📚 Formula & Theorem Quick Reference | 公式定理速查手册
This quick reference guide collects the essential formulas, theorems and rules you need for Year 8 SQA Advanced Mathematics. Use it to revise key concepts in number, algebra, geometry, statistics and probability, and keep it beside you when you work through practice questions or past papers.
本速查手册收集了 Year 8 SQA 进阶数学所需的基本公式、定理和法则。你可以用它来复习数、代数、几何、统计与概率等关键内容，并在做练习题或历年真题时把它放在手边随时查阅。
1. Number Operations and BIDMAS | 数的运算与运算顺序
When adding or subtracting negative numbers, two signs next to each other combine into a single sign. This is often called the "sign rules" and it is the first thing examiners expect you to apply correctly.
加减负数时，相邻的两个符号可以合并成一个符号。这就是常说的"符号法则"，也是考官最希望你首先正确运用的地方。
a – (–b) = a + b
a + (–b) = a – b
Multiplying or dividing two numbers with the same sign gives a positive answer; opposite signs give a negative answer. Remember that a negative sign in front of a bracket reverses every sign inside it.
两数相乘或相除，同号得正，异号得负。要记住，括号前面的负号会把括号内每一项的符号都反过来。
(–a) × (–b) = a × b
(–a) × b = –(a × b)
Always follow BIDMAS: Brackets, Indices, Division & Multiplication (left to right), Addition & Subtraction (left to right). Working strictly from left to right within the same priority level avoids the most common careless errors.
始终遵循 BIDMAS 运算顺序：括号、指数、乘除（从左到右）、加减（从左到右）。在同一优先级内严格从左到右计算，可以避免最常见的粗心错误。
Example: 10 – 2 × 3² = 10 – 2 × 9 = 10 – 18 = –8.
例如：10 – 2 × 3² = 10 – 2 × 9 = 10 – 18 = –8。
2. Fractions and Decimals | 分数与小数
To add or subtract fractions, find a common denominator first, then combine the numerators. The denominator itself never changes when you add or subtract.
分数加减时，先找出公分母，再把分子相加或相减。加减法中分母本身保持不变。
a/b + c/d = (ad + bc) / bd
Multiply fractions by multiplying numerators and denominators separately. Simplify the answer by cancelling any common factor, and remember that cancelling before multiplying often keeps the numbers small.
分数相乘，分子乘分子，分母乘分母。最后要约分去掉公因数；记住，在相乘之前先约分通常能让数字更小、更好算。
a/b × c/d = (a × c) / (b × d)
the reciprocal, which means turning the second fraction upside down and changing ÷ into ×.
分数相除，乘以倒数，也就是把第二个分数上下翻转，并把 ÷ 改成 ×。
a/b ÷ c/d = a/b × d/c
the denominator. Recurring decimals can be shown with a dot or bar, for example 1/3 = 0.3̇ and 5/6 = 0.83̇.
将分数化为小数，用分子除以分母。循环小数可用点或横线表示，例如 1/3 = 0.3̇，5/6 = 0.83̇。
3. Percentages | 百分数
A percentage is a fraction out of 100. To find a percentage of an amount, multiply by the percentage and divide by 100, or equivalently multiply by the decimal equivalent.
百分数表示分母为 100 的分数。求一个数的百分之几，用这个数乘以百分数再除以 100，等价地也可以直接乘以对应的小数。
p% of x = (p/100) × x
To increase an amount by p%, multiply by (1 + p/100). Use this method rather than finding the increase first, because it works in a single step for repeated percentage changes.
将一个数增加 p%，乘以 (1 + p/100)。建议直接用这个方法，而不要先算增加量，因为它在连续百分数变化中可以一步完成。
New value = original × (1 + p/100)
To decrease an amount by p%, multiply by (1 – p/100). A percentage change compares the change with the original value, so always divide by the original amount.
将一个数减少 p%，乘以 (1 – p/100)。百分数变化是把变化量与原来的值相比较，所以务必除以原来的数值。
New value = original × (1 – p/100)
Percentage change = (change ÷ original) × 100%
lue and must find the original, divide instead of multiply. If a price of 72 includes a 20% increase, the original is 72 ÷ 1.2 = 60.
