Year 12 OCR Further Maths: Quick Terminology Guide | Year 12 OCR 进阶数学：词汇术语速记指南

📚 Year 12 OCR Further Maths: Quick Terminology Guide | Year 12 OCR 进阶数学：词汇术语速记指南

Mastering the precise language of OCR Further Mathematics unlocks exam confidence and deepens your conceptual grasp. This guide pairs every technical term with a concise definition in English and Chinese, followed by a memory hook to make the vocabulary stick — from imaginary numbers to recurrence relations.

掌握 OCR 进阶数学的精确语言，能让你在考试中更加自信，并加深对概念的理解。本指南将每个专业术语与英中双语定义配对，并附加记忆窍门，帮助你把词汇牢牢记住——从虚数到递推关系，一应俱全。

1. Complex Numbers | 复数

Complex number — a number of the form z = a + bi, with real part a, imaginary part b, and i² = –1. Think of it as a two-dimensional number living on the Argand diagram.

复数——形如 z = a + bi 的数，实部为 a，虚部为 b，且 i² = –1。可以将其想像为生活在阿干特图上的二维数。

Conjugate — denoted z̅ or z*, where the sign of the imaginary part is flipped: a – bi. Multiplying a complex number by its conjugate gives a real number a² + b².

共轭——记作 z̅ 或 z*，将虚部符号反转：a – bi。复数与其共轭相乘得到实数 a² + b²。

Modulus — the distance from the origin to the point representing z on the Argand plane: |z| = √a² + b². Picture it as the “length” of the complex vector.

模——原点到表示 z 的点在阿干特平面上的距离：|z| = √a² + b²。把它视为复数向量的“长度”。

Argument — the angle θ measured from the positive real axis to the line joining the origin to z, usually in radians. Keep an eye on the quadrant for the correct sign.

辐角——从正实轴到连接原点与 z 的线段的夹角 θ，通常以弧度为单位。需注意象限以确定正确符号。

Euler’s formula — e^iθ θ + i sin θ. This is the bridge that turns an exponential into a point on the unit circle.

欧拉公式——e^iθ = cos θ + i sin θ。这座桥将指数函数转化为单位圆上的一个点。

Root of unity — he n distinct roots are equally spaced around the unit circle, each separated by an angle of 2π/n.

单位根——满足 ωⁿ = 1 的复数 ω。n 个不同的根等间距地分布在单位圆上，相邻根之间的夹角为 2π/n。

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2. Matrices | 矩阵

Order of a matrix — atrix has two rows and three columns. Order determines whether multiplication is possible.

矩阵的阶——记作 行数 × 列数。一个 2×3 矩阵有两行三列。阶决定了乘法是否可行。

Identity matrix — diagonal and 0s elsewhere, acting like the number 1 in matrix multiplication: AI = IA = A.

单位矩阵——主对角线全为 1、其余元素为 0 的方阵 I，在矩阵乘法中起着数字 1 的作用：AI = IA = A。

Determinant — (det ≠ 0) and relates to area/volume scale factor. For a 2×2 matrix, det([a b; c d]) = ad – bc.

行列式——由方阵算出的纯量，告知矩阵是否可逆（det ≠ 0），并与面积/体积缩放因子有关。对 2×2 矩阵，det([a b; c d]) = ad – bc。

Inverse matrix — ¹, satisfies AA⁻¹ = A⁻¹A = I. Only square matrices with nonzero determinant have inverses.

逆矩阵——记作 A⁻¹，满足 AA⁻¹ = A⁻¹A = I。只有行列式非零的方阵才有逆矩阵。

Singular matrix — a square matrix whose determinant is zero; it has no inverse and its columns are linearly dependent.

奇异矩阵——行列式为零的方阵；它没有逆阵，且其列向量线性相关。

Transformation matrix — a matrix that represents a linear transformation in the plane or space. Common examples include reflections, rotations, stretches and shears.

