📚 Year 7 CIE Computing: Formula & Theorem Quick Reference | Year 7 CIE 计算机：公式定理速查手册
This quick reference handbook brings together all the essential formulae and theorems you need for Year 7 CIE Computing. From binary conversions and data units to Boolean logic and simplification laws, every concept is explained with clear examples and paired English–Chinese explanations. Use it to revise, check your understanding, and build confidence for your assessments.
这本速查手册汇集了 Year 7 CIE 计算机所需的所有核心公式与定理。无论是二进制转换、数据单位，还是布尔逻辑和简化定律，每个概念都配有清晰的例子和中英文对照解释。用它来复习、检验你的理解，并为考试建立信心。
1. Binary to Decimal Conversion | 二进制转十进制
Binary numbers use base 2. Each bit (binary digit) holds a place value that is a power of 2, starting from 2⁰ on the rightmost side. Multiply each bit by its place value and add the results to get the decimal equivalent.
二进制数使用以 2 为基数的系统。每个比特（二进制位）具有 2 的幂的位值，最右边从 2⁰ 开始。将每位乘以它的位值并求和，即可得到对应的十进制数。
Example: 1011₂ = (1 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰) = 8 + 0 + 2 + 1 = 11.
示例：1011₂ = (1 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰) = 8 + 0 + 2 + 1 = 11。
2. Decimal to Binary Conversion | 十进制转二进制
To convert a decimal integer to binary, repeatedly divide the number by 2 and record the remainders. The binary equivalent is the remainders read in reverse order (from the last remainder to the first).
要将十进制整数转换为二进制，反复将数字除以 2 并记录余数。将这些余数从最后一个到第一个倒序读出，就得到二进制数。
Example: Convert 13 to binary. 13 ÷ 2 = 6 remainder 1; 6 ÷ 2 = 3 remainder 0; 3 ÷ 2 = 1 remainder 1; 1 ÷ 2 = 0 remainder 1. Reading the remainders upwards gives 1101₂.
示例：将 13 转换为二进制。13 ÷ 2 = 6 余 1；6 ÷ 2 = 3 余 0；3 ÷ 2 = 1 余 1；1 ÷ 2 = 0 余 1。从下往上读出余数得到 1101₂。
3. Binary Addition Rules | 二进制加法规则
Binary addition follows four simple rules and uses a carry when the sum exceeds 1. Mastering these rules is essential for arithmetic in computer systems.
二进制加法遵循四条简单规则，当和超过 1 时会产生进位。掌握这些规则对理解计算机系统的算术至关重要。
Rule 1: 0 + 0 = 0
规则 1：0 + 0 = 0
Rule 2: 0 + 1 = 1
规则 2：0 + 1 = 1
Rule 3: 1 + 0 = 1
规则 3：1 + 0 = 1
Rule 4: 1 + 1 = 0, with a carry of 1 into the next column (result is 10₂).
规则 4：1 + 1 = 0，并向下一列进位 1（结果为 10₂）。
Example: Add 1011₂ (11) and 1101₂ (13). Working from right to left: 1+1 = 0 carry 1; 1+0+carry1 = 0 carry 1; 0+1+carry1 = 0 carry 1; 1+1+carry1 = 1 carry 1. The final result is 11000₂ (24).
示例：将 1011₂ (11) 与 1101₂ (13) 相加。从右向左计算：1+1 = 0 进位 1；1+0+进位 1 = 0 进位 1；0+1+进位 1 = 0 进位 1；1+1+进位 1 = 1 进位 1。最终结果为 11000₂ (24)。
4. Data Storage Units | 数据存储单位
Digital data is measured in bits and bytes. Calculating file sizes and storage capacities requires knowing the exact conversion factors between units. All conversions are based on powers of 2.
数字数据以比特和字节衡量。计算文件大小和存储容量需要知道各单位之间的精确换算关系。所有换算都基于 2 的幂。
1 byte = 8 bits
1 字节 = 8 比特
1 kilobyte (KB) = 2¹⁰ bytes = 1024 bytes
1 千字节 (KB) = 2¹⁰ 字节 = 1024 字节
1 megabyte (MB) = 2²⁰ bytes = 1,048,576 bytes ≈ 1024 KB
1 兆字节 (MB) = 2²⁰ 字节 = 1,048,576 字节 ≈ 1024 KB
1 gigabyte (GB) = 2³⁰ bytes = 1,073,741,824 bytes ≈ 1024 MB
1 吉字节 (GB) = 2³⁰ 字节 = 1,073,741,824 字节 ≈ 1024 MB
1 terabyte (TB) = 2⁴⁰ bytes ≈ 1024 GB
1 太字节 (TB) = 2⁴⁰ 字节 ≈ 1024 GB
5. Logic Gates: Symbols and Truth Tables | 逻辑门：符号与真值表
The three fundamental logic gates are NOT, AND, and OR. Their behaviour is fully described by truth tables, which list all possible input combinations and the corresponding output.
三种基本逻辑门是 NOT、AND 和 OR。它们的运作完全由真值表描述，真值表列出所有可能的输入组合及对应的输出。
NOT Gate (Inverter)
非门（反相器）
| A | NOT A |
|---|---|
| 0 | 1 |
| 1 | 0 |
Boolean expression: Q = NOT A
布尔表达式：Q = NOT A
AND Gate
与门
| A | B | A AND B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
Boolean expression: Q = A AND B
布尔表达式：Q = A AND B
OR Gate
或门
| A | B | A OR B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
Boolean expression: Q = A OR B
布尔表达式：Q = A OR B
6. Commutative, Associative and Distributive Laws | 交换律、结合律与分配律
Boolean algebra has several fundamental laws that allow you to rearrange and simplify logic expressions. The commutative, associative and distributive laws mirror those in normal algebra but are proven with truth tables.
布尔代数有几条基本定律，可以用来重新排列和简化逻辑表达式。交换律、结合律和分配律与普通代数中的对应定律相似，但需要由真值表来证明。
A AND B = B AND A
与运算交换律：A 与 B 等于 B 与 A。
A OR B = B OR A
或运算交换律：A 或 B 等于 B 或 A。
(A AND B) AND C = A AND (B AND C)
与运算结合律：先运算 A 与 B 再与 C，等同于 A 与 (B 与 C) 的结果。
(A OR B) OR C = A OR (B OR C)
或运算结合律：先运算 A 或 B 再或 C，等同于 A 或 (B 或 C) 的结果。
A AND (B OR C) = (A AND B) OR (A AND C)
与对或的分配律：A 与 (B 或 C) 等于 (A 与 B) 或 (A 与 C)。
A OR (B AND C) = (A OR B) AND (A OR C)
或对与的分配律：A 或 (B 与 C) 等于 (A 或 B) 与 (A 或 C)。
7. Identity, Annulment and Idempotent Laws | 恒等律、零律与幂等律
These simple laws help you fix the value of an expression when one input is a constant (0 or 1) or when the same variable is repeated. They are extremely useful for simplifying logic circuits.
这些简单的定律在输入为常量 (0 或 1) 或重复使用同一变量时，帮助你确定表达式的值。它们在简化逻辑电路中非常有用。
A AND 1 = A
恒等律 (AND)：任何变量与 1 进行 AND 运算，结果仍为该变量。
A OR 0 = A
恒等律 (OR)：任何变量与 0 进行 OR 运算，结果仍为该变量。
A AND 0 = 0
零律 (AND)：任何变量与 0 进行 AND 运算，结果总是 0。
A OR 1 = 1
零律 (OR)：任何变量与 1 进行 OR 运算，结果总是 1。
A AND A = A
幂等律 (AND)：同一变量进行
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