Year 13 OCR Further Maths: Vocabulary & Terminology Quick Reference | Year 13 OCR 进阶数学：词汇术语速记指南

📚 Year 13 OCR Further Maths: Vocabulary & Terminology Quick Reference | Year 13 OCR 进阶数学：词汇术语速记指南

This guide presents the essential vocabulary and terminology of the OCR Year 13 Further Mathematics specification (H245). Every term is given with a precise definition, the symbolic form you must be able to write, and a short memory aid so that you can revise rapidly before a mock or a final exam. The topics span Core Pure (complex numbers, matrices, vectors, further calculus, polar coordinates, differential equations), Further Statistics, Further Mechanics and Discrete Mathematics.

本指南梳理 OCR Year 13 进阶数学（H245）大纲中的核心词汇与术语。每个术语都给出精确定义、必须会写的符号形式，以及一句速记提示，方便你在模考或正式考试前快速复习。内容涵盖纯数核心（复数、矩阵、向量、进阶微积分、极坐标、微分方程）、进阶统计、进阶力学以及离散数学。

---

1. Complex Numbers: Core Vocabulary | 复数核心词汇

Imaginary unit i – i² = −1. It allows the square root of any negative number to be written down, for example √−9 = 3i.

虚数单位 i – 由 i² = −1 定义，使任何负数的平方根都可以写出，例如 √−9 = 3i。

Complex number – Im(z). Two complex numbers are equal only when both real parts and both imaginary parts are equal.

复数 – 形如 z = x + iy 的数，其中 x、y 为实数，记 x = Re(z)，y = Im(z)。两个复数相等，当且仅当实部与虚部分别相等。

Purely imaginary / purely real – ary when Re(z) = 0, and purely real when Im(z) = 0. A common exam trap is to assume that a "large" modulus means a "large" number; on the Argand diagram, only distance from the origin matters.

纯虚数 / 纯实数 – 当 Re(z) = 0 时 z 为纯虚数；当 Im(z) = 0 时 z 为纯实数。常见失分点是误以为模大就代表数大——在阿尔冈图上，只有到原点的距离才有意义。

Complex conjugate – y. Key results: z + z* = 2Re(z) is real, z − z* = 2i Im(z) is purely imaginary, and z z* = |z| negative. Conjugation distributes over addition, subtraction, multiplication and division.

共轭复数 – z = x + iy 的共轭为 z* = x − iy。关键结论：z + z* = 2Re(z) 为实数；z − z* = 2i Im(z) 为纯虚数；z z* = |z|² 为非负实数。共轭对加、减、乘、除均满足分配律。

Complex conjugate root theorem – if a polynomial has real coefficients and a + ib is a root, then a − ib is also a root. This is how you reconstruct a quadratic factor from one complex root.

共轭复根定理 – 若实系数多项式有根 a + ib，则 a − ib 也必为根。这是由一个复根反推二次因式的关键方法。

z z* = (x + iy)(x − iy) = x² + y² = |z|²

---

2. Modulus, Argument and Polar Form | 模、辐角与极坐标形式

Modulus |z| = √x² + y² om the origin to the point representing z on the Argand diagram. It behaves like a length: |z₁ z₂| = |z₁||z₂| and |z₁ / z₂| = |z₁| / |z₂|.

模 |z| = √x² + y²，即阿尔冈图上 z 所对应的点到原点的距离。它具有长度性质：|z₁ z₂| = |z₁||z₂|，|z₁ / z₂| = |z₁| / |z₂|。

Argument ment lies in the interval (−π, π], so you must always check the quadrant before quoting θ.

辐角 arg(z) = θ 是从正实轴逆时针量得的角度，满足 tan θ = y / x。主辐角位于区间 (−π, π] 内，因此引用 θ 前必须先判断象限。

Polar form – z = r(cos θ + i sin θ), where r = |z| horthand z = r cis θ is acceptable in working. Arguments combine beautifully in this form.

