📚 Year 10 Eduqas Further Maths: Glossary & Mnemonics | Year 10 Eduqas 进阶数学：词汇术语速记指南
Mastering the language of Further Mathematics is half the battle. This guide breaks down essential Year 10 terminology with clear definitions and memory tricks, so you can recall them quickly in your Eduqas exams.
掌握进阶数学的语言是成功的一半。本指南将Year 10关键术语配上清晰定义与记忆窍门，助你在Eduqas考试中迅速记起它们。
1. Algebra & Polynomial Terminology | 代数与多项式术语
Variable: A symbol, usually a letter, that represents an unknown number. Think of it as a 'varying' quantity – it can change.
变量：通常用字母表示未知数。不妨把它想成一个可以“变”的量。
Polynomial: An expression with terms formed from variables with non-negative integer exponents and coefficients. 'Poly' means many – imagine a many-term creature.
多项式：由变量非负整数次幂和系数构成的表达式。“poly”意为“多”，可想像成一只多肢生物。
Degree of a polynomial: The highest exponent of the variable. Mnemonic: 'degree' as in temperature – the highest 'power' of heat.
多项式的次数：变量的最高指数。联想：“degree”既有“次数”也有“度数”的意思，像体温的最高热度。
Leading coefficient: The coefficient of the term with the highest degree. Picture the leader of a pack – it goes first.
首项系数：次数最高项的系数。想象领头羊走在最前面。
Factor theorem: For a polynomial f(x), if f(a)=0, then (x−a) is a factor. Remember: if plugging in 'a' makes it zero, (x−a) is a zero-maker.
因式定理：对多项式f(x)来说，若f(a)=0，则(x−a)是因式。记法：代入a得零，则(x−a)就是“归零因子”。
Remainder theorem: When a polynomial f(x) is divided by (x−a), the remainder is f(a). Hold onto the remainder – it's just the function evaluated at a.
余式定理：多项式f(x)除以(x−a)时，余数为f(a)。把余数抓牢——它其实就是函数在a处的值。
2. Functions & Graphs | 函数与图像
Function: A rule that assigns exactly one output to each input. Think 'Function machine' – every x gets precisely one y.
函数：为每一输入分配唯一输出的规则。想像一台“函数机器”——每个x唯一产生一个y。
Domain e : The set of all possible input values (x-values). Domain ≈ Dominion, the territory of inputs.
定义域：所有可能输入值（x值）的集合。domain就像“领地”，是输入的地盘。
Range: The set of all actual output values (y-values). The Range is what you get out of the function's gun (domain is ammunition).
值域：所有实际输出值（y值）的集合。range像射程，是函数这把“枪”打出的结果。
One-to-one function: Each y-value comes from exactly one x-value. Passes both vertical and horizontal line tests. 'Unique pairing' both ways.
一一函数：每个y值恰对应一个x值。同时通过垂直线和水平线测试。像双向“唯一配对”。
Inverse function: Reverses the effect of the original function, written f⁻¹(x). Swap x and y, then solve. Think of undoing your steps.
反函数：与原函数作用相反，记为f⁻¹(x)。交换x与y再解出y。就像撤销操作。
Composite function: Applying one function to the result of another, e.g. f(g(x)). Remember 'function inside function', like Russian dolls.
复合函数：将一个函数作用于另一函数的结果，如f(g(x))。记忆：函数套函数，像俄罗斯套娃。
Graph transformations: f(x+a) shifts left, f(x)+a shifts up, −f(x) reflects in x-axis. Learn as: inside bracket affects x opposite, outside affects y directly.
图像变换：f(x+a)左移，f(x)+a上移，−f(x)关于x轴反射。口诀：括号内对x反向，括号外对y直接。
3. Inequalities | 不等式
Inequality notation: Symbols <, >, ≤, ≥. The smaller end points to the smaller number – 'the alligator eats the bigger number'.
不等号记法：<, >, ≤, ≥。小的一端指向小数——“鳄鱼吃大数”。
Strict inequality: Uses < or >, excludes the boundary value. Strict like a bouncer who won't let you in.
