📚 Common Mistakes in International AS Pure Mathematics Unit 1 (MA01) | 国际AS纯数学单元1 (MA01) 常见易错点总结
The International AS Pure Mathematics Unit 1 (MA01) paper tests core algebraic, trigonometric, coordinate geometry, and basic calculus skills. Many students lose marks not because they do not understand the concepts, but because they fall into predictable traps. This article summarises the most frequent errors candidates make in each topic area, explains why they occur, and shows how to avoid them. Mastering these subtleties can transform a good score into an excellent one.
国际AS纯数学单元1（MA01）试卷考察核心代数、三角、坐标几何和基础微积分技能。许多学生失分并非因为不理解概念，而是陷入了可预见的陷阱。本文总结了考生在各个专题中最常犯的错误，解释了原因，并展示了如何避免。掌握这些细节可以将好成绩提升为优异成绩。
1. Algebraic Expansions and Factorising | 代数展开与因式分解
Students often lose a sign when expanding brackets containing negative terms. For example, expanding (x − 3)(x + 2) incorrectly as x² − x − 6 instead of the correct x² − x − 6 — wait, that example is correct; the common error is sign confusion in products like (2x − 5)(x + 4). A typical mistake would be writing the x-term as 2x² + 8x − 5x − 20 and then wrongly combining to 2x² + 3x − 20. The correct combination is 2x² + 3x − 20, actually that is also correct. Let’s use a better example: (3x − 2)(x − 5). The error: 3x² − 15x − 2x − 10 = 3x² − 17x − 10? But that is correct! The real error is forgetting that minus times minus equals plus. For (x − 4)(x − 3), some write x² − 7x + 12 because they incorrectly take −4 × −3 = −12. Emphasise: negative × negative = positive. When factorising, the classic blunder is incomplete factorisation. Candidates factorise x² − 9 as (x − 3)(x + 3) but miss the common factor in 2x² − 18, leaving it as 2(x² − 9) without further factorising to 2(x − 3)(x + 3). Always check for a common factor first.
学生在展开含负项的括号时经常弄错符号。例如展开 (x − 4)(x − 3)，有人会错误地写成 x² − 7x − 12，因为他们把 −4 × −3 算成了 −12。正确结果应为 x² − 7x + 12，因为负负得正。因式分解时，典型的错误是分解不彻底。考生将 x² − 9 分解为 (x − 3)(x + 3)，却忽略了 2x² − 18 中的公因子，只写成 2(x² − 9) 而不继续分解为 2(x − 3)(x + 3)。应力争先提取公因子。
Another frequent error occurs when factorising quadratics with a leading coefficient not equal to 1, such as 2x² + 7x + 3. Students often guess (2x + 3)(x + 1) which gives the wrong middle term. The correct method is systematic: find two numbers that multiply to ac = 6 and add to 7, i.e., 6 and 1, split the middle term and factor by grouping.
另一个常见错误发生在二次项系数不为1的因式分解上，例如 2x² + 7x + 3。学生常常猜测 (2x + 3)(x + 1)，得到错误的中间项。正确的方法是系统的：找到两个数乘积为 ac = 6 且和为 7，即 6 和 1，然后分裂中间项并用分组分解。
2. Surds and Indices | 根式与指数
Misunderstanding the rules of indices leads to errors such as √(a + b) = √a + √b. This is not valid; √(9 + 16) = 5, but √9 + √16 = 3 + 4 = 7. The square root of a sum is not the sum of square roots. Similarly, (a + b)² ≠ a² + b²; students must apply the full expansion. With surds, rationalising the denominator of 1/(√2 + 1) is often attempted by multiplying top and bottom by √2 instead of the conjugate √2 − 1, leaving a surd in the denominator unchanged. Always use the conjugate when the denominator is a binomial with surds.
