📚 Mastering Upper and Lower Bounds | 掌握上界与下界
In IGCSE and A-Level mathematics, upper and lower bounds describe the interval within which a rounded or measured value must lie. Whenever a number is rounded, truncated, or read from a measuring instrument, its exact value is lost, yet we can always state the smallest and largest values that would still produce the quoted figure. This topic appears frequently on Paper 2 and Paper 4, and it is often linked with significant figures, decimal places, error intervals, and real-world measurement problems.
在 IGCSE 与 A-Level 数学中，上界和下界描述的是经过舍入或测量的数值必须落在的区间。只要一个数被舍入、截断或从测量工具上读出，它的精确值就无法得知，但我们总可以说出仍会得到该记录值的最小值与最大值。这个主题经常出现在 Paper 2 和 Paper 4 中，并且常与有效数字、小数位、误差区间以及实际测量问题结合考查。
1. What Are Upper and Lower Bounds? | 什么是上界和下界？
When a number is rounded, the true value is not exactly known. The lower bound is the smallest possible value, and the upper bound is the largest possible value, that would round to the given number. Together the two bounds define the error interval of the measurement.
当一个数被舍入后，真实值并不完全已知。下界是可能的最小值，上界是可能的最大值，它们都能舍入到给定的数。两个界限合在一起，就定义了这个测量值的误差区间。
For example, if a length is given as 8 cm to the nearest centimetre, any actual length from 7.5 cm up to but not including 8.5 cm would round to 8 cm.
例如，如果某长度以厘米为单位精确到整数给出为 8 cm，那么任何从 7.5 cm 到但不包括 8.5 cm 的实际长度都会舍入为 8 cm。
Lower bound = 7.5 cm, Upper bound = 8.5 cm
下界 = 7.5 cm，上界 = 8.5 cm
as an inequality: 7.5 ≤ x < 8.5. This means x can equal 7.5 but cannot quite reach 8.5.
我们通常用不等式表示为：7.5 ≤ x < 8.5。这意味着 x 可以等于 7.5，但不能等于或超过 8.5。
- The lower bound is included (≤) because 7.5 itself rounds up to 8. — 下界取等号（≤），因为 7.5 本身会进位到 8。
- The upper bound is excluded (<) because 8.5 itself rounds up to 9, not 8. — 上界不取等号（<），因为 8.5 本身会进位到 9，而不是 8。
2. Why Bounds Matter in Measurement | 测量中为什么需要界限？
All measurements are approximate because measuring tools have limited precision. When a ruler shows 8 cm, the real object might be slightly more or slightly less, depending on the smallest division on the ruler and on how carefully the reading was taken.
所有测量都是近似的，因为测量工具的精度有限。当尺子显示 8 cm 时，实际物体可能略大或略小，这取决于尺子的最小刻度以及读数时的仔细程度。
If a value is given to the nearest unit, the maximum error is half of that unit.
如果一个数值精确到某一单位，则最大误差是该单位的一半。
s 0.5 cm. This gives the error interval from 8 − 0.5 to 8 + 0.5, that is 7.5 ≤ x < 8.5.
对于精确到整数的 8 cm，半个厘米是 0.5 cm。由此得到误差区间为 8 − 0.5 到 8 + 0.5，即 7.5 ≤ x < 8.5。
Maximum error = ½ × (smallest unit of measurement)
最大误差 = ½ × （测量的最小单位）
This "half-unit rule" is the single most useful idea in the whole topic. It converts any rounded value instantly into a pair of bounds, and it works for centimetres, grams, seconds, metres, litres or any other unit.
这条“半单位法则”是整个主题中最有用的一个想法。它能立刻把任何舍入值转换成一对界限，并且对厘米、克、秒、米、升或任何其他单位都同样适用。
3. Finding Bounds from Rounding | 如何由舍入确定界限
To find the bounds of a rounded value, first identify the place value to which the number was rounded, then halve it. Adding and subtracting this half-unit gives the upper and lower bounds.
要求一个舍入值的界限，首先要确定这个数被舍入到了哪一位数位，然后把它除以二。用原数加减这个半单位，就得到上界与下界。
Worked examples for several levels of accuracy are shown in the table below.
