📚 Year 8 Cambridge Computing: Formula & Theorem Quick Reference Guide | 剑桥八年级计算机：公式定理速查手册
This quick reference guide compiles essential formulas, theorems, and concepts for the Year 8 Cambridge Computing curriculum. It covers number systems, Boolean logic, data representation, file size calculations, and more to support revision and practice. Each entry pairs an English explanation with a Chinese translation, making it easy to check key points before tests or homework.
本速查手册汇编了剑桥八年级计算机课程的核心公式、定理和概念。涵盖数制、布尔逻辑、数据表示、文件大小计算等内容，助力复习与练习。每个条目均提供中英双语解释，便于考前查漏补缺。
1. Binary to Decimal Conversion | 二进制与十进制转换
In the binary system (base 2), each digit represents a power of 2. To convert a binary number to decimal, multiply each bit by 2 raised to the exponent of its position (starting from 0 on the far right) and add all the products together.
在二进制系统（基数为2）中，每一位数字代表 2 的幂。将二进制数转换为十进制时，将每一位乘以 2 的位权（从最右侧的 0 次方开始），再把所有乘积相加。
Decimal = dn-1 × 2n-1 + dn-2 × 2n-2 + ... + d1 × 21 + d0 × 20
Example: Convert binary 1011₂ to decimal.
1011₂ = (1 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰) = 8 + 0 + 2 + 1 = 11₁₀
示例：将二进制 1011₂ 转为十进制。
1011₂ = (1 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰) = 8 + 0 + 2 + 1 = 11₁₀
2. Decimal to Binary Conversion | 十进制与二进制转换
Repeatedly divide the decimal number by 2. Write down the remainder (0 or 1) each time, then read the remainders from bottom to top as the binary equivalent. Stop when the quotient becomes 0.
将十进制数反复除以 2，每次记下余数（0 或 1），直到商为 0。从下往上读取所有余数，即得到对应的二进制数。
Example: Convert 13₁₀ to binary.
13 ÷ 2 = 6 remainder 1
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Reading upwards: 1101₂
示例：将 13₁₀ 转为二进制。
13 ÷ 2 = 6 余 1
6 ÷ 2 = 3 余 0
3 ÷ 2 = 1 余 1
1 ÷ 2 = 0 余 1
自下而上读取：1101₂
3. Hexadecimal Conversions | 十六进制转换
Hexadecimal (base 16) uses digits 0–9 and letters A– rsion between hex and binary is straightforward because each hex digit maps to exactly four binary bits.
十六进制（基数为16）使用数字 0–9 和字母 A–F，其中 A=10，B=11，...，F=15。十六进制与二进制的转换非常直接，因为每一位十六进制数字恰好对应四位二进制数。
group binary digits into sets of four (starting from the right), then convert each group.
Hex → Binary: replace each hex digit with its 4-bit binary equivalent.
Example: 1011 1100₂ → BC₁₆ (1011 = B, 1100 = C).
二进制转十六进制：从右向左将二进制数四位一组分组，再将每组转为十六进制。
十六进制转二进制：将每位十六进制数替换为对应的四位二进制数。
示例：1011 1100₂ → BC₁₆（1011 即 B，1100 即 C）。
Hex to decimal: multiply each digit by 16 raised to its position, then sum.
Example: 2F₁₆ = (2 × 16¹) + (15 × 16⁰) = 32 + 15 = 47₁₀
十六进制转十进制：每位乘以 16 的位权并求和。
示例：2F₁₆ = (2 × 16¹) + (15 × 16⁰) = 32 + 15 = 47₁₀
4. Binary Shifts | 二进制移位
A logical shift moves every bit left or right by a specified number of places. Vacant positions are filled with zeros. Left shifts multiply the number by powers of two; right shifts perform integer division by powers of two (discarding the remainder).
逻辑移位将所有位向左或向右移动指定位数，空位补 0。左移相当于乘以 2 的幂次；右移相当于整数除以 2 的幂次（丢弃余数）。
Logical Left Shift: a << n = a × 2n
Logical Right Shift: a >> n = a ÷ 2n (floor division)
Example: 00001100₂ (12) shifted left 2 places: 00110000₂ (12 × 2² = 48).
Example: 00110000₂ (48) shifted right 2 places: 00001100₂ (48 ÷ 2² = 12).
示例：00001100₂（12）左移两位：00110000₂（12 × 2² = 48）。
示例：00110000₂（48）右移两位：00001100₂（48 ÷ 2² = 12）。
Note: bits shifted beyond the register boundary are lost; overflow may occur if the result exceeds the bit width.
注意：移出寄存器边界的位会丢失；若结果超出位宽则可能发生溢出。
5. Boolean Algebra Laws | 布尔代数定律
Boolean algebra uses binary variables (0 and 1) and operators AND ( · ), OR ( + ), NOT ( ¬ ). The following laws simplify logic expressions. Each law is shown in AND form and OR form.
布尔代数使用二进制变量（0 和 1）与运算符 与（·）、或（+）、非（¬）。以下定律可简化逻辑表达式，每条均列出与形式和或形式。
Identity Law: A · 1 = A ; A + 0 = A
同一律：A · 1 = A ； A + 0 = A
Null Law: A · 0 = 0 ; A + 1 = 1
零一律：A · 0 = 0 ； A + 1 = 1
Idempotent Law: A · A = A ; A + A = A
幂等律：A · A = A ； A + A = A
Complement Law: A · ¬A = 0 ; A + ¬A = 1
互补律：A · ¬A = 0 ； A + ¬A = 1
Commutative Law: A · B = B · A ; A + B = B + A
交换律：A · B = B · A ； A + B = B + A
Associative Law: (A · B) · C = A · (B · C) ; (A + B) + C = A + (B + C)
结合律：(A · B) · C = A · (B · C) ； (A + B) + C = A + (B + C)
Distributive Law: A · (B + C) = (A · B) + (A · C) ; A + (B · C) = (A + B) · (A + C)
分配律：A · (B + C) = (A · B) + (A · C) ； A + (B · C) = (A + B) · (A + C)
Absorption Law: A + (A · B) = A ; A · (A + B) = A
吸收律：A + (A · B) = A ； A · (A + B) = A
De Morgan's Laws: ¬(A · B) = ¬A + ¬B ; ¬(A + B) = ¬A · ¬B
德·摩根定律：¬(A · B) = ¬A + ¬B ； ¬(A + B) = ¬A · ¬B
6. Logic Gate Truth Tables | 逻辑门真值表
Logic gates implement Boolean functions. The truth table lists all possible input combinations and the resulting output. Below are the standard truth tables for the six basic gates.
逻辑门实现布尔函数。真值表列出了所有可能的输入组合及其对应输出。以下是六种基本门的标准真值表。
AND Gate (与门): output is 1 only when both inputs are 1.
与门：仅当两个输入均为 1 时，输出为 1。
| A | B | A AND B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
OR Gate (或门): output is 1 if at least one input is 1.
或门：至少一个输入为 1 时，输出为 1。
| A | B | A OR B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
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