严格不等式：使用<或>，边界值不包括在内。严格得像个铁面门卫，绝不让进。
Interval notation mbers greater than 2 and up to 5, including 5. Parenthesis = excluded, bracket = included.
区间记法：(2,5] 表示从大于2到5的所有数，含5。圆括号表示不含，方括号表示含。
Quadratic inequality ssion = 0, then test intervals. Sketch the parabola to see which parts are above or below.
二次不等式：如 x² − 5x + 6 > 0。先找出表达式为零的临界值，再分段检验。画抛物线草图看哪部分在上在下。
Sign diagram: A number line showing where a function is positive or negative. Mark zeros and undefined points, then test signs between them.
符号图：标出函数正负的数轴。在零点和无定义点处画界，然后检验区间符号。
4. Sequences & Series | 数列与级数
Sequence: An ordered list of numbers according to a rule. 'SeQUEnce' – think 'queue' of terms.
数列：按规则排列的一串数。“sequence”发音似“sequ”——想象成排队的一列项。
Series: The sum of the terms of a sequence. Series sounds like 'serial addition'.
级数：数列各项之和。series听起来像“连续加法”。
Arithmetic progression (AP): A sequence where the difference between consecutive terms is constant (common difference d). Mnemonic: 'Arithmetic' involves adding the same difference.
等差数列：相邻两项差为常数（公差d）的数列。记法：“等差”就是等距地加差值。
Geometric progression (GP): A sequence where each term is multiplied by a constant ratio r. 'Geometric' – think of multiplying like geometry scaling.
等比数列：每一项乘以常数公比r的数列。“等比”像几何缩放。
Sum to infinity of a GP: S∞ = a/(1−r) for |r|<1. It 'approaches' a finite sum if the terms shrink. Remember 'if r is small, the sum is tall (finite)'.
等比数列无穷和：当|r|<1时，S∞ = a/(1−r)。若项越来越小，和将趋于有限值。口诀：r若小于1，和便有个底。
Sigma notation (Σ): Compact way to write a sum using the Greek capital sigma. Σ for 'Sum'. The puppy's tail is a sigma shape.
求和符号 (Σ)：用希腊大写西格玛表示和的简写法。Σ 代表 “Sum”。可以想象西格玛形状像求和的小尾巴。
5. Logarithms & Exponentials | 对数与指数
Exponent: The power to which a number is raised. In bⁿ, n is the exponent. It says how many times to multiply the base.
指数：数字的乘方次数。bⁿ中n是指数，表示基数自乘的次数。
Logarithm: The inverse of exponentiation. logₐ(b) asks 'what power of a gives b?'. 'Log' is the exponent lifter.
对数：指数运算的逆运算。logₐ(b) 问：“a的几次幂等于b？”。“log”像是把指数找出来的工具。
Natural logarithm (ln): Log with base e (≈2.71828). 'Natural' because it grows naturally from continuous compounding. e is Euler's number.
自然对数 (ln)：以e为底的对数。e≈2.718，之所以“自然”是因为源自连续复利，e是欧拉数。
Log laws ogs turn multiplication into addition, division into subtraction, powers into multipliers.
对数法则：log(ab)=log a+log b；log(a/b)=log a−log b；log(aⁿ)=n log a。记住：对数化乘为加，化除为减，化幂为乘。
Change of base gₐ(b) = log_c(b) / log_c(a). The base you want moves to the bottom. Mnemonic: bottom is a.
换底公式：logₐ(b) = log_c(b) / log_c(a)。你想换的底跑到分母。记法：底在底下。
6. Trigonometry & Identities | 三角学与恒等式
Radian: A unit of angle where π rad = 180°. It's the angle subtended by an arc length equal to the radius. 'Radius-an' – radian.
弧度：角度单位，π rad = 180°。它是弧长等于半径所对的圆心角。联想：radian — radius（半径）的变体。
Sine, cosine, tangent: In a right triangle: sinθ=opp/hyp, cosθ=adj/hyp, tanθ=opp/adj. SOH CAH TOA – Some Old Horses Can Always Hear Their Owners Approach.