对指数法则的误解会导致像 √(a + b) = √a + √b 这样的错误。这是不成立的；√(9 + 16) = 5，但 √9 + √16 = 7。和的平方根不等于平方根之和。同理，(a + b)² ≠ a² + b²；必须完整展开。处理根式时，有理化分母 1/(√2 + 1) 时，考生常常只乘以 √2 而不是共轭根式 √2 − 1，导致分母中的根式依然存在。当分母为二项根式时，务必使用共轭根式。
Negative and fractional indices are a common source of mistakes. The expression x⁻² is often misread as −x², but it means 1/x². Also, 8^(2/3) is incorrectly evaluated by taking the cube root of 8 and then squaring, but many square first and then cube root, which can lead to larger numbers. The order does not matter mathematically, but students often forget that the denominator is the root. For 25^(3/2), a common error is to write 125 instead of 125? No, 25^(3/2) = (√25)³ = 5³ = 125. That is correct. The error is thinking 25^(3/2) = 25^(3/2) = 25^1.5 = 25×√25? That’s actually correct too. The mistake is more simple: writing 25^(3/2) = 25^(3) / 25^(2) or something nonsensical. Emphasise the meaning: a^(m/n) = (ⁿ√a)^m.
负指数和分数指数是常见的出错点。表达式 x⁻² 常被误解为 −x²，实际上它是 1/x²。此外，8^(2/3) 应该先取立方根再平方，但许多人先平方再开立方，虽然数学上结果一样，但数字变大容易出错。关键是记住分母代表根指数。对于 25^(3/2)，有的学生会错误地计算为 25 ÷ 2 × 3，完全混淆了运算。必须强调：a^(m/n) = (ⁿ√a)^m。
3. Quadratics: Discriminant and Completing the Square | 二次方程：判别式与配方法
The discriminant, b² − 4ac, is frequently calculated incorrectly due to sign errors in substituting a, b, c. For 2x² − 3x + 1, a = 2, b = −3, c = 1, so b² = 9, 4ac = 8, discriminant = 1. But if a student writes b = 3, they get 9 − 8 = 1, still correct by luck? Actually if b is mis-signed, say for x² − 5x + 6, b = −5, b² = 25, correct. If they use b = 5, they’d get 25 − 24 = 1, also correct. The real problem arises when c is negative. For x² + 4x − 5, c = −5, so 4ac = 4(1)(−5) = −20, then b² − 4ac = 16 − (−20) = 36. Students often forget to subtract a negative, giving 16 − 20 = −4, wrongly concluding no real roots. So careful substitution is essential.
判别式 b² − 4ac 常因代入 a, b, c 时的符号错误而计算错误。对于 x² + 4x − 5，c = −5，因此 4ac = −20，判别式 = 16 − (−20) = 36。许多学生会忘掉减去负数，算成 16 − 20 = −4，从而错误地得出无实根的结论。代入时务必小心符号。
Completing the square for x² + 6x + 10: a common mistake is to write (x + 3)² + 10, forgetting to subtract the square of half the x-coefficient. The correct form is (x + 3)² + 1. The error stems from not squaring 3 and subtracting it: (x+3)² = x² + 6x + 9, so x² + 6x + 10 = (x+3)² + 1. Another classic error is when the leading coefficient is not 1, e.g., 2x² + 8x + 5. Students often half-completely square it as 2(x+4)²… which is wrong. First factor out the 2: 2[x² + 4x] + 5, then complete the square inside the bracket.
配方法对于 x² + 6x + 10：常见错误是写成 (x + 3)² + 10，忘记减去 x 系数一半的平方。正确形式为 (x + 3)² + 1。错误源于没有把 (x+3)² 展开得 9，然后从常数项中扣去。当二次项系数不为1时，例如 2x² + 8x + 5，学生经常错误地写成 2(x+4)²… 必须先提取系数2：2[x² + 4x] + 5，再在括号内配方。
4. Simultaneous Equations | 联立方程组
When solving linear and quadratic simultaneous equations by substitution, candidates often substitute correctly but then expand the squared bracket incorrectly. For y = 2x − 1 and y = x² + 2, substituting gives 2x − 1 = x² + 2, which is straightforward. But with y = x + 3 and x² + y² = 9, substituting yields x² + (x+3)² = 9. The expansion (x+3)² is frequently written as x² + 9 instead of x² + 6x + 9, losing the 6x term and leading to an incomplete equation. Always expand (a + b)² as a² + 2ab + b².
