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 called scientific notation) is a compact way of writing very large or very small numbers. In the Edexcel IGCSE Mathematics syllabus, you are expected to convert between ordinary numbers and standard form, perform calculations, and compare values written in this form.

标准形式（也叫科学记数法）是一种简洁地表示非常大或非常小的数的方式。在 Edexcel IGCSE 数学考纲中，你需要在普通数和标准形式之间进行转换、进行运算，并比较用这种形式表示的数的大小。

1. What is Standard Form? | 什么是标准形式？

A number is written in standard form as: A × 10ⁿ, where 1 ≤ A < 10 and n is an integer. The value A is called the coefficient or mantissa, and n is the exponent or power of 10.

一个数以标准形式写成：A × 10ⁿ，其中 1 ≤ A < 10，n 是整数。A 称为系数（或尾数），n 称为指数（或 10 的幂）。

3.2 × 10⁵ = 320,000

Here 3.2 is between 1 and 10, and 5 is a positive integer. The exponent tells us how many places the decimal point has moved.

这里 3.2 在 1 和 10 之间，5 是正整数。指数告诉我们小数点移动了多少位。

2. Converting Large Numbers to Standard Form | 将较大的数转换为标准形式

For large numbers, move the decimal point to the left until there is exactly one non-zero digit before the decimal point. Count how many positions you moved; this becomes the positive exponent n.

对于较大的数，将小数点向左移动，直到小数点前只剩下一个非零数字。数一数移动了多少位，这个位数就是正整数指数 n。

Example: Write 45,600 in standard form.

例子：把 45,600 写成标准形式。

Move the decimal point from the end: 4.5600. That is 4 places to the left. So 45,600 = 4.56 × 10⁴.

把小数点从末尾向左移动 4 位后得到 4.5600。因此 45,600 = 4.56 × 10⁴。

Example: 1,200,000 = 1.2 × 10⁶ because the decimal point moves 6 places left.

例子：1,200,000 = 1.2 × 10⁶，因为小数点向左移动了 6 位。

3. Converting Small Numbers to Standard Form | 将较小的数转换为标准形式

For numbers less than 1, move the decimal point to the right until there is one non-zero digit on the left. The number of moves becomes a negative exponent.

对于小于 1 的数，将小数点向右移动，直到左边只有一个非零数字。移动的次数就是负指数。

Example: Write 0.00072 in standard form.

例子：把 0.00072 写成标准形式。

Move the decimal point 4 places right to get 7.2. Hence 0.00072 = 7.2 × 10⁻⁴.

将小数点向右移动 4 位得到 7.2。因此 0.00072 = 7.2 × 10⁻⁴。

Example: 0.000000035 = 3.5 × 10⁻⁸.

例子：0.000000035 = 3.5 × 10⁻⁸。

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

If the exponent is positive, move the decimal point to the right by that many places, adding zeros as needed. If the exponent is negative, move the decimal point to the left.

如果指数为正，将小数点向右移动相应位数，必要时补零。如果指数为负，则将小数点向左移动。

Standard Form	Ordinary Number	过程
2.5 × 10³	2500	Move decimal 3 places right / 右移3位
6.02 × 10⁻⁴	0.000602	Move decimal 4 places left / 左移4位
1 × 10⁰	1	No movement / 不动

5. Comparing Numbers in Standard Form | 比较标准形式的数

To compare two numbers in standard form, first compare the exponents (powers of 10). The larger the exponent, the larger the number, regardless of A. If the exponents are equal, then compare the coefficients A.

比较两个标准形式的数时，先比较指数（10 的幂）。指数越大，数越大，和 A 无关。如果指数相同，再比较系数 A。

Example: Which is larger: 3.2 × 10⁴ or 9.8 × 10³?

例子：3.2 × 10⁴ 和 9.8 × 10³ 哪个更大？

The exponent 4 is greater than 3, so 3.2 × 10⁴ is larger, even though 9.8 > 3.2.

