10⁷) = (2 × 3) × 10⁴⁺⁷ = 6 × 10¹¹. Here the decimal part is already between 1 and 10, so no further adjustment is needed.

例题：(2 × 10⁴) × (3 × 10⁷) = (2 × 3) × 10⁴⁺⁷ = 6 × 10¹¹。这里小数部分已经介于 1 和 10 之间，因此无需进一步调整。

However, if the product of the A values is 10 or more, you must adjust the answer. For example, (5 × 10³) × (4 × 10⁶) = 20 × 10⁹ = 2 × 10¹⁰. Since 20 is not less than 10, divide 20 by 10 and add 1 to the exponent.

然而，如果 A 部分的乘积大于或等于 10，则需要对答案进行调整。例如，(5 × 10³) × (4 × 10⁶) = 20 × 10⁹ = 2 × 10¹⁰。因为 20 不小于 10，所以将 20 除以 10，并把指数加 1。

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

To divide one number in standard form by another, divide the A values, and subtract the exponent of the divisor from the exponent of the dividend. This follows from the index law aᵐ ÷ aⁿ = aᵐ⁻ⁿ.

标准形式的数相除时，将 A 部分相除，并用被除数的指数减去除数的指数。这来源于指数法则 aᵐ ÷ aⁿ = aᵐ⁻ⁿ。

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

Worked example: (9 × 10⁸) ÷ (3 × 10²) = (9 ÷ 3) × 10⁸⁻² = 3 × 10⁶. Again, check that A is in the correct range.

例题：(9 × 10⁸) ÷ (3 × 10²) = (9 ÷ 3) × 10⁸⁻² = 3 × 10⁶。同样要检查 A 是否在正确的范围内。

If the quotient is less than 1, adjust it: (4 × 10⁵) ÷ (8 × 10²) = 0.5 × 10³ = 5 × 10². Since 0.5 is below 1, multiply it by 10 and subtract 1 from the exponent.

如果商小于 1，则需调整：(4 × 10⁵) ÷ (8 × 10²) = 0.5 × 10³ = 5 × 10²。由于 0.5 小于 1，将其乘以 10，并对指数减 1。

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

You cannot add or subtract the A values directly when the powers of 10 are different. First rewrite both numbers so that they share the same power of 10, then add or subtract the A values. Afterwards, check that the result is still in standard form.

当两个数的 10 的指数不同时，不能直接对 A 部分进行加减。首先应将两个数改写成相同的 10 的幂，然后对 A 部分进行加减。最后检查结果是否仍为标准形式。

Worked example (addition): 5.2 × 10⁴ + 3.8 × 10³. Write 3.8 × 10³ as 0.38 × 10⁴, so the sum becomes 5.2 × 10⁴ + 0.38 × 10⁴ = 5.58 × 10⁴. The answer is already in standard form.

加法例题：5.2 × 10⁴ + 3.8 × 10³。将 3.8 × 10³ 写成 0.38 × 10⁴，于是得到 5.2 × 10⁴ + 0.38 × 10⁴ = 5.58 × 10⁴。答案已经符合标准形式。

Worked example (subtraction): 6.4 × 10⁵ – 8.1 × 10⁴. Rewrite 8.1 × 10⁴ as 0.81 × 10⁵, so 6.4 × 10⁵ – 0.81 × 10⁵ = 5.59 × 10⁵. Alternatively, convert both numbers to ordinary form, subtract, then convert back.

减法例题：6.4 × 10⁵ – 8.1 × 10⁴。将 8.1 × 10⁴ 改写成 0.81 × 10⁵，则 6.4 × 10⁵ – 0.81 × 10⁵ = 5.59 × 10⁵。或者先将两个数都转换成普通形式，相减后再转换回标准形式。

8. Using a Calculator | 使用计算器

Most scientific calculators have a dedicated key labelled “×10ˣ”, “ENG”, or

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Standard Form | 标准形式（科学记数法）