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 way of writing very large or very small numbers compactly. In IGCSE Mathematics, this topic appears in both Paper 1 and Paper 2, and it is essential for calculations involving distances in space, sizes of particles, and many scientific contexts.

标准形式，又称科学记数法，是一种将非常大或非常小的数简洁书写的方式。在IGCSE数学考试中，这一考点在Paper 1和Paper 2中都会出现，并且在涉及太空距离、粒子尺寸以及许多科学情境的计算中至关重要。

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

A number is written in standard form when it is expressed as \(A \times 10^n\), where \(1 \le A < 10\) and \(n\) is an integer. The value of \(A\) must have exactly one non-zero digit before the decimal point.

当一个数被表示为 \(A \times 10^n\) 的形式时，它就是标准形式，其中 \(1 \le A < 10\)，且 \(n\) 是整数。\(A\) 的值必须在小数点前恰好有一位非零数字。

The coefficient \(A\) is always between 1 and 10 (including 1, excluding 10).
The exponent \(n\) tells us how many places the decimal point has been moved.
If \(n\) is positive, the original number is large; if \(n\) is negative, the original number is small.

系数 \(A\) 总是在1到10之间（包含1，不包含10）。
指数 \(n\) 表示小数点移动的位数。
如果 \(n\) 是正数，原数很大；如果 \(n\) 是负数，原数很小。

Standard form: A × 10ⁿ, where 1 ≤ A < 10 and n ∈ ℤ

标准形式：A × 10ⁿ，其中 1 ≤ A < 10，n 为整数

2. Writing Large Numbers in Standard Form | 将大数写成标准形式

To write a large number such as 45,000 in standard form, first identify the first non-zero digit (4), then place the decimal point after it to get 4.5. Count how many places the decimal point has moved from its original position to this new position.

要将一个较大的数如 45,000 写成标准形式，首先找出第一个非零数字（即4），然后将小数点放在它后面得到4.5。接着数一数小数点从原来的位置移动到新位置所经过的位数。

For 45,000, the decimal point starts after the last zero and moves 4 places to the left, so the exponent is 4. Therefore, 45,000 = 4.5 × 10⁴.

对于45,000，小数点原本在最后一个零之后，向左移动4位，因此指数为4。所以，45,000 = 4.5 × 10⁴。

320,000 = 3.2 × 10⁵

7,500,000,000 = 7.5 × 10⁹

Remember that the exponent for large numbers is always positive, and it equals the number of digits after the first digit in the original number.

请记住，对于大数，指数总是正数，它等于原数中第一个数字之后的数字个数。

3. Writing Small Numbers in Standard Form | 将小数写成标准形式

For small numbers such as 0.00072, move the decimal point to the right until there is exactly one non-zero digit before it. Here, the decimal point moves 4 places to the right to obtain 7.2.

对于像0.00072这样较小的数，需要将小数点向右移动，直到小数点前恰好有一个非零数字。在这里，小数点向右移动4位，得到7.2。

Since the decimal point moved to the right, the exponent is negative: 0.00072 = 7.2 × 10⁻⁴. This makes sense because dividing 7.2 by 10,000 gives the original number.

由于小数点向右移动，指数为负数：0.00072 = 7.2 × 10⁻⁴。这是合理的，因为7.2除以10000就得到原来的数。

0.000045 = 4.5 × 10⁻⁵

0.0000009 = 9 × 10⁻⁷

Count the zeros before the first non-zero digit (including the zero before the decimal point) to determine the absolute value of the negative exponent.
For 0.000003, there are 6 places to move: 0.000003 = 3 × 10⁻⁶.

数一数第一个非零数字之前的零的个数（包括小数点前的零），以确定负指数的绝对值。
对于0.000003，需要移动6位：0.000003 = 3 × 10⁻⁶。

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

To convert \(6.02 \times 10^{23}\) back to an ordinary number, the positive exponent 23 tells us to move the decimal point 23 places to the right. Since 6.02 has only one digit before the decimal point, we need to add zeros as placeholders.

要将 \(6.02 \times 10^{23}\) 转换回普通数，正指数23告诉我们把小数点向右移动23位。由于6.02在小数点前只有一位数字，我们需要添加零作为占位符。

Similarly, a negative exponent means moving the decimal point to the left. For \(4.5 \times 10^{-3}\), move the decimal point 3 places left, giving 0.0045.

类似地，负指数意味着将小数点向左移动。对于 \(4.5 \times 10^{-3}\)，将小数点向左移动3位，得到0.0045。

2.1 × 10⁵ = 210,000

8.9 × 10⁻² = 0.089

When converting, be careful to fill empty spaces with zeros. The total number of digits on the correct side of the decimal point must equal the absolute value of the exponent.

转换时，注意用零填充空位。小数点正确一侧的总位数必须等于指数的绝对值。

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

To multiply two numbers in standard form, multiply the coefficients together and then multiply the powers of 10 separately. Use the law of indices: \(10^a \times 10^b = 10^{a+b}\).

要将两个标准形式的数相乘，先乘系数，再分别处理10的幂。使用指数律：\(10^a \times 10^b = 10^{a+b}\)。

(3 × 10⁵) × (2 × 10⁷) = 3 × 2 × 10⁵ × 10⁷ = 6 × 10¹²

(4 × 10⁻²) × (5 × 10³) = 20 × 10¹ = 2 × 10²

Notice in the second example, the product of the coefficients is 20, which is not less than 10. In such cases, convert the result into proper standard form by adjusting the coefficient down by a factor of 10 and increasing the exponent by 1.

