📚 Mastering Upper and Lower Bounds: Limits of Accuracy for Edexcel IGCSE Maths | 掌握上下限：Edexcel IGCSE 数学中的精确度极限
In Edexcel IGCSE Mathematics, upper and lower bounds – also called limits of accuracy – are essential whenever a measurement has been rounded. This topic tests your ability to find the maximum and minimum possible values hidden behind a rounded number and to carry those bounds through calculations, from simple sums to compound measures like speed or area.
在 Edexcel IGCSE 数学中，只要测量值经过四舍五入，上界与下界（也称精确度极限）就变得至关重要。这一主题考察你从舍入后的数字中找出最大与最小可能值，并在加、减、乘、除乃至速度或面积等复合量计算中正确使用这些界限的能力。
Understanding limits of accuracy is not just an exam skill – it connects maths to real-world engineering, manufacturing and science, where safe design often depends on knowing the worst‑case scenario. The following guide will walk you through the concepts, rules and common exam traps so you can tackle any bounds question with confidence.
理解精确度极限不仅是应试技巧——它还把数学与现实工程、制造和科学联系起来；在这些领域中，安全设计往往取决于对最坏情况的了解。下面的指南将带你梳理相关概念、规则和常见考试陷阱，让你能够从容应对任何有关上下限的问题。
1. What Are Upper and Lower Bounds? | 什么是上下限？
When a numerical value is rounded to a certain degree of accuracy, the original measurement is not known exactly. Instead, it lies inside an interval. The upper bound is the greatest possible true value, and the lower bound is the smallest possible true value.
当一个数值被四舍五入到某个精确度时，最初的测量值并不精确——它落在某个区间内。上界是真实值可能的最大值，下界则是真实值可能的最小值。
1.5 cm and the upper bound is 12.5 cm. We express this as 11.5 cm ≤ true length < 12.5 cm.
例如，长度给出为 12 cm（精确到厘米），那么自 11.5 cm 起、直到（但不包含）12.5 cm 的任何测量值都会四舍五入为 12 cm。因此下界是 11.5 cm，上界是 12.5 cm。我们记为 11.5 cm ≤ 真实长度 < 12.5 cm。
Note the strict inequality for the upper bound: the true value can be exactly equal to the lower bound but must always be strictly less than the upper bound. This distinction is vital when interpreting inequality signs in exam answers.
注意上界用的是严格不等号：真实值可以恰好等于下界，但必须始终严格小于上界。在考试答案中解释不等号时，这一区别至关重要。
2. Understanding Accuracy and Rounding | 理解精确度与四舍五入
The key to finding bounds is the half-unit rule: bounds are obtained by adding and subtracting half of the rounding unit to the stated value. If a number is rounded to the nearest 10, the half unit is 5; to the nearest 0.1, it is 0.05; to the nearest integer, it is 0.5.
求上下界的关键是半单位法则：在给定值上加减半个舍入单位即得界限。如果数字舍入到最近的 10，半单位是 5；舍入到最近的 0.1，半单位是 0.05；舍入到最近的整数，半单位是 0.5。
an error of ±0.05, so lower bound = 3.65, upper bound = 3.75, satisfying 3.65 ≤ x < 3.75.
示例：数值 3.7 （精确到 1 位小数）的误差为 ±0.05，故下界 = 3.65，上界 = 3.75，满足 3.65 ≤ x < 3.75。
nt figures has a rounding unit of 10 m, giving an error of ±5 m and bounds 245 m to 255 m.
当精确度用有效数字表示时，舍入单位取最后一个有效数字所在的数位。例如，250 m 给出 2 位有效数字，舍入单位是 10 m，误差为 ±5 m，上下界分别为 245 m 和 255 m。
3. Finding Bounds of a Given Measurement | 求给定测量值的上下限
To find bounds systematically, identify the smallest place value to which the number is rounded. Halve that place value and add/subtract to the stated value. Always check whether the question uses decimal places, significant figures, or phrases like
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