📚 Standard Form and Significant Figures | 标准形式与有效数字
Standard form and significant figures appear throughout the IGCSE Mathematics course, especially in questions on estimation, measurement and calculator work. Standard form gives a compact way to write very large or very small numbers, while significant figures help you describe the precision of a rounded value. This revision guide explains the core rules, worked examples and common exam techniques for both topics.
标准形式和有效数字贯穿 IGCSE 数学课程，尤其在估算、测量和计算器相关题目中频繁出现。标准形式提供了一种简洁的方法来表示非常大或非常小的数，而有效数字则帮助你描述一个四舍五入后数值的精确程度。本篇复习指南将讲解这两个主题的核心规则、典型例题和常见考试技巧。
1. What Is Standard Form? | 什么是标准形式？
Standard form, also called scientific notation, is a way of writing a number as a product of a decimal part and a power of 10. The decimal part must be at least 1 but less than 10, and the power of 10 must be an integer. This makes it easy to compare very large and very small quantities at a glance.
标准形式又称科学记数法，是将一个数写成一个小数部分与 10 的幂相乘的形式。小数部分必须大于或等于 1 且小于 10，10 的指数必须是整数。这样就能一眼比较非常大和非常小的量。
N = A × 10ⁿ, where 1 ≤ A < 10 and n ∈ ℤ
umbers are much easier to read and use in standard form than in ordinary decimal notation.
例如，光速约为 3.0 × 10⁸ 米每秒，而电子所带电荷约为 1.6 × 10⁻¹⁹ 库仑。这两个数用标准形式表示后，比普通小数记法更容易阅读和使用。
2. Converting into Standard Form | 转换为标准形式
To convert a number into standard form, move the decimal point until the remaining number is between 1 and 10. Count how many places the decimal point moves. If the original number is 10 or greater, the power of 10 is positive. If the original number is between 0 and 1, the power of 10 is negative.
要将一个数转换为标准形式，请移动小数点，直到剩余数字介于 1 和 10 之间。数出小数点移动的位数。如果原数大于或等于 10，10 的指数为正；如果原数在 0 和 1 之间，10 的指数为负。
s three places to the right. Always check that the decimal part A satisfies 1 ≤ A < 10.
例如，45000 变为 4.5 × 10⁴，因为小数点向左移动了四位。同理，0.0032 变为 3.2 × 10⁻³，因为小数点向右移动了三位。务必检查小数部分 A 是否满足 1 ≤ A < 10。
3. Converting out of Standard Form | 从标准形式展开
To expand a number written in standard form, move the decimal point in the opposite direction. A positive power of 10 moves the decimal point to the right, making the number larger. A negative power of 10 moves the decimal point to the left, making the number smaller.
要将标准形式表示的数展开，请向相反方向移动小数点。10 的正指数使小数点向右移动，数值变大；10 的负指数使小数点向左移动，数值变小。
7.1 × 10⁻⁴ means 7.1 with the decimal point moved four places to the left, giving 0.00071.
例如，2.6 × 10⁵ 表示将 2.6 的小数点向右移动五位，得到 260000。另一方面，7.1 × 10⁻⁴ 表示将 7.1 的小数点向左移动四位，得到 0.00071。
4. Significant Figures | 有效数字
Significant figures are the digits in a number that carry meaning about its precision. The first significant figure is the first non-zero digit when reading from the left. Leading zeros before the first non-zero digit are not significant, but zeros between non-zero digits and trailing zeros after a decimal point are significant.
有效数字是一个数中表示精确程度的数字。第一个有效数字是从左往右读时遇到的第一个非零数字。第一个非零数字之前的零不计入有效数字，但非零数字之间的零以及小数点后的末尾零计入有效数字。
For example, 0.00456 has three significant figures: 4, 5 and 6. The number 0.05020 has four significant figures: 5, 0, 2 and 0, because the final zero after the decimal point shows measured precision. The number 3004 has four significant figures because the zeros are between non-zero digits.
例如，0.00456 有三位有效数字：4、5 和 6。0.05020 有四位有效数字：5、0、2 和 0，因为小数点后的末尾零体现了测量精度。3004 有四位有效数字，因为其中的零位于非零数字之间。
5. Rounding to Significant Figures | 四舍五入到有效数字
To round a number to a given number of significant figures, first identify the required significant digit. Look at the next digit to the right. If this next digit is 5 or more, round the significant digit up by 1. Otherwise, leave it unchanged. Replace the remaining digits with zeros if they are needed to keep the place value.
要将一个数四舍五入到指定数量的有效数字，首先要确定所需的那个有效数字位。再看它右边的一位数字。如果右边这位数字是 5 或更大，则将有效数字位加 1；否则保持不变。其余数字如果需要保留数位，就用零代替。
For example, 0.006837 rounded to 2 significant figures is 0.0068. The first two significant digits are 6 and 8, and the next digit is 3, so no rounding up occurs. The number 48256 rounded to 3 significant figures is 48300, because the first three significant digits are 4, 8 and 2, and the next digit is 5, so the 2 rounds up to 3.