注意在第二个例子中，系数的乘积是20，它并不小于10。在这种情况下，需要将结果调整为标准形式，即将系数缩小10倍，同时指数加1。

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

For division, divide the coefficients and subtract the exponents: \(10^a \div 10^b = 10^{a-b}\).

做除法时，将系数相除，指数相减：\(10^a \div 10^b = 10^{a-b}\)。

(9 × 10⁶) ÷ (3 × 10²) = 3 × 10⁴

(8 × 10⁻³) ÷ (2 × 10⁻⁵) = 4 × 10²

Check the second example carefully: \(-3 – (-5) = 2\), so the exponent becomes 2. Always pay attention to the sign of the exponents when subtracting.

仔细检查第二个例子：\(-3 – (-5) = 2\)，所以指数变为2。在相减时，始终注意指数的正负号。

If the coefficient after division is not between 1 and 10, adjust it using the same method as in multiplication.

如果除法后得到的系数不在1到10之间，采用与乘法相同的方法进行调整。

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

To add or subtract numbers in standard form, both numbers must have the same power of 10. If the exponents differ, rewrite one number so that the exponents match before performing the operation.

要加减标准形式的数，两个数必须具有相同的10的幂。如果指数不同，需要先将其中一个数重写为相同的指数，再进行运算。

3 × 10⁴ + 4 × 10⁴ = 7 × 10⁴

2.5 × 10³ + 1.5 × 10⁴ = 2.5 × 10³ + 15 × 10³ = 17.5 × 10³ = 1.75 × 10⁴

In the second example, \(1.5 \times 10^4\) is rewritten as \(15 \times 10^3\).
After adding, the coefficient 17.5 is adjusted to 1.75 and the exponent increases by 1.

在第二个例子中，\(1.5 \times 10^4\) 被重写为 \(15 \times 10^3\)。
相加后，系数17.5调整成1.75，指数增加1。

Subtraction works in exactly the same way: align the exponents first, then subtract the coefficients.

减法也完全相同：先对齐指数，再相减系数。

8. Standard Form on a Calculator | 计算器上的标准形式

Most scientific calculators have an “EXP” button or a “×10ˣ” button. To enter \(3.2 \times 10^7\), press 3.2, then EXP, then 7. The calculator displays 3.2×10⁷ or 32000000 depending on the model and its display settings.

大多数科学计算器都带有“EXP”按键或“×10ˣ”按键。要输入 \(3.2 \times 10^7\)，先按3.2，再按EXP，然后按7。计算器会显示3.2×10⁷或32000000，具体显示取决于型号及其显示设置。

Use the ENG mode on some calculators to display results in scientific notation.
If the answer exceeds the screen limit, the calculator automatically switches to scientific notation.
Do not include a multiplication symbol between numbers and 10 when using the EXP key.

在某些计算器上，可使用ENG模式使结果显示为科学记数法。
如果结果超出屏幕限制，计算器会自动切换为科学记数法。
使用EXP键时，不要在数字与10之间输入乘号。

When checking your work, always estimate the approximate size of the result to catch errors in the exponent.

检查答案时，始终估算结果的大致量级，以发现指数上的错误。

9. Real-World Applications | 实际应用

Standard form is widely used in science and engineering. The distance from the Earth to the Sun is approximately \(1.5 \times 10^{11}\) metres, and the mass of a hydrogen atom is about \(1.67 \times 10^{-27}\) kilograms.

标准形式在科学和工程领域被广泛使用。地球到太阳的距离约为 \(1.5 \times 10^{11}\) 米，氢原子的质量约为 \(1.67 \times 10^{-27}\) 千克。

Speed of light: \(3 \times 10^8\) m/s. Population of the world: approximately \(8 \times 10^9\) people. Diameter of a red blood cell: about \(8 \times 10^{-6}\) metres.

光速为 \(3 \times 10^8\) 米/秒。世界人口约为 \(8 \times 10^9\) 人。红血球的直径约为 \(8 \times 10^{-6}\) 米。

When using standard form in calculations, always complete the operation first and present the final answer in correct standard form.

在计算中使用标准形式时，先完成运算，再将最终答案表示为正确的标准形式。

10. Common Mistakes and Exam Tips | 常见错误与考试技巧

Students often make the mistake of writing 45 × 10⁴ instead of 4.5 × 10⁵. Always check that the coefficient falls between 1 and 10. If it does not, you must adjust it.

学生经常犯把45 × 10⁴写成而不是4.5 × 10⁵的错误。始终检查系数是否在1到10之间。如果不在，必须进行调整。

When multiplying, add exponents only if the base is the same (10). Do not add the coefficients’ powers.
When adding or subtracting, exponents must be identical before you combine coefficients.
Write every intermediate step explicitly to avoid small arithmetic errors.
On a calculator, use scientific notation mode and record the displayed exponent carefully.

乘法中只有在底数相同（都是10）时才能将指数相加。不要对系数添加指数。
加减时，必须先使指数相同，再合并系数。
写出每一步中间过程，避免微小算术错误。
在计算器上，使用科学记数法模式并仔细记录显示的指数。

Finally, always read the question carefully. Some questions require you to rearrange a calculation into standard form, while others ask you to compare the sizes of numbers written in standard form. Understanding the exponent is the key to comparing them quickly.

最后，一定要仔细阅读题目。有些题目要求你将计算结果改写为标准形式，而另一些则要求你比较以标准形式书写的大小。理解指数是关键，可帮助你快速比较这些数的大小。

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Standard Form | 标准形式（科学记数法）