下表给出了几个不同精确度下的例子。
| Rounded to | 精确到 | Half-unit | 半单位 | Value | 数值 | Error interval | 误差区间 |
| Nearest 1 | 精确到个位 | 0.5 | 8 | 7.5 ≤ x < 8.5 |
| Nearest 10 | 精确到十位 | 5 | 40 | 35 ≤ x < 45 |
| Nearest 0.1 (1 dp) | 精确到 0.1（1 位小数） | 0.05 | 6.3 | 6.25 ≤ x < 6.35 |
| Nearest 0.01 (2 dp) | 精确到 0.01（2 位小数） | 0.005 | 2.47 | 2.465 ≤ x < 2.475 |
| Nearest 0.5 | 精确到 0.5 | 0.25 | 7.5 | 7.25 ≤ x < 7.75 |
The pattern is always the same: locate the last digit that was kept, halve the value of that digit's place, and move up and down by that amount.
规律始终一致：找到被保留的最后一位数字，把它所在数位的值减半，再上下各移动这个量。
4. Bounds and Significant Figures | 界限与有效数字
Significant figures cause the most confusion, because the half-unit depends on where the first significant figure sits, not on how many digits are written. Zeros at the front of a decimal are never significant; they only fix the position of the decimal point.
有效数字最容易引起混淆，因为半单位取决于第一个有效数字所在的位置，而不是写了多少位数字。小数开头的零永远不算有效数字，它们只决定小数点的位置。
n the hundredths place, so the half-unit is 0.005 and the bounds are 4.555 ≤ x < 4.565.
以 4.56 精确到 3 位有效数字为例。最后一个有效数字是百分位上的 6，因此半单位是 0.005，界限为 4.555 ≤ x < 4.565。
housandths place, so the half-unit is 0.00005 and the bounds are 0.00455 ≤ x < 0.00465.
再看 0.00456 精确到 2 位有效数字。有效数字是 4 和 5，其中最后一位在万分位上，因此半单位是 0.00005，界限为 0.00455 ≤ x < 0.00465。
- Rounding to 3 sf is not the same as rounding to 3 dp. — 精确到 3 位有效数字与精确到 3 位小数并不相同。
- Count significant figures from the first non-zero digit. — 有效数字从第一个非零数字开始数起。
- In a number such as 0.00456, only 4, 5 and 6 count. — 在 0.00456 这样的数中，只有 4、5、6 算有效数字。
5. Truncation and Other Intervals | 截断与其他区间
Truncation means cutting the number off rather than rounding it. If a value is truncated to 2 decimal places, the true value lies from the given number up to the next value of that place, so the interval is one-sided and open at the top.
截断是指把数字直接切掉，而不是四舍五入。如果一个数值被截断到 2 位小数，真实值就在给定数值到该数位的下一个数值之间，因此这个区间是单侧的，且上端为开区间。
ncating to 2 dp satisfies 4.56 ≤ x < 4.57, because 4.569 would also be written as 4.56.
一个长度截断到 2 位小数后记为 4.56 m，则满足 4.56 ≤ x < 4.57，因为 4.569 也会被写成 4.56。
give the interval the other way round and ask you to deduce the rounded value, for example "a mass m satisfies 12.35 ≤ m < 12.45" unit here is 0.05, so m was given to the nearest 0.1, that is to 1 decimal place, as 12.4.
考题有时会反过来给出区间，要求你推断舍入值，例如“某质量 m 满足 12.35 ≤ m < 12.45”。这里的半单位是 0.05，所以 m 是精确到 0.1，也就是 1 位小数，即 12.4。
6. Bounds in Addition and Subtraction | 计算中的界限：加法与减法
When two rounded quantities are added, the smallest possible total comes from adding the two lower bounds, and the largest possible total comes from adding the two upper bounds.
当两个舍入值相加时，可能的最小总和来自两个下界相加，可能的最大总和来自两个上界相加。
Subtraction is different and is a very common source of lost marks. To make a difference as large as possible, subtract the smaller value from the larger one, so the upper bound of a − b uses the upper bound of a and the lower bound of b.
