AS Cambridge Computer Science: Formula & Theorem Quick Reference | AS剑桥计算机公式定理速查手册

📚 AS Cambridge Computer Science: Formula & Theorem Quick Reference | AS剑桥计算机公式定理速查手册

Welcome to the AS Cambridge Computer Science Quick Reference Handbook. This concise guide brings together the essential formulas, theorems, and calculation methods you need for the exam. From binary conversions and Boolean algebra to performance metrics and error detection, each section presents the key facts in a clear, paired English‑Chinese format.

欢迎使用AS剑桥计算机科学公式定理速查手册。这份简明指南汇集了考试必备的基本公式、定理与计算方法，涵盖二进制转换、布尔代数、性能指标和错误检测等主题，每个部分都采用清晰的英中对照方式呈现关键知识。

1. Number Systems and Conversions | 数制与转换

Computers use binary (base 2), and we frequently work with denary (base 10) and hexadecimal (base 16). The value of an n‑bit binary number bₙ₋₁…b₀ is given by Σ (bᵢ × 2ⁱ), where each bit is multiplied by its place weight.

计算机使用二进制（基数为2），而我们常需处理十进制（基数为10）和十六进制（基数为16）。一个n位二进制数 bₙ₋₁…b₀ 的值为 Σ (bᵢ × 2ⁱ)，每位乘以其位权后求和。

To convert denary to binary, repeatedly divide the number by 2, writing down the remainder each time. The binary result is the sequence of remainders read from the last one obtained back to the first.

将十进制转换为二进制，只需反复将数字除以2并记录余数。二进制结果就是余数序列从后往前读取。

a compact shorthand: each hex digit represents exactly four binary bits. The digits use 0‑9 and A‑F (where A=10, B=11, …, F=15). To convert hex to binary, simply replace each hex digit with its 4‑bit equivalent. The reverse process groups binary bits into nibbles of four.

十六进制是一种紧凑的表示法：每个十六进制数字代表恰好四个二进制位。数字使用0‑9和A‑F（A=10，B=11，…，F=15）。十六进制转二进制只需将每位替换为对应的4位二进制值；反向转换则把二进制位每四位分为一组。

广告 / Sponsored

2. Two's Complement and Signed Integer Range | 二进制补码与有符号整数范围

Two’s complement is the standard way to represent signed integers in binary. In an n‑bit system, the most significant bit acts as the sign bit (0 for positive, 1 for negative). The range of representable integers is from −2ⁿ⁻¹ to 2ⁿ⁻¹ − 1.

二进制补码是表示有符号整数的标准方法。在n位系统中，最高位为符号位（0为正，1为负）。可表示的整数范围是 −2ⁿ⁻¹ 到 2ⁿ⁻¹ − 1。

lement), and then add 1 to the least significant bit. For example, to represent −6 in an 8‑bit register: +6 is 0000 0110; invert bits → 1111 1001; add 1 → 1111 1010.

求负数的补码：先写出对应正数的二进制，将每一位取反（得反码），然后加1。例如在8位寄存器中表示−6：+6为 0000 0110；取反 → 1111 1001；加1 → 1111 1010。

Overflow occurs when the result of an operation exceeds the representable range. For instance, adding two large positive numbers may produce a negative sign bit, indicating overflow. You can detect overflow by checking if the carry into the sign bit differs from the carry out.

当运算结果超出表示范围时会发生溢出。例如两个大正数相加可能产生负的符号位，表明溢出。可通过检查进入符号位的进位与出符号位的进位是否不同来检测溢出。

3. Floating‑Point Representation | 浮点数表示

Floating‑point numbers are stored in the form mantissa × 2^exponent, allowing a trade‑off between range and precision. In binary, the mantissa is a fractional binary number with an implied binary point, and the exponent is a two’s complement integer. The value is computed as Value = mantissa × 2^exponent.

浮点数以 尾数 × 2^指数 的形式存储，实现了范围与精度的折衷。在二进制中，尾数为带隐含二进制点的分数形式，指数则是采用补码的整数。数值计算为 值 = 尾数 × 2^指数。

nificant bit is 1, i.e. it lies in the range 0.5 ≤ mantissa < 1 (for binary). For example, 0.1010₂ × 2³ is normalised, while 0.0101₂ × 2⁴ is not.

规格化确保了数值表示的唯一性并最大化精度。对于正尾数，若最高有效位为1，即处于0.5 ≤ 尾数 < 1（二进制），则为规格化形式。例如 0.1010₂ × 2³ 是规格化的，而 0.0101₂ × 2⁴ 则不是。

Increasing the exponent bits extends the range, while increasing the mantissa bits improves precision. The total bit width is fixed, so you must balance range and precision for the application.

增加指数位可扩大表示范围，而增加尾数位可提高精度。由于总位宽固定，必须根据应用在范围与精度之间做出平衡。

4. Boolean Algebra and De Morgan’s Laws | 布尔代数与德摩根定律

Boolean algebra provides a formal system for manipulating logical expressions. The fundamental laws include: Commutative (A·B = B·A, A+B = B+A), Associative (A·(B·C) = (A·B)·C), Distributive (A·(B+C) = A·B + A·C), Absorption (A + A·B = A), Idempotent (A + A = A, A·A = A), and Double Negation ( (A')' = A).

布尔代数提供了处理逻辑表达式的形式体系。基本定律包括：交换律（A·B = B·A, A+B = B+A）、结合律（A·(B·C) = (A·B)·C）、分配律（A·(B+C) = A·B + A·C）、吸收律（A + A·B = A）、幂等律（A + A = A, A·A = A）以及双重否定律（(A')' = A）。

De Morgan’s laws are crucial for simplifying and transforming logic circuits. They state: (A·B)' = A' + B' and (A+B)' = A' · B'. In words: the complement of an AND is the OR of the complements; the complement of an OR is the AND of the complements.

德摩根定律对简化和转换逻辑电路至关重要。其形式为：(A·B)' = A' + B' 和 (A+B)' = A' · B'。即：与运算的补等于各补的或；或运算的补等于各补的与。

Example: F = (A + B)' · C → apply De Morgan → F = (A'·B') · C

例子：F = (A + B)' · C → 应用德摩根 → F = (A'·B')·C

5. Logic Gates and Truth Tables | 逻辑门与真值表

A logic gate is an electronic circuit that implements a Boolean function. The basic gates are AND, OR, NOT, NAND, NOR, XOR, and XNOR. Each has a distinct symbol and truth table. The output of an AND gate is 1 only if all inputs are 1; an OR gate outputs 1 if at least one input is 1; XOR (exclusive‑OR) outputs 1 when an odd number of inputs are 1.

逻辑门是实现布尔函数的电子电路。基本门包括与门、或门、非门、与非门、或非门、异或门和同或门，每个都有独特的符号和真值表。与门仅在所有输入均为1时输出1；或门只要至少一个输入为1就输出1；异或门在1的个数为奇数时输出1。

Gate	Symbol	Boolean Expression	Truth Table (A,B → Output)
AND	A·B	Q = A·B	00→0, 01→0, 10→0, 11→1
OR	A+B	Q = A+B	00→0, 01→1, 10→1, 11→1
广告 / Sponsored 现货 Cambridge International AS & ALevel IT Coursebook剑桥国际IGC… 计算机 备考教辅 · 正版现货 · 与本文内容相关 查看详情 / View on Taobao参考价 ¥467.0 更多咨询请联系16621398022（同微信） 备考资讯每日更新，关注不错过 考纲变化 · 科目分析 · 学习方法 在微信里长按二维码识别关注 或截图后用「扫一扫 → 相册」选择本图 公众号：tutorhao