Year 12 CIE Further Mathematics: Vocabulary & Terminology Quick Memorisation Guide | Year 12 CIE 进阶数学：词汇术语速记指南

📚 Year 12 CIE Further Mathematics: Vocabulary & Terminology Quick Memorisation Guide | Year 12 CIE 进阶数学：词汇术语速记指南

Mastering the technical vocabulary of CIE Further Mathematics is crucial for interpreting exam questions accurately and justifying solutions. This guide presents key terms grouped by topic, with clear English definitions immediately followed by their Chinese equivalents, helping you build a solid bilingual understanding and speed up revision.

掌握 CIE 进阶数学的专业术语对于准确理解考题和论证解题至关重要。本指南按主题分组列出关键术语，每个英文定义后紧跟中文释义，帮助你建立扎实的双语理解，加快复习速度。

1. Complex Numbers | 复数

the terminology below is essential for working with Argand diagrams, polar forms and transformations.

复数扩展了实数系，使得像 x² + 1 = 0 这样的方程有解。熟悉下列术语对运用阿根图、极坐标形式和变换至关重要。

Imaginary Unit i: Defined by i² = -1. It is not a real number and is the foundation of all complex numbers.

虚数单位 i： 定义为 i² = -1。它不是实数，是一切复数的基础。

Complex Number: A number of the form z = a + bi, where a, b ∈ ℝ.

复数： 形如 z = a + bi 的数，其中 a 和 b 为实数。

Real Part Re(z): The value a in z = a + bi.

实部 Re(z)： 复数 z = a + bi 中的 a。

Imaginary Part Im(z): The coefficient b, not including i.

虚部 Im(z)： 系数 b，不包括单位 i。

Complex Conjugate: For z = a + bi, the conjugate is z* = a - bi. It reflects the number in the real axis.

共轭复数： 若 z = a + bi，其轭为 z* = a - bi，在实轴上对称。

Modulus |z|: The distance from the origin to z on the Argand diagram; |z| = √a² + b².

模 |z|： 在阿根图上原点到 z 的距离，|z| = √a² + b²。

Argument arg(z): The angle θ measured from the positive real axis to the line representing z, typically in radians.

辐角 arg(z)： 从正实轴到表示 z 的射线所成的角 θ，通常用弧度。

Principal Argument: The unique value of the argument in the interval (-π, π].

辐角主值： 辐角在区间 (-π, π] 内的唯一值。

Argand Diagram: A plane where the horizontal axis is Re(z) and the vertical axis is Im(z).

阿根图： 以 Re(z) 为横轴、Im(z) 为纵轴的平面。

Polar Form: z = r(cosθ + i sinθ), where r = |z| and θ = arg(z).

极坐标形式： z = r(cosθ + i sinθ)，其中 r = |z|，θ = arg(z)。

Euler's Formula: e^iθ = cosθ + i sinθ, linking exponential and trigonometric functions.

欧拉公式： e^iθ = cosθ + i sinθ，将指数函数与三角函数联系起来。

De Moivre's Theorem: (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ) for integer n.

棣莫弗定理： 对于整数 n，(cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ)。

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2. Matrices | 矩阵

Matrices provide a compact way to handle linear transformations and systems of equations. Knowing the precise terms helps with operations, determinants and inverses.

矩阵为处理线性变换和方程组提供了紧凑的方式。准确掌握术语有助于进行运算、求行列式和逆。

Matrix: A rectangular array of numbers, arranged in rows and columns.

矩阵： 由数字按行和列排列成的矩形阵列。

Element: An individual number in a matrix, denoted by aᵢⱼ for row i, column j.

元素： 矩阵中的单个数字，记作 aᵢⱼ，表示第 i 行第 j 列。

Order of a Matrix: The dimensions expressed as 'rows × columns', e.g. 2×3.

矩阵的阶： 用 '行 × 列' 表示的维数，如 2×3。

Square Matrix: A matrix with equal number of rows and columns (n × n).

