📚 AS Edexcel Further Mathematics: A Quick-Memorisation Guide to Key Terms | AS Edexcel 进阶数学：词汇术语速记指南
Struggling to remember what an argument, a determinant, or a hyperbolic function actually mean? In AS Edexcel Further Mathematics, precise vocabulary is not just a matter of language—it is the foundation for solving problems and scoring marks in reasoning questions. This bilingual guide pairs every critical term with a clear English definition, its Chinese translation, and a memory shortcut to make exam preparation faster and more effective.
还在为“辐角”、“行列式”或“双曲函数”这些词发愁吗？在AS Edexcel进阶数学中，准确的术语不仅是语言问题，更是解题和证明题得分的基石。这份中英双语速记指南将每个核心术语配上清晰的英文定义、中文翻译和记忆捷径，帮助你更高效地复习备考。
1. Complex Numbers – Fundamental Concepts | 复数——基本概念
A complex number is of the form a + bi where a, b ∈ ℝ and i² = −1. a is the real part Re(z), and b is the imaginary part Im(z). Think of a two‑dimensional number living on the x‑y plane.
复数 是一个形如 a + bi 的数，其中 a, b ∈ ℝ，i² = −1。a 称为 实部 Re(z)，b 称为 虚部 Im(z)。可以想象它是一个生活在x‑y平面上的二维数字。
The modulus |z| = √a² + b² gives the distance from the origin. Memorise it as “modulus = magnitude”. The argument arg z is the angle θ it makes with the positive real axis, measured in radians. Mnemonic: arg = angle.
模 |z| = √a² + b² 表示到原点的距离。记住：“模就是大小”。辐角 arg z 是该向量与正实轴所成的角 θ，单位为弧度。记忆技巧：arg = angle（角度）。
A complex conjugate z* = a − bi reflects the point across the real axis. Useful rule: z × z* = |z|², a real number. Use this to divide complex numbers — multiply top and bottom by the conjugate of the denominator.
共轭复数 z* = a − bi，就是将点关于实轴对称。实用法则：z × z* = |z|²，是一个实数。复数除法时，分子分母同乘分母的共轭即可。
2. Polar Form and De Moivre’s Theorem | 极坐标形式与棣莫弗定理
The polar form (or modulus‑argument form) of a complex number is z = r(cos θ + i sin θ), where r = |z| and θ = arg z. This converts multiplication into a simple rotation‑scaling operation. A shorter notation is z = r cis θ.
复数的极坐标形式（或称模‑辐角形式）为 z = r(cos θ + i sin θ)，其中 r = |z|，θ = arg z。它将乘法变成了简单的旋转‑缩放操作。简写为 z = r cis θ。
De Moivre’s theorem . Use this to find powers, roots, and to derive trigonometric identities. Key memory hook: “power becomes multiple of angle”.
棣莫弗定理：对任意整数 n，(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)。用它计算幂、求根以及推导三角恒等式。关键的速记点：“幂变成角的倍数”。
To find the nth roots of unity, solve zⁿ = 1. The solutions are zk = cis(2kπ/n), for k = 0, 1, … ed around the unit circle. For the nth roots of any number w, write w in polar form first.
求n次单位根时，解 zⁿ = 1，解为 zk = cis(2kπ/n)，k = 0, 1, …, n−1。它们在单位圆上均匀分布。求任意数 w 的n次根时，先写出 w 的极坐标形式。
3. Matrices – Operations and Transformations | 矩阵——运算与变换
A matrix is a rectangular array of numbers. The order qual the number of rows of the second. Remember: (p × q) × (q × r) gives a (p × r) matrix.
矩阵 是一个矩形数字阵列。阶（或维度）是行数×列数。两个矩阵相乘时，第一个矩阵的列数必须等于第二个的行数。记住：(p × q) × (q × r) 得到 (p × r) 矩阵。
The determinant of a 2×2 matrix M = [[a, b], [c, d]] is det(M) = ad − bc. It determines whether the matrix has an inverse (non‑zero det) and describes the area scale factor of the transformation. Think: “determinant = determinant of invertibility”.
