📚 Year 11 CCEA Further Mathematics: Vocabulary & Terminology Quick Memory Guide | Year 11 CCEA 进阶数学：词汇术语速记指南
Mastering the language of Further Mathematics is half the battle won. When you can instantly recall that 'discriminant' means the expression under the square root sign or that 'differentiation' comes from the idea of finding small differences, you unlock deeper understanding and speed in exams. This guide breaks down essential terms by topic and equips you with clever memory tricks.
掌握进阶数学的语言就赢得了一半的胜利。当你瞬间想起“判别式”指的是平方根号下的表达式，或者“微分”源自寻找微小差异的概念时，你便打开了更深的理解之门，并在考试中赢得速度。本指南按主题分解基本术语，并为你配备了巧妙的记忆诀窍。
1. Algebra Basics & Indices | 代数基础与指数
Algebra is written logic using symbols. An index (plural: indices) tells you how many times to multiply a base by itself. In 5³, 3 is the index and 5 is the base. Think of your index finger pointing: it shows how many repeats.
代数是使用符号的书面逻辑。指数（index，复数为 indices）告诉你将底数自身相乘多少次。在 5³ 中，3 是指数，5 是底数。想到你的食指（index finger）在指：它展示重复的次数。
A surd is a root that stays irrational, like √2 or ∛7. The word sounds like 'absurd' because these numbers cannot be written neatly as fractions. You must leave them as roots for exactness.
根式（surd）是保持无理形式的根，例如 √2 或 ∛7。它听起来像“荒谬的”（absurd），因为这些数字无法整洁地写成分数。为了精确，你必须保留根号形式。
Rationalising the denominator means converting a fraction like 1/√2 into √2/2 by multiplying top and bottom by √2. The aim is to make the denominator 'rational' again – free of surds.
分母有理化指通过上下同乘 √2，将 1/√2 转化为 √2/2。目的是让分母重新成为“有理的”数——不再含有根式。
Standard form writes very large or small numbers as a × 10ⁿ, where 1 ≤ a < 10. Memory: it is the 'standard' way to tame many zeros.
标准形式（科学计数法）将极大或极小的数写成 a × 10ⁿ，其中 1 ≤ a < 10。记忆：这是驯服大量零的“标准”方式。
2. Equations & Inequalities | 方程与不等式
A quadratic equation has the shape ax² + bx + c = 0. 'Quad' means square – think of a square's four sides, though here it is the power two.
二次方程具有 ax² + bx + c = 0 的形状。“Quad”意为平方——想到正方形的四条边，但这里指的是二次方。
The discriminant is the part b² - 4ac under the square root in the quadratic formula. It 'discriminates' between two real roots, one repeated root, or no real roots. Label it D and remember: positive D = 2 roots, zero D = 1 root, negative D = 0 real roots.
判别式是二次公式中根号下的 b² - 4ac 部分。它“判别”出两个实根、一个重根或无实根。标记为 D，记住：正 D = 2 根，零 D = 1 根，负 D = 0 个实根。
Simultaneous equations are a set of equations that share variables. They must be true at the same time – hence 'simultaneous'. Methods: elimination (cancel a variable) or substitution (replace one variable).
联立方程是一组共享变量的方程。它们必须同时成立——因而成为“联立”（simultaneous）。解法：消元法（抵消一个变量）或代入法（替换一个变量）。
An inequality value, while ≤ or ≥ includes it. Reading tip: the open mouth always faces the larger side.
不等式使用 <、>、≤、≥ 等符号。严格不等式（< 或 >）不包含边界值，而 ≤ 或 ≥ 包含。阅读窍门：张开的嘴巴总是朝向更大的一边。
3. Polynomials & Factorisation | 多项式与因式分解
A polynomial is a sum of terms made of variables raised to non‑negative integer powers. 'Poly' means many, 'nomial' means term. Memory: think of a polygon (many sides) – a polynomial has many terms.
多项式是变量非负整数次幂构成的各项之和。“Poly”意为多，“nomial”意为项。记忆：联想多边形（polygon，多条边）——多项式有许多项。
The degree of a polynomial is the highest power of the variable. For 4x⁵ - 2x³ + 7, the degree is 5. It tells you the 'strength' of the leading term.
多项式的次数（degree）是变量的最高次幂。对于 4x⁵ - 2x³ + 7，次数为 5。它告诉你首项的“强度”。
Factor theorem: x). It is a quick test: plug a value into the polynomial; if you get zero, you have found a factor. Link with the remainder theorem: when f(x) is divided by (x - p), the remainder equals f(p).
因式定理：若 f(p) = 0，则 (x - p) 是 f(x) 的一个因式。这是快速检测：将某个值代入多项式；若得到零，你就找到了因式。联系余式定理：当 f(x) 除以 (x - p) 时，余数等于 f(p)。
Completing the square rewrites a quadratic x² + bx + c into the form (x + p)² + q. It reveals the vertex and helps solve equations. Memorise the steps: halve b, square it, add and subtract.