在通过代入法解一次与二次联立方程时，考生常能正确代入，但展开平方项时出错。对于 y = x + 3 与 x² + y² = 9，代入得 x² + (x+3)² = 9。很多人将 (x+3)² 错误地展开为 x² + 9，丢失了 6x 这一项，导致方程不完整。务必记住 (a + b)² = a² + 2ab + b²。
Elimination method errors often stem from aligning terms incorrectly. When adding or subtracting equations to eliminate a variable, signs matter. For example: 3x + 2y = 8 and 2x − 2y = 2. Adding eliminates y smoothly. But with 5x + 3y = 11 and 2x − y = 4, students multiply the second equation by 3 but forget to multiply the constant term, writing 6x − 3y = 4 instead of 6x − 3y = 12. Every term must be multiplied.
加减消元法的错误常源于对齐不当。当用加减消元时，符号很重要。例如方程 5x + 3y = 11 和 2x − y = 4，学生将第二个方程乘以3，但忘记了乘以常数项，写成 6x − 3y = 4，而不是 6x − 3y = 12。必须乘以每一项。
5. Inequalities | 不等式
The most notorious mistake is reversing the inequality sign only when multiplying or dividing by a negative number, but forgetting to do so when the variable is multiplied by a negative coefficient. For −2x > 6, many students write x > −3 instead of flipping the sign to x < −3. The rule: if you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. This also applies when rearranging; e.g., from 3 − x ≥ 7, subtract 3: −x ≥ 4, then multiply by −1: x ≤ −4.
最常犯的错误是只在乘以或除以负数时才翻转不等号，但在变量系数为负时却忘记了。对于 −2x > 6，很多学生写成 x > −3，而没有将符号翻转为 x < −3。规则是：当你对不等式两边乘或除一个负数时，必须反转不等号。在移项时也适用；比如 3 − x ≥ 7，减3得 −x ≥ 4，再乘以 −1 得 x ≤ −4。
Quadratic inequalities such as x² − 5x + 6 > 0 require a critical values approach. The error is to write the solution as 2 < x < 3 after factorising to (x − 2)(x − 3) > 0. That interval actually corresponds to < 0. For > 0, we need x < 2 or x > 3. A sketch of the parabola helps avoid confusion. Also, with inequalities involving fractions like (x+1)/(x−2) ≤ 0, multiplying both sides by (x−2)² is safe, but students often multiply by just (x−2) and forget to consider when it is negative, leading to sign errors. Always use a sign table or multiply by the square of the denominator.
二次不等式如 x² − 5x + 6 > 0 需要找临界值。错误是将解写成 2 < x < 3，而那实际上对应 < 0 的情况。对于 > 0，需取 x < 2 或 x > 3。画抛物线草图有助于避免混淆。同样，对于含分式的不等式如 (x+1)/(x−2) ≤ 0，两边乘以 (x−2)² 最安全，但学生常只乘以 (x−2)，忘记考虑其为负时的不等号方向变化，导致符号错误。务必使用符号表或乘以分母的平方。
6. Functions: Domain, Range, and Inverse | 函数：定义域、值域与反函数
Domain and range are often confused. The domain is the set of allowed inputs (x-values), while the range is the set of possible outputs (y-values). For f(x) = √(x − 2), the domain is x ≥ 2 because the expression under the square root must be non-negative. A common mistake is to write the domain as x > 2 or f(x) ≥ 2. The range is f(x) ≥ 0. Understanding the effect of transformations on domain/range is also vital. For f(x) = 1/(x−3), the domain excludes x = 3. Students sometimes write x ≠ 3 but then, when finding the inverse, forget that the range of f is the domain of f⁻¹. The inverse of f(x) = (x−2)/(x+3) requires solving for x and then swapping variables; a frequent slip is not correctly handling the algebra when multiplying both sides by the denominator.
定义域和值域常被混淆。定义域是允许的输入（x值）集合，而值域是可能输出（y值）的集合。对于 f(x) = √(x − 2)，定义域为 x ≥ 2，因为平方根下的表达式必须非负。常犯的错误是写成 x > 2 或 f(x) ≥ 2。值域为 f(x) ≥ 0。理解变换对定义域/值域的影响也至关重要。对于 f(x) = 1/(x−3)，定义域排除 x = 3。学生有时正确写出 x ≠ 3，但在求反函数时忘了 f 的值域就是 f⁻¹ 的定义域。求 f(x) = (x−2)/(x+3) 的反函数时，需解出 x 再交换变量；常见的疏忽是两边乘以分母时没有正确处理代数式。
Composite functions: fg(x) means apply g first, then f. Order is critical. If f(x) = 2x + 1 and g(x) = x², then fg(x) = 2x² + 1, but gf(x) = (2x+1)² = 4x² + 4x + 1. Students often misinterpret the notation and reverse the order. Also, when evaluating fg(3), they might calculate f(3) first.
