📚 1.20 Upper and Lower Bounds | 1.20 上下界
In Edexcel IGCSE Mathematics, the concept of upper and lower bounds is fundamental whenever measurements are given to a certain level of accuracy. This topic, often labelled as 1.20 in the legacy specification, teaches us how to find the interval within which an original, unrounded value must lie. Whether a number has been rounded to the nearest integer, decimal place or significant figure, bounds give us the maximum and minimum possible true values.
在Edexcel IGCSE数学中，只要测量值是按一定精度给出的，上限与下限的概念就是基础。该专题在旧大纲中常被称为1.20，它教会我们如何找到原始未舍入数值必在的区间。不论一个数字是被舍入到最接近的整数、小数位还是有效数字，界限都能给出可能真实值的最大值和最小值。
1. What Are Upper and Lower Bounds? | 什么是上限和下限？
nd. Note the strict inequality for the upper bound because a rounded value can never quite reach the next half-unit.
一个舍入数的上限是指原始测量在舍入前可能的最大值。下限是指可能的最小原始值。两者共同构成了误差区间，通常用不等式表示：下限 ≤ 真实值 < 上限。请注意上限用严格不等号，是因为舍入值永远达不到下一个半单位。
2. Finding Bounds from Rounding | 从舍入确定界限
isfies 115 ≤ L < 125. The same principle applies no matter how large the rounding unit is.
要找出按给定单位舍入的数的界限，只需将舍入单位减半。比如某测量值为120米，精确到最近10米，舍入单位是10米，一半为5米。下限是120 – 5 = 115米，上限是120 + 5 = 125米。因此真实长度L满足115 ≤ L < 125。无论舍入单位多大，原理都相同。
3. Bounds and Significant Figures | 界限与有效数字
005 g. The bounds are: lower bound = 0.1195 g, upper bound = 0.1205 g. The true mass m is given by 0.1195 ≤ m < 0.1205.
当一个数被舍入到一定数量的有效数字时，要找到最后一位有效数字及其所在的数位。例如0.120 g舍入到3位有效数字，最后一位有效数字在千分位，舍入单位是0.001 g，一半为0.0005 g。界限是：下限=0.1195 g，上限=0.1205 g。真实质量m满足0.1195 ≤ m < 0.1205。
4. Bounds for Decimal Places | 小数位的界限
Similar logic applies to decimal places. A length of 12.0 cm correct to 1 decimal place has a rounding unit of 0.1 cm. Halving gives 0.05 cm. The lower bound is 11.95 cm and the upper bound is 12.05 cm. However, if the measurement were given as 12.00 cm correct to 2 decimal places, the rounding unit is 0.01 cm and bounds become 11.995 cm and 12.005 cm. The number of zeros matters because it indicates the degree of accuracy.
类似的逻辑适用于小数位。长度12.0 cm精确到1位小数，舍入单位是0.1 cm，减半后为0.05 cm。下限是11.95 cm，上限是12.05 cm。但若测量值给为12.00 cm，精确到2位小数，舍入单位就是0.01 cm，界限变为11.995 cm和12.005 cm。零的个数很重要，因为它表明了精度级别。
5. Bounds for Truncated Values | 截断值的界限
1 dp), the lower bound is 3.2 kg, but the upper bound is 3.3 kg, since any value up to but not including 3.3 would truncate to 3.2. Thus 3.2 ≤ m < 3.3.
截断与舍入不同。当一个数被截断时，超出某位的数字直接被切去。误差区间更宽：下限就是截断值本身，而上限是下一个尚未达到的可能值。例如某质量截断为3.2 kg（1位小数），下限为3.2 kg，但上限为3.3 kg，因为任何小于3.3的值截断后都得到3.2。因此3.2 ≤ m < 3.3。
6. Continuous and Discrete Values | 连续值与离散值
Bounds only apply to continuous data, such as length, mass and time, which can take any value within an interval. Discrete data, such as the number of students in a class, cannot have a fractional true value, so bounds are not defined in the same way. If a count is given as 30 to the nearest 10, the true number could be 25, 26, ..., 34, and it must be an integer. The idea of an upper bound being 34.999... is meaningless here.