减法不同，而且是很容易丢分的地方。要让差最大，就要用大数减小数，因此 a − b 的上界要用 a 的上界减去 b 的下界。
UB(a + b) = UB(a) + UB(b), LB(a + b) = LB(a) + LB(b)
UB(a − b) = UB(a) − LB(b), LB(a − b) = LB(a) − UB(b)
b < 3.5. Then a + b satisfies 10 ≤ a + b < 12, and a − b satisfies 4 ≤ a − b < 6.
设 a = 8、b = 3，两者都精确到个位，因此 a 满足 7.5 ≤ a < 8.5，b 满足 2.5 ≤ b < 3.5。于是 a + b 满足 10 ≤ a + b < 12，而 a − b 满足 4 ≤ a − b < 6。
- Add bounds for addition: LB + LB and UB + UB. — 加法用界限相加：下界加下界，上界加上界。
- For subtraction, pair the outer bounds: UB − LB and LB − UB. — 减法要配最外侧的界限：上界减下界，下界减上界。
- Write the final answer as an inequality, not as two separate numbers. — 最终答案要写成不等式，而不是两个孤立的数。
7. Bounds in Multiplication and Division | 计算中的界限：乘法与除法
For multiplication and division with positive quantities, the largest result always comes from pairing the two largest values, and the smallest result from pairing the two smallest values.
对于正数的乘法与除法，最大的结果总是来自最大的两个值相配，最小的结果则来自最小的两个值相配。
Division reverses the pairing on the bottom: to make a quotient large, divide the largest numerator by the smallest denominator.
除法在分母上要把配对反过来：要让商最大，就用最大的被除数除以最小的除数。
UB(a × b) = UB(a) × UB(b), LB(a × b) = LB(a) × LB(b)
UB(a ÷ b) = UB(a) ÷ LB(b), LB(a ÷ b) = LB(a) ÷ UB(b)
tient a ÷ b satisfies 2.14 ≤ a ÷ b < 3.4, since 7.5 ÷ 3.5 ≈ 2.1429 and 8.5 ÷ 2.5 = 3.4.
仍以 a = 8（精确到个位）和 b = 3（精确到个位）为例，乘积 ab 满足 18.75 ≤ ab < 29.75，因为 7.5 × 2.5 = 18.75，8.5 × 3.5 = 29.75。商 a ÷ b 满足 2.14 ≤ a ÷ b < 3.4，因为 7.5 ÷ 3.5 ≈ 2.1429，8.5 ÷ 2.5 = 3.4。
- Always test all four pairings if you are unsure which gives the extreme values. — 如果不确定哪种配对给出极值，就把四种配对都算一遍。
- Keep extra decimal places during the working and round only at the end. — 计算过程中保留更多小数位，只在最后一步舍入。
- If a quantity could be negative, the rule changes and each case must be checked separately. — 如果量可能是负数，规则会改变，必须逐一检验每种情况。
8. Discrete Quantities and Rounding Counts | 离散量与计数取整
Not every quantity is continuous. Numbers of people, coins, buses or marks are discrete: they must be whole numbers. Bounds on a count therefore have to be adjusted to integers.
并非每个量都是连续的。人数、硬币数、公交车数或分数都是离散量，必须是整数。因此计数的界限需要调整到整数。
interval is 35 ≤ n < 45, but since n must be an integer the possible counts are 35, 36, … , 44.
如果一个数量精确到十位给出为 40，连续区间是 35 ≤ n < 45，但由于 n 必须是整数，可能的取值为 35、36、……、44。
0.01 gives 4.555 ≤ x < 4.565, but a real money amount would be limited to whole pence.
同样的思路适用于精确到分的金额：4.56 精确到 0.01 得到 4.555 ≤ x < 4.565，但真实的金额只能取整数分。
- Ask yourself whether a fraction of the unit is physically possible. — 先问自己这个单位的零头在现实中是否可能。
- People, items and vehicles are always integers. — 人、物品和车辆永远是整数。
- Time recorded to the nearest minute behaves as a discrete quantity in many exam questions. — 在许多考题中，精确到分钟的时间会按离散量处理。
9. A Worked Exam-Style Example | 一道完整的考题示例
The length l of a rectangle is 12 cm and its width w is 5 cm, both measured to the nearest centimetre. Find the error intervals of l and w, and hence the bounds of the perimeter and the area.
某矩形的
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