📚 Upper and Lower Bounds | 上下界与误差
Welcome to this IGCSE Mathematics 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, how to write error intervals correctly, and how to use bounds in calculations so that you can always find the maximum or minimum possible result.
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，如何正确书写误差区间，以及如何在计算中使用上下界，从而求出结果可能的最大值或最小值。
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 ound is not included (<), because a value of exactly 255 cm would round up to 260 cm, not 250 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。请注意：下界是包含的（≤），而上界不包含（<），因为恰好等于 255 cm 的数值会舍入到 260 cm，而不是 250 cm。
The key vocabulary in this topic is: lower bound (the smallest possible true value), upper bound (the largest value that still rounds to the recorded value), error interval (the whole range between them), and degree of accuracy (the place value or significant figure the value was rounded to).
本主题的核心术语有：下界（真实值可能的最小值）、上界（仍然能舍入到该记录值的最大数值）、误差区间（两者之间的整个范围），以及精确度（该数值被舍入到的数位或有效数字位数）。
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 up to, but not including, 6.45 cm. The error interval is written as 6.35 ≤ L < 6.45.
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（不含 6.45）。误差区间写作 6.35 ≤ L < 6.45。
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, because every later calculation depends on it.
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，因为后续每一个计算都依赖于它。
Bounds also matter in engineering and science. If a machine part is specified as 12.5 mm to the nearest 0.1 mm, then a part measuring 12.56 mm may be rejected even though it looks correct on paper. Bounds tell you how much tolerance is acceptable.
上下界在工程与科学中同样重要。如果某个机器零件规定为 12.5 mm（精确到 0.1 mm），那么一个实际尺寸为 12.56 mm 的零件可能就会被判定不合格，尽管它在纸面上看起来是对的。上下界能告诉你可接受的容差是多少。
3. Degree of Accuracy and Half the Place Value | 精确度与半个位值
The width of the error interval depends entirely 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. If a value is given to the nearest 10, the error is ±5. If a value is given to the nearest 0.1, the error is ±0.05.
误差区间的宽度完全取决于精确度。如果数值精确到个位，区间宽度为 1，因此误差为 ±0.5。若数值精确到十位，误差为 ±5。若数值精确到 0.1，误差为 ±0.05。
The rule is simple: always halve the place value of the last rounded digit. For 1 decimal place the place value is 0.1, so half is 0.05. For 2 decimal places the place value is 0.01, so half is 0.005. For the nearest 100 the place value is 100, so half is 50.
规则很简单：始终将最后一位舍入数字的位值减半。精确到 1 位小数时，位值为 0.1，一半是 0.05；精确到 2 位小数时，位值为 0.01，一半是 0.005；精确到百位时，位值为 100，一半是 50。
Error = ½ × Place value of last digit
| Recorded value | Accuracy | Lower bound | Upper bound |
|---|---|---|---|
| 7.3 | 1 decimal place | 7.25 | 7.35 |
| 250 | Nearest 10 | 245 | 255 |
| 4.72 | 2 decimal places | 4.715 | 4.725 |
| 0.068 | 2 significant figures | 0.0675 | 0.0685 |
| 1300 | 2 significant figures | 1250 | 1350 |
e 3 in the hundreds column, so the interval is 1250 ≤ x < 1350, not 1295 ≤ x < 1305.
有效数字需要格外小心。对于精确到 2 位有效数字的 1300，最后一位有效数字是百位上的 3，因此区间为 1250 ≤ x < 1350，而不是 1295 ≤ x < 1305。
4. Writing the Error Interval Correctly | 正确书写误差区间
The standard IGCSE way to write an error interval is lower bound ≤ x < upper bound. The lower bound takes the "less than or equal to" sign because it is the smallest value that still rounds to the recorded number. The upper bound takes the strict "less than" sign because the bound itself would round up to the next value.
IGCSE 书写误差区间的标准方式是 下界 ≤ x < 上界。下界使用"小于等于"符号，因为它仍然能舍入到所记录数值的最小值；上界使用严格的"小于"符号，因为上界本身会舍入到下一个数值。
For example, if x = 5.4 correct to 1 decimal place, the error interval is 5.35 ≤ x < 5.45 or convenience, but the strict version is more accurate and safer in exams that ask for the upper bound explicitly.