正弦、余弦、正切：直角三角形中：sinθ=对/斜，cosθ=邻/斜，tanθ=对/邻。口诀SOH CAH TOA（国内常说“对斜邻斜对邻”）。
CAST diagram: Shows which trig functions are positive in each quadrant. All, Sin, Tan, Cos starting from first quadrant. Remember 'All Students Take Calculus'.
CAST图：表示各象限三角函数的正负。从第一象限开始：全正、Sin正、Tan正、Cos正。记法：All Students Take Calculus。
Pythagorean identity: sin²θ + cos²θ = 1. Born from Pythagoras' theorem on the unit circle.
勾股恒等式：sin²θ + cos²θ = 1。源自单位圆上的勾股定理。
Sine rule: a/sinA = b/sinB = c/sinC. Use for non‑right triangles when you know two angles and a side, or two sides and a non‑included angle.
正弦定理：a/sinA=b/sinB=c/sinC。用于已知两角一边或两边一非夹角时的斜三角形。
Cosine rule: a² = b² + c² − 2bc cosA. Generalised Pythagoras; use when you have SAS or SSS.
余弦定理：a² = b² + c² − 2bc cosA。勾股定理的推广，用于SAS或SSS情形。
Double angle formulas 2θ = 2sinθ cosθ; cos2θ = cos²θ − sin²θ = 2cos²θ−1 = 1−2sin²θ. Notice cos2θ has three forms – choose the one matching your problem.
倍角公式：sin2θ=2sinθcosθ；cos2θ=cos²θ−sin²θ=2cos²θ−1=1−2sin²θ。注意cos2θ有三种形式，根据题目选用。
7. Basics of Calculus | 微积分基础
Differentiation: Finding the derivative, or rate of change. From 'difference' – it measures how a tiny change in x affects y.
微分：求导数或变化率。源于“微小差异”，衡量x的微小改变如何影响y。
Derivative (dy/dx): The gradient of the tangent to a curve at a point. Think 'dy over dx' – a small rise over a small run.
导数 (dy/dx)：曲线在某点切线的斜率。想象“dy除以dx”——微小增量之比。
Power rule: If f(x)=xⁿ, then f'(x)=n xⁿ⁻¹. Bring the power down and reduce exponent by 1. 'Power comes down, one less for the crown.'
幂法则：若f(x)=xⁿ，则f'(x)=n xⁿ⁻¹。把指数拖下来，指数减一。“指数当系数，头上减一”。
Stationary point: Where dy/dx = 0. It could be a maximum, minimum, or point of inflection. 'Stationary' means the curve is momentarily 'standing still'.
驻点：dy/dx=0的点。可能是极大、极小或拐点。stationary就是曲线“停驻”的地方。
Integration: The reverse of differentiation (anti‑derivative). It finds the area under a curve. Think 'integrate' as 'gathering slices' to make whole.
积分：微分的逆运算（反导数），求曲线下方面积。integrate像是把许多薄片“整合”成一个整体。
Indefinite integral: ∫ f(x) dx = F(x) + C, where F'(x)=f(x). Don't forget the constant C – 'C for constant, always there after integration.'
不定积分：∫ f(x) dx = F(x) + C，其中F'(x)=f(x)。别忘了常数C——积分后总有个“C for constant”。
Definite integral: ∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b)−F(a). It gives a number (area). The limits a to b are 'definite' ends.
定积分：∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b)−F(a)。结果是一个数值（面积）。a到b是“确定的”界限。
8. Matrices | 矩阵
Matrix (plural matrices): A rectangular array of numbers, arranged in rows and columns. Think 'matrix' like a grid of data.
矩阵：按行列排列的矩形数字阵列。矩阵就像数据表格。
Order of a matrix: Given as rows × columns, e.g. 2×3 has 2 rows and 3 columns. 'R×C' – rows come first.
矩阵的阶：表示为行数×列数，如2×3表示2行3列。row × column，先行后列。
Identity matrix (I): Square matrix with 1s on the main diagonal and 0s elsewhere. Multiplying by I leaves a matrix unchanged, like 1 in numbers.
