Year 12 CCEA Further Mathematics: Key Terminology Quick Guide | CCEA 进阶数学：词汇术语速记指南

📚 Year 12 CCEA Further Mathematics: Key Terminology Quick Guide | CCEA 进阶数学：词汇术语速记指南

Mastering the language of Further Mathematics is half the battle. This guide breaks down the essential terminology you will encounter in Year 12 CCEA Further Mathematics, with concise definitions and memory tips. From complex numbers to hyperbolic functions, get ready to boost your confidence and exam performance.

掌握进阶数学的语言是成功的一半。本指南为你梳理 CCEA 进阶数学 Year 12 中必考的核心词汇，提供清晰定义和速记窍门。从复数到双曲函数，助你提升解题信心与考试成绩。

---

1. Complex Numbers: Basic Components | 复数基本构成

Imaginary unit i: The number defined by i² = -1. It is not a real number, but forms the foundation of complex numbers.

虚数单位 i： 定义为满足 i² = -1 的数。它不是实数，但构成了复数的基础。

i² = -1

Complex number: A number of the form z = a + bi, where a, b ∈ ℝ. Here a is the real part, written Re(z), and b is the imaginary part, Im(z).

复数： 形如 z = a + bi 的数，其中 a 和 b 为实数。a 称为实部 Re(z)，b 称为虚部 Im(z)。

Equality of complex numbers: Two complex numbers a + bi and c + di are equal if and only if a = c and b = d. Real and imaginary parts must match independently.

复数相等： 两个复数 a + bi 与 c + di 相等当且仅当 a = c 且 b = d。实部与虚部必须分别相等。

💡 Memory tip: i is the imaginary square root of -1. Its powers cycle every four: i¹ = i, i² = -1, i³ = -i, i⁴ = 1, then repeat.

💡 记忆窍门：i 是 -1 的虚平方根，其幂次以 4 为周期循环：i¹ = i, i² = -1, i³ = -i, i⁴ = 1，之后重复。

---

2. Complex Conjugate and Modulus | 共轭复数与模

Complex conjugate: For z = a + bi, its conjugate is z* = a – bi (sometimes written as z̄). Reflecting across the real axis.

共轭复数： 若 z = a + bi，则其共轭为 z* = a – bi（也记作 z̄），在复平面上关于实轴对称。

Modulus (absolute value): The distance from the origin to the point (a, b), given by |z| = √(a² + b²). It is always non-negative.

模（绝对值）： 从原点到点 (a, b) 的距离，|z| = √(a² + b²)，总是非负的。

|z| = √(a² + b²)

Key property: z · z* = (a + bi)(a – bi) = a² + b² = |z|². This is extremely useful for dividing complex numbers and finding real denominators.

核心性质： z · z* = (a + bi)(a – bi) = a² + b² = |z|²。这在复数除法及有理化分母时极其实用。

💡 Think of conjugate as the mirror image, and modulus as the length of the vector.

💡 可以将共轭视为镜像，模视为向量的长度。

---

3. Argument and Polar Form | 辐角与极坐标形式

Argument of z: The angle θ measured from the positive real axis to the line segment representing z, usually in radians. The principal argument Arg(z) is typically taken in the range (-π, π].

辐角： 从正实轴到表示 z 的线段所成的角度 θ，通常以弧度为单位。主辐角 Arg(z) 一般取在 (-π, π] 范围内。

θ = arg(z), -π < Arg(z) ≤ π

Polar (modulus-argument) form: z = r(cos θ + i sin θ), where r = |z| and θ = arg(z). Also written as r cis θ.

极坐标形式（模-辐角形式）： z = r(cos θ + i sin θ)，其中 r = |z|，θ = arg(z)。也常记作 r cis θ。

Euler’s relation: e^{iθ} = cos θ + i sin θ, hence z = r e^{iθ}. This compact form is powerful for multiplication, division and powers.

欧拉公式： e^{iθ} = cos θ + i sin θ，因此 z = r e^{iθ}。这一简洁形式在复数乘除和乘幂运算中极为强大。

💡 Remember ‘CIS’ as Cos + i Sin. Multiplication in polar form: multiply moduli, add arguments.

💡 记住 ‘CIS’ 对应 Cos + i Sin。极坐标下的乘法：模相乘，辐角相加。

---

4. Matrices: Order, Types and Determinant | 矩阵：阶、类型与行列式

Order of a matrix: Described as m × n, where m is the number of rows and n is the number of columns. A square matrix has m = n.

矩阵的阶： 记作 m × n，其中 m 为行数，n 为列数。方阵满足 m = n。

Identity matrix I: A square matrix with 1’s on the leading diagonal and 0’s elsewhere. Acts like the number 1 in matrix multiplication: AI = IA = A.

单位矩阵 I： 主对角线元素全为 1，其余元素为 0 的方阵。它在矩阵乘法中扮演数字 1 的角色：AI = IA = A。

Transpose of A, Aᵀ: Obtained by interchanging rows and columns. If A is symmetric, Aᵀ = A.

矩阵的转置 Aᵀ： 将矩阵的行与列互换而得到。若 A 为对称矩阵，则 Aᵀ = A。

Determinant of a 2×2 matrix: For A = [[a, b], [c, d]], det(A) = ad – bc. Denoted det A or |A|.

2×2 矩阵的行列式： 对于 A = [[a, b], [c, d]]，det(A) = ad – bc，记作 det A 或 |A|。

det [[a, b], [c, d]] = ad – bc

💡 For determinant, think ‘cross-multiply and subtract’. A zero determinant means the matrix is singular.

💡 求行列式时，记住”交叉相乘再相减”。行列式为零意味着矩阵是奇异的。

---

5. Inverse Matrix, Singular Matrices and Transformations | 逆矩阵、奇异矩阵与几何变换

Inverse of a 2×2 matrix: If det(A) ≠ 0, then A⁻¹ = 1/(ad – bc) [[d, -b], [-c, a]]. It satisfies AA⁻¹ = I.

2×2 逆矩阵： 若 det(A) ≠ 0，则 A⁻¹ = 1/(ad – bc) [[d, -b], [-c, a]]，且满足 AA⁻¹ = I。

Singular matrix: A square matrix with determinant zero; it has no inverse. Non-singular matrices have inverses.

奇异矩阵：Published by TutorHao | Year 12 进阶数学 Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）