2×2 矩阵 M = [[a, b], [c, d]] 的行列式为 det(M) = ad − bc。它决定矩阵是否可逆（行列式非零），并描述变换的面积比例因子。速记：“行列式决定是否为可逆矩阵”。
An inverse matrix M⁻¹ satisfies M M⁻¹ = I. For 2×2, M⁻¹ = (1/det) [[d, −b], [−c, a]]. The identity matrix I has 1s on the leading diagonal and 0s elsewhere; it leaves points unchanged under transformation.
逆矩阵 M⁻¹ 满足 M M⁻¹ = I。对于2×2，M⁻¹ = (1/det) [[d, −b], [−c, a]]。单位矩阵 I 主对角线为1，其余为0；用它做变换不会改变任何点的位置。
Common transformations origin), and enlargements (scaling). Each has a characteristic matrix; the determinant tells you the area change (negative indicates a reflection).
FP1中常见的变换 有反射（关于 y = ±x、坐标轴）、旋转（绕原点）和放大（缩放）。每种变换都有特征矩阵；行列式表示面积变化（负值表示包含一次反射）。
4. Summation of Series | 级数求和
You need to know standard results: Σr=1ⁿ 1 = n, Σ r = ½ n(n+1), Σ r² = ⅙ Σ r³ = ¼ n²(n+1)². A quick check: Σ r³ = (Σ r)², so cube sum is square of the linear sum.
需要熟记标准结果：Σr=1ⁿ 1 = n，Σ r = ½ n(n+1)，Σ r² = ⅙ n(n+1)(2n+1)，Σ r³ = ¼ n²(n+1)²。速查：Σ r³ = (Σ r)²，因此立方和就是线性和的平方。
The method of differences is used when a series can be expressed as f(r) − f(r+1) or f(r+1) − f(r). Terms cancel in pairs, leaving only the first and last parts. Look for partial fractions to set this up.
差项法 用于能把级数表示为 f(r) − f(r+1) 或 f(r+1) − f(r) 的情况。各项成对相消，只留下首尾部分。通常先用部分分式将其分解。
m telescopes. Always underline the first few terms and the last few to spot cancellations.
对于 Σ 1/(r(r+1)) 这样的级数，写成 1/(r(r+1)) = 1/r − 1/(r+1)。然后求和时会“望远镜式”消去。始终划出前几项和后几项来观察相消规律。
5. Proof by Mathematical Induction | 数学归纳法证明
Mathematical induction has four key stages: (1) Basis case — prove true for n = 1 (or smallest value). (2) Induction hypothesis — assume true for n = k. (3) Induction step — prove true for n = k + 1 using the assumption. (4) Conclusion — r n = 1 and if true for n = k then true for n = k+1, it is true for all positive integers.
数学归纳法 有四个关键步骤：(1) 奠基步——证明 n = 1（或最小取值）时命题成立。(2) 归纳假设——假设 n = k 时成立。(3) 归纳递推——利用假设证明 n = k + 1 时也成立。(4) 结论——由于 n=1 成立且 n=k 成立推出 n=k+1 成立，因此对全体正整数成立。
Typical FP1 induction topics: summation of series, divisibility (e.g., 3²ⁿ − 1 divisible by 8), matrix powers, recurrence relations. For divisibility, express f(k+1) in terms of f(k) plus a multiple of the divisor.
FP1常见的归纳法题目：级数求和、整除性（如 3²ⁿ − 1 可被8整除）、矩阵的幂、递推关系。对于整除性问题，将 f(k+1) 用 f(k) 加上除数的倍数来表示。
Memory aid: “Assume, add, and check.” Always write the induction hypothesis explicitly — many marks are awarded for stating the assumption clearly.
记忆口诀：“假设、添加、验证。”始终明确写出归纳假设——清晰地陈述假设就能获得许多分数。
6. Coordinate Systems – Parametric Equations | 坐标系——参数方程
A parametric equation expresses x and y in terms of a third variable, t. The gradient ng the chain rule: dy/dx = (dy/dt) ÷ (dx/dt). To find the cartesian equation, eliminate t.