指数 4 大于 3，所以 3.2 × 10⁴ 更大，尽管 9.8 > 3.2。

Example: 6.1 × 10⁵ and 6.1 × 10⁵ are equal; but 6.1 × 10⁵ > 5.9 × 10⁵ because coefficients differ.

例子：6.1 × 10⁵ 与 6.1 × 10⁵ 相等；而 6.1 × 10⁵ > 5.9 × 10⁵，因为系数不同。

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

To add or subtract numbers in standard form, you must first rewrite them so they have the same power of 10. Then add/subtract the coefficients and keep that common power.

在标准形式下做加减时，必须先改写成相同的 10 的幂。然后将系数相加或相减，保留这一共同的幂。

(2.4 × 10⁶) + (3.1 × 10⁵) = (2.4 × 10⁶) + (0.31 × 10⁶) = 2.71 × 10⁶

Notice that 3.1 × 10⁵ = 0.31 × 10⁶ because 0.31 × 10 × 10⁵ = 3.1 × 10⁵. Always adjust the smaller exponent to match the larger one.

注意 3.1 × 10⁵ = 0.31 × 10⁶，因为 0.31 × 10 × 10⁵ = 3.1 × 10⁵。通常将较小的指数调整为较大的那个。

If the result has A ≥ 10, you must adjust the answer back into standard form. For example: 6.7 × 10⁸ + 5.4 × 10⁸ = 12.1 × 10⁸ = 1.21 × 10⁹.

如果结果的 A ≥ 10，需要再调整为标准形式。例如：6.7 × 10⁸ + 5.4 × 10⁸ = 12.1 × 10⁸ = 1.21 × 10⁹。

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

For multiplication, multiply the coefficients A values together and add the exponents. For division, divide the coefficients and subtract the exponents.

乘法：将系数 A 相乘，指数相加。除法：将系数相除，指数相减。

(2 × 10⁴) × (3 × 10⁵) = 6 × 10⁹

(8 × 10⁷) ÷ (2 × 10²) = 4 × 10⁵

After multiplying, you may need to convert the coefficient. Example: (4 × 10³) × (5 × 10²) = 20 × 10⁵ = 2 × 10⁶.

相乘后可能需要对系数进行转换。例如：(4 × 10³) × (5 × 10²) = 20 × 10⁵ = 2 × 10⁶。

8. Using Your Calculator for Standard Form | 在计算器上使用标准形式

Most scientific calculators have an “EXP” or “×10ˣ” key. To enter 3.2 × 10⁵, press 3 . 2 then EXP (or ×10ˣ) then 5. For a negative exponent, use the (-) key before the number.

大多数科学计算器都有 EXP 或 ×10ˣ 键。输入 3.2 × 10⁵ 时，依次按 3 . 2、EXP（或 ×10ˣ）、5。负指数则先按 (-) 键再输入数字。

When the calculator displays something like 3.2⁵, it actually means 3.2 × 10⁵. Never write “3.2⁵” in your answer.

如果计算器显示类似 3.2⁵，它实际表示 3.2 × 10⁵。千万不要在答案中写“3.2⁵”。

9. Common IGCSE Exam Tips and Errors | IGCSE 常见考点与易错点

Always ensure 1 ≤ A < 10. Writing 12 × 10³ is not standard form; it must be changed to 1.2 × 10⁴.

始终确保 1 ≤ A < 10。写 12 × 10³ 不是标准形式，必须改为 1.2 × 10⁴。
Pay attention to the negative sign in the exponent. A small number always has a negative power of 10.

注意指数中的负号。较小的数一定对应一个负的 10 的幂。
Count the decimal positions carefully. One careless move causes a full mark to be lost.

仔细数小数点移动的位数。粗心移动一次就会导致整题失分。
When adding or subtracting, do not add the exponents. Only add exponents in multiplication.

加减时不要把指数相加。只有乘法才需要指数相加。
Before giving a final answer, check if the coefficient is in the allowed range. If not, adjust by moving the decimal one more place and changing the exponent by 1.

给出最终答案前，检查系数是否在允许范围内。如果不在，再多移动一次小数点，并把指数相应改变 1。

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Standard Form | 标准形式（科学记数法）