复合函数：fg(x) 表示先作用 g，再作用 f。顺序至关重要。若 f(x) = 2x + 1，g(x) = x²，则 fg(x) = 2x² + 1，而 gf(x) = (2x+1)²。学生常误解符号，把顺序搞反。在求 fg(3) 时，他们有可能先算 f(3)。
7. Coordinate Geometry: Straight Lines | 坐标几何：直线
Gradient calculation errors occur when coordinates are subtracted in the wrong order. For points (x₁, y₁) and (x₂, y₂), gradient m = (y₂ − y₁)/(x₂ − x₁). If a student reverses the numerator and denominator, they get the reciprocal. Using (2, 5) and (4, 9) as an example, m = 4/2 = 2. If they do (9−5)/(2−4) they get 4/(−2) = −2, which is wrong but only off by a sign; if they do (5−9)/(4−2) = −4/2 = −2, also wrong sign. The correct approach is to be consistent with the chosen point as ‘second’. Many errors stem from mixing up the coordinates. Also, the equation of a line: y − y₁ = m(x − x₁). If the gradient is 3 and the line passes through (1, 4), the equation is y − 4 = 3(x − 1). A common mistake is to write y − 1 = 3(x − 4), using x₁ and y₁ swapped.
计算梯度时，常因坐标相减的顺序错误而得出错误结果。对于点 (x₁, y₁) 和 (x₂, y₂)，梯度 m = (y₂ − y₁)/(x₂ − x₁)。若学生颠倒分子分母，会得到倒数。例如 (2, 5) 和 (4, 9)，m = 4/2 = 2。若他们用 (5−9)/(4−2) = −2，符号错误；若用 (9−5)/(2−4) 也得 −2。关键要保持一致，选定一个点作为“第二”。还有许多错误源于混淆了点的坐标。直线方程 y − y₁ = m(x − x₁) 中，若梯度为 3 且过点 (1, 4)，方程应为 y − 4 = 3(x − 1)。常见错误是写成 y − 1 = 3(x − 4)，颠倒了 x₁ 和 y₁。
Parallel and perpendicular lines: gradient of perpendicular is the negative reciprocal, m₂ = −1/m₁. Students often forget the negative sign and just take the reciprocal. For a line with gradient 2/3, the perpendicular gradient is −3/2, not 3/2. Also, when finding the equation of a perpendicular bisector, they might find the midpoint but then use the gradient of the original line instead of the negative reciprocal.
平行与垂直线：垂直直线的梯度是负倒数，m₂ = −1/m₁。学生常常忘记负号，只取倒数。对于梯度为 2/3 的直线，其垂线的梯度为 −3/2，而不是 3/2。此外，求垂直平分线方程时，他们可能求出了中点，却用了原直线的梯度而非其负倒数。
8. Circles | 圆
The equation of a circle with centre (a, b) and radius r is (x − a)² + (y − b)² = r². When the centre is at (−2, 5), the equation becomes (x + 2)² + (y − 5)² = r². Many students incorrectly write (x − 2)², losing the sign. Remember that (x − (−2))² simplifies to (x + 2)². When completing the square to find the centre and radius from a general equation like x² + y² − 4x + 6y − 3 = 0, the error is in handling the constant term after grouping. Group x terms: x² − 4x; complete square: (x − 2)² − 4. For y: y² + 6y = (y + 3)² − 9. Then circle becomes (x − 2)² + (y + 3)² = 16, so centre (2, −3), radius 4. If they forget to move the constants to the right side, they might get the radius squared wrong.