变换矩阵——表示平面或空间中线性变换的矩阵。常见例子包括反射、旋转、拉伸和剪切。

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3. Vectors | 向量

Position vector — the vector from the origin to a point P. If P has coordinates (x,y,z), its position vector is xi + yj + zk.

位置向量——从原点到点 P 的向量。若 P 坐标为 (x,y,z)，则其位置向量为 xi + yj + zk。

Scalar product (dot product) — for vectors a and b, a·b = |a||b| cos θ. The result is a scalar; it tests perpendicularity (a·b = 0) and finds angles.

数量积（点积）——对向量 a 和 b，a·b = |a||b| cos θ。结果为纯量；可用来检验垂直性（a·b = 0）和求夹角。

Vector product (cross product) — for three-dimensional vectors, a×b gives a vector perpendicular to both a and b, magnitude |a||b| sin θ. Direction obeys the right-hand rule.

向量积（叉积）——对三维向量，a×b 得到一个垂直于 a 和 b 的向量，大小为 |a||b| sin θ。方向遵循右手定则。

Unit vector — a vector with magnitude 1. Often used to indicate direction, such as the standard basis vectors i, j, k.

单位向量——长度为 1 的向量。常用于表示方向，如标准基向量 i、j、k。

Normal vector — perpendicular to a plane. Essential in writing the Cartesian equation of a plane: r·n = d.

法向量——垂直于平面的向量。在书写平面笛卡尔方程 r·n = d 时必不可少。

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4. Polar Coordinates | 极坐标

Polar coordinates — θ is the angle from the initial line. Converts to Cartesian via x = r cos θ, y = r sin θ.

极坐标——(r, θ)，其中 r 为到极点（原点）的距离，θ 为从极轴出发的角度。通过 x = r cos θ，y = r sin θ 转换为直角坐标。

Cardioid — m r = a(1 + cos θ). Its symmetry about the initial line is a time-saver when finding area.

心脏线——形如 r = a(1 + cos θ) 的心形极坐标曲线。它关于极轴对称，这在求面积时能省下不少时间。

Area in polar coordinates — = f(θ) between θ = α and θ = β is ½ ∫α→β r² dθ. The ½ r² dθ comes from the sector formula.

极坐标下的面积——极坐标曲线 r = f(θ) 在 θ = α 到 θ = β 之间围成的面积为 ½ ∫α→β r² dθ。½ r² dθ 来源于扇形面积公式。

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5. Hyperbolic Functions | 双曲函数

Hyperbolic sine & cosine — sinh x = (eˣ – e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2. They behave like ordinary sine and cosine but lack the periodic wave — the graphs are instead exponential in shape.

双曲正弦与双曲余弦——sinh x = (eˣ – e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2。它们的性质类似普通正弦余弦，但没有周期性波动——图像呈指数形态。

Osborn’s rule — he sign whenever a product of two sines appears. This links sin² x + cos² x = 1 to cosh² x – sinh² x = 1.

奥斯本法则——把三角恒等式转化为双曲恒等式时，将正弦换成 sinh，但每当出现两个正弦之积时改变符号。这就把 sin² x + cos² x = 1 联系到 cosh² x – sinh² x = 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. Each is expressible purely in natural logs.

反双曲函数——arsinh x = ln(x + √x²+1)，arcosh x = ln(x + √x²–1) 对 x ≥ 1，artanh x = ½ ln((1+x)/(1–x)) 对 |x| < 1。每一个都可以纯粹用自然对数表达。

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6. Sequences and Series | 数列与级数

Arithmetic sequence — a sequence with a constant difference d between consecutive terms: u_n = a + (n–1)d. Sum to n terms: S_n = n/2 [2a + (n–1)d].

等差数列——相邻两项之差为常数 d 的数列：u_n = a + (n–1)d。前 n 项和：S_n = n/2 [2a + (n–1)d]。

Geometric sequence — consecutive terms have a constant ratio r: u_n = ar^n–1. Sum to n terms: S_n = a(1–rⁿ)/(1–r) for r ≠ 1. Converges to sum to infinity a/(1–r) if |r| < 1.