配方法将二次式 x² + bx + c 重写为 (x + p)² + q 的形式。它揭示了顶点并帮助解方程。记住步骤：取 b 的一半，平方，先加后减。
4. Functions & Graphs | 函数与图像
A function f(x) assigns exactly one output for each input. The domain is the set of allowed inputs, and the range is the set of possible outputs. Think of a vending machine: button (input) → snack (output); each button gives one snack.
函数 f(x) 为每个输入分配唯一一个输出。定义域是允许输入的集合，值域是可能输出的集合。想象自动售货机：按钮（输入）→ 零食（输出）；每个按钮只给出一种零食。
The inverse function phically, it is a reflection in the line y = x. Caution: f⁻¹(x) is not the same as 1/f(x).
反函数 f⁻¹(x) 逆转 f(x) 的作用。它交换定义域和值域。图像上是关于直线 y = x 的反射。注意：f⁻¹(x) 不是 1/f(x)。
A composite function (f ◦ g)(x) means first apply g, then apply f to the result. It is like a two‑stage machine. Order matters: f(g(x)) is usually different from g(f(x)).
复合函数 (f ◦ g)(x) 表示先作用 g，然后将结果代入 f。这就像两阶段机器。顺序很重要：f(g(x)) 通常不同于 g(f(x))。
Common transformations: translation shifts the graph, reflection flips it over an axis, and stretch pulls it away from an axis. Remember: inside the bracket affects x‑direction (opposite sign), outside affects y‑direction directly.
常见变换：平移移动图像，反射将其翻过坐标轴，伸缩将其拉离坐标轴。记住：括号内影响 x 方向（符号相反），括号外直接影响 y 方向。
5. Trigonometry | 三角学
Trigonometry links angles and side lengths in triangles. The key ratios – sine, cosine, tangent – OH CAH TOA: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent.
三角学联系三角形中的角与边长。关键比率——正弦（sine）、余弦（cosine）、正切（tangent）——用 SOH CAH TOA 记忆：Sin = 对边/斜边，Cos = 邻边/斜边，Tan = 对边/邻边。
The unit circle has radius 1 centred at (0,0). It extends trigonometric values to all angles and defines radians : one radian is the angle subtended by an arc equal to the radius. 180° = π radians. Use this to convert.
单位圆半径为 1，中心在 (0,0)。它将三角值扩展到所有角，并定义了弧度：一弧度是弧长等于半径时所对的角。180° = π 弧度。用此转换。
The period is the interval over which a trigonometric function repeats. For sinθ and cosθ it is 2π or 360°; for tanθ it is π or 180°. Amplitude is half the difference between maximum and minimum values; for asin(bθ) it is |a|.
周期是三角函数重复自身的间隔。对 sinθ 和 cosθ 为 2π 或 360°；对 tanθ 为 π 或 180°。振幅是最大值与最小值之差的一半；对于 asin(bθ) 为 |a|。
6. Exponentials & Logarithms | 指数与对数
An exponential function takes the form aˣ where the variable is in the exponent. The most important base is e ≈ 2.718. eˣ is its own derivative – a unique property. Grow your memory like 'exponential growth': it explodes quickly.
指数函数形式为 aˣ，其中变量在指数位置。最重要的底数是 e ≈ 2.718。eˣ 的导数等于它本身——独特性质。像“指数增长”一样增长你的记忆：它爆发式增加。
A logarithm is the inverse operation: if aˣ = y, then logₐ y = x. The natural logarithm ster the three laws: log(ab) = log a + log b, log(a/b) = log a - log b, log(aᵏ) = k log a.
对数是指数的逆运算：若 aˣ = y，则 logₐ y = x。自然对数 ln x 表示 logₑ x。对数将乘法化为加法，除法化为减法。掌握三个法则即可。
7. Calculus Terms | 微积分术语
Differentiation finds the instantaneous rate of change. The derivative f'(x) or dy/dx gives the gradient of the curve. The word comes from 'difference' – it measures tiny differences. A tangent touches a curve at one point with gradient equal to the derivative; a normal is perpendicular to the tangent.
微分求瞬时变化率。导数 f'(x) 或 dy/dx 给出曲线的梯度。此词源于“差异”（difference）——它衡量微小差异。切线（tangent）在一点接触曲线，梯度等于导数；法线（normal）垂直于切线。
A stationary point is where dy/dx = 0. It can be a maximum, minimum or a point of inflection. Think of a car stopping (stationary) at a hilltop, valley bottom, or flat road.
驻点是 dy/dx = 0 的地方。它可能是极大值、极小值或拐点。想象汽车在山顶、谷底或平路停下（驻留）。
Integration of is the reverse of differentiation. The indefinite integral ∫ f(x) dx gives a family of functions plus the constant of integration The definite integral ∫ₐᵇ f(x) dx computes the exact area under the curve from x=a to x=b. 'Integrate' means to bring together – integration sums up infinitely many tiny slices.
积分是微分的逆运算。不定积分 ∫ f(x) dx 给出一族函数加上积分常数 +C。定积分 ∫ₐᵇ f(x) dx 计算从 x=a 到 x=b 曲线下的精确面积。“积分”（integrate）意为汇集——积分将无穷多个微小切片累加起来。
8. Vectors & Matrices | 向量与矩阵
A vector has both magnitude (size
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