在反向百分数问题中，已知最终值而要求原值，应该做除法而不是乘法。若某价格 72 是增加了 20% 之后的结果，则原价为 72 ÷ 1.2 = 60。
4. Algebraic Expressions and Like Terms | 代数表达式与同类项
An algebraic expression is made of terms separated by + or – signs. Each term has a coefficient and possibly a variable, and a term with no variable is called a constant.
代数表达式由被 + 或 – 号分隔的项组成。每一项都有系数，可能还有变量；不含变量的项称为常数项。
Like terms contain exactly the same variable or variables raised to the same powers. Only like terms can be added or subtracted, so 3x and 5x combine to 8x, but 3x and 5x² cannot be combined.
同类项指所含变量及其指数完全相同的项。只有同类项才能相加减，所以 3x 与 5x 合并为 8x，但 3x 与 5x² 不能合并。
3a + 5a = 8a (identical variable) / 3a + 5a = 8a（变量相同）
7b – 2b = 5b / 7b – 2b = 5b
4x + 3y – x + 2y = 3x + 5y (collect separately) / 4x + 3y – x + 2y = 3x + 5y（分别合并）
6m² + 2m² = 8m² / 6m² + 2m² = 8m²
When multiplying terms, multiply the coefficients and add the indices of the same variable. When dividing, divide the coefficients and subtract the indices.
乘法运算时，系数相乘，同一变量的指数相加。除法运算时，系数相除，指数相减。
aᵐ × aⁿ = aᵐ⁺ⁿ aᵐ ÷ aⁿ = aᵐ⁻ⁿ (aᵐ)ⁿ = aᵐⁿ
each variable with a given number and evaluating carefully using BIDMAS. If x = 4 and y = –2, then 3x² – 2y = 3 × 16 – 2 × (–2) = 48 + 4 = 52.
代入求值是指把每个变量换成给定的数，并严格按 BIDMAS 计算。若 x = 4，y = –2，则 3x² – 2y = 3 × 16 – 2 × (–2) = 48 + 4 = 52。
5. Expanding Brackets and Factorising | 展开括号与因式分解
To expand a single bracket, multiply the term outside by every term inside. Watch the signs carefully, especially when the outside term is negative.
展开单括号时，用括号外的项分别乘以括号内的每一项。要特别注意符号，尤其当括号外的项为负时。
a(b + c) = ab + ac –a(b – c) = –ab + ac
To expand double brackets, multiply every term in the first bracket by every term in the second. A common check is to remember the four products: first, outside, inside, last.
展开双括号时，用第一个括号中的每一项去乘第二个括号中的每一项。常见的检查方法是记住四组乘积：首项、外项、内项、末项。
(x + a)(x + b) = x² + (a + b)x + ab
(x + a)² = x² + 2ax + a² (x – a)² = x² – 2ax + a²
(x + a)(x – a) = x² – a²
The last line is the difference of two squares, an identity worth memorising because it appears constantly in later algebra and in simplifying surds.
最后一行是平方差公式，这个恒等式值得牢记，因为它在后续代数和根式化简中反复出现。
Factorising reverses expansion. Always remove the highest common factor first, then check whether the remaining expression is a difference of two squares or a simple quadratic trinomial.
因式分解是展开的逆过程。先提取最大公因式，再检查剩下的表达式是否为平方差或简单的二次三项式。
6x + 9 = 3(2x + 3) – common factor / 6x + 9 = 3(2x + 3)（公因式）
x² – 25 = (x + 5)(x – 5) – difference of squares / x² – 25 = (x + 5)(x – 5)（平方差）
x² + 7x + 12 = (x + 3)(x + 4) – trinomial / x² + 7x + 12 = (x + 3)(x + 4)（三项式）
6. Linear Equations and Inequalities | 一元一次方程与不等式
Solving a linear equation means finding the value of the variable that makes both sides equal. Do the same operation to both sides and work in reverse BIDMAS order: undo addition and subtraction first, then multiplication and division.
解一元一次方程就是求出使等式两边相等的未知数的值。两边同时做相同的运算，并按与 BIDMAS 相反的顺序操作：先抵消加减，再抵消乘除。
If the variable appears on both sides, collect the variable terms on one side and the numbers on the other. Always check your answer by substituting it back into the original equation.
若未知数出现在两边，把含未知数的项移到一边，把常数移到另一边。务必把答案代回原方程检验。
3x + 5 = 2x + 11 → x = 6
Inequalities are solved in exactly the same way, but there is one vital extra rule: multiplying or dividing both sides by a negative number reverses the inequality sign.