界限只适用于连续数据，如长度、质量和时间，它们可以取区间内的任何值。离散数据，比如一个班级的学生人数，真实值不可能是分数，因此界限不能以相同方式定义。如果一个人数按最近10人给出为30，真实人数可能是25、26……34，且必须是整数。上限为34.999…在此毫无意义。
7. Calculations with Bounds: Addition and Subtraction | 界限运算：加法与减法
When adding two measurements, the maximum possible sum comes from adding their upper bounds, and the minimum sum from adding their lower bounds. For subtraction, the maximum difference is obtained by subtracting the lower bound of the second number from the upper bound of the first, while the minimum difference uses the opposite pairing. This ensures we capture the full range of possible results.
当两个测量值相加时，最大可能和来自各自上限相加，最小可能和来自各自下限相加。对于减法，最大差值是用第一个数的上限减去第二个数的下限，最小差值则用相反配对。这样才能完整捕获所有可能结果的区间。
8. Calculations with Bounds: Multiplication and Division | 界限运算：乘法与除法
For multiplication, the maximum product uses the upper bounds of both numbers, and the minimum product uses the lower bounds, provided all numbers are positive. When division is involved, to maximise a quotient, divide the upper bound of the numerator by the lower bound of the denominator. To minimise it, divide the lower bound of the numerator by the upper bound of the denominator. With negative values or mixed signs, careful case analysis is needed.
对于乘法，如果所有数字都是正数，则最大乘积使用两者的上限，最小乘积使用两者的下限。涉及除法时，为使商最大，用分子的上限除以分母的下限；要使商最小，用分子的下限除以分母的上限。若出现负值或符号混合，需仔细进行情况分析。
9. Finding Maximum and Minimum Possible Values of a Formula | 求算式的最大与最小可能值
nimum area is 15.15 × 8.65 = 131.0475 cm². So the true area lies between these two values.
一矩形的测量长度是15.2 cm，宽度是8.7 cm，各精确到1位小数。长度上限为15.25 cm，下限为15.15 cm；宽度上限为8.75 cm，下限为8.65 cm。最大可能面积是15.25 × 8.75 = 133.4375 cm²，最小面积是15.15 × 8.65 = 131.0475 cm²。因此真实面积在这两个值之间。
10. Bounds in Compound Measures | 复合测量中的界限
d = 120.5 ÷ 1.45 ≈ 83.1 km/h, minimum = 119.5 ÷ 1.55 ≈ 77.1 km/h. Compare the range to the calculated value 80 km/h.
当速度由距离和时间计算得来，且两者均按某精度给出时，必须先求出距离和时间的界限。假设一辆车行驶120 km（精确到最近km），用时1.5小时（到1位小数）。距离界限为119.5 km和120.5 km；时间界限为1.45 h和1.55 h。最大可能速度=120.5 ÷ 1.45 ≈ 83.1 km/h，最小=119.5 ÷ 1.55 ≈ 77.1 km/h。将此区间与计算值80 km/h做对比。
11. Expressing Final Answers with Appropriate Accuracy | 用适当精度表达最终答案
After performing a calculation with bounds, the answer must be given to a sensible degree of accuracy. Usually we round the final maximum and minimum to the same number of significant figures or decimal places as the original measurements. For the rectangle area, stating the upper bound as 133.4375 cm² is too precise; we should round to 133.4 cm² and the lower to 131.0 cm², reflecting that the original data were given to 3 significant figures.
在使用界限完成计算后，答案必须以合理的精度给出。通常我们将最终的最大值和最小值舍入到与原始测量相同数量的有效数字或小数位。对于矩形面积，把上限写成133.4375 cm²就过精了；应舍入为133.4 cm²，下限为131.0 cm²，以反映原始数据给到3位有效数字。
12. Common Mistakes and Exam Tips | 常见错误与应试技巧
Another is applying the half-unit rule to truncated numbers instead of using the next integral boundary. In exams, always identify the rounding unit first, halve it, and then construct the interval. When the problem involves several steps, keep all bounds in your calculator without intermediate rounding to preserve accuracy. Finally, check whether the context demands the answer to be an integer, such as the number of items that can be packed, where the upper bound may need truncation downwards.
常见错误是忘记上界要用严格不等号，而两端都写成≤。另一个是把截断值错误地套用半单位法则，而不知道要用下一个整数边界。在考试中，要始终先确定舍入单位，减半，然后构造区间。当题目涉及多个步骤时，所有界限都要在计算器中保留，不进行中间舍入以保持精度。最后，检查题意是否要求答案为整数，比如能打包的物品数量，此时上限可能需要向下截断。
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