Key treaties and agreements | 数学关键约定与协议

📚 Key treaties and agreements | 数学关键约定与协议

In mathematics, precision depends on shared conventions. Although we do not sign formal treaties, mathematicians worldwide agree on notation, order of operations, angle units, and principal values. This article reviews the key agreements that Edexcel A-Level Maths candidates must apply consistently.

在数学中，精确性依赖于共同的约定。虽然我们并不签署正式条约，但全世界的数学家对符号、运算次序、角度单位以及主值都有共识。本文回顾 Edexcel A-Level 数学考生必须一致应用的关键约定与协议。

1. Order of Operations: BIDMAS/BODMAS | 运算次序：BIDMAS/BODMAS

The agreed order is Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right). This prevents ambiguity in expressions such as 3 + 2 × 4.

约定的运算次序是括号、指数、除法和乘法（从左到右）、加法和减法（从左到右）。这避免了诸如 3 + 2 × 4 这类表达式的歧义。

3 + 2 × 4 = 3 + 8 = 11, not 20

Division and multiplication have equal priority, so you work from left to right. For example, 12 ÷ 3 × 2 = 4 × 2 = 8, not 12 ÷ 6 = 2.

除法和乘法具有相同的优先级，因此应从左到右计算。例如 12 ÷ 3 × 2 = 4 × 2 = 8，而不是 12 ÷ 6 = 2。

2. Angle Measurement: Degrees and Radians | 角度度量：角度制与弧度制

A-Level work uses both degrees and radians. The convention is that radians are dimensionless, and π rad = 180°. You must set your calculator to the correct mode for each question.

A-Level 学习中角度制和弧度制都会使用。约定弧度是无量纲的，且 π rad = 180°。每道题都必须将计算器设置为正确模式。

π rad = 180°, 360° = 2π rad

When no unit is specified in calculus, trigonometric functions are assumed to have inputs in radians. This is especially important in differentiation and integration of sin x and cos x.

在微积分中，如果没有指定单位，三角函数的角度默认以弧度为单位。这在 sin x 和 cos x 的求导与积分中尤其重要。

3. Function Notation and Domain Conventions | 函数记号与定义域约定

f(x) denotes the output of function f for input x. The domain is the set of allowed inputs, and the range is the set of possible outputs. If no domain is stated, we assume the natural domain over real numbers unless the context says otherwise.

f(x) 表示函数 f 对输入 x 的输出。定义域是允许的输入集合，值域是可能的输出集合。如果没有说明定义域，除非题目另有说明，通常假设实数范围内的自然定义域。

f: x ↦ x², x ∈ ℝ

The natural domain of f(x) = 1/(x – 2) is all real numbers except x = 2. This is because division by zero is undefined under standard mathematical agreement.

f(x) = 1/(x – 2) 的自然定义域是所有实数除去 x = 2。这是因为根据标准数学约定，除以零是未定义的。

4. Logarithm Conventions | 对数约定

log₁₀ x is often written simply as log x, and logₑ x is written as ln x. The base e is the natural logarithm. This shorthand is universally recognised in Edexcel exams.

log₁₀ x 常简单地写作 log x，logₑ x 写作 ln x。以 e 为底的是自然对数。这种简写在 Edexcel 考试中被普遍认可。

ln x = logₑ x, log x = log₁₀ x

The change of base formula is agreed as logₐ b = log b / log a. You can use base 10 or base e in the calculator for this, but the formula must be applied correctly.

换底公式约定为 logₐ b = log b / log a。计算时可以使用以 10 或以 e 为底，但公式必须正确应用。

logₐ b = log b / log a

5. Square Roots and Principal Values | 平方根与主值

√x denotes the non-negative square root for x ≥ 0. The symbol √ always means the principal square root, never the negative one. Therefore √4 = 2, not ±2.

√x 表示 x ≥ 0 时的非负平方根。符号 √ 始终表示主平方根，绝不是负的那个。因此 √4 = 2，而不是 ±2。

√4 = 2, but x² = 4 ⇒ x = ±2

When solving an equation like x² = 4, both the positive and negative roots must be included. This distinction between the principal root and the equation solutions is a common A-Level marking point.

在解像 x² = 4 这样的方程时，必须同时包含正根和负根。主值根与方程解之间的区别是 A-Level 评分中的常见要点。

6. Coordinate System Conventions | 坐标系约定

The Cartesian plane uses a horizontal x-axis and a vertical y-axis. Positive x is to the right, positive y is upwards. The origin has coordinates (0, 0).

笛卡尔平面使用水平的 x 轴和垂直的 y 轴。x 轴正方向向右，y 轴正方向向上。原点坐标为 (0, 0)。

P(x, y), origin O(0, 0)

Anti-clockwise rotation is positive for angles in standard position. Angles measured clockwise from the positive x-axis are negative. This sign convention is essential for trigonometry and polar coordinates.