参数方程 用第三个变量 t 来表达 x 和 y。斜率 dy/dx 由链式法则求得：dy/dx = (dy/dt) ÷ (dx/dt)。要得到直角坐标方程，只需消去参数 t。
The parabola is given parametrically by (x = at², y = 2at). The rectangular hyperbola epresented by (x = ct, y = c/t). For the hyperbola, remember x y = c², which is a constant — a useful trick for tangents and normals.
抛物线 的参数形式为 (x = at², y = 2at)。等轴双曲线 用 (x = ct, y = c/t) 表示。对于双曲线，记住 x y = c² 是常数——这个性质在切线、法线的计算中非常有用。
When finding the equation of a tangent or normal at a parameter t, first find dy/dx, then use point‑slope form y − y(t) = m (x − x(t)). For a normal, the gradient is −1/m.
求参数 t 对应点处的切线或法线方程时，先求 dy/dx，再使用点斜式 y − y(t) = m (x − x(t))。法线的斜率为 −1/m。
7. Inequalities and Modulus | 不等式与绝对值
Linear inequalities are solved like equations, but multiplying or dividing by a negative number reverses the inequality sign. For quadratic inequalities, sketch the graph to see where the expression is >0 or <0.
线性不等式 的解法与方程类似，但乘以或除以负数时需要反转不等号方向。二次不等式 则借助图形判断表达式大于0或小于0的区间。
The modulus function |x| removes any sign. |x − a| < b means the distance from x to a is less than b, so a − b < x < a + b. For |x − a| > b, you get two intervals x < a − b or x > a + b.
绝对值 函数 |x| 消除符号。|x − a| < b 表示 x 到 a 的距离小于 b，因此 a − b < x < a + b。对于 |x − a| > b，则得到两个区间：x < a − b 或 x > a + b。
To solve modulus equations like |2x − 1| 3x, then check solutions satisfy the original equation. Always eliminate spurious answers.
解 绝对值方程 如 |2x − 1| = 3x 时，考虑两种情况：2x − 1 = 3x 或 2x − 1 = −3x，然后检验解是否满足原方程。要剔除增根。
8. Vectors – Scalar Product and Lines | 向量——标量积与直线
The scalar product (dot product) of two vectors a and b is a·b = |a||b| m. In component form, a·b = a₁b₁ + a₂b₂ + a₃b₃. If a·b = 0, the vectors are perpendicular.
两个向量 a 和 b 的标量积（点积）为 a·b = |a||b| cos θ，θ 为夹角。分量形式中，a·b = a₁b₁ + a₂b₂ + a₃b₃。若 a·b = 0，则两向量垂直。
The vector equation of a line is r = a + λb, where a is a position vector on the line and b is a direction vector. To convert to cartesian form, write x, y, z in terms of λ, then eliminate λ.
直线的向量方程 为 r = a + λb，其中 a 是线上一点的位矢，b 是方向向量。要转换为直角坐标形式，写出 x, y, z 关于 λ 的表达式，然后消去 λ。
To find the angle between two lines, use the direction vectors b₁ and b₂ in cos θ = |b₁·b₂| / (|b₁||b₂|). The absolute value ensures the acute angle is obtained.
求 两直线的夹角 时，利用方向向量 b₁ 和 b₂，cos θ = |b₁·b₂| / (|b₁||b₂|)。绝对值确保得到的是锐角。
9. Complex Loci | 复平面的轨迹
An equation like |z − z₁| = r is a circle with centre z₁ (a fixed complex number) and radius r. |z − z₁| = |z − z₂| describes the perpendicular bisector of the segment joining z₁ and z₂.
方程 |z − z₁| = r 表示一个 圆，圆心为 z₁（固定复数），半径为 r。|z − z₁| = |z − z₂| 表示线段 z₁z₂ 的垂直平分线。
arg(z − z₁) = θ represents a half‑line starting at z₁ and making an angle θ with the positive real axis. The point z₁ is usually excluded (open circle). To sketch these, always identify the starting point and direction.
arg(z − z₁) = θ 表示一条以 z₁ 为起点、与正实轴夹角为 θ 的射线。起点 z₁ 通常用空心圆标记。画图时，务必先确定起点和方向。
Combinations of inequalities, e.g., {z: |z − 3| < 2} ∩ define regions. Shade the required region and use solid or dashed boundaries accordingly.
