📚 5 Place Studies | A-Level数学中的位置与数值研究
In Edexcel A-Level Mathematics, place-related skills appear throughout pure mathematics, statistics and mechanics. This article groups five essential ‘place studies’ for revision: decimal places, significant figures, coordinate positions, loci as sets of points, and locating roots by sign change. Each section gives the core rule, an exam-style example and common pitfalls.
在 Edexcel A-Level 数学中，与“位置”和“精度”相关的技能贯穿纯数学、统计和力学。本文围绕五个关键“位置研究”主题展开复习：小数位数、有效数字、坐标位置、轨迹点集以及通过符号变化确定根的位置。每节给出核心规则、考试型例题和常见错误。
1. Decimal Places and Rounding | 小数位数与四舍五入
Decimal places (d.p.) count digits after the decimal point. To round to n decimal places, look at the (n+1)th decimal digit. If this digit is 5 or more, increase the nth digit by 1; otherwise leave the nth digit unchanged. This rule is tested directly in Edexcel questions and is also required when using numerical methods.
小数位数指小数点后的数字个数。保留 n 位小数时，看第 n+1 位小数。如果该位数字大于等于 5，则第 n 位加 1；否则第 n 位保持不变。Edexcel 考试中不仅直接考查这一规则，数值方法中也要求按此规则保留答案。
- π = 3.14159… to 3 d.p. becomes 3.142 because the fourth decimal digit is 5.
- 2.3562 to 2 d.p. becomes 2.36 because the third decimal digit is 6.
中文要点：π = 3.14159… 保留到小数点后 3 位为 3.142，因为小数点后第 4 位是 5。2.3562 保留到小数点后 2 位为 2.36，因为第 3 位是 6。
2. Significant Figures in Calculations | 有效数字在计算中的使用
Significant figures (s.f.) are counted from the first non-zero digit on the left. Leading zeros are not significant, but zeros between non-zero digits and trailing zeros after a decimal point are significant. When rounding to n significant figures, locate the nth significant digit and inspect the next digit.
有效数字从左边第一个非零数字开始计数。前导零不算有效数字，但非零数字之间的零以及小数点后的末尾零都算有效数字。保留 n 位有效数字时，找到第 n 位有效数字并检查下一位数字。
| Number | To 3 s.f. | Reason |
|---|---|---|
| 0.004056 | 0.00406 | Start at 4; next digit 6 rounds 5 to 6 |
| 52,470 | 52,500 | Third sig. digit 4; next digit 7 rounds up |
中文要点：0.004056 保留 3 位有效数字为 0.00406，因为有效数字从 4 开始，下一位 6 将 5 进位为 6。52,470 保留 3 位有效数字为 52,500，因为第 3 位有效数字是 4，下一位 7 使其进 1。
3. Place Value and Standard Form | 位值制与标准形式
Place value gives each digit a value according to its position. Moving one place to the left multiplies by 10; moving one place to the right divides by 10. Standard form writes a number as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. This is especially useful for very large or very small numbers in Edexcel modelling and statistics questions.
位值制根据数字所在位置赋予其数值。向左移动一位乘以 10，向右移动一位除以 10。标准形式将数字写成 a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。这在 Edexcel 建模和统计题中处理非常大或非常小的数时尤为有用。
0.000503 = 5.03 × 10⁻⁴
Multiplying by 10⁻⁴ shifts the decimal point four places to the left; dividing by 10⁻⁴ shifts it four places to the right. Checking the leading digit position helps avoid sign errors with n.
乘以 10⁻⁴ 相当于将小数点向左移动四位；除以 10⁻⁴ 则向右移动四位。检查首位数字的位置有助于避免指数 n 的正负号错误。
4. Coordinates and the Cartesian Plane | 坐标与笛卡尔平面
A point’s place in the Cartesian plane is described by an ordered pair (x, y). Edexcel Pure 1 requires confident use of distance and midpoint formulae. These place relationships are central to straight line graphs, circles and parametric equations.
点在笛卡尔平面中的位置用有序数对 (x, y) 表示。Edexcel Pure 1 要求熟练使用距离公式和中点公式。这些位置关系是直线图形、圆和参数方程的核心。
Distance = √((x₂ − x₁)² + (y₂ − y₁)²)
Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
For example, the distance from (1, 2) to (4, 6) is √((4−1)² + (6−2)²) = √(9 + 16) = √25 = 5. The midpoint is (2.5, 4).
例如，从 (1, 2) 到 (4, 6) 的距离为 √((4−1)² + (6−2)²) = √(9 + 16) = √25 = 5。中点为 (2.5, 4)。
5. Loci as Sets of Places | 轨迹作为点的集合
A locus is a set of points that satisfy a given condition. In coordinate geometry, the circle is the most common locus: all points (x, y) at a fixed distance r from a centre (a, b). Its equation is (x − a)² + (y − b)² = r².