极坐标形式 – z = r(cos θ + i sin θ)，其中 r = |z|，θ = arg(z)。由于 cos θ + i sin θ 常写作 cis θ，故 z = r cis θ 的简写可用于解题过程。此形式下辐角的运算规则非常简洁。

Exponential form – z = r e^iθ by Euler's relation e^iθ plies the moduli and adds the arguments; division divides the moduli and subtracts the arguments.

指数形式 – 由欧拉关系 e^iθ = cos θ + i sin θ 得 z = r e^iθ。乘法：模相乘、辐角相加；除法：模相除、辐角相减。

z₁ z₂ = r₁ r₂ cis(θ₁ + θ₂)  |  z₁ / z₂ = (r₁ / r₂) cis(θ₁ − θ₂)

Loci – |z − a| = r describes a circle of radius r centred on the point a; |z − a| = |z − b| cribes a half-line from a at angle α. Sketching the locus before doing algebra saves time.

轨迹 – |z − a| = r 表示以点 a 为圆心、r 为半径的圆；|z − a| = |z − b| 表示连接 a、b 的线段的垂直平分线；arg(z − a) = α 表示从 a 出发、倾角为 α 的射线。先画图再做代数能省大量时间。

---

3. De Moivre's Theorem and Roots of Unity | 棣莫弗定理与单位根

De Moivre's theorem – for all integers n, (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ). Written compactly, (cis θ)ⁿ = cis(nθ). It converts powers into simple multiplication of angles.

棣莫弗定理 – 对任意整数 n，(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)，简写为 (cis θ)ⁿ = cis(nθ)。它把幂运算转化为角度的简单倍数运算。

Roots of a complex number – the n distinct n-th roots of r cis θ are r^1/n cis((θ + 2kπ) / n) for k = 0, 1, …, n − 1. They form a regular n-gon on a circle of radius r^1/n.

复数的根 – r cis θ 的 n 个互不相同的 n 次根为 r^1/n cis((θ + 2kπ) / n)，k = 0, 1, …, n − 1。它们在半径为 r^1/n 的圆上构成正 n 边形。

Roots of unity – the solutions of zⁿ sum is always zero for n > 1, and the roots are equally spaced around the unit circle.

单位根 – 方程 zⁿ = 1 的解，即 z = cis(2kπ / n)。当 n > 1 时其和恒为 0，各根在单位圆上等距分布。

Trigonometric identities via De Moivre – expand (cos θ + i sin θ)ⁿ and compare real and imaginary parts to obtain expressions for cos nθ and sin nθ in terms of powers of cos θ and sin θ. Pairing the expansion with the binomial theorem is standard in the exam.

用棣莫弗定理推三角恒等式 – 展开 (cos θ + i sin θ)ⁿ，比较实部与虚部，即可用 cos θ、sin θ 的幂表示 cos nθ 与 sin nθ。与二项式定理配合展开是标准考法。

Applications – De Moivre's theorem is also used to evaluate integrals of powers of sine and cosine, to sum series such as Σ cos(kθ), and to solve polynomial equations that have no real roots.

应用 – 棣莫弗定理还可用于计算正弦、余弦幂的积分，求 Σ cos(kθ) 之类的级数和，以及求解无实根的整式方程。

---

4. Matrices and Linear Transformations | 矩阵与线性变换

Order of a matrix – The order tells you immediately whether two matrices can be multiplied: the number of columns of the first must equal the number of rows of the second.

矩阵的阶 – m × n 矩阵有 m 行 n 列。阶数直接决定两矩阵能否相乘：前者的列数必须等于后者的行数。

Identity matrix I – sewhere. It satisfies AI = IA = A, so it plays the role of the number 1 in matrix algebra.

单位矩阵 I – 主对角线元素为 1、其余为 0 的方阵，满足 AI = IA = A，在矩阵代数中相当于数字 1。

Matrix multiplication – in general AB ≠ BA. Matrix multiplication is associative and distributive but not commutative, which is the single most examinable difference from ordinary arithmetic.

