AS CCEA Computer Science: Formula & Theorem Quick Reference Handbook | AS CCEA 计算机科学：公式定理速查手册

📚 AS CCEA Computer Science: Formula & Theorem Quick Reference Handbook | AS CCEA 计算机科学：公式定理速查手册

This handbook provides a concise collection of essential formulas, theorems, and quick rules for AS-Level CCEA Computer Science. It covers number systems, Boolean logic, data representation, compression, encryption, and performance metrics. Use it as a daily revision companion.

本手册为AS CCEA计算机科学课程汇集了关键公式、定理和速查要点，涵盖数制、布尔逻辑、数据表示、压缩、加密以及性能指标，是日常复习的得力助手。

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1. Number Systems & Conversions | 数制与转换

In computer science, numbers are represented in bases 2, 10 and 16. Converting between these bases is a fundamental skill. The value of a binary number bₙ₋₁…b₁b₀ is Σ bᵢ × 2ⁱ.

计算机科学中，数值用二、十、十六进制表示，相互转换是基本技能。二进制数bₙ₋₁…b₁b₀的值为Σ bᵢ × 2ⁱ。

To convert a decimal integer to binary, repeatedly divide by 2 and record the remainders in reverse order.

十进制整数转二进制时，不断除以2并逆序记录余数。

N = Q × 2 + R, stack remainders backwards

Binary to hexadecimal conversion groups bits into nibbles of four, each mapping to a hex digit (0–9, A–F).

二进制转十六进制将每四位分成一组，对应一个十六进制数字（0–9, A–F）。

Hex digit = 4-bit group value (0d–15d)

A hexadecimal number hₙ₋₁…h₀ has decimal value hₙ₋₁×16ⁿ⁻¹ + … + h₀×16⁰.

十六进制数hₙ₋₁…h₀的十进制值为hₙ₋₁×16ⁿ⁻¹ + … + h₀×16⁰。

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2. Binary Arithmetic & Overflow | 二进制算术与溢出

Binary addition works like decimal addition with carry propagation. Overflow is a concern in signed two’s complement arithmetic when the result exceeds the representable range.

二进制加法与十进制加法原理相同，须处理进位。在有符号二进制补码运算中，当结果超出表示范围时发生溢出。

Overflow can be detected by comparing the carry into the sign bit and the carry out of the sign bit.

通过比较进入符号位的进位和离开符号位的进位可以检测溢出。

Overflow = Cₒᵤₜ ⊕ Cᵢₙ

For two’s complement addition, an overflow occurs if two positive numbers sum to a negative, or two negative numbers sum to a positive.

在二进制补码加法中，两个正数相加得负，或两个负数相加得正，则发生溢出。

0111₂ (+7) + 0001₂ (+1) = 1000₂ (−8) → overflow

Subtraction is performed by adding the two’s complement of the subtrahend.

减法通过加上减数的二进制补码来实现。

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3. Boolean Algebra Laws | 布尔代数定律

Boolean algebra provides a set of rules to simplify logic expressions, helping to minimise gate counts in circuits.

布尔代数提供一组规则来简化逻辑表达式，有助于最小化电路中的门数量。

Law	Expression	中文名称
Commutative	A + B = B + A ; A · B = B · A	交换律
Associative	(A + B) + C = A + (B + C) ; (A · B) · C = A · (B · C)	结合律
Distributive	A · (B + C) = A · B + A · C	分配律
Absorption	A + (A · B) = A ; A · (A + B) = A	吸收律
Complement	A + ¬A = 1 ; A · ¬A = 0	互补律
Identity	A + 0 = A ; A · 1 = A	同一律
Idempotent	A + A = A ; A · A = A	幂等律

These laws are the foundation for simplifying combinational logic and for converting between SOP and POS forms.

这些定律是化简组合逻辑、实现SOP与POS形式转换的基础。

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4. De Morgan’s Theorems & Logic Gate Equivalents | 德摩根定理与逻辑门等价

De Morgan’s theorems allow transformation between AND and OR operations by adding inverters. They are crucial for NAND/NOR-only circuit implementations.

德摩根定理通过添加反相器实现与、或运算的变换，对纯NAND/NOR电路实现至关重要。

The first theorem states that the negation of a conjunction is the disjunction of the negations.

第一个定理指出，与运算的否定等于否定的或运算。

¬(A ∧ B) = ¬A ∨ ¬B

The second theorem transforms a negated disjunction into a conjunction of negations.

第二个定理将否定的或运算转化为否定的与运算。

¬(A ∨ B) = ¬A ∧ ¬B

These equivalences mean a NAND gate is an

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