圆心为 (a, b)、半径为 r 的圆的方程是 (x − a)² + (y − b)² = r²。当圆心为 (−2, 5) 时，方程变为 (x + 2)² + (y − 5)² = r²。许多学生错误地写成 (x − 2)²，丢失了符号。记住 (x − (−2))² 化简为 (x + 2)²。在由一般式 x² + y² − 4x + 6y − 3 = 0 通过配方法求圆心和半径时，易错点在处理分组后的常数项。x 组：x² − 4x 配方得 (x − 2)² − 4；y 组：y² + 6y = (y + 3)² − 9。然后圆方程变为 (x − 2)² + (y + 3)² = 16，所以圆心 (2, −3)，半径 4。若忘记将常数移到右边，可能会算错半径的平方。
Finding tangents to circles: the key fact that the tangent is perpendicular to the radius at the point of contact is often overlooked. Students sometimes try to use differentiation before covering it or use the gradient of the chord instead. Always find the centre, then the gradient of the radius, then the perpendicular gradient for the tangent, then use the point to get the line equation.
求圆的切线：切线垂直于切点处的半径这一关键事实常被忽略。学生有时在未学微分前尝试用微分，或用弦的梯度代替。务必先求圆心，再求半径的梯度，然后取负倒数得切线的梯度，最后用点斜式求切线方程。
9. Trigonometry: Basic Ratios and Graphs | 三角学：基本比值与图像
Exact values of sin, cos, tan for 30°, 45°, 60° are frequently memorised incorrectly. Mixing up sin 30° and sin 60° is common: sin 30° = 1/2, sin 60° = √3/2. Cos 30° = √3/2, cos 60° = 1/2. Tan 45° = 1. Knowing these and the related angles in other quadrants using CAST or graphs is essential. A typical error is assuming sin(θ) = sin(θ − 180°) without considering the sign. For example, sin 210° = −1/2, not +1/2.
30°、45°、60° 的正弦、余弦和正切的精确值经常被记错。混淆 sin 30° 和 sin 60° 很常见：sin 30° = 1/2，sin 60° = √3/2。cos 30° = √3/2，cos 60° = 1/2。tan 45° = 1。记住这些及利用 CAST 图或图像推导其他象限的相关角至关重要。典型错误是假设 sin(θ) = sin(θ − 180°) 而不考虑符号。例如，sin 210° = −1/2，不是 +1/2。
Solving trig equations like sin x = 0.5 for 0° ≤ x ≤ 360°: the first solution is 30°, the second is 180° − 30° = 150°. Students often only give 30° or wrongly give 210° and 330° (which are for negative sine). Using the sine graph or CAST is vital to get all solutions in the interval. For cos x = 0.5, solutions are 60° and 300° (or 360° − 60°). For tan x = 1, solutions are 45° and 225°. Many candidates forget to add or subtract the period (180° for tan, 360° for sin/cos) until they are outside the range, or they include values outside the given domain.
解三角方程如 sin x = 0.5 在 0° ≤ x ≤ 360° 内：第一个解是 30°，第二个是 180° − 30° = 150°。学生往往只给出 30°，或错误地给出 210° 和 330°（那是负正弦的解）。使用正弦图像或 CAST 图对于求区间内所有解至关重要。对于 cos x = 0.5，解为 60° 和 300°。对于 tan x = 1，解为 45° 和 225°。许多考生忘记加减周期（tan 周期 180°，sin/cos 周期 360°）直到超出范围，或者包含了不在给定定义域内的值。
Trigonometric identities: the identity sin²θ + cos²θ = 1 is used to simplify or solve equations. An error arises when rearranging: sinθ = √(1 − cos²θ) is not always valid because sinθ can be negative. The correct is sinθ = ± √(1 − cos²θ). Similarly, when dividing by cosθ to get tanθ, students often forget to consider cosθ = 0 cases separately, losing potential solutions.
三角恒等式：sin²θ + cos²θ = 1 用于化简或解方程。一个常见错误出现在移项时：sinθ = √(1 − cos²θ) 并不总是成立，因为 sinθ 可以是负的。正确的应为 sinθ = ±√(1 − cos²θ)。同样，当除以 cosθ 得到 tanθ 时，学生常忘记单独考虑 cosθ = 0 的情形，从而丢失可能的解。
10. Differentiation: Basic Rules and Tangents/Normals | 微分：基本法则与切线/法线
Power rule: d/dx (xⁿ) = n xⁿ⁻¹. A frequent mistake is forgetting to reduce the power by 1, or incorrectly reducing it. For example, differentiating x³ gives 3x², not 3x³ or 3x. For a constant term, the derivative is zero. Students sometimes differentiate π or √2 as if they were variables. Another mistake is when the coefficient is a fraction: d/dx ( (1/2)x⁴ ) = (1/2)×4x³ = 2x³. Some students erroneously apply the power rule to the fraction’s denominator separately, leading to 1/(2×4x³) which is nonsense.