矩阵乘法 – 一般来说 AB ≠ BA。矩阵乘法满足结合律与分配律，但不满足交换律，这是它与普通算术最常考的差别。

Singular and non-singular matrices – ar when det A = 0 and it has no inverse; it is non-singular when det A ≠ 0 and A⁻¹ exists.

奇异矩阵与非奇异矩阵 – 方阵满足 det A = 0 时为奇异矩阵，没有逆矩阵；满足 det A ≠ 0 时为非奇异矩阵，A⁻¹ 存在。

广告 / Sponsored

Linear transformation – apping of the plane (or space) given by a matrix, for example the column-vector equation r' = A r. Geometrically it maps straight lines to straight lines and keeps the origin fixed.

线性变换 – 由矩阵给出的平面（或空间）映射，例如列向量方程 r' = A r。几何上它把直线映为直线，并保持原点不动。

Transformation 变换	Matrix 矩阵	Key idea 要点
Rotation about O through θ	[[cos θ, −sin θ], [sin θ, cos θ]]	det = 1, area preserved 面积不变
Reflection in the x-axis	[[1, 0], [0, −1]]	det = −1, orientation reversed 方向反转
Enlargement, scale factor k	[[k, 0], [0, k]]	determinant = k² 行列式为 k²
Shear parallel to the x-axis	[[1, k], [0, 1]]	det = 1, area preserved 面积不变

Area scale factor – ion matrix gives the factor by which areas are multiplied; a negative determinant indicates that the orientation has been reversed.

面积伸缩因子 – 2 × 2 变换矩阵的行列式就是面积被放大的倍数；行列式为负说明图形的方向发生了反转。

---

5. Determinants, Inverses and Eigenvalues | 行列式、逆矩阵与特征值

Determinant – ad − bc. For a 3 × 3 matrix, expand along a row or column using cofactors and signs + − +.

行列式 – 对 2 × 2 矩阵 A = [[a, b], [c, d]]，det A = ad − bc。对 3 × 3 矩阵，按某行或某列用余子式展开，符号规律为 + − +。

Inverse of a 2 × 2 matrix – vided det A ≠ 0. Swapping the leading diagonal and negating the other two entries is the fastest route in an exam.

2 × 2 矩阵的逆 – 当 det A ≠ 0 时，A⁻¹ = (1 / det A) [[d, −b], [−c, a]]。考试中最快的做法就是主对角线互换、副对角线变号。

Singular systems – has no solution or infinitely many. Geometrically, the transformation collapses the plane onto a line or a point.

奇异方程组 – 当 det A = 0 时，方程 A x = b 无解或有无穷多解。几何上，该变换把整个平面压缩成一条直线或一个点。

Eigenvalue and eigenvector – nvectors are the directions that are simply stretched, not rotated, by the transformation.

特征值与特征向量 – 若存在非零向量 v 使 A v = λ v，则 λ 为 A 的特征值，v 为对应的特征向量。特征向量就是被变换"只拉伸、不旋转"的方向。

Characteristic equation – eigenvalues; substituting each λ back into (A − λI)v = 0 gives the matching eigenvectors.

特征方程 – det(A − λI) = 0。解此多项式方程可得特征值；再把每个 λ 代回 (A − λI)v = 0 即可求出相应的特征向量。

p;→  λ₁, λ₂  →  (A − λI)v = 0  →  v₁, v₂

Diagonalisation – l matrix of eigenvalues and P has the eigenvectors as columns. This makes computing large powers such as Aⁿ straightforward.

对角化 – 若 A 有两个线性无关的特征向量，则 A = P D P⁻¹，其中 D 是由特征值构成的对角矩阵，P 的列向量为特征向量。这样计算 Aⁿ 之类的高次幂就变得非常简单。

---

6. Vectors and 3D Geometry | 向量与三维几何

Vector, scalar and unit vector – de only. A unit vector has modulus 1, and the unit vector in the direction of a is written â = a / |a|.