例如，0.006837 保留 2 位有效数字为 0.0068。前两位有效数字是 6 和 8，下一位是 3，因此不需要进位。48256 保留 3 位有效数字为 48300，因为前三位有效数字是 4、8 和 2，下一位是 5，所以 2 进位为 3。
6. Estimation Using Significant Figures | 使用有效数字进行估算
Estimation questions often ask you to round each number in a calculation to 1 significant figure before working out an approximate answer. This keeps the arithmetic simple and gives a sensible check against the exact result. The estimated answer does not need to be exact, but it should be reasonably close.
估算题通常要求先将计算中的每个数保留 1 位有效数字，再计算出近似答案。这样可以使运算保持简单，并作为精确结果的一个合理检验。估算答案不需要完全精确，但应当比较接近。
| Original expression | Approximation | Estimated result |
| 48.7 × 2.13 ÷ 0.52 | 50 × 2 ÷ 0.5 | 200 |
In this example, 48.7 rounds to 50, 2.13 rounds to 2, and 0.52 rounds to 0.5. Multiplying 50 by 2 gives 100, and dividing by 0.5 gives 200. The exact value is approximately 199.48, so the estimate is very useful.
在这个例子中，48.7 四舍五入为 50，2.13 四舍五入为 2，0.52 四舍五入为 0.5。50 乘以 2 得 100，再除以 0.5 得 200。精确值约为 199.48，因此这个估算非常有用。
7. Adding and Subtracting in Standard Form | 标准形式的加减
When adding or subtracting numbers in standard form, the powers of 10 must be the same. If they already match, simply add or subtract the decimal parts and keep the same power of 10. If they do not match, first convert one number so that both powers of 10 are equal.
对标准形式表示的数进行加减时，10 的指数必须相同。如果指数已经相同，只需将小数部分相加或相减，并保持 10 的指数不变。如果指数不同，应先将其中一个数转换，使两个 10 的指数相等。
(a × 10ⁿ) + (b × 10ⁿ) = (a + b) × 10ⁿ
2.5 × 10³. Expanding to ordinary numbers, 2000 + 500 also gives 2500, which is 2.5 × 10³.
例如，(3 × 10⁵) + (4 × 10⁵) = 7 × 10⁵。对于不同指数的情形，(2 × 10³) + (5 × 10²) 可以写成 2 × 10³ + 0.5 × 10³，得到 2.5 × 10³。展开成普通数字，2000 + 500 也得到 2500，即 2.5 × 10³。
8. Multiplying and Dividing in Standard Form | 标准形式的乘除
Multiplication and division in standard form are straightforward. To multiply, multiply the decimal parts and add the powers of 10. To divide, divide the decimal parts and subtract the powers of 10. After the operation, check that the decimal part is still at least 1 and less than 10; if not, adjust it.
标准形式的乘法和除法比较简单。乘法是将小数部分相乘，同时将 10 的指数相加。除法是将小数部分相除，同时将 10 的指数相减。运算后要检查小数部分是否仍在 1 到 10 之间；如果不是，需要调整。
(A × 10ᵐ) × (B × 10ⁿ) = (A × B) × 10ᵐ⁺ⁿ
(A × 10ᵐ) ÷ (B × 10ⁿ) = (A ÷ B) × 10ᵐ⁻ⁿ
ication gives 12 × 10⁶, this must be adjusted to 1.2 × 10⁷ because A must be less than 10.
例如，(3 × 10⁴) × (2 × 10³) = 6 × 10⁷。另外，(8 × 10⁵) ÷ (2 × 10²) = 4 × 10³。如果乘法得到 12 × 10⁶，则必须调整为 1.2 × 10⁷，因为 A 必须小于 10。
9. Calculator Use and Common Errors | 计算器使用与常见错误
, such as 3.2E5 for 3.2 × 10⁵. To enter a number in standard form, look for the EXP or ×10x er separately unless you are confident, because this can cause bracket mistakes in longer calculations.
许多计算器使用 E 表示法显示标准形式，例如 3.2E5 表示 3.2 × 10⁵。要输入标准形式的数，请查找 EXP 或 ×10x 键。除非很有把握，否则不要单独输入 × 10 再输入指数，因为在较长计算中这会导致括号错误。
- Forgetting to adjust the decimal part so that 1 ≤ A < 10.
- Confusing significant figures with decimal places.
- Moving the decimal point in the wrong direction when converting.
- Changing the size of a number when rounding to significant figures.
常见错误包括：忘记调整小数部分，使 1 ≤ A < 10；将有效数字与小数位数混淆；转换
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