📚 Error Analysis: Some Practical Considerations | 误差分析：一些实际考虑
In AQA A-level Mathematics, numerical answers often have to be given to a specified accuracy. Error analysis is the study of how uncertainty in data, rounding, and numerical methods affects final results.
在 AQA A-level 数学中，数值答案通常需要以指定精度给出。误差分析研究数据中的不确定性、舍入以及数值方法如何影响最终结果。
1. What Is Error Analysis? | 什么是误差分析
Error analysis quantifies the difference between an approximate value and the true or accepted value. At A-level this appears in measurement problems, numerical integration, iteration, and when using rounded values.
误差分析量化近似值与真实值或公认值之间的差异。在 A-level 中，这出现在测量问题、数值积分、迭代以及使用舍入值时。
An error is not necessarily a mistake. In applied mathematics, every measurement and every finite decimal representation carries some uncertainty, so we use error bounds to describe how reliable a result is.
误差并不一定是错误。在应用数学中，每次测量和每个有限小数表示都带有一定不确定性，因此我们用误差界来描述结果的可靠程度。
Learning to estimate and control error is a practical skill: it helps you decide how many decimal places to keep, how accurate a measurement needs to be, and whether a numerical method has produced a trustworthy answer.
学习估计和控制误差是一项实用技能：它帮助你决定应保留多少位小数、测量需要多精确，以及数值方法是否产生了可信的答案。
2. Absolute and Relative Error | 绝对误差与相对误差
Absolute error = |approximate value − true value| s the quantity being measured and tells us how far an estimate is from the accepted value.
绝对误差 = |近似值 − 真实值|。它与被测量具有相同单位，表明估计值与公认值相差多远。
Relative error = absolute error ÷ |true value| s dimensionless and is useful when comparing errors in quantities of very different sizes.
相对误差 = 绝对误差 ÷ |真实值|。它是无量纲的，在比较大小差异很大的量时非常有用。
Percentage error = relative error × 100% en requested when a measurement has been rounded or when a formula gives an approximation.
百分比误差 = 相对误差 × 100%。在 AQA 题目中，当测量值被舍入或公式给出近似值时，通常会要求计算百分比误差。
Absolute error = |approximate value − true value|
绝对误差 = |近似值 − 真实值|
For example, if a length is recorded as 8.2 cm to 1 decimal place, the true length is expected to lie between 8.15 cm and 8.25 cm, so the maximum absolute error is 0.05 cm.
例如，如果长度以 1 位小数记录为 8.2 cm，则真实长度预计位于 8.15 cm 与 8.25 cm 之间，因此最大绝对误差为 0.05 cm。
3. Significant Figures and Decimal Places | 有效数字与小数位
Significant figures count from the first non-zero digit, while decimal places count digits after the decimal point. A value such as 0.004060 has four significant figures but six decimal places if written as 0.004060.
有效数字从第一个非零数字开始计数，而小数位计算小数点后的数字。例如 0.004060 有四位有效数字，但如果写作 0.004060 则有六位小数。
When a final answer is rounded to, say, 3 significant figures, the implied absolute error is at most half a unit in the last significant place. This convention is central to upper and lower bound questions.
当最终答案舍入到例如 3 位有效数字时，隐含的绝对误差最多为最后一个有效位上的半个单位。这一约定是上界和下界问题的核心。
You must also be careful with zeros. In 2500, the zeros may or may not be significant unless the question states an accuracy such as 'to the nearest 10' or 'to 3 significant figures'.
你还必须注意零。在 2500 中，除非题目说明了诸如“精确到 10”或“保留 3 位有效数字”这样的精度，否则零可能是也可能不是有效数字。
4. Rounding vs Truncation Errors | 舍入误差与截断误差
Rounding chooses the nearest value at a given precision, so the error is at most half of the last retained unit. Truncation simply cuts off digits, so the error can be up to one full unit in the last retained place.
舍入在给定精度下选择最接近的值，因此误差最多为最后保留单位的半个单位。截断只是去掉数字，因此误差最多可达最后保留位的一个完整单位。
In numerical algorithms, truncation error also refers to the error caused by stopping an infinite process, such as using finitely many terms of a series or a finite number of iterations.
在数值算法中，截断误差也指停止无限过程所引起的误差，例如使用级数的有限项或有限次迭代。
For instance, approximating sin x by x − x³/6 is a truncation of the full Maclaurin series. The missing higher-order terms produce a truncation error that is small when x is close to 0.
例如，用 x − x³/6 近似 sin x 就是截断了完整的麦克劳林级数。缺失的高阶项会产生截断误差，当 x 接近 0 时该误差很小。
5. Error Propagation: Sums and Differences | 误差传播：和与差
If two measurements have absolute errors ∆x and ∆y, the maximum absolute error in x + y or x − y is ∆x + ∆y. Errors always add in the worst case, even when the quantities are subtracted.
如果两个测量的绝对误差为 ∆x 和 ∆y，那么 x + y 或 x − y 的最大绝对误差为 ∆x + ∆y。在最坏情况下误差总是相加，即使两个量相减也是如此。
This explains why subtracting two nearly equal rounded numbers can produce a result with a very large relative error. We call this subtractive cancellation and it is a key practical issue in error analysis.
这解释了为什么两个几乎相等的舍入数相减会产生相对误差很大的结果。我们称之为相减抵消，这是误差分析中的一个关键实际问题。
x − y = 0.02, but each absolute error is 0.005, so the error in the difference could be 0
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