📚 Year 10 WJEC Further Maths: Terminology Quick-Memorisation Guide | Year 10 WJEC 进阶数学：词汇术语速记指南
Mastering the essential terminology in Further Mathematics is like learning the vocabulary of a new language – once you know the words, the sentences (problems) suddenly make sense. This guide pairs every key term with a clear definition and a quick memory trick, presented in bite‑sized English–Chinese pairs so you can lock them into long‑term memory fast.
掌握进阶数学的核心术语就像学习一门新语言的词汇，一旦你记住了这些“单词”，那些“句子”（题目）立刻变得清晰易懂。本指南将每个关键术语与简明定义和速记窍门配对，并以英—中双语短句呈现，帮助你快速把它们刻入长期记忆。
1. Number Systems and Sets | 数系与集合
Natural numbers (ℕ): The positive whole numbers we use for counting: 1, 2, 3, … Think ‘Nature starts at one’ – no zero, no negatives.
自然数 (ℕ): 我们用来计数的正整数：1、2、3……联想“自然从1开始”——没有零，没有负数。
Integers (ℤ): All whole numbers, including zero and negatives: … −2, −1, 0, 1, 2 … ‘Z’ comes from the German Zahlen (numbers) – it is the whole family.
整数 (ℤ): 所有正整数、负整数和零：…… −2, −1, 0, 1, 2 ……“Z”源自德语Zahlen（数字）——这是一个完整的大家庭。
Rational numbers (ℚ): Any number that can be written as a fraction a/b, where a and b are integers and b ≠ 0. ‘Q’ stands for Quotient – a fraction is a quotient.
有理数 (ℚ): 任何可以写成 a/b 分数形式的数，其中 a 和 b 是整数且 b ≠ 0。“Q”代表商（Quotient）——分数就是一个商。
Irrational numbers: Numbers that cannot be expressed as simple fractions – their decimal expansions go on forever without repeating, e.g. √2, π. They ‘irritate’ the rational world with their endlessness.
无理数: 无法表示为简单分数的数——小数部分无限不循环，如 √2、π。它们用无穷无尽“扰乱”了有理数的世界。
Real numbers (ℝ): All rational and irrational numbers together, filling the entire number line. ‘Real’ = the complete, unbroken line.
实数 (ℝ): 所有有理数和无理数的集合，占满整条数轴。“实”意味着完整、连续的数线。
Set notation: ∈ means ‘is an element of’, ∉ means ‘is not an element of’. E.g. 2 ∈ ℤ, ½ ∉ ℤ. Memorise by seeing ∈ as a stylised E for Element.
集合符号: ∈ 表示“属于”，∉ 表示“不属于”。例如 2 ∈ ℤ，½ ∉ ℤ。把 ∈ 看成花体字母 E，代表元素（Element）。
2. Algebraic Expressions and Manipulation | 代数表达式与化简
Term: A single number, variable, or product of numbers and variables, separated by + or − signs. In 3x² + 5x − 2, there are three terms.
项（Term）: 一个单独的数、变量或数与变量的乘积，用加号或减号分隔。在 3x² + 5x − 2 中有三个项。
Coefficient: The number part attached to a variable. In 7x³, 7 is the coefficient. Think ‘co‑pilot’ – the coefficient ‘flies’ with the variable.
系数: 变量前面的数字部分。在 7x³ 中，7 是系数。联想“co‑pilot（副驾驶）”——系数与变量一起“驾驶”。
Like terms: Terms with identical variable parts (same variable and exponent), e.g. 4x² and −3x² are like terms; 4x² and 4x are not. Only like terms can be combined.
同类项: 变量部分完全相同的项（变量及其指数都相同），例如 4x² 和 −3x² 是同类项；4x² 和 4x 不是。只有同类项才能合并。
Expanding brackets (FOIL): When multiplying two binomials, use First, Outer, Inner, Last. For (x+2)(x+3): First x·x, Outer x·3, Inner 2·x, Last 2·3 → x²+5x+6.
去括号展开（FOIL 法则）: 两个二项式相乘时，使用首项、外项、内项、尾项。比如 (x+2)(x+3)：首 x·x，外 x·3，内 2·x，尾 2·3 → x²+5x+6。
Factorising: The reverse of expanding – writing an expression as a product. Look for a common factor first, then try grouping or quadratic patterns. Mnemonic: ‘Factorise’ = ‘Find A Common Term Or Reverse It Simply, Every time’.
因式分解: 与展开相反——把表达式写成乘积形式。先找公因式，再用分组或二次三项式模式。速记口诀：“分解”就是“找出公因子，反过来写”。
3. Equations and Inequalities | 方程与不等式
Linear equation: An equation of the first degree, meaning the variable’s highest power is 1, e.g. 2x+3=7. Solve by isolating x step‑by‑step. It’s a ‘line’ because its graph is straight.
线性方程: 一次方程，即变量的最高指数为 1，例如 2x+3=7。通过逐步分离 x 求解。之所以叫“线性”，是因为它的图像是一条直线。
Quadratic equation: An equation of the form ax²+bx+c=0, a≠0. ‘Quad’ means square – the variable is squared.
二次方程: 形如 ax²+bx+c=0 的方程，a≠0。“Quad”表示平方——变量被平方了。
Discriminant (Δ): Δ = b² − 4ac. It discriminates the nature of roots: Δ>0 gives two real roots, Δ=0 one repeated root, Δ<0 no real roots (two complex). Remember: Discriminant Decides.
判别式 (Δ): Δ = b² − 4ac。它判别根的性质：Δ>0 有两个不等实根，Δ=0 有一个重根，Δ<0 无实根。记住：判别式决定一切。
Quadratic formula: The solution of ax²+bx+c=0 is x = (−b ± √(b²−4ac))/(2a). Sing it to ‘Pop Goes the Weasel’!
求根公式: ax²+bx+c=0 的解为 x = (−b ± √(b²−4ac))/(2a)。可用旋律记忆。
Inequality notation: Symbols <, >, ≤, ≥. The small end points to the smaller number, the open mouth to the larger – think of a crocodile eating the bigger meal.
不等号: 符号 <、 >、 ≤、 ≥。小端指向较小的数，开口指向较大的数——想象鳄鱼张大嘴吃大餐。
4. Functions and Graphs | 函数与图像
Function f(x): A rule that assigns exactly one output for each input. Write f(x)=2x+3, read as ‘f of x’. Every x has exactly one y – like a vending machine: press a button, get exactly one snack.
函数 f(x): 一种法则，每个输入对应唯一输出。写作 f(x)=2x+3，读作“f of x”。每个 x 只产生一个 y——像自动售货机：按一个按钮，只出一个商品。
Domain and range: Domain = all possible input (x) values. Range = all possible output (y) values. D for x, R for y; think ‘Dose’ of x gives a ‘Result’ of y.
定义域与值域: 定义域 = 所有可能的输入 (x) 值。值域 = 所有可能的输出 (y) 值。D 对 x，R 对 y；联想：输入 x 的“剂量”得到 y 的“结果”。
Composite function: Combining two functions: (f∘g)(x) = f(g(x)). Apply the inner function first. ‘Compose’ two stages: first g, then f.
复合函数: 把两个函数合并：(f∘g)(x) = f(g(x))。先套用里面的函数。“复合”是两个阶段：先 g 后 f。
Inverse function f⁻¹(x): The function that reverses the effect of f(x). Write y = f(x), swap x and y, solve for y. The −1 is not an exponent; it signals ‘opposite’.
反函数 f⁻¹
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