不等式的解法完全相同，但有一条关键附加规则：两边同时乘以或除以负数时，不等号方向要反转。
–2x > 8 → x < –4
ution set on a number line: an open circle for < and >, a filled circle for ≤ and ≥.
要记住四个不等号符号，以及如何在数轴上表示解集：< 和 > 用空心圆，≤ 和 ≥ 用实心圆。
7. Rearranging Formulae | 公式变形
Rearranging a formula, sometimes called changing the subject, uses exactly the same skills as solving an equation. The letter you want becomes the subject on the left-hand side.
公式变形（也称"换主元"）所用的技巧与解方程完全相同。你要求的那个字母最终成为等号左边的"主元"。
Undo the operations in reverse order, keeping the formula balanced at every step. If the required variable appears twice, collect those terms together and factorise.
按相反顺序逐步抵消运算，每一步都保持等式两边平衡。如果所求变量出现两次，先把这些项归到一起再提取公因式。
P = 2(l + w) → l = P/2 – w
v = u + at → t = (v – u)/a
A = πr² → r = √A/π
Note that taking a square root is the inverse of squaring, and that only the positive root is usually required in geometric formulae because lengths and radii cannot be negative.
注意，开平方是平方的逆运算；在几何公式中通常只取正根，因为长度和半径不可能为负。
A useful habit is to write down the formula, then draw a small arrow or note beside each term showing the operation you are about to undo. This reduces sign and order errors dramatically.
一个实用的习惯是：先抄下公式，然后在每一项旁边用小箭头或注释标出你接下来要抵消的运算。这样能大大减少符号和顺序错误。
8. Sequences and Number Patterns | 数列与数字规律
A sequence is an ordered list of numbers, and each number in the list is called a term. The first term is written u₁ or a, the second u₂, and the nth term uₙ.
数列是按顺序排列的一列数，其中的每个数称为项。第一项记作 u₁ 或 a，第二项记作 u₂，第 n 项记作 uₙ。
In a linear (arithmetic) sequence, the difference between consecutive terms is constant. This constant is called the common difference, d, and the sequence can be built by adding d repeatedly.
在线性（等差）数列中，相邻两项的差是常数。这个常数称为公差 d，数列可以通过不断加上 d 得到。
uₙ = a + (n – 1)d
For a simple linear sequence such as 5, 8, 11, 14, … e common difference is 3 and the nth term is 3n + 2. Check by substituting n = 1, 2 and 3.
对于 5, 8, 11, 14, … 这样的简单线性数列，公差为 3，通项为 3n + 2。可代入 n = 1、2、3 来检验。
Some sequences are not linear. Squares are 1, 4, 9, 16, 25, …, cubes are 1, 8, 27, 64, …, the triangular numbers are 1, 3, 6, 10, 15, …, and the Fibonacci sequence starts 1, 1, 2, 3, 5, 8, … with each term being the sum of the two before it.
有些数列不是线性的。平方数为 1, 4, 9, 16, 25, …；立方数为 1, 8, 27, 64, …；三角数为 1, 3, 6, 10, 15, …；斐波那契数列以 1, 1, 2, 3, 5, 8, … 开头，每项都是前两项之和。
Triangular number Tₙ = n(n + 1)/2
9. Angles, Triangles and the Circle | 角、三角形与圆
Angle facts are used in almost every geometry question, so they must be quoted accurately in your working. Give the reason every time you find an angle.
角的知识几乎出现在每一道几何题中，因此作答时必须准确写出理由。每求出一个角，都要给出依据。
Angles on a straight line add to 180°. / 同一直线上的角之和为 180°。
Angles around a point add to 360°. / 围绕一点的角之和为 360°。
Vertically opposite angles are equal. / 对顶角相等。
Angles in a triangle add to 180°; in a quadrilateral they add to 360°. / 三角形内角和为 180°；四边形内角和为 360°。
The exterior angle of a triangle equals the sum of the two opposite interior angles. / 三角形的外角等于不相邻的两个内角之和。
Alternate, corresponding and co-interior angles appear when a transversal crosses parallel lines. / 当截线穿过平行线时，会出现内错角、同位角和同旁内角。
The circle has two essential formulae for Year 8: the circumference, which is the distance around the circle, and the area, which is the space inside it.