轨迹是满足给定条件的点的集合。在坐标几何中，圆是最常见的轨迹：所有到圆心 (a, b) 的距离为固定值 r 的点 (x, y) 组成圆。其方程为 (x − a)² + (y − b)² = r²。
(x − 3)² + (y + 2)² = 25
Here the centre is (3, −2) and the radius is 5. Edexcel questions often give the expanded form and require completing the square to identify the centre and radius.
该方程中圆心为 (3, −2)，半径为 5。Edexcel 题目常给出展开式，要求通过配方法找出圆心和半径。
6. Position Vectors and Displacement | 位置向量与位移
In mechanics and vectors, the position vector of a point P(x, y) relative to the origin O is OP = xi + yj. Its magnitude gives the distance from the origin: |OP| = √(x² + y²). Displacement vectors describe the movement from one place to another.
在力学和向量中，点 P(x, y) 相对于原点 O 的位置向量为 OP = xi + yj。其大小表示该点到原点的距离：|OP| = √(x² + y²)。位移向量描述从一处到另一处的运动。
For P(3, 4), OP = 3i + 4j and |OP| = √(3² + 4²) = √25 = 5. If Q(1, 2), then the displacement PQ = OQ − OP = −2i − 2j.
例如 P(3, 4)，OP = 3i + 4j，|OP| = √(3² + 4²) = √25 = 5。若 Q(1, 2)，则位移 PQ = OQ − OP = −2i − 2j。
7. Locating Roots by Sign Change | 通过符号变化确定根的位置
A continuous function f(x) has at least one root in the interval [a, b] if f(a) and f(b) have opposite signs. This is the foundation of decimal search and iterative methods in Edexcel numerical methods.
若连续函数 f(x) 在区间 [a, b] 两端取值异号，则该区间内至少存在一个根。这是 Edexcel 数值方法中十进制搜索和迭代方法的基础。
f(a) × f(b) < 0 ⇒ root exists in [a, b]
Example: f(x) = x³ − 2x − 5. f(2) = 8 − 4 − 5 = −1 and f(3) = 27 − 6 − 5 = 16. Since the sign changes, a root lies between 2 and 3. Repeated interval subdivision can locate the root to 3 d.p.
例如：f(x) = x³ − 2x − 5。f(2) = 8 − 4 − 5 = −1，f(3) = 27 − 6 − 5 = 16。由于符号改变，根位于 2 和 3 之间。反复缩小区间可将根确定到小数点后 3 位。
8. Bounds and Error Intervals | 界限与误差区间
When a measurement is rounded, the true value lies within an error interval. For a value given to 1 d.p., the lower bound is half a unit below the rounded value and the upper bound is half a unit above. The upper bound is not included if the original value was continuous.
测量值经过四舍五入后，真实值位于误差区间内。对于保留 1 位小数的值，下界是舍入值减 0.05，上界是舍入值加 0.05。若原值是连续量，上界通常不包含。
5.6 to 1 d.p. means 5.55 ≤ x < 5.65
In area calculations using bound values, use lower bounds for both dimensions to find the lower area and upper bounds for the upper area. For sides 3.4 cm and 5.6 cm rounded to 1 d.p., the lower area is 3.35 × 5.55 = 18.5925 cm² and the upper area is 3.45 × 5.65 = 19.4925 cm².
在使用界限值计算面积时，两个维度都用下界可得面积下界，两个维度都用上界可得面积上界。例如边长 3.4 cm 和 5.6 cm 均保留 1 位小数，面积下界为 3.35 × 5.55 = 18.5925 cm²，面积上界为 3.45 × 5.65 = 19.4925 cm²。
9. Exam Strategy for Place Accuracy | 考试中位置精度的策略
Edexcel papers often state ‘give your answer to 3 significant figures unless otherwise specified’. This instruction is not a style choice; marks may depend on using the correct degree of accuracy. Carry exact values or at least several extra decimal places through intermediate steps, then round only at the final answer.
Edexcel 试卷通常要求“除特别说明外，答案保留 3 位有效数字”。这一要求不是风格偏好，分数可能取决于精度是否正确。中间步骤应保留精确值或至少多保留几位小数，最后一步才四舍五入。
- Use exact forms such as √2 or π until the final line.
- In trigonometry questions, check whether the angle is in degrees or radians.
- For decimal search, record enough decimal places to show a clear sign change.
中文要点：在最后一行之前使用精确形式，如 √2 或 π。三角题中要检查角度单位是度还是弧度。十进制搜索中要记录足够的小数位以显示明显的符号变化。
10. Common Mistakes and How to Avoid Them | 常见错误与避免方法
Many marks are lost through avoidable accuracy errors. The most common mistakes include miscounting decimal places
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