📚 AS Eduqas Computer Science: Formula & Theorem Quick Reference Handbook | AS Eduqas 计算机：公式定理速查手册
This handbook provides a concise reference for all essential formulas, theorems, and performance metrics covered in the AS Eduqas Computer Science specification. From Boolean logic laws to CPU performance equations, this guide helps you quickly review the mathematical foundations required for the exam. Use it alongside your practice to strengthen your problem-solving skills.
本手册提供了 AS Eduqas 计算机科学课程中涵盖的所有核心公式、定理和性能指标的简明参考。从布尔逻辑定律到 CPU 性能方程式，本指南可帮助你快速复习考试所需的数学基础。配合练习使用，能强化你的解题能力。
1. Boolean Algebra Laws | 布尔代数定律
Boolean algebra operates on binary variables. The fundamental laws govern logical operations AND (.), OR (+), and NOT ('). These laws enable simplification of complex logic expressions.
布尔代数运算基于二进制变量。基本定律规定了逻辑与（.）、逻辑或（+）和逻辑非（'）的运算规则。这些定律可用于简化复杂的逻辑表达式。
1 and A . 0 = 0. The Idempotent Laws: A + A = A and A . A = A. The Complement Laws: A + A' = 1 and A . A' = 0. The Double Complement Law: (A')' B) + (A . C) and A + (B . C) = (A + B) . (A + C). The Absorption Laws: A + (A . B) = A and A . (A + B) = A.
恒等律：A + 0 = A，A . 1 = A。零一律：A + 1 = 1，A . 0 = 0。幂等律：A + A = A，A . A = A。补码律：A + A' = 1，A . A' = 0。双重补码律：(A')' = A。交换律：A + B = B + A，A . B = B . A。结合律：(A + B) + C = A + (B + C)，(A . B) . C = A . (B . C)。分配律：A . (B + C) = (A . B) + (A . C)，A + (B . C) = (A + B) . (A + C)。吸收律：A + (A . B) = A，A . (A + B) = A。
2. De Morgan's Theorems | 德摩根定理
De Morgan's theorems provide a method to transform conjunctions into disjunctions and vice versa, using negation. They are vital for converting logic gates between NAND/NOR implementations.
德摩根定理提供了一种利用否定将合取式转化为析取式的方法，反之亦然。它们在将逻辑门转换为纯与非或纯或非实现时至关重要。
Theorem 1: (A . B)' = A' + B' This states that the complement of an AND is the OR of the complements. Theorem 2: (A + B)' = A' . B' OR is the AND of the complements. These can be extended to multiple variables: (A . B . C)' = A' + B' + C'.
定理1：(A . B)' = A' + B'。这表示与门的补等于补的或。定理2：(A + B)' = A' . B'。这表示或门的补等于补的与。这些定理可扩展到多个变量：(A . B . C)' = A' + B' + C'。
3. Boolean Simplification & Karnaugh Maps | 布尔化简与卡诺图
ressions by grouping adjacent cells containing 1s. The simplification uses the rule that A' + A = 1, thus allowing a variable to be eliminated.
卡诺图通过将含 1 的相邻方格分组，提供了化简布尔表达式的直观方法。化简利用了 A' + A = 1 这一规则，从而消去变量。
For a 2-variable K-map, groups of size 2, 4 result in term elimination. For a 3-variable map, groups must be rectangular and wrap-around edges are allowed. The simplified expression is formed by the sum of terms corresponding to each group, where a variable is omitted if it changes within the group. Don't-care conditions (X) can be treated as 0 or 1 to maximise grouping.
对于双变量卡诺图，大小为 2 或 4 的分组可消去项。对于三变量卡诺图，分组必须为矩形且允许边缘环绕。化简表达式由每个分组对应项的或构成，若变量在组内发生变化则省略该项。无关条件（X）可视为 0 或 1 以最大化分组。
4. Binary Arithmetic & Two's Complement | 二进制算术与补码
Two's complement is used to represent negative integers in binary. To find the two's complement of a number, invert all bits (one's complement) and add 1. For an n-bit number, the range of representable integers is –2ⁿ⁻¹ to 2ⁿ⁻¹ – 1.
补码用于表示二进制中的负整数。求一个数的补码，需将所有位取反（反码）后加 1。对于 n 位二进制数，可表示的整数范围为 –2ⁿ⁻¹ 至 2ⁿ⁻¹ – 1。
e, indicated by a carry into the sign bit that differs from the carry out. Subtraction is performed by adding the two's complement of the subtrahend: A – B = A + (–B).
二进制加法遵循与十进制相同的规则，进位方式为：0+0=0，0+1=1，1+0=1，1+1=0 进位 1。当结果超出范围时会发生溢出，表现为进入符号位的进位与出符号位的进位不同。减法通过加上减数的补码实现：A – B = A + (–B)。
5. Floating Point Representation & Normalisation | 浮点表示与规格化
ent. The mantissa is usually normalised to maximise precision, meaning the binary point is positioned so that the most significant bit of the mantissa is 1 for positive numbers, or 0 for negative numbers (in two's complement).
实数使用带尾数和阶码的浮点记数法存储：数值 = 尾数 × 2^阶码。尾数通常进行规格化以最大化精度，即正数的尾数最高位为 1，负数的尾数最高位为 0（补码表示）。
The formula for a normalised positive floating point number: mantissa is 0.1xxxx... (with the leading bit 1). For negative numbers: mantissa is 1.0xxxx... The exponent is biased to avoid storing a separate sign. For a given number of bits in mantissa (m) and exponent (e), the precision and range are determined. Trade-off: more bits in exponent increase range; more bits in mantissa increase precision.
正数规格化浮点数的形式为：尾数 = 0.1xxxx...（首位为 1）。负数为：尾数 = 1.0xxxx...。阶码采用移码表示以避免存储单独符号。给定尾数位数 m 和阶码位数 e，可确定精度和范围。权衡：阶码位数越多范围越大；尾数位数越多精度越高。
6. Error Detection: Parity Bits & Checksums | 错误检测：奇偶校验位与校验和
Error detection codes help identify transmission errors. A parity bit is added to make the total number of 1s even (
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