例如，若 x = 5.4（精确到 1 位小数），误差区间为 5.35 ≤ x < 5.45。有些教材为了方便会写成 5.35 ≤ x ≤ 5.45，但严格写法更准确，在明确要求写出上界的题目中也更安全。
If the question asks for the upper bound as a single number, give the value itself, such as 5.45, not the whole inequality. If the question says "write down the error interval", give the full inequality. Always reread the command word.
如果题目要求单独给出上界的数值，就写这个数值本身，例如 5.45，而不是整个不等式。如果题目说"写出误差区间"，则要给出完整的不等式。务必重新读一遍题干中的指令词。
5. Bounds for Sums and Differences | 和与差的上下界
When two rounded values are added, the maximum sum uses the two upper bounds and the minimum sum uses the two lower bounds. This is because adding larger quantities always gives a larger total.
当两个舍入值相加时，最大和取两个上界相加，最小和取两个下界相加。这是因为加上更大的数量总是得到更大的总量。
Max (a + b) = UB(a) + UB(b) Min (a + b) = LB(a) + LB(b)
For subtraction the pattern changes. The maximum difference comes from the largest possible first value minus the smallest possible second value, while the minimum difference comes from the smallest first value minus the largest second value.
而对于减法，规律就变了。最大差值来自"第一个数可能的最大值减去第二个数可能的最小值"，最小差值则来自"第一个数的最小值减去第二个数的最大值"。
Max (a − b) = UB(a) − LB(b) Min (a − b) = LB(a) − UB(b)
t; 2.35. The maximum of a + b is 4.65 + 2.35 = 7.00 and the minimum is 4.55 + 2.25 = 6.80.
例题：a = 4.6、b = 2.3，两者都精确到 1 位小数。则 a 位于 4.55 ≤ a < 4.65，b 位于 2.25 ≤ b < 2.35。a + b 的最大值为 4.65 + 2.35 = 7.00，最小值为 4.55 + 2.25 = 6.80。
ce that we never use 4.65 − 2.35 to find a maximum; mixing the two is a very common error.
同样这两个数，a − b 的最大值为 4.65 − 2.25 = 2.40，最小值为 4.55 − 2.35 = 2.20。注意我们绝不用 4.65 − 2.35 去求最大值；把两者混用是非常常见的错误。
6. Bounds for Products | 乘积的上下界
For multiplication of positive quantities, the maximum product uses both upper bounds and the minimum product uses both lower bounds. This is because multiplying by a larger positive number always gives a larger result.
对于正数的乘法，最大乘积取两个上界相乘，最小乘积取两个下界相乘。这是因为乘以一个更大的正数总是得到更大的结果。
Max (a × b) = UB(a) × UB(b) Min (a × b) = LB(a) × LB(b)
lace. Then the length lies in 8.35 ≤ l < 8.45 and the width lies in 3.15 ≤ w < 3.25.
例题：一个长方形的长 8.4 cm、宽 3.2 cm，均为精确到 1 位小数的测量值。则长位于 8.35 ≤ l < 8.45，宽位于 3.15 ≤ w < 3.25。
he question asks for the area to a suitable degree of accuracy, you would normally give 27.5 cm² and 26.3 cm² to 1 decimal place.
可能的最大面积为 8.45 × 3.25 = 27.4625 cm²。可能的最小面积为 8.35 × 3.15 = 26.3025 cm²。若题目要求给出适当精确度的面积，通常写成 27.5 cm² 和 26.3 cm²（保留 1 位小数）。
oose the largest and smallest of the four results. IGCSE questions rarely do this, but it is worth knowing.
如果其中一个量可能为负数，规律就会改变。此时必须检验四种组合：上界 × 上界、上界 × 下界、下界 × 上界、下界 × 下界，然后从四个结果中选出最大值与最小值。IGCSE 题目很少这样考，但了解这一点很有价值。
7. Bounds for Quotients | 商的上下界
Division follows the opposite pattern to multiplication, because dividing by a smaller positive number gives a larger result. For positive quantities, the maximum quotient uses the largest numerator and the smallest denominator.
除法的规律与乘法相反，因为除以一个更小的正数会得到更大的结果。对于正数而言，最大商取最大的分子除以最小的分母。
Max (a ÷ b) = UB(a) ÷ LB(b) Min (a ÷ b) = LB(a) ÷ UB(b)
est second. The distance lies in 245 ≤ d < 255 and the time lies in 11.5 ≤ t < 12.5.