单位矩阵 (I)：主对角线为1其余为0的方阵。乘上I矩阵不变，就像数字1。
Determinant of 2×2 matrix: For A = [a b; c d], det(A) = ad − bc. Mentally say 'ad minus bc', the cross‑product difference.
2×2矩阵行列式：对A=[a b; c d]，det(A)=ad−bc。心中默念“ad减bc”，交叉乘积之差。
Inverse of 2×2 matrix: A⁻¹ = 1/(ad−bc) [d −b; −c a]. Only exists if det≠0. 'Swap a and d, change signs of b and c, divide by determinant.'
2×2逆矩阵：A⁻¹ = 1/(ad−bc) [d −b; −c a]。仅当行列式不为零时存在。口诀：交换a和d，b和c变号，除以行列式。
Transformation matrix: Matrices represent geometric transformations: rotation, reflection, enlargement. Apply by multiplying matrix by column vector.
变换矩阵：矩阵可表示几何变换：旋转、反射、放大。使用矩阵乘列向量实现。
9. Vectors | 向量
Vector: A quantity with both magnitude (size) and direction. Represented as a directed line segment or column vector. Think 'arrow'.
向量：既有大小又有方向的量。用有向线段或列向量表示。想象箭头。
Magnitude of a vector: Length, found by Pythagoras: |v| = √x² + y² for a 2D column vector (x over y). 'Magnitude' is how 'big' the arrow is.
向量的模：长度，用勾股定理求：对2D列向量(x y)ᵀ，|v|=√x²+y²。magnitude就是箭头的“大小”。
Unit vector ten denoted with a hat: ^i, ^j. To make a unit vector, divide the vector by its magnitude.
单位向量：模为1的向量。常戴“帽子”记作^î, ^ĵ。将向量除以其模即得单位向量。
Position vector: A vector from the origin to a point. It gives the coordinates as a vector. 'Position' – where the point sits.
位置向量：从原点指向某点的向量。用向量表示坐标。position就是点的“位置”。
Parallel vectors: Vectors that are scalar multiples of each other. They have the same or opposite direction. Check if one equals k times the other.
平行向量：彼此成标量倍数的向量。它们方向相同或相反。检验是否一个等于k倍另一个。
Resultant vector: The sum of two or more vectors. Found by adding corresponding components. 'Resultant' is the combined result.
合向量：两个或多个向量之和。将对应分量相加即得。resultant就是合成的“结果”。
10. Complex Numbers | 复数
Imaginary unit i: Defined as i² = −1. i is not real; it's the foundation of complex numbers. Say 'i is the square root of −1'.
虚数单位 i：定义i²=−1。i不是实数；它是复数的基础。记作“i是负一的平方根”。
Complex number: z = a + bi, where a is the real part and b is the imaginary part. A 'complex' is a combination of real and imaginary.
复数：z = a + bi，a为实部，b为虚部。complex意为“复合”，实虚结合。
Complex conjugate: If z = a + bi, the conjugate is a − bi, denoted ̄z or z*. The conjugate has the same real part but opposite imaginary part.
共轭复数：若z=a+bi，则共轭为a−bi，记作 ̄z 或 z*。实部相同，虚部反号。
Modulus of z: |z| = √a² + b². It represents the distance from the origin on the Argand diagram. 'Modulus' measures how 'large' the complex number is.
复数的模：|z| = √a² + b²。表示在阿干特图上到原点的距离。模度量复数的“大小”。
Argand diagram: A plane where the x-axis is the real part and y-axis is the imaginary part. Every complex number is a point or vector.
阿干特图：x轴为实部，y轴为虚部的平面。每个复数是一个点或向量。
Solving quadratics with complex roots: When discriminant < 0, use i. For x²+1=0, x = ±i. Remember 'negative discriminant → imaginary roots'.
解含复根的二次方程：判别式 < 0 时，用 i。如x²+1=0，得x = ±i。记住“判别式为负→虚根”。
Addition/Subtraction = (a±c) + (b±d)i. Real and imaginary parts stay separate, like sorting apples and oranges.
加减法：(a+bi) ± (c+di) = (a±c) + (b±d)i。实部和虚部分别运算，好比苹果归苹果，橘子归橘子。
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