等比数列——相邻两项之比为常数 r：u_n = ar^n–1。前 n 项和：S_n = a(1–rⁿ)/(1–r)，r ≠ 1。若 |r| < 1，无穷项和为 a/(1–r)。

Recurrence relation — defines a term based on previous terms, e.g. u_n+1 = 2u_n – 3. A closed-form solution turns this into a direct formula for the n-th term.

递推关系——根据前几项来定义某一项，例如 u_n+1 = 2u_n – 3。闭式解将其转化为第 n 项的直接公式。

Method of differences — a technique for summing a series by writing each term as a difference f(r) – f(r+1) so that most parts cancel, leaving only the first and last pieces.

差分法——一种求级数和的方法：把每一项写成差分 f(r) – f(r+1)，使得大部分项相消，只留下首尾两段。

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7. Further Mechanics – Momentum & Impulse | 进阶力学——动量与冲量

Momentum — ty (p = mv). In a closed system, total momentum is conserved before and after a collision.

动量——质量乘以速度的向量量（p = mv）。在封闭系统中，碰撞前后总动量守恒。

Impulse — he change in momentum, also equal to force × time when the force is constant. Impulse = mv – mu. The vector direction matters.

冲量——动量的变化，也等于力 × 时间（当力恒定时）。冲量 = mv – mu。方向至关重要。

Restitution coefficient e — f approach). e = 1 for a perfectly elastic collision; e = 0 for a perfectly inelastic one.

恢复系数 e——对于两个物体的直接碰撞，e =（分离速度）/（接近速度）。完全弹性碰撞时 e = 1；完全非弹性碰撞时 e = 0。

Conservation of momentum in two dimensions — resolve velocities into perpendicular directions and equate total momentum before and after impact separately for each direction.

二维动量守恒——将速度沿相互垂直的方向分解，再对每个方向分别列出碰撞前后的总动量等式。

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8. Further Mechanics – Work, Energy & Power | 进阶力学——功、能与功率

Work done by a constant force — on of the force = Fd cos θ. The component of force perpendicular to motion does zero work.

恒力做功——功 = 力 × 沿力方向移动的距离 = Fd cos θ。垂直于运动方向的力分量不做功。

Kinetic energy — KE = ½ mv². The work– principle states that the net work done on a particle equals its change in kinetic energy.

动能——KE = ½ mv²。功–能原理指出，合外力对质点所做的功等于其动能的变化量。

Gravitational potential energy — ’s surface. Only changes in GPE matter, so you can set the zero level wherever convenient.

重力势能——近地表 GPE = mgh。只有重力势能的变化才是重要的，因此可随意设定零势能面。

Power — Fv for constant force and velocity in the same direction. Average power = total work/time.

功率——做功的速率：力与速度同向且恒定时，P = Fv。平均功率 = 总功 / 时间。

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9. Further Statistics – Discrete Random Variables | 进阶统计——离散随机变量

Random variable — a variable whose value depends on the outcome of a random experiment. Denoted by capital letters like X; its specific values are lower case x.

随机变量——其值取决于随机试验结果的变量。常用大写字母 X 表示，其具体取值则为小写 x。

Probability mass function — ty for each possible value of a discrete random variable. The sum over all x must equal 1.

概率质量函数——P(X = x) 定义了离散随机变量每个可能取值的概率。所有 x 的概率之和必须等于 1。

Expected value E(X) — Σ x·P(X = x). It represents the long-run average if the experiment is repeated many times.

期望值 E(X)——随机变量的均值，由 Σ x·P(X = x) 给出。它表示若大量重复实验所得的长期平均值。

Variance Var(X) — measures spread; Var(X) = E(X²) – [E(X)]². It is the expected value of the squared deviation from the mean.

方差 Var(X)——度量离散程度；Var(X) = E(X²) – [E(X)]²。它是偏离均值的平方的期望值。

Discrete uniform distribution — each of the n outcomes is equally likely: P(X = x_i) = 1/n. A fair dice roll is a classic example.