圆在 Year 8 有两个基本公式：周长，即绕圆一周的长度；面积，即圆内部的区域大小。
C = 2πr = πd A = πr²
or the angle at the centre. So arc length = (θ/360) × 2πr and sector area = (θ/360) × πr².
对于弧或扇形，取整个圆的相应分数：用圆心角 θ/360。因此弧长 = (θ/360) × 2πr，扇形面积 = (θ/360) × πr²。
10. Pythagoras and Mensuration | 勾股定理与周长面积体积
Pythagoras' theorem states that in any right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides. The hypotenuse is the longest side and is always opposite the right angle.
勾股定理指出：在任何直角三角形中，斜边的平方等于另外两条边的平方和。斜边是最长边，且永远对着直角。
a² + b² = c²
To find the hypotenuse, add the squares and take the square root. To find a shorter side, subtract the known square from the hypotenuse square before taking the root.
求斜边时，把两个平方相加再开方。求较短边时，先用斜边的平方减去已知的平方，再开方。
c = √a² + b² a = √c² – b²
triangle, then the triangle must be right-angled. This is a common justification question.
该定理的逆定理同样成立：若三角形三边满足 a² + b² = c²，则该三角形必为直角三角形。这是常见的说理题考点。
| Shape / 图形 | Formula / 公式 |
| Rectangle / 长方形 | A = l × w |
| Triangle / 三角形 | A = ½ × b × h |
| Parallelogram / 平行四边形 | A = b × h |
| Trapezium / 梯形 | A = ½(a + b)h |
| Cuboid / 长方体 | V = l × w × h |
| Prism / 棱柱 | V = area of cross-section × length |
| Cylinder / 圆柱 | V = πr²h |
Always check that all lengths are in the same unit before substituting into any area or volume formula, and give volume answers in cubic units such as cm³.
代入任何面积或体积公式之前，务必检查所有长度单位是否一致；体积答案要用立方单位，如 cm³。
11. Statistics: Mean, Median, Mode and Range | 统计：平均数、中位数、众数与极差
The mean is the total of all values divided by the number of values. It uses every piece of data, so it is affected by extreme values.
平均数等于所有数值之和除以数值的个数。它用到了每一个数据，因此会受到极端值的影响。
Mean = Σx / n
The median is the middle value when the data are arranged in order. If there is an even number of values, the median is the mean of the two middle values.
中位数是把数据按顺序排列后的中间值。若数据个数为偶数，中位数是中间两个数的平均数。
The mode is the value that appears most often, and there may be no mode or more than one mode. The range is the largest value minus the smallest and measures the spread, not an average.
众数是出现次数最多的数值，可能没有众数，也可能不止一个。极差等于最大值减最小值，它衡量的是数据的离散程度，而不是平均值。
Range = highest – lowest
For data given in a frequency table, the mean is found by multiplying each value by its frequency, adding these products, then dividing by the total frequency.
对于以频数表给出的数据，平均数求法为：每个数值乘以其频数，把这些乘积相加，再除以总频数。
Mean = Σfx / Σf
12. Probability | 概率
Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). It can be written as a fraction, decimal or percentage.
概率衡量某个事件发生的可能性，取值从 0（不可能）到 1（必然）。它可以用分数、小数或百分数表示。
P(E) = number of favourable outcomes ÷ total number of equally likely outcomes
The probabilities of all possible outcomes add up to 1, so the probability of an event not happening is 1 minus the probability that it does. This is called the complement.
所有可能结果的概率之和为 1，因此某事件不发生的概率等于 1 减去它发生的概率。这称为对立事件（补事件）。
P(not E) = 1 – P(E)
For two independent events, multiply the probabilities to find the probability that both happen. For mutually exclusive events, which cannot happen at the same time, add the probabilities.
对于两个相互独立的事件，把概率相乘即可得到两者同时发生的概率。对于互斥事件（不可能同时发生），把概率相加。
P(A and B) = P(A) × P(B) P(A or B) = P(A) + P(B)
When a question involves two stages, such as rolling two dice or taking two counters without replacement, a sample space diagram or tree diagram makes the calculation much clearer and helps you list all outcomes systematically.
当题目涉及两个阶段时，例如掷两枚骰子或先后取两个球且不放回，用样本空间图或树状图会让计算清晰得多，也能帮助你系统地列出所有结果。
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