📚 Year 7 AQA Further Maths Formula & Theorem Quick Reference Handbook | Year 7 AQA 进阶数学：公式定理速查手册
This handbook is a focused revision tool for Year 7 pupils studying AQA Further Mathematics. It brings together all the key formulas, theorems and definitions you need to memorise and apply with confidence. Each entry is written in clear bilingual form, making it ideal for quick reference both in class and during independent revision. Mastering these fundamentals will help you tackle problem-solving questions accurately and build a solid foundation for higher-level study.
本手册是为学习 AQA 进阶数学的 Year 7 学生量身定制的复习工具。它汇集了你需要记忆并熟练运用的所有关键公式、定理和定义。每一条目均以清晰的双语形式呈现，非常适合在课堂和自主复习中快速查阅。掌握这些基础知识有助于你精准解决各类问题，并为更高级的学习打下坚实基础。
1. Number Types and Properties | 数的类型与性质
A number system classifies values into different sets. Natural numbers start from 1 and go upwards. Whole numbers include zero alongside the naturals. Integers extend to negative whole numbers as well.
数系将数值划分为不同的集合。自然数从1开始向上延伸。非负整数在自然数的基础上包含了零。整数则进一步包含了负的整数。
Prime numbers have exactly two distinct factors: 1 and the number itself. Composite numbers have more than two factors. The number 1 is neither prime nor composite.
质数恰好有两个不同的因数，即1和它本身。合数拥有多于两个因数。数字1既不是质数也不是合数。
Square of a number: n² = n × n
一个数的平方：n² = n × n
Cube of a number: n³ = n × n × n
一个数的立方：n³ = n × n × n
quared, gives a. The radical symbol √ represents the principal (non-negative) square root.
一个数 a 的平方根是指某个数值，其平方等于 a。根号 √ 表示主平方根（非负）。
√a × √b = √ab
√a × √b = √ab
∛a × ∛b = ∛(ab)
∛a × ∛b = ∛(ab)
Index laws simplify expressions with powers. For any non-zero base a and integers m, n:
指数律可化简幂的表达式。对于任意非零底数 a 和整数 m、n：
aᵐ × aⁿ = aᵐ⁺ⁿ
aᵐ × aⁿ = aᵐ⁺ⁿ
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
(aᵐ)ⁿ = aᵐⁿ
(aᵐ)ⁿ = aᵐⁿ
a⁰ = 1 (a ≠ 0)
a⁰ = 1 (a ≠ 0)
writes very large or very small numbers as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer.
标准形式（科学记数法）将极大或极小的数表示为 a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
Fractions express parts of a whole: numerator over denominator. Equivalent fractions are formed by multiplying or dividing both terms by the same non-zero number.
分数表示整体的一部分：分子在分母之上。等值分数通过将分子分母同时乘或除以同一个非零数得到。
To add or subtract fractions, first rewrite them with a common denominator. To multiply fractions, multiply numerators and denominators separately. To divide by a fraction, multiply by its reciprocal.
加减分数时，先通分化为同分母。分数相乘，分子乘分子，分母乘分母。除以一个分数等于乘它的倒数。
a/b × c/d = (a×c) / (b×d)
a/b × c/d = (a×c) / (b×d)
a/b ÷ c/d = a/b × d/c = (a×d) / (b×c)
a/b ÷ c/d = a/b × d/c = (a×d) / (b×c)
Decimals are another way to represent fractions. To convert a fraction to a decimal, divide the numerator by the denominator. Terminating decimals have a finite number of digits; recurring decimals repeat a pattern endlessly.
小数是分数的另一种表示形式。将分子除以分母即可将分数化为小数。有限小数位数有限；循环小数则无限重复某一数位模式。
Percentages mean “out of 100”. To convert a decimal to a percentage multiply by 100; to convert a percentage to a decimal divide by 100.
