📚 Year 9 CIE Computer Science: Formula & Theorem Quick Reference Guide | Year 9 CIE 计算机：公式定理速查手册
Welcome to your go-to revision guide for all the essential formulas and theorems in Year 9 CIE Computer Science. This concise handbook covers number systems, binary arithmetic, Boolean logic, data storage calculations, and more, providing you with fast access to the key concepts you need for exams.
欢迎使用这份 Year 9 CIE 计算机科学的公式定理速查手册。本手册涵盖了数制、二进制运算、布尔逻辑、数据存储计算等核心知识点，帮助你快速查阅考试必备的重要公式和定理。
1. Number System Conversions | 数制转换
Conversions between decimal (denary), binary, and hexadecimal are fundamental. You must be confident converting values in both directions. The place‑value system underpins all these transformations.
十进制（denary）、二进制和十六进制之间的转换是基础，必须熟练掌握双向转换。所有转换都基于位权（place‑value）的概念。
Binary to Denary: Multiply each bit by its place value (power of 2) and sum the results. For an n‑bit binary number bₙ₋₁bₙ₋₂...b₀, the denary value is calculated as:
Denary = bₙ₋₁ × 2ⁿ⁻¹ + bₙ₋₂ × 2ⁿ⁻² + ... + b₀ × 2⁰
二进制转十进制：将每一位乘以其对应的位权（2 的幂），然后求和。对于 n 位二进制数 bₙ₋₁bₙ₋₂...b₀，十进制值计算公式如上所示。
Denary to Binary: Repeatedly divide the denary number by 2, recording the remainders. The binary equivalent is the sequence of remainders read from bottom to top (last remainder is the most significant bit).
十进制转二进制：不断将十进制数除以 2，记录每次的余数。从下往上读取余数（最后的余数为最高有效位）即得到二进制值。
Binary to Hexadecimal: Group the binary digits into nibbles (groups of 4 bits) from the right. Convert each nibble into the corresponding hex digit using the table below. Hexadecimal to Binary is the reverse process: replace each hex digit with its 4‑bit binary equivalent.
二进制与十六进制互转：将二进制位从右至左每 4 位一组（不足补 0），每组转换为对应的十六进制数字（见下表）。十六进制转二进制则反向操作，每位十六进制数字替换为 4 位二进制数。
| Denary | Binary (4‑bit) | Hex |
|---|---|---|
| 0 | 0000 | 0 |
| 1 | 0001 | 1 |
| 2 | 0010 | 2 |
| 3 | 0011 | 3 |
| 4 | 0100 | 4 |
| 5 | 0101 | 5 |
| 6 | 0110 | 6 |
| 7 | 0111 | 7 |
| 8 | 1000 | 8 |
| 9 | 1001 | 9 |
| 10 | 1010 | A |
| 11 | 1011 | B |
| 12 | 1100 | C |
| 13 | 1101 | D |
| 14 | 1110 | E |
| 15 | 1111 | F |
This reference table is essential for any conversion involving hexadecimal. Hex digits A to F represent denary values 10 to 15.
这张对照表对于任何涉及十六进制的转换都至关重要。十六进制数字 A 到 F 分别表示十进制值 10 至 15。
2. Binary Arithmetic and Overflow | 二进制算术与溢出
arry of 1 to the next column. If the carry propagates beyond the most significant bit and we are limited to a fixed number of bits (e.g. 8 bits), an overflow error occurs.
二进制加法遵循简单规则：0+0=0，0+1=1，1+0=1，1+1=0 并向高位进 1。当进位超出最高有效位，而我们限定在固定位宽（如 8 位）时，就会发生溢出错误。
For unsigned binary numbers with n bits, the valid range is 0 to (2ⁿ − 1). If a sum falls outside this range, the result cannot be represented correctly using only n bits, and the overflow flag is set.
对于 n 位无符号二进制数，合法范围是 0 到 (2ⁿ − 1)。若计算结果超出此范围，便无法用 n 位正确表示，溢出标志会被置位。
Example: Adding the 8‑bit values 1010 1101 (173) and 0111 0011 (115) gives 1 0010 0000. The extra carry beyond the 8 bits indicates an overflow, because the true sum (288) exceeds 255.
示例：8 位值 1010 1101（173）与 0111 0011（115）相加得到 1 0010 0000。超出 8 位的额外进位表明发生了溢出，因为实际和（288）超过了 255。
To detect overflow in unsigned addition, check whether there is a carry out of the most significant bit. In signed representations, overflow detection is more nuanced, but at Year 9 level you will mostly deal with unsigned arithmetic.
检测无符号加法溢出只需检查最高位是否有进位输出。有符号数的溢出检测更复杂，但在 Year 9 阶段大多处理无符号运算。
3. Logical Shifts | 逻辑移位
A logical shift moves all bits in a binary number to the left or right by a specified number of positions. Vacated positions are filled with zeros, and bits shifted out are discarded.
逻辑移位将二进制数中的所有位向左或向右移动指定的位数。空出的位置补 0，移出的位直接丢弃。
Left logical shift: Shifting left by n positions multiplies the original number by 2ⁿ. This is a very efficient way to perform multiplication by powers of two.
Result = number × 2ⁿ
左逻辑移位：左移 n 位相当于将原数乘以 2ⁿ。这是进行 2 的幂次乘法的极高效方式。
Right logical shift: Shifting right by n positions divides the original number by 2ⁿ, discarding any fractional part (integer division).
Result = number ÷ 2ⁿ
右逻辑移位：右移 n 位相当于将原数除以 2ⁿ，并舍弃小数部分（即整数除法）。
Be aware that if a '1' bit is shifted out on the left during a left shift, the result will be incorrect unless the register is wide enough. Similarly, right shifts lose the least significant bits.
注意，若左移过程中有 “1” 被移出左侧，除非寄存器宽度足够，否则结果将是错误的。同样，右移会丢失最低有效位。
4. Boolean Algebra Theorems | 布尔代数定理
Boolean algebra provides a set of laws that allow you to simplify logic expressions, reducing the number of gates required in a circuit. The most important theorems are summarised below.
布尔代数提供了一系列定律，可用来化简逻辑表达式，从而减少电路中所需的门数量。最重要的定理总结如下。
| Law | AND form | OR form |
|---|---|---|
| Identity | A · 1 = A | A + 0 = A |
| Null (Annulment) | A · 0 = 0 | A + 1 = 1 |
| Idempotent | A · A = A | A + A = A |
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