📚 AS Eduqas Further Mathematics: Key Terminology Quick Reference | AS Eduqas 进阶数学：词汇术语速记指南
Mastering the precise terminology is essential for success in AS Eduqas Further Mathematics. This guide provides a bilingual quick-reference to key terms, helping you to build confidence in both conceptual understanding and examination language.
掌握精确的术语对于在 AS Eduqas 进阶数学中取得成功至关重要。本指南提供关键术语的双语速查，帮助您在概念理解和考试语言方面建立信心。
1. Complex Numbers | 复数
Complex Number: A number of the form z = a + ib, where a and b are real numbers, and i² = –1.
复数： 形如 z = a + ib 的数，其中 a 和 b 是实数，且 i² = –1。
Real Part: In a + ib, the real part is a, written as Re(z).
实部： 在 a + ib 中，实部为 a，记作 Re(z)。
Imaginary Part: In a + ib, the imaginary part is b, written as Im(z). Note that b is real.
虚部： 在 a + ib 中，虚部为 b，记作 Im(z)。请注意 b 是实数。
Imaginary Unit i: Defined as √–1, so i² = –1, i³ = –i, i⁴ = 1.
虚数单位 i： 定义为 √–1，因此 i² = –1，i³ = –i，i⁴ = 1。
Complex Conjugate: For z = a + ib, the conjugate is z* or z̄ = a – ib.
共轭复数： 对于 z = a + ib，其共轭为 z* 或 z̄ = a – ib。
Equality of Complex Numbers: a + ib = c + id implies a = c and b = d.
复数相等： a + ib = c + id 意味着 a = c 且 b = d。
2. Argand Diagrams and Polar Form | 阿根图与极坐标形式
Argand Diagram: A plane where the horizontal axis is the real part and the vertical axis is the imaginary part.
阿根图： 一个平面，其中水平轴表示实部，垂直轴表示虚部。
Modulus of z: For z = a + ib, the modulus is |z| = √a² + b², representing the distance from the origin.
复数的模： 对于 z = a + ib，模为 |z| = √a² + b²，表示到原点的距离。
Argument of z: The angle θ measured from the positive real axis to the line joining the origin to the point, usually given in radians in the range –π < θ ≤ π.
复数的辐角： 从正实轴到原点到该点连线的夹角 θ，通常以弧度表示，范围为 –π < θ ≤ π。
Polar Form: z = r(cos θ + i sin θ), where r = |z| and θ = arg(z).
极坐标形式： z = r(cos θ + i sin θ)，其中 r = |z| 且 θ = arg(z)。
Exponential Form: z = r eiθ, often used with Euler's formula.
指数形式： z = r eiθ，常与欧拉公式一起使用。
3. Matrices: Definitions and Operations | 矩阵：定义与运算
Matrix: A rectangular array of numbers (elements) arranged in rows and columns.
矩阵： 一个由数（元素）排列成的矩形阵列，有行和列。
Order of a Matrix: Described as 'm × n' where m is the number of rows and n is the number of columns.
矩阵的阶： 用 m × n 描述，其中 m 为行数，n 为列数。
Element aij: The entry in the i-th row and j-th column.
元素 aij： 位于第 i 行、第 j 列的元素。
Square Matrix: A matrix with the same number of rows and columns.
方阵： 行数和列数相等的矩阵。
Zero Matrix: A matrix in which every element is 0, denoted by O.
零矩阵： 所有元素均为 0 的矩阵，记作 O。
Identity Matrix: A square matrix with 1s on the leading diagonal and 0s elsewhere, denoted by I.
单位矩阵： 主对角线元素为 1，其余元素为 0 的方阵，记作 I。
Matrix Addition: A + B adds corresponding elements; requires matrices of the same order.
矩阵加法： A + B 将对应元素相加；要求矩阵同阶。
Matrix Multiplication: AB is defined if the number of columns of A equals the number of rows of B. In general AB ≠ BA.
矩阵乘法： 若 A 的列数等于 B 的行数，则 AB 有定义。通常 AB ≠ BA。
4. Determinants and Inverses | 行列式与逆矩阵
Determinant of a 2×2 Matrix: For A = [a b; c d], det(A) = ad – bc. It is denoted |A|.