百分数表示“每一百份中的份数”。将小数乘以100化为百分数；将百分数除以100化为小数。
Fraction → Decimal: divide numerator by denominator
分数化小数：分子除以分母
Decimal → Percentage: × 100%
小数化百分数：× 100%
Percentage increase/decrease = (change / original) × 100%
百分比增减 = (变化量 / 原值) × 100%
3. Algebraic Expressions | 代数表达式
Algebra uses letters to stand for unknown or variable numbers. Terms are separated by + or − signs. Like terms have identical variable parts and can be combined.
代数用字母代表未知数或变量。项由加号或减号分隔。同类项具有完全相同的字母部分，可以合并。
The distributive law is fundamental when expanding brackets:
分配律是展开括号的基础：
a(b + c) = ab + ac
a(b + c) = ab + ac
(a + b)(c + d) = ac + ad + bc + bd
(a + b)(c + d) = ac + ad + bc + bd
Factorising reverses expansion: writing an expression as a product of its factors. Always look for the highest common factor (HCF) first.
因式分解是展开的逆过程：将一个表达式写成其因式的乘积。务必先提取最高公因式（HCF）。
ab + ac = a(b + c)
ab + ac = a(b + c)
The square of a binomial yields a special pattern that is useful to recognise:
二项式的平方会产生一种特殊模式，值得熟记：
(a + b)² = a² + 2ab + b²
(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
(a − b)² = a² − 2ab + b²
Substitution means replacing variables with given numerical values and then evaluating using the order of operations (BIDMAS/BODMAS).
代入法是指用给定的数值替换变量，然后按照运算顺序（括号、指数、乘除、加减）求值。
4. Linear Equations and Inequalities | 一元一次方程与不等式
An equation states that two expressions are equal. Solving an equation means finding the value(s) of the variable that make the statement true. Maintain balance by performing the same operation on both sides.
方程表示两个表达式相等。解方程就是求出使等式成立的变量的值。通过对方程两边执行相同运算来保持平衡。
orm of a linear equation in one variable is ax + b = c. Isolate the variable step by step.
一元一次方程的一般形式为 ax + b = c。逐步分离变量即可求解。
If ax + b = c, then x = (c − b) / a, provided a ≠ 0.
若 ax + b = c，则 x = (c − b) / a，前提是 a ≠ 0。
greater than or equal to). When multiplying or dividing by a negative number, the inequality sign must be reversed.
不等式使用符号比较表达式：<（小于）、>（大于）、≤（小于等于）、≥（大于等于）。当乘以或除以一个负数时，不等号方向必须改变。
If a > b, then a + c > b + c
若 a > b，则 a + c > b + c
If a > b and c < 0, then a × c < b × c
若 a > b 且 c < 0，则 a × c < b × c
Solutions to inequalities are often shown on a number line with open or closed circles.
不等式的解集通常用数轴表示，空心圆表示不包含端点，实心圆表示包含端点。
5. Sequences | 数列
A sequence is an ordered list of numbers following a rule. Each number is called a term. An arithmetic sequence has a constant difference between consecutive terms.
数列是按一定规则排列的一列数。每个数称为一项。等差数列中，相邻两项的差为常数。
nth term of an arithmetic sequence: T(n) = a + (n − 1)d
等差数列的第 n 项：T(n) = a + (n − 1)d
where a is the first term, d is the common difference, and n is the position number.
其中 a 为首项，d 为公差，n 为项数。
Other common sequences include square numbers (1, 4, 9, 16, …) and triangular numbers (1, 3, 6, 10, …). Recognising the pattern helps find any term.
其他常见数列有平方数数列（1, 4, 9, 16, …）和三角形数数列（1, 3, 6, 10, …）。识别模式可帮助求出任意项。
A term-to-term rule describes how to move from one term to the next, while the position-to-term rule (nth term) gives the value directly from its position.
