📚 KS3 CIE Computer Science: Formula & Theorem Quick Reference Handbook | KS3 CIE 计算机：公式定理速查手册
This quick reference handbook brings together all the essential formulas, conversions, and logic theorems you will encounter in the KS3 CIE Computer Science curriculum. Use it for revision, homework, and exam preparation.
本速查手册汇集了KS3 CIE计算机科学课程中所有基本公式、转换和逻辑定理。可用于复习、作业和考试准备。
1. Binary to Denary Conversion | 二进制转十进制
Every binary digit (bit) has a place value that is a power of 2. Starting from the rightmost bit (2⁰), the place values double as you move left. Multiply each bit by its place value and add the results to get the denary equivalent.
每个二进制数字（位）都有一个以2为底的位值。从最右边的位（2⁰）开始，向左移动时位值加倍。将每个位乘以其位值，再将结果相加，即可得到十进制等效值。
Denary = bₙ₋₁ × 2ⁿ⁻¹ + bₙ₋₂ × 2ⁿ⁻² + … + b₁ × 2¹ + b₀ × 2⁰
For example, the binary number 1011₂ has place values 8, 4, 2, 1. Therefore, (1×8) + (0×4) + (1×2) + (1×1) = 8 + 0 + 2 + 1 = 11 in denary.
例如，二进制数 1011₂ 的位值为 8、4、2、1。因此，(1×8) + (0×4) + (1×2) + (1×1) = 8 + 0 + 2 + 1 = 11（十进制）。
| Binary Bit | 1 | 0 | 1 | 1 |
|---|---|---|---|---|
| Place value (2ⁿ) | 2³ = 8 | 2² = 4 | 2¹ = 2 | 2⁰ = 1 |
| Bit × Value | 1 × 8 = 8 | 0 × 4 = 0 | 1 × 2 = 2 | 1 × 1 = 1 |
The total sum 8+0+2+1 = 11 confirms the conversion.
总和 8+0+2+1 = 11 证实了转换结果。
2. Denary to Binary Conversion | 十进制转二进制
Divide the denary number repeatedly by 2, keeping track of the remainders (0 or 1). Stop when the quotient reaches 0. Then read the remainders from bottom to top to form the binary equivalent.
将十进制数反复除以 2，记录余数（0 或 1）。当商为 0 时停止。然后从下往上读取余数，得到二进制数。
Denary value → Divide by 2 → Record remainder → Repeat with quotient
Example: Convert 29 to binary. 29 ÷ 2 = 14 remainder 1; 14 ÷ 2 = 7 remainder 0; 7 ÷ 2 = 3 remainder 1; 3 ÷ 2 = 1 remainder 1; 1 ÷ 2 = 0 remainder 1. Reading upwards: 11101₂.
示例：将 29 转换为二进制。29 ÷ 2 = 14 余 1；14 ÷ 2 = 7 余 0；7 ÷ 2 = 3 余 1；3 ÷ 2 = 1 余 1；1 ÷ 2 = 0 余 1。向上读取：11101₂。
To verify, use the binary-to-denary method: 1×16 + 1×8 + 1×4 + 0×2 + 1×1 = 29.
验证时，使用二进制转十进制方法：1×16 + 1×8 + 1×4 + 0×2 + 1×1 = 29。
3. Binary Addition | 二进制加法
Binary addition follows four simple rules. When two 1s are added, the result is 0 with a carry of 1 into the next column on the left. These rules are the foundation of arithmetic inside the CPU.
二进制加法遵循四条简单规则。两个 1 相加时，结果为 0，并向左进 1。这些规则是 CPU 内部算术的基础。
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, carry 1 to the next most significant bit.
规则4：1 + 1 = 0，向更高有效位进 1。
Example: Add 1011 (11) and 0110 (6). Column 0: 1+0=1; column 1: 1+1=0 carry 1; column 2: 0+1+carry 1=0 carry 1; column 3: 1+0+carry 1=0 carry 1; final carry is 1. Result: 10001₂ (17).
示例：计算 1011 (11) 加 0110 (6)。第0列：1+0=1；第1列：1+1=0 进位 1；第2列：0+1+进位1=0 进位 1；第3列：1+0+进位1=0 进位 1；最终进位为 1。结果：10001₂ (17)。
4. Hexadecimal Conversions | 十六进制转换
Hexadecimal (base 16) uses digits 0-9 and letters A-F to represent values 10 to 15. Each hex digit corresponds exactly to a 4-bit binary nibble, making it a compact way to represent binary.
十六进制（基数为 16）使用数字 0-9 和字母 A-F 表示 10 到 15 的值。每个十六进制数字完全对应一个 4 位二进制半字节，因此它是一种紧凑的二进制表示方式。
Hex value = dₙ₋₁ × 16ⁿ⁻¹ + … + d₀ × 16⁰
To convert binary to hex, group bits into nibbles from the right, then replace each nibble with its hex symbol.
要将二进制转换为十六进制，从右边开始将位分成每 4 位一组，然后用相应的十六进制符号替换每组。
| Binary Nibble | Hex | Denary |
|---|---|---|
| 0000 | 0 | 0 |
| 0001 | 1 | 1 |
| 0010 | 2 | 2 |
| 0011 | 3 | 3 |
| 0100 | 4 | 4 |
| 0101 | 5 | 5 |
| 0110 | 6 | 6 |
| 0111 | 7 | 7 |
| 1000 | 8 | 8 |
| 1001 | 9 | 9 |
| 1010 | A | 10 |
| 1011 | B | 11 |
| 1100 | C | 12 |
| 1101 | D | 13 |
| 1110 | E | 14 |
| 1111 | F | 15 |
Example: 11010011₂ becomes 1101 0011, which is D and 3, giving D3₁₆. Conversely, hex A2F = 1010 0010 1111₂.
示例：11010011₂ 分成 1101 和 0011，对应 D 和 3，得到 D3₁₆。反过来，十六进制 A2F = 1010 0010 1111₂。
5. Data Storage Units | 数据存储单位
Data size in computers is measured using a hierarchy of units. The smallest unit is the bit (binary digit). Larger units are powers of 2, not 10. Remember these conversion factors for all calculations.
计算机中的数据大小使用一系列单位来衡量。最小单位是位（二进制数字）。更大的单位是 2 的幂次，而非 10 的幂次。请牢记这些换算系数以应对所有计算。
| Unit | Equivalent | Power of 2 |
|---|---|---|
| 1 bit (b) | Smallest unit of data | – |
| 1 nibble | 4 bits | 2² |
| 1 byte (B) | 8 bits | 2³ |
| 1 kilobyte (KB) |
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