2×2 矩阵的行列式： 对于 A = [a b; c d]，det(A) = ad – bc，记作 |A|。
Singular Matrix: A square matrix whose determinant is zero; it has no inverse.
奇异矩阵： 行列式为零的方阵；没有逆矩阵。
Non-singular Matrix: A square matrix with a non-zero determinant; it possesses an inverse.
非奇异矩阵： 行列式不为零的方阵；存在逆矩阵。
Inverse of a 2×2 Matrix: If A = [a b; c d] is non-singular, A⁻¹ = (1/(ad – bc)) [d –b; –c a].
2×2 逆矩阵： 若 A = [a b; c d] 非奇异，则 A⁻¹ = (1/(ad – bc)) [d –b; –c a]。
Properties: AA⁻¹ = A⁻¹A = I, and (AB)⁻¹ = B⁻¹A⁻¹.
性质： AA⁻¹ = A⁻¹A = I，且 (AB)⁻¹ = B⁻¹A⁻¹。
5. Linear Transformations Using Matrices | 矩阵的线性变换
Linear Transformation: A mapping that can be represented by a matrix M such that the image of a point (x,y) is given by M [x; y].
线性变换： 可用矩阵 M 表示的映射，点 (x,y) 的像由 M [x; y] 给出。
Reflection in the x-axis: Matrix [1 0; 0 –1].
关于 x 轴的反射： 矩阵 [1 0; 0 –1]。
Reflection in the line y = x: Matrix [0 1; 1 0].
关于直线 y = x 的反射： 矩阵 [0 1; 1 0]。
Rotation by angle θ (anticlockwise): Matrix [cos θ –sin θ; sin θ cos θ].
逆时针旋转 θ 角： 矩阵 [cos θ –sin θ; sin θ cos θ]。
Enlargement scale factor k: Matrix [k 0; 0 k].
放大（比例因子 k）： 矩阵 [k 0; 0 k]。
Invariant Points: Points that are unchanged under a transformation, satisfying M [x; y] = [x; y].
不变点： 在变换下保持不变的点，满足 M [x; y] = [x; y]。
Invariant Lines: Lines mapped onto themselves; points on the line may move along it.
不变直线： 被映射到自身的直线；直线上的点可沿该线移动。
6. Vectors in Two and Three Dimensions | 二维与三维向量
Vector: A quantity with both magnitude and direction; can be written in column form, as a position vector, or using i, j, k notation.
向量： 既有大小又有方向的量；可用列向量、位置向量或 i, j, k 形式表示。
Magnitude: Length of a vector; for v = a i + b j, |v| = √a² + b².
模： 向量的长度；对于 v = a i + b j，|v| = √a² + b²。
Unit Vector: A vector of magnitude 1; given by v/|v|.
单位向量： 模为 1 的向量；由 v/|v| 给出。
Position Vector: A vector from the origin to a point P; usually denoted OP→ or r.
位置向量： 从原点到点 P 的向量；常记作 OP→ 或 r。
Direction Vector: Any vector parallel to a given line; used in vector equations of lines.
方向向量： 与给定直线平行的任一向量；用于直线的向量方程。
Scalar (Dot) Product: a · b = |a||b| cos θ = a₁b₁ + a₂b₂ (+ a₃b₃).
标量积（点积）： a · b = |a||b| cos θ = a₁b₁ + a₂b₂ (+ a₃b₃)。
Perpendicular Vectors: a · b = 0.
垂直向量： a · b = 0。
7. Vector Equations of Lines | 直线的向量方程
Vector Equation (2D/3D): r = a + λ b, where a is a position vector of a point on the line, b is a direction vector, and λ is a scalar parameter.
向量方程（二维/三维）： r = a + λ b，其中 a 是直线上某点的位置向量，b 是方向向量，λ 是标量参数。
Parametric Equations: Derived from the vector equation, e.g. x = a₁ + λ b₁, y = a₂ + λ b₂.
参数方程： 由向量方程导出，例如 x = a₁ + λ b₁, y = a₂ + λ b₂。
Cartesian Form (2D): Eliminate λ to obtain (x – a₁)/b₁ = (y – a₂)/b₂, provided b₁, b₂ ≠ 0.