幂法则：d/dx (xⁿ) = n xⁿ⁻¹。常见错误是忘记将指数减1，或者减错。例如，x³ 的导数是 3x²，而不是 3x³ 或 3x。常数项的导数为零。学生有时对 π 或 √2 像变量一样求导。另一个错误是系数为分数时：d/dx ( (1/2)x⁴ ) = (1/2)×4x³ = 2x³。有些学生错误地将幂法则分别应用到分母的分数上，得出 1/(2×4x³)，这毫无意义。
Tangents and normals: after differentiating to find the gradient function dy/dx, the gradient at a point is found by substituting the x-coordinate. To find the tangent equation, use y − y₁ = m(x − x₁). The normal gradient is −1/m. A typical error is using the same gradient for normal, or forgetting to find the y-coordinate of the point on the curve when only x is given, then using the wrong point. Ensuring the point lies on the curve is crucial; substituting x into the original equation yields the correct y.
切线与法线：求导得梯度函数 dy/dx 后，代入 x 坐标得到该点梯度。切线方程用 y − y₁ = m(x − x₁)。法线梯度为 −1/m。常见错误是法线用了和切线相同的梯度，或当只给 x 坐标时，忘记求曲线上对应点的 y 坐标，导致使用了错误的点。确保点在曲线上至关重要；将 x 代入原方程才能得到正确的 y。
11. Integration: Basic Integration and Area | 积分：基本积分与面积
Integration as the reverse of differentiation: ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + c, for n ≠ −1. The most common error is increasing the power but forgetting to divide by the new power. Integrating x² gives (1/3)x³, not 3x³. Also, the constant of integration ‘+ c’ is frequently omitted, which loses marks in indefinite integrals. When dealing with definite integrals, students sometimes forget to apply the limits or apply them incorrectly, subtracting in the wrong order: F(b) − F(a). Some do F(a) − F(b).
积分是微分的逆运算：∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + c，n ≠ −1。最常见的错误是增加了幂次却忘记除以新指数。积分 x² 得 (1/3)x³，而不是 3x³。另外，积分常数 ‘+ c’ 经常被遗漏，在不定期积分中扣分。处理定积分时，学生有时候忘记代入上下限，或代入后相减顺序颠倒：F(b) − F(a)。有的算成 F(a) − F(b)。
Finding area under a curve: the definite integral gives the signed area. When the curve crosses the x-axis, integrating across the root without splitting will yield net area (cancelling), not total area. For y = x(x − 2) between 0 and 3, the curve is below the axis from 0 to 2, and above from 2 to 3. To find total area, compute |∫₀² f(x) dx| + ∫₂³ f(x) dx. Many students simply integrate from 0 to 3 and get a smaller number, which is wrong for ‘total area’. Always sketch the graph to identify intervals where the function is negative.
求曲线下方面积：定积分给出带符号的面积。当曲线穿过 x 轴时，不分割而直接对跨根区间积分会得到净面积（正负相抵），而非总面积。对于 y = x(x − 2) 在 0 到 3 之间，曲线在 0 至 2 位于轴下方，2 至 3 位于上方。求总面积需计算 |∫₀² f(x) dx| + ∫₂³ f(x) dx。许多学生简单地从 0 积到 3，得到一个较小的数字，这对于“总面积”是错误的。务必先画草图以识别函数为负的区间。
Integration of expressions like (2x + 1)⁴ benefits from the reverse chain rule, but candidates often forget to divide by the derivative of the inner function. ∫ (2x + 1)⁴ dx = (1/5)(2x + 1)⁵ × (1/2) + c = (1/10)(2x + 1)⁵ + c. Missing the division by 2 is a recurring mistake.
对像 (2x + 1)⁴ 这样的表达式积分可以利用逆链式法则，但考生常忘记除以内部函数的导数。∫ (2x + 1)⁴ dx = (1/5)(2x + 1)⁵ × (1/2) + c = (1/10)(2x + 1)⁵ + c。遗漏除以 2 是非常常见的错误。
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