标准位置角的逆时针旋转为正。从 x 轴正方向顺时针测量的角度为负。这一符号约定对三角学和极坐标至关重要。

7. Vector Notation and Unit Vectors | 向量记号与单位向量

Vectors can be written as bold letters, underlined, or with arrows. Unit vectors i, j, k point along the x, y, z axes respectively. In two dimensions, a position vector is often written as a column vector.

向量可以写成粗体字母、下划线或带箭头。单位向量 i、j、k 分别指向 x、y、z 轴。在二维中，位置向量常写成列向量。

a = xi + yj, |a| = √(x² + y²)

The magnitude of vector a is always non-negative and uses the principal square root. Direction is given by the angle made with the positive x-axis, measured anti-clockwise.

向量 a 的模长总是非负的，并且使用主平方根。方向由与 x 轴正方向所成的角给出，按逆时针测量。

8. Differentiation and Integration Notation | 微分与积分记号

Leibniz notation dy/dx represents the derivative of y with respect to x. Lagrange notation f'(x) is interchangeable with dy/dx. Both are accepted in Edexcel solutions.

莱布尼茨记号 dy/dx 表示 y 对 x 的导数。拉格朗日记号 f'(x) 与 dy/dx 可以互换。在 Edexcel 解答中两者都接受。

dy/dx = f'(x), ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ -1

The integral sign ∫ f(x) dx denotes an antiderivative or indefinite integral, and ∫ₐᵇ f(x) dx denotes the definite integral with lower limit a and upper limit b. The constant of integration C is required for indefinite integrals.

积分号 ∫ f(x) dx 表示原函数或不定积分，∫ₐᵇ f(x) dx 表示下限为 a、上限为 b 的定积分。不定积分必须加上积分常数 C。

9. Set Notation and Interval Notation | 集合记号与区间记号

A-Level uses set-builder notation such as {x : x > 2} and interval notation such as (2, ∞). Square brackets include endpoints, round brackets exclude them.

A-Level 使用集合构建符号如 {x : x > 2} 和区间符号如 (2, ∞)。方括号包含端点，圆括号不包含端点。

{x : a ≤ x < b} = [a, b)

The union symbol ∪ combines disjoint intervals, and the intersection symbol ∩ shows common values. For example, {x : x < 1 or x > 3} = (-∞, 1) ∪ (3, ∞).

并集符号 ∪ 用于合并不相交的区间，交集符号 ∩ 表示公共值。例如 {x : x < 1 或 x > 3} = (-∞, 1) ∪ (3, ∞)。

10. Trigonometric Conventions and Principal Values | 三角学约定与主值

sin⁻¹ x returns the principal value in [-π/2, π/2], cos⁻¹ x returns values in [0, π], and tan⁻¹ x returns values in (-π/2, π/2). These restricted ranges make inverse functions well-defined.

sin⁻¹ x 返回 [-π/2, π/2] 内的主值，cos⁻¹ x 返回 [0, π] 内的值，tan⁻¹ x 返回 (-π/2, π/2) 内的值。这些受限制的范围使反函数有良好定义。

sin 30° = 1/2, cos 45° = √2/2, tan 60° = √3

Exact values for 30°, 45°, and 60° must be memorised and used in surd form unless the question asks for a decimal. For angles in radians, the equivalent values are π/6, π/4, and π/3.

必须记住 30°、45° 和 60° 的精确值，并以根式形式使用，除非题目要求小数。对于弧度制角度，等价值为 π/6、π/4 和 π/3。

11. Algebraic Index and Surd Conventions | 代数指数与根式约定

Negative indices mean reciprocals, and fractional indices mean roots: a^(m/n) = ⁿ√(aᵐ). These index laws are agreed across all A-Level algebra topics.

负指数表示倒数，分数指数表示根：a^(m/n) = ⁿ√(aᵐ)。这些指数法则在所有 A-Level 代数主题中都得到公认。

a⁻ⁿ = 1/aⁿ, a^(1/n) = ⁿ√a

Surds should be simplified as far as possible, and rationalising denominators is expected in final answers. For example, 1/√2 should be written as √2/2 in exact form.

根式应尽可能化简，并且最终答案通常要求分母有理化。例如 1/√2 应以精确形式写作 √2/2。

12. Calculator and Rounding Conventions | 计算器与舍入约定

Unless told otherwise, give exact answers or round to 3 significant figures. Use the degree or radian mode as required by the question. Show your working even when using a calculator.

除非另有说明，给出精确答案或保留 3 位有效数字。根据题目要求使用角度制或弧度制模式。即使使用计算器也要展示计算过程。

π ≈ 3.14159, √2 ≈ 1.414, e ≈ 2.718

Intermediate working should not be rounded prematurely, otherwise the final answer may lose accuracy. Store exact values in the calculator memory and only round at the final step.

中间计算过程不应过早舍入，否则最终答案可能会失去精度。将精确值存入计算器存储器中，仅在最后一步进行舍入。

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Key treaties and agreements | 数学关键约定与协议