方阵： 行数与列数相等的矩阵 (n × n)。

Identity Matrix I: A square matrix with 1s on the leading diagonal and 0s elsewhere; acts as the multiplicative identity.

单位矩阵 I： 主对角线为 1、其余为 0 的方阵；起着乘法单位元的作用。

Zero Matrix: A matrix in which every element is 0.

零矩阵： 所有元素均为 0 的矩阵。

Transpose Aᵀ: The matrix obtained by interchanging rows and columns of A.

转置 Aᵀ： 交换 A 的行和列所得的矩阵。

Symmetric Matrix: A square matrix such that Aᵀ = A.

对称矩阵： 满足 Aᵀ = A 的方阵。

Determinant det(A): A scalar value computed from a square matrix, revealing whether the matrix is invertible.

行列式 det(A)： 从方阵算出的标量，可判断矩阵是否可逆。

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，无逆矩阵。

Inverse A⁻¹: The unique matrix such that AA⁻¹ = A⁻¹A = I, provided det(A) ≠ 0.

逆矩阵 A⁻¹： 若 det(A) ≠ 0，满足 AA⁻¹ = A⁻¹A = I 的方阵。

Orthogonal Matrix: A square matrix with Aᵀ = A⁻¹; its rows and columns are orthonormal vectors.

正交矩阵： 满足 Aᵀ = A⁻¹ 的方阵，其行向量和列向量为标准正交向量组。

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3. Vectors in 3D | 三维向量

Vectors describe quantities with both magnitude and direction. In Year 12 Further Mathematics, you often work in three dimensions, using components and scalar products to solve geometric problems.

向量描述既有大小又有方向的量。在 Year 12 进阶数学中，经常在三维空间中用分量和标量积解决几何问题。

Vector: A quantity having both magnitude (length) and direction, often written as a column matrix or in i, j, k form.

向量： 既有大小（长度）又有方向的量，常写作列矩阵或 i, j, k 形式。

Magnitude of a: The length |a| = √x² + y² + z² for a = xi + yj + zk.

向量 a 的模： 长度 |a| = √x² + y² + z²，其中 a = xi + yj + zk。

Position Vector: The vector from the origin O to a point P, denoted p.

位置向量： 从原点 O 到点 P 的向量，记为 p。

Unit Vector: A vector of magnitude 1, often written as â = a/|a|.

单位向量： 模为 1 的向量，常记作 â = a/|a|。

Scalar (Dot) Product: For a = x₁i + y₁j + z₁k and b = x₂i + y₂j + z₂k, a·b = x₁x₂ + y₁y₂ + z₁z₂ = |a||b|cosθ.

标量积（点积）： 对于 a 和 b，a·b = x₁x₂ + y₁y₂ + z₁z₂ = |a||b|cosθ。

Angle Between Vectors: cosθ = (a·b) / (|a||b|).

两向量夹角： cosθ = (a·b) / (|a||b|)。

Direction Cosines: The cosines of the angles a vector makes with the positive x, y, z axes.

方向余弦： 向量与正 x、y、z 轴夹角的余弦值。

Vector Equation of a Line: r = a + λb, where a is a point on the line and b is the direction vector.

直线的向量方程： r = a + λb，其中 a 是直线上一点，b 是方向向量。

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4. Hyperbolic Functions | 双曲函数

Hyperbolic functions are analogues of trigonometric functions but based on the hyperbola rather than the circle. They appear in integration, differential equations and FP1 topics.

双曲函数类似于三角函数，但以双曲线而非圆为基础。它们出现在积分、微分方程和 FP1 各个专题中。

Hyperbolic Sine sinh x: sinh x = (e^x - e^-x) / 2.

双曲正弦 sinh x： sinh x = (e^x - e^-x) / 2。

Hyperbolic Cosine cosh x: cosh x = (e^x + e^-x) / 2.

双曲余弦 cosh x： cosh x = (e^x + e^-x) / 2。

Hyperbolic Tangent tanh x: tanh x = sinh x / cosh x = (e^x - e^-x) / (e^x + e^-x).