不等式组合，如 {z: |z − 3| < 2} ∩ {z: 0 ≤ arg(z) ≤ π/2}，会定义区域。将所求区域涂上阴影，并根据不等式是否含等号使用实线或虚线边界。
10. Numerical Methods – Solving Equations | 数值方法——方程求解
To show that a root exists in an interval [a,b], use the sign change rule: if f(a) and f(b) have opposite signs and f is continuous on [a,b], there is at least one root. Always state that f is continuous.
要证明区间 [a,b] 内存在根，使用符号变化法则：若 f(a) 与 f(b) 反号且 f 在 [a,b] 上连续，则至少存在一个根。务必说明 f 是连续的。
The Newton‑Raphson method uses the iterative formula xn+1 = xn − f(xn) / f '(xn). Choose a starting value close to the root. The method can fail if f '(x) = 0 or the tangent sends you away from the root.
牛顿‑拉弗森法 使用迭代公式 xn+1 = xn − f(xn) / f '(xn)。选择靠近根的初值。如果 f '(x) = 0 或切线将点带到远离根的地方，该方法可能失效。
Linear interpolation a)] / [f(b) − f(a)]. It replaces the curve with a chord. Draw a sketch to avoid sign errors.
线性插值（有时会考查）利用了相似三角形。公式为 x ≈ a − [f(a)(b − a)] / [f(b) − f(a)]。它用弦来替代曲线。画简图可避免符号错误。
11. Hyperbolic Functions – Quick Introduction | 双曲函数——快速入门
For AS Further Mathematics, you may encounter the definitions: sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ − sinh² x = 1. Notice the similarity to trigonometric identities but with different signs.
在AS进阶数学中，你会接触到定义：sinh x = (eˣ − e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2。关键恒等式是 cosh² x − sinh² x = 1。注意它与三角恒等式的相似性，但符号有所不同。
Osborn’s rule n² with −sinh². For example, cos 2x = cos² x − sin² x becomes cosh 2x = cosh² x + sinh² x.
奥斯本法则：将三角恒等式转换为双曲恒等式时，把 cos² 换为 cosh²，sin² 换为 −sinh²。例如，cos 2x = cos² x − sin² x 转换为 cosh 2x = cosh² x + sinh² x。
rea hyperbolic sine), arcosh x, etc. Understand their logarithmic forms: arsinh x = ln(x + √x² + 1), arcosh x = ln(x + √x² − 1) for x ≥ 1.
反双曲函数写作 arsinh x（表示双曲正弦面积函数）、arcosh x 等。要理解它们的对数形式：arsinh x = ln(x + √x² + 1)，arcosh x = ln(x + √x² − 1) (x ≥ 1)。
12. Quick-Fire Glossary of Additional Terms | 附加术语速查表
Here is a compact table of other terms you must know, with memory triggers.
以下是一个紧凑的表格，列有其它必知术语及其记忆触发器。
| English Term | 中文 | Memory Shortcut |
|---|---|---|
| Invariant point | 不动点 | Stays the same after transformation |
| Invariant line | 不动线 | Points on the line stay on the line (but may move along it) |
| Singular matrix | 奇异矩阵 | det = 0, no inverse → “singularly useless” |
| Characteristic equation | 特征方程 | det(M − λI) = 0, used to find eigenvalues (beyond AS but worth knowing) |
| Eigenvalue | 特征值 | The λ that makes Mv = λv, where v is eigenvector |
| Cartesian form | 直角坐标形式 | x and y equation, no parameter |
| Parametric form | 参数形式 | x(t), y(t) — parameter t |
| Perpendicular | 垂直 | Dot product = 0 |
| Parallel | 平行 | Direction vectors are scalar multiples |
| Collinear | 共线 | Points lie on the same straight line → vectors are multiples |
| Linear combination | 线性组合 | aa + bb, used in solving vector problems |
| Discriminant | 判别式 | Δ = b² − 4ac; use to find number of real roots |
| Asymptote | 渐近线 | A line the curve approaches but never touches |
Published by TutorHao | AS Further Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图