直角坐标形式（二维）： 消去 λ 得到 (x – a₁)/b₁ = (y – a₂)/b₂，前提是 b₁, b₂ ≠ 0。
Angle Between Two Lines: Uses the dot product of their direction vectors: cos θ = |b₁ · b₂| / (|b₁||b₂|).
两条直线的夹角： 利用它们方向向量的点积：cos θ = |b₁ · b₂| / (|b₁||b₂|)。
8. Summation of Series | 级数求和
Summation Notation Σ: Σ_r=1n ur denotes u₁ + u₂ + … + un.
求和符号 Σ： Σ_r=1n ur 表示 u₁ + u₂ + … + un。
Standard Results (for integer n): Σ_r=1n r = ½ n(n+1), Σ_r=1n r² = ⅙ n(n+1)(2n+1), Σ_r=1n r³ = ¼ n²(n+1)².
标准公式（对于整数 n）： Σ_r=1n r = ½ n(n+1), Σ_r=1n r² = ⅙ n(n+1)(2n+1), Σ_r=1n r³ = ¼ n²(n+1)²。
Linear Combinations: Break sums into known parts, e.g. Σ (2r² + 3) = 2 Σ r² + Σ 3.
线性组合： 将求和分解为已知部分，例如 Σ (2r² + 3) = 2 Σ r² + Σ 3。
Method of Differences: Express term ur as f(r) – f(r+1); the sum telescopes to f(1) – f(n+1).
拆项法（差分法）： 将项 ur 表示为 f(r) – f(r+1)；和式相消为 f(1) – f(n+1)。
9. Proof by Induction | 数学归纳法
Principle of Mathematical Induction: A method to prove statements P(n) for all positive integers n: show P(1) is true, then prove P(k) ⇒ P(k+1).
数学归纳法原理： 一种用于证明对所有正整数 n 成立的命题 P(n) 的方法：证明 P(1) 为真，再证明 P(k) ⇒ P(k+1)。
Base Case: Verify the statement for the smallest value (usually n = 1).
基础步骤： 验证命题对最小值（通常 n = 1）成立。
Inductive Hypothesis: Assume the statement is true for n = k.
归纳假设： 假设命题对 n = k 为真。
Inductive Step: Show that if the statement holds for n = k, then it must also hold for n = k+1.
归纳递推： 证明若命题对 n = k 成立，则其对 n = k+1 也必定成立。
Conclusion: By mathematical induction, P(n) is true for all positive integers n.
结论： 根据数学归纳法，P(n) 对所有正整数 n 成立。
10. Roots of Polynomials | 多项式的根
Relationship Between Roots and Coefficients (Quadratic): For ax² + bx + c = 0 with roots α, β: α+β = –b/a, αβ = c/a.
根与系数的关系（二次）： 对于 ax² + bx + c = 0 且根为 α, β：α+β = –b/a，αβ = c/a。
Cubic Equation: For ax³ + bx² + cx + d = 0 with roots α, β, γ: Σα = –b/a, Σαβ = c/a, αβγ = –d/a.
三次方程： 对于 ax³ + bx² + cx + d = 0 且根为 α, β, γ：Σα = –b/a，Σαβ = c/a，αβγ = –d/a。
Quartic Equation: For ax⁴ + bx³ + cx² + dx + e = 0: Σα = –b/a, Σαβ = c/a, Σαβγ = –d/a, αβγδ = e/a.
四次方程： 对于 ax⁴ + bx³ + cx² + dx + e = 0：Σα = –b/a，Σαβ = c/a，Σαβγ = –d/a，αβγδ = e/a。
Complex Conjugate Root Theorem: If a polynomial has real coefficients, then complex roots occur in conjugate pairs (if a + ib is a root, so is a – ib).
共轭复根定理： 若多项式的系数为实数，则复根成共轭对出现（若 a + ib 是根，则 a – ib 也是根）。
Forming Equations from Roots: Given roots, the polynomial can be written as a(x – α)(x – β)… = 0.
由根构造方程： 给定根，多项式可写为 a(x – α)(x – β)… = 0。
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