逐项规则描述如何从一项得到下一项，而位置公式（第 n 项公式）则直接根据位置给出项的值。
6. Coordinates and Linear Graphs | 坐标与线性图像
The Cartesian plane uses two perpendicular axes: the x‑axis (horizontal) and the y‑axis (vertical). A point is identified by an ordered pair (x, y).
笛卡尔平面使用两条互相垂直的轴：x 轴（水平）和 y 轴（垂直）。一个点由有序数对 (x, y) 标识。
tion can be written in the form y = mx + c, where m is the gradient (slope) and c is the y‑intercept (the value of y when x = 0).
线性方程可以写成 y = mx + c 的形式，其中 m 为斜率（梯度），c 为 y 轴截距（即 x = 0 时的 y 值）。
Gradient m = (change in y) / (change in x) = (y₂ − y₁) / (x₂ − x₁)
斜率 m = (y 的变化量) / (x 的变化量) = (y₂ − y₁) / (x₂ − x₁)
have equation y = k (gradient 0). Vertical lines have equation x = h (gradient undefined).
水平线的方程为 y = k（斜率为0）。竖直线的方程为 x = h（斜率无定义）。
To plot a linear graph, generate at least three points by substituting x‑values into the equation, then draw a straight line through them.
绘制线性图像时，通过将 x 值代入方程生成至少三个点，然后过这些点画一条直线。
7. Perimeter, Area and Volume | 周长、面积与体积
Perimeter is the total distance around a 2D shape. Area measures the surface inside the shape. Volume measures the space a 3D object occupies.
周长是二维图形一周的总长度。面积测量图形内部的表面大小。体积测量三维物体所占的空间。
Key formulas for common shapes:
常见图形的关键公式：
Rectangle: Area = l × w, Perimeter = 2(l + w)
矩形：面积 = 长 × 宽，周长 = 2(长 + 宽)
Triangle: Area = (base × height) / 2
三角形：面积 = (底 × 高) / 2
Parallelogram: Area = base × perpendicular height
平行四边形：面积 = 底 × 垂直高
Trapezium: Area = ½(a + b)h (a, b are parallel sides, h is height)
梯形：面积 = ½(a + b)h（a、b 为平行边，h 为高）
Circle: Circumference = πd = 2πr, Area = πr²
圆：周长 = πd = 2πr，面积 = πr²
a rectangular prism; its volume = l × w × h. Surface area is the total area of all faces.
对于三维图形，棱柱的体积 = 横截面积 × 长度。长方体是一种矩形棱柱，其体积 = 长 × 宽 × 高。表面积是所有面的总面积。
Volume of a cuboid = lwh
长方体体积 = lwh
8. Angles and Polygons | 角与多边形
Angles are measured in degrees (°). Key angle facts:
角度以度（°）为单位。重要角度事实：
- Angles on a straight line sum to 180°.
- 补角：直线上的邻角之和为180°。
- Angles around a point sum to 360°.
- 周角：一点周围的角之和为360°。
- Vertically opposite angles are equal.
- 对顶角相等。
- Angles in a triangle sum to 180°.
- 三角形内角和为180°。
Parallel lines create special angle relationships: alternate angles are equal, corresponding angles are equal, and co‑interior (allied) angles sum to 180°.
平行线产生特殊的角度关系：内错角相等，同位角相等，同旁内角（协内角）互补（和为180°）。
A polygon is a closed shape with straight sides. A regular polygon has all sides and angles equal.
多边形是由直线段组成的封闭图形。正多边形的所有边和角都相等。
Sum of interior angles of an n‑sided polygon = (n − 2) × 180°
n 边形的内角和 = (n − 2) × 180°
Each interior angle of a regular n‑gon = (n − 2) × 180° / n
正 n 边形的每个内角 = (n − 2) × 180° / n
Sum of exterior angles of any convex polygon = 360°
任何凸多边形的外角和 = 360° Published by TutorHao | Year 7 进阶数学 Revision Series | aleveler.com
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