向量、标量与单位向量 – 向量既有大小又有方向，标量只有大小。单位向量的模为 1，与 a 同向的单位向量记作 â = a / |a|。

Scalar (dot) product – a · b = |a||b| = a₁b₁ + a₂b₂ + a₃b₃. It is a scalar, and a · b = 0 is the test for perpendicular vectors.

标量（点）积 – a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃。结果是一个标量，a · b = 0 是两向量垂直的判据。

Vector (cross) product – a × b is perpendicular to both a and b, with |a × b| = |a||b| sin θ. It is essential for finding normals to planes and areas of triangles.

向量（叉）积 – a × b 同时垂直于 a 与 b，且 |a × b| = |a||b| sin θ。它是求平面法向量和三角形面积的关键工具。

Line in vector form – nt on the line and b is a direction vector. The scalar λ is a parameter, not a coordinate.

直线的向量形式 – r = a + λb，其中 a 为直线上一点，b 为方向向量。标量 λ 是参数，不是坐标。

Plane in vector form – e plane. The cartesian version ax + by + cz = d uses the same normal vector n = (a, b, c).

平面的向量形式 – r · n = d，其中 n 为平面的法向量。笛卡尔形式 ax + by + cz = d 使用的是同一个法向量 n = (a, b, c)。

Skew lines – two lines in 3D that are neither parallel nor intersecting. Equating the two vector equations and finding an inconsistent system is the standard proof that they are skew.

异面直线 – 三维空间中既不平行也不相交的两条直线。令两向量方程相等、得到无解方程组，是证明异面直线的标准方法。

---

7. Further Calculus and Hyperbolic Functions | 进阶微积分与双曲函数

Hyperbolic functions – cosh x = (e^x + e^−x) / 2 and sinh x = (e^x − e^−x) / 2. They are defined through exponential functions, which is why their calculus is so clean.

双曲函数 – cosh x = (e^x + e^−x) / 2，sinh x = (e^x − e^−x) / 2。它们由指数函数定义，这也是其微积分形式如此简洁的原因。

Hyperbolic identities – 2x = cosh²x + sinh²x. Note the sign difference from the circular trigonometric identities.

双曲恒等式 – cosh²x − sinh²x = 1，sinh 2x = 2 sinh x cosh x，cosh 2x = cosh²x + sinh²x。注意与圆三角恒等式相比的符号差异。

Derivatives – x)/dx = sinh x. Unlike the circular functions, there is no sign change on the second one.

导数 – d(sinh x)/dx = cosh x，d(cosh x)/dx = sinh x。与圆函数不同，第二个导数没有变号。

Inverse hyperbolic functions – arsinh x = ln(x + √x² + 1) and arcosh x = ln(x + √x² − 1 se logarithmic forms come directly from solving the exponential definition as a quadratic.

反双曲函数 – 当 x ≥ 1 时，arsinh x = ln(x + √x² + 1)，arcosh x = ln(x + √x² − 1)。这些对数形式可直接由指数定义解二次方程得到。

Inverse trigonometric derivatives – d(arcsin x)/dx = 1 / √1 − x² ). Recognising these in reverse is the standard route to integrating 1 / (a² + x²) and 1 / √a² − x².

反三角函数的导数 – d(arcsin x)/dx = 1 / √1 − x²，d(arctan x)/dx = 1 / (1 + x²)。反向识别它们是把 1 / (a² + x²)、1 / √a² − x² 积分出来的标准思路。

Improper integrals – integrals with an infinite limit or an integrand that becomes infinite at an endpoint. Evaluate as a limit; if the limit exists the integral converges, otherwise it diverges.

反常积分 – 积分上限无限或 integrand 在端点趋于无穷的积分。用极限来求值：极限存在则积分收敛，否则发散。

<

更多咨询请联系16621398022（同微信）