例题：一辆汽车行驶了 250 m（精确到 10 m），用时 12 s（精确到秒）。距离位于 245 ≤ d < 255，时间位于 11.5 ≤ t < 12.5。
ed is 245 ÷ 12.5 = 19.6 m/s exactly. So the true speed lies between 19.6 m/s and 22.2 m/s.
可能的最大速度为 255 ÷ 11.5 = 22.173 913... m/s，保留 1 位小数得 22.2 m/s。可能的最小速度为 245 ÷ 12.5 = 19.6 m/s（准确值）。因此真实速度介于 19.6 m/s 与 22.2 m/s 之间。
Always keep the full calculator display until the final step. Rounding the intermediate quotient too early can shift your answer beyond the accepted range, especially in multi-step problems.
务必在计算器上保留完整位数直到最后一步。过早对中间商进行舍入，可能使答案超出可接受范围，尤其是在多步计算题中。
8. Choosing the Right Combination | 选择正确的组合方式
The single most useful skill in this topic is knowing which bound to use for which operation. The table below summarises the four basic cases for positive quantities and is worth memorising.
本主题中最有用的单项技能，就是知道每种运算该用哪个界。下表总结了正数情形下的四种基本组合，值得牢记。
| Operation | Maximum | Minimum |
|---|---|---|
| a + b | UB + UB | LB + LB |
| a − b | UB − LB | LB − UB |
| a × b | UB × UB | LB × LB |
| a ÷ b | UB ÷ LB | LB ÷ UB |
A quick sanity check helps: ask yourself whether making a quantity bigger should make the final answer bigger or smaller. If bigger, put the upper bound there for a maximum. If smaller, put the lower bound there instead.
一个快速的合理性检验很有帮助：问自己，把某个量变大，应该让最终结果变大还是变小。如果变大，求最大值时就放上界；如果变小，求最大值时就放下界。
ep by step. First find the maximum and minimum of (a + b), then divide the maximum of the sum by the minimum of c, and the minimum of the sum by the maximum of c.
对于含两个以上量的表达式，例如 p = (a + b) ÷ c，要分步处理。先求出 (a + b) 的最大值与最小值，再用和的最大值除以 c 的最小值，用和的最小值除以 c 的最大值。
9. Truncation Bounds | 截断的上下界
Truncation means cutting off digits rather than rounding them. A truncated value is always smaller than or equal to the true value if the number is positive, so the error interval is not symmetrical.
截断指的是直接砍掉后面的数字，而不是四舍五入。对于正数，截断值总是小于或等于真实值，因此误差区间并不是对称的。
If x = 7.3 after truncation to 1 decimal place, then the true value satisfies 7.3 ≤ x < 7.4. The lower bound is the recorded value itself, and the upper bound is one place value higher. There is no subtraction of half a unit here.
若 x 截断到 1 位小数后为 7.3，则真实值满足 7.3 ≤ x < 7.4。下界就是记录值本身，上界则比它高一个位值。这里不存在减去半个单位的情况。
Truncated to place value p: value ≤ x < value + p
t; 7.4. Examiners like to test both in the same paper, so read the wording very carefully.
对比一下：7.3 四舍五入到 1 位小数给出 7.25 ≤ x < 7.35，而 7.3 截断到 1 位小数给出 7.3 ≤ x < 7.4。出题人喜欢在同一份试卷中同时考查这两种情况，所以一定要仔细读题。
10. Worked Exam Examples | 典型考题精讲
ls. Each parcel has mass less than 3.65 kg, so the total is less than 7 × 3.65 = 25.55 kg.
例题 1：一个包裹的质量为 3.6 kg（精确到 1 位小数）。求 7 个相同包裹总质量的上界。每个包裹质量小于 3.65 kg，因此总质量小于 7 × 3.65 = 25.55 kg。
5, so the minimum area is 9.65 × 9.65 = 93.1225 cm², which is 93.1 cm² to 1 decimal place.
例题 2：一个正方形的边长为 9.7 cm（精确到 1 位小数）。求可能的最小面积。边长满足 9.65 ≤ s < 9.75，因此最小面积为 9.65 × 9.65 = 93.1225 cm²，保留 1 位小数为 93.1 cm²。
because the last significant digit is the 4 in the hundreds column and half of 100 is 50.
例题 3：某数 n 精确到 1 位有效数字为 400。则误差区间为 350 ≤ n < 450，因为最后一位有效数字是百位上的 4，而 100 的一半是 50。
Example 4: Given a = 6.4 and b = 2.8
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