📚 Mastering Number: Integers, Powers and Roots | 掌握数字：整数、幂与根
This unit builds the number skills that underpin algebra, geometry and data handling. You will work with directed numbers, order of operations, powers, roots, prime factors, and conversions between fractions, decimals and percentages. Each skill is common in Year 9 exams and in everyday problem solving.
本单元将构建支撑代数、几何和数据处理的数字技能。你将学习有向数、运算顺序、幂、根、质因数以及分数、小数和百分比之间的转换。这些技能在九年级考试和日常问题解决中都很常见。
1. Directed Numbers and the Number Line | 有向数与数轴
Directed numbers are positive and negative numbers shown on a number line. Negative numbers lie to the left of zero, and positive numbers lie to the right. The further left a number is, the smaller its value; the further right, the larger its value.
有向数是数轴上表示的正数和负数。负数位于零的左侧，正数位于零的右侧。一个数越靠左，它的值越小；越靠右，值越大。
en you subtract a negative number, move right. For example, 5 + (−3) = 2 and 4 − (−2) = 6.
加一个负数时在数轴上向左移动；减一个负数时向右移动。例如，5 + (−3) = 2，而 4 − (−2) = 6。
Multiplication and division follow sign rules: same signs give a positive result, and different signs give a negative result.
乘法和除法遵循符号法则：同号得正，异号得负。
- English: (+5) × (+2) = +10 | Chinese: (+5) × (+2) = +10
- English: (−5) × (+2) = −10 | Chinese: (−5) × (+2) = −10
- English: (−5) × (−2) = +10 | Chinese: (−5) × (−2) = +10
otation, −3² means −(3 × 3) = −9, while (−3)² = 9. Brackets change the meaning completely.
在多步计算中请始终清楚地写出负号。一个常见错误是把 −3² 误读为 (−3)²。在标准记法中，−3² 表示 −(3 × 3) = −9，而 (−3)² = 9。括号会完全改变含义。
2. Order of Operations (BIDMAS) | 运算顺序 (BIDMAS)
BIDMAS tells us the order in which operations must be carried out: Brackets, Indices, Division and Multiplication from left to right, then Addition and Subtraction from left to right. Some schools use BODMAS, BEDMAS or PEMDAS, but the logic is the same.
BIDMAS 告诉我们运算必须按什么顺序进行：括号、指数、从左到右的除法和乘法，然后是从左到右的加法和减法。有些学校使用 BODMAS、BEDMAS 或 PEMDAS，但逻辑相同。
dition, so 4 × 2 = 8, then 3 + 8 = 11. If you add first, you would get 14, which is wrong.
示例：计算 3 + 4 × 2。乘法先于加法，所以 4 × 2 = 8，然后 3 + 8 = 11。如果先相加，你会得到 14，这是错误的。
so come before multiplication: in 2 × 3², first find 3² = 9, then multiply by 2 to get 18.
括号会改变运算顺序。计算 (3 + 4) × 2：括号内得 7，然后 7 × 2 = 14。指数也在乘法之前：在 2 × 3² 中，先求 3² = 9，再乘以 2 得到 18。
10 − 2 × (3 + 1)² = 10 − 2 × 4² = 10 − 2 × 16 = 10 − 32 = −22
Work step by step and show each line. This reduces mistakes and helps you gain method marks even if a calculation slips at the final stage.
一步一步计算并写出每一行。这可以减少错误，即使最后一步计算失误，也能帮助你获得方法分。
3. Powers and Indices | 幂与指数
The expression 3⁴ means 3 × 3 × 3 × 3, so 3⁴ = 81. Here 3 is the base and 4 is the index.
一个幂包含底数和指数。表达式 3⁴ 表示 3 × 3 × 3 × 3，所以 3⁴ = 81。这里 3 是底数，4 是指数。
elf once: 5² = 5 × 5 = 25. Cubing means using the number three times: 4³ = 4 × 4 × 4 = 64.
常见的幂包括平方和立方。平方表示一个数自乘一次：5² = 5 × 5 = 25。立方表示这个数使用三次：4³ = 4 × 4 × 4 = 64。
and (−12)⁰ = 1. This rule is useful for simplifying expressions such as x⁵ ÷ x⁵ = x⁰ = 1.
任何非零数的 0 次幂都等于 1。例如，7⁰ = 1，(−12)⁰ = 1。这一规则在化简表达式时非常有用，例如 x⁵ ÷ x⁵ = x⁰ = 1。
- English: 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36, 7² = 49, 8² = 64, 9² = 81, 10² = 100 | Chinese: 2² = 4，3² = 9，4² = 16，5² = 25，6² = 36，7² = 49，8² = 64，9² = 81，10² = 100
- English: 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125 | Chinese: 2³ = 8，3³ = 27，4³ = 64，5³ = 125
Memorise these common powers. They appear often in roots, volume questions and index law problems.
请记住这些常见幂。它们经常出现在根、体积问题和指数法则题目中。
4. Square Roots and Cube Roots | 平方根与立方根
The square root of a number reverses squaring. Since 7² = 49, we write √49 quare root of a number asks which positive value multiplies by itself to give that number.
平方根是平方的逆运算。因为 7² = 49，所以写作 √49 = 7。一个数的平方根是在问哪一个正值与自身相乘能得到该数。
The cube root reverses cubing. Since 4³ = 64, the cube root of 64 is 4. We write ∛ . Unlike square roots, cube roots of negative numbers are possible because (−4)³ = −64, so ∛(−64) = −4.
立方根是立方的逆运算。因为 4³ = 64，所以 64 的立方根是 4，写作 ∛64 = 4。与平方根不同，负数的立方根是存在的，因为 (−4)³ = −64，所以 ∛(−64) = −4。
For numbers that are not perfect squares or cubes, estimate between known values. Since √36 = 6 and √49 = 7, the value of √40 is between 6 and 7, closer to 6.3. A calculator gives √40 ≈ 6.324555...
对于不是完全平方数或完全立方数的数，可以在已知值之间进行估算。因为 √36 = 6 且 √49 = 7，所以 √40 的值在 6 和 7 之间，更接近 6.3。计算器给出 √40 ≈ 6.324555...
You should also know that √a × b = √a × √b for non-negative a and b. For example, √12 = √4 × 3 = √4 × √3 = 2√3. This is called simplifying a surd.
你还应知道，对于非负的 a 和 b，√a × b = √a × √b。例如，√12 = √4 × 3 = √4 × √3 = 2√3。这称为化简根式。
√50 = √25 × 2 = √25 × √2 = 5√2
This simplification is tested regularly in higher-level number and algebra questions.
这种化简在较高要求的数字和代数题目中经常考查。
5. Prime Factorisation | 质因数分解
A prime number has exactly two distinct factors: 1 and itself. The first prime numbers are 2, 3, 5, 7, 11, 13, 17, 19. Note that 1 is not prime because it has only one factor.
质数恰好有两个不同的因数：1 和它本身。最初的几个质数是 2、3、5、7、11、13、17、19。注意 1 不是质数，因为它只有一个因数。
tree helps break a number down step by step. For example, 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5.
质因数分解将一个数写成质数的乘积。因数树可以帮助逐步分解一个数
Published by TutorHao | Mathematics Revision Series | aleveler.com
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