📚 8H Unit 1: Number and Arithmetic Review | 8H 第一单元：数与算术复习
This unit reviews the number skills needed for higher-level mathematics: integers, fractions, decimals, percentages, powers, roots and standard form. Mastering these ideas allows you to solve problems accurately before algebra becomes more abstract.
本单元复习更高层次数学所需的数感技能：整数、分数、小数、百分数、幂、根和标准形式。掌握这些概念可以让你在代数变得更加抽象之前准确解题。
1. Order of Operations and Integer Arithmetic | 运算顺序与整数运算
The BIDMAS rule tells you the order in which operations must be carried out: Brackets, Indices (powers), Division/Multiplication, Addition/Subtraction. Division and multiplication have equal priority, so work from left to right. The same is true for addition and subtraction.
BIDMAS 法则说明计算的顺序：括号、指数、除法/乘法、加法/减法。除法和乘法优先级相同，因此从左到右计算；加法和减法同理。
3 + 4 × 5 = 3 + 20 = 23
With negative integers, remember that adding a negative is the same as subtracting, and multiplying two negatives gives a positive. For example, (−3) × (−2) = 6, while (−3) + (−2) = −5. Always apply the sign rules after deciding the operation order.
对于负整数，记住加一个负数等于减去它，两个负数相乘得正。例如 (−3) × (−2) = 6，而 (−3) + (−2) = −5。在确定运算顺序后，始终套用符号规则。
2. Factors, Multiples and Prime Factorisation | 因数、倍数与质因数分解
A factor divides a number exactly; a multiple is the result of multiplying a number by an integer. A prime number has exactly two factors: 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19. Prime factorisation writes any integer as a product of prime powers.
因数能整除一个数；倍数是一个数乘以整数所得的结果。质数恰好有两个因数：1 和它本身。前几个质数是 2、3、5、7、11、13、17、19。质因数分解将任意整数写成质数幂的乘积。
72 = 2³ × 3²
Use a factor tree: split 72 into 8 × 9, then 2 × 2 × 2 × 3 × 3. Group equal factors and write them as powers. You can then use this form for fraction work, HCF, LCM and index questions.
使用因数树：将 72 分成 8 × 9，再分成 2 × 2 × 2 × 3 × 3。把相同的因数合并写成幂。之后便可把这种形式用于分数、HCF、LCM 和指数题。
3. HCF and LCM Using Prime Factors | 利用质因数求最大公因数与最小公倍数
The highest common factor (HCF) is the largest number that divides all given numbers. The lowest common multiple (LCM) is the smallest number that all given numbers divide into. Prime factorisation gives a reliable method for both.
最大公因数 (HCF) 是能整除所有给定数的最大数。最小公倍数 (LCM) 是给定数都能整除的最小数。质因数分解为求这两者提供了可靠方法。
24 = 2³ × 3 and 36 = 2² × 3²
For HCF, take the lowest power of each common prime factor: 2² × 3 = 12. For LCM, take the highest power of every prime factor: 2³ × 3² = 72. This avoids listing multiples, especially with larger numbers.
求 HCF 时，取每个公共质因数的最低次幂：2² × 3 = 12。求 LCM 时，取每个质因数的最高次幂：2³ × 3² = 72。这种方法避免了列出所有倍数，尤其在数较大时更方便。
4. Fractions: Equivalence, Addition and Subtraction | 分数：等价、加法与减法
Equivalent fractions represent the same value: ½ = 2/4 = 8/16. To add or subtract fractions, first find a common denominator, usually the LCM of the denominators. Only add the numerators; keep the denominator unchanged.
等价分数表示相同的值：½ = 2/4 = 8/16。进行分数加减时，先找到一个公分母，通常是各分母的最小公倍数。只加分子，分母保持不变。
⅓ + ¼ = 4/12 + 3/12 = 7/12
For mixed numbers, convert to improper fractions, perform the operation, then convert back if needed. Simplify your final answer by dividing the numerator and denominator by their highest common factor.
对于带分数，先转换为假分数，再进行运算，必要时再转换回来。最后答案要通过分子和分母同时除以它们的最大公因数来化简。
5. Fractions: Multiplication and Division | 分数：乘法与除法
Multiply fractions by multiplying numerators and multiplying denominators. Simplify before or after multiplying to keep numbers small. To divide by a fraction, multiply by its reciprocal, which means swapping the numerator and denominator of the divisor.
分数相乘时，分子乘分子、分母乘分母。在乘法前或后约分，以使数字较小。分数相除时，乘以除数的倒数，也就是把除数的分子与分母互换。
⅔ × 5/7 = 10/21 and ⅔ ÷ 5/7 = ⅔ × 7/5
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