双曲正切 tanh x： tanh x = sinh x / cosh x = (e^x - e^-x) / (e^x + e^-x)。

Osborn's Rule: A rule for converting trigonometric identities into hyperbolic ones: replace cos by cosh, sin by i sinh, and change the sign of any product of two sines.

奥斯本法则： 将三角恒等式转换为双曲恒等式的法则：用 cosh 替换 cos，用 i sinh 替换 sin，并改变任何两个正弦乘积的符号。

Fundamental Identity: cosh²x - sinh²x = 1.

基本恒等式： cosh²x - sinh²x = 1。

Inverse Hyperbolic Functions: arsinh x = ln(x + √x² + 1), arcosh x = ln(x + √x² - 1) for x ≥ 1, artanh x = ½ ln((1+x)/(1-x)) for |x| < 1.

反双曲函数： arsinh x = ln(x + √x² + 1)，arcosh x = ln(x + √x² - 1) 且 x ≥ 1，artanh x = ½ ln((1+x)/(1-x)) 且 |x| < 1。

Derivatives: d/dx (sinh x) = cosh x; d/dx (cosh x) = sinh x; d/dx (tanh x) = sech²x.

导数： d/dx (sinh x) = cosh x，d/dx (cosh x) = sinh x，d/dx (tanh x) = sech²x。

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5. Differential Equations | 微分方程

First-order differential equations are a core part of FP1. Knowing the classification and solution methods is vital for modelling and problem-solving.

一阶微分方程是 FP1 的核心内容。了解其分类和解法对建模和解题至关重要。

Differential Equation: An equation that relates a function with its derivatives, e.g. dy/dx + P(x)y = Q(x).

微分方程： 联系函数及其导数的方程，如 dy/dx + P(x)y = Q(x)。

Order: The highest derivative present in the equation. First-order involves only dy/dx.

阶： 方程中出现的最高阶导数。一阶只含 dy/dx。

Separable Equation: A first-order equation that can be written as f(y) dy = g(x) dx.

可分离方程： 可写成 f(y) dy = g(x) dx 的一阶微分方程。

Integrating Factor: For a linear ODE dy/dx + P(x)y = Q(x), the integrating factor is e^(∫ P(x) dx).

积分因子： 对于线性常微分方程 dy/dx + P(x)y = Q(x)，积分因子为 e^(∫ P(x) dx)。

General Solution: A family of solutions containing an arbitrary constant.

通解： 包含任意常数的解族。

Particular Solution: A solution obtained from the general solution by applying an initial condition.

特解： 由通解代入初始条件后得到的解。

Initial Condition: A given value of y at a specific x, used to find the constant of integration.

初始条件： 给定某特定 x 处的 y 值，用于确定积分常数。

Boundary Conditions: Conditions specified at different values of the independent variable, often used in second-order ODEs.

边界条件： 在自变量的不同值处指定的条件，常用于二阶常微分方程。

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6. Series and Expansions | 级数与展开

Maclaurin series, binomial expansions and standard summation formulas allow functions to be approximated and series to be summed quickly.

麦克劳林级数、二项式展开和标准求和公式可用于函数逼近与快速求和。

Maclaurin Series: f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + ...

麦克劳林级数： f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + …

Series for e^x: e^x = 1 + x + x²/2! + x³/3! + ... valid for all x.

e^x 的级数： e^x = 1 + x + x²/2! + x³/3! + …，对所有 x 成立。

Series for sin x: sin x = x - x³/3! + x⁵/5! - ... valid for all x.

sin x 的级数： sin x = x - x³/3! + x⁵/5! - …，对所有 x 成立。

Series for cos x: cos x = 1 - x²/2! + x⁴/4! - ... valid for all x.

cos x 的级数： cos x = 1 - x²/2! + x⁴/4! - …，对所有 x 成立。

Binomial Expansion (1+x)ⁿ: (1+x)ⁿ = 1 + nx + n(n-1)x²/2! + ... , valid for |x| < 1 when n is not a positive integer.

