Upper and Lower Bounds | 上下界与误差

📚 Upper and Lower Bounds | 上下界与误差

Welcome to this IGCSE Maths revision note on upper and lower bounds, linked to Topic 1 Number (T-1-1062). You will learn how to find the range of possible true values for a rounded measurement and how to use bounds in calculations.

欢迎学习本篇 IGCSE 数学复习笔记，主题为数字中的上下界（T-1-1062）。你将学习如何求一个经过舍入的测量值可能对应的真实值范围，以及如何在计算中使用上下界。

1. What are bounds? | 什么是上下界？

When a measurement is rounded, the recorded value is not exact. The true value lies between a lower bound and an upper bound. For example, if a length is given as 250 cm to the nearest 10 cm, the true length L is at least 245 cm and less than 255 cm.

当测量值被舍入时，记录的数据并不精确。真实值位于下界和上界之间。例如，若长度记录为 250 cm（精确到 10 cm），真实长度 L 至少为 245 cm，且小于 255 cm。

We write the bound inequality as 245 ≤ L < 255. The lower bound is 245 cm and the upper bound is 255 cm.

我们可以用不等式表示为 245 ≤ L < 255。下界是 245 cm，上界是 255 cm。

2. Why bounds matter | 为什么上下界重要

Bounds are essential in real-life measurement because instruments are never perfectly exact. When a ruler shows 6.4 cm to the nearest millimetre, the actual length might be anywhere from 6.35 cm to 6.45 cm.

上下界在现实测量中非常重要，因为仪器永远不可能完全精确。当尺子显示 6.4 cm（精确到毫米）时，实际长度可能介于 6.35 cm 到 6.45 cm 之间。

In IGCSE exams, bounds questions often ask for the minimum or maximum possible value of a result calculated from rounded inputs. Understanding the error interval is the first step.

在 IGCSE 考试中，上下界题目通常会要求根据舍入的输入值，求出计算结果可能的最小值或最大值。理解误差区间是第一步。

3. Degree of accuracy and place value | 精确度与数位

The lower and upper bounds depend on the degree of accuracy. If a value is given to the nearest 1, the interval width is 1, so the error is ±0.5.

下界和上界取决于精确度。如果数值精确到个位，区间宽度为 1，因此误差为 ±0.5。

If a value is given to the nearest 10, the error is ±5. If it is given to the nearest 0.1, the error is ±0.05. Always halve the place value of the last rounded digit.

若数值精确到十位，误差为 ±5；若精确到 0.1，误差为 ±0.05。始终将最后一位舍入数字的位值减半，即可得到误差。

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