离散均匀分布——n 种结果中的每一种概率相等：P(X = x_i) = 1/n。公平骰子掷出是其典型例子。

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10. Further Statistics – Special Distributions | 进阶统计——特殊分布

Binomial distribution B(n, p) — of successes in n independent trials each with success probability p. P(X = k) = ⁿCₖ pᵏ (1–p)ⁿ⁻ᵏ.

二项分布 B(n, p)——建模 n 次独立试验中成功次数，每次成功概率为 p。P(X = k) = ⁿCₖ pᵏ (1–p)ⁿ⁻ᵏ。

Poisson distribution Po(λ) — space when events happen independently at a constant average rate λ. P(X = k) = e⁻λ λᵏ / k!.

泊松分布 Po(λ)——建模在固定时间或空间区间内事件发生的次数，事件独立且以恒定平均速率 λ 发生。P(X = k) = e⁻λ λᵏ / k!。

Geometric distribution Geo(p) — models the number of trials up to and including the first success. P(X = k) = (1–p)ᵏ⁻¹ p.

几何分布 Geo(p)——建模直到首次成功（含）为止的试验次数。P(X = k) = (1–p)ᵏ⁻¹ p。

Poisson approximation to binomial — when n is large and p is small, B(n, p) ≈ Po(np). Use this if n > 50 and np < 5.

泊松近似二项分布——当 n 大、p 小时，B(n, p) ≈ Po(np)。适用条件常为 n > 50 且 np < 5。

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11. Decision Mathematics – Graphs & Networks | 决策数学——图与网络

Graph — a collection of vertices (nodes) connected by edges (arcs). A simple graph has no loops or multiple edges.

图——由顶点（节点）和连接它们的边（弧）组成的集合。简单图不含环和多重边。

Walk, trail, path — a walk is a sequence of edges. A trail has no repeated edges. A path has no repeated vertices.

步道、迹、路径——步道是边的序列。迹无边重复。路径无顶点重复。

Cycle — a closed path where the start and end vertices are the same, with no other vertex repeated.

圈——一条起点和终点为同一顶点、且无其他顶点重复的闭合路径。

Tree — a connected graph with no cycles. A spanning tree of a graph connects all vertices using a subset of edges without forming any cycles.

树——一个连通而无圈的图。图的生成树使用边子集连接所有顶点而不形成任何圈。

Minimum spanning tree — a spanning tree with the smallest total weight. Kruskal’s or Prim’s algorithm finds it efficiently.

最小生成树——總边权重最小的生成树。克鲁斯卡尔或普里姆算法可高效求之。

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12. Decision Mathematics – Algorithms & Linear Programming | 决策数学——算法与线性规划

Bubble sort, quick sort — basic sorting algorithms. Bubble sort repeatedly compares adjacent items and swaps them if out of order; quick sort uses a pivot and partitions the list.

冒泡排序、快速排序——基本排序算法。冒泡排序反复比较相邻项并在乱序时交换；快速排序使用枢轴并分割列表。

Bin packing — fitting objects of given sizes into bins of fixed capacity. First-fit, first-fit decreasing are heuristic algorithms that do not always give the optimal solution.

装箱问题——将给定尺寸的物品装入固定容量的箱子。首次适应、首次适应递减是启发式算法，并不总能得到最优解。

Linear programming — a method to maximise or minimise a linear objective function subject to linear inequalities (constraints). The feasible region is a convex polygon; the optimal solution lies at a vertex.

线性规划——在满足线性不等式（约束条件）的前提下，最大化或最小化线性目标函数的方法。可行域是一个凸多边形；最优解位于某个顶点处。

Simplex method — an algebraic algorithm for solving more complex linear programming problems beyond two variables, using slack variables and pivot operations.

单纯形法——一种代数算法，用于求解变量数超过两个的更复杂线性规划问题，引入松弛变量并进行枢轴运算。

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