二项式展开 (1+x)ⁿ： (1+x)ⁿ = 1 + nx + n(n-1)x²/2! + …，当 n 非正整数时，|x| < 1 收敛。

Sum of First n Integers: Σr = n(n+1)/2.

前 n 个整数之和： Σr = n(n+1)/2。

Sum of Squares: Σr² = n(n+1)(2n+1)/6.

平方和： Σr² = n(n+1)(2n+1)/6。

Sum of Cubes: Σr³ = {n(n+1)/2}².

立方和： Σr³ = {n(n+1)/2}²。

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7. Polar Coordinates | 极坐标

Polar coordinates (r, θ) describe a point by its distance from the pole and its angle from the polar axis. This system simplifies curves like circles and cardioids.

极坐标 (r, θ) 通过点到极点的距离及与极轴的夹角来描述点，可简化圆和心脏线等曲线的表达。

Pole: The origin O in the polar coordinate system.

极点： 极坐标系中的原点 O。

Polar Axis: The ray from the pole in the direction of the positive x-axis, corresponding to θ = 0.

极轴： 从极点出发、沿正 x 轴方向的射线，对应 θ = 0。

Polar Equation r = f(θ): A curve defined by the distance r as a function of angle θ.

极坐标方程 r = f(θ)： 由距离 r 随角度 θ 变化而定义的曲线。

Cardioid: A curve of the form r = a(1 + cosθ) or r = a(1 + sinθ).

心脏线： 形如 r = a(1 + cosθ) 或 r = a(1 + sinθ) 的曲线。

Limacon: Curve r = a + b cosθ or r = a + b sinθ; shape depends on the ratio a/b.

蚶线： r = a + b cosθ 或 r = a + b sinθ，其形状取决于 a/b 的比值。

Area in Polar Coordinates: A = ½ ∫_θ₁^θ₂ r² dθ.

极坐标下的面积： A = ½ ∫_θ₁^θ₂ r² dθ。

Tangent at the Pole: The line θ = α where r = 0 and the curve touches the pole.

极点处的切线： 满足 r = 0 的射线 θ = α，曲线在此与极点相切。

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8. Roots of Polynomials | 多项式根

Relations between roots and coefficients allow you to form equations without actually solving them. This is a powerful tool in CIE Further Maths.

根与系数的关系能让你不求根而直接构造方程，是 CIE 进阶数学中的有力工具。

Root of an Equation: A value that satisfies the polynomial equation P(x) = 0.

方程的根： 满足多项式方程 P(x) = 0 的值。

Sum of Roots (Quadratic): For ax² + bx + c = 0 with roots α, β: α + β = -b/a.

二次根之和： 若 ax² + bx + c = 0 有两根 α, β，则 α + β = -b/a。

Product of Roots (Quadratic): αβ = c/a.

二次根之积： αβ = c/a。

Cubic Roots α, β, γ: For ax³ + bx² + cx + d = 0: Σα = -b/a, Σαβ = c/a, αβγ = -d/a.

三次根 α, β, γ： 对于 ax³ + bx² + cx + d = 0，Σα = -b/a, Σαβ = c/a, αβγ = -d/a。

Quartic Roots (Extension): Similar relationships exist for higher-degree polynomials, with alternating signs.

四次根（拓展）： 高次多项式也有类似关系，符号交替变化。

Complex Conjugate Roots: If a polynomial with real coefficients has a complex root p + qi, then p - qi is also a root.

共轭复根： 若实系数多项式有一个复根 p + qi，则 p - qi 也是根。

Forming an Equation from Roots: Given Σα, Σαβ, etc., you can write the polynomial in factored or expanded form.

由根构造方程： 已知 Σα、Σαβ 等，可写出因式形式或展开式。

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9. Further Integration Techniques | 进阶积分方法

Year 12 Further Mathematics extends integration skills with techniques like reduction formulae and volumes of revolution, essential for complex areas and solids.

Year 12 进阶数学用递推

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