📚 Year 12 AQA Computer Science: Formula & Theorem Quick Reference | AQA 计算机 公式定理速查手册
This quick-reference handbook consolidates all the essential formulas, theorems, and performance equations required for the Year 12 AQA Computer Science specification (7516/7517). It covers Boolean algebra, data representation, multimedia calculations, network delays, algorithmic complexity, hardware performance metrics, and storage conversion rules. Each section presents the key relationships in a clear, exam-ready format followed by concise explanations.
本速查手册汇总了 Year 12 AQA 计算机科学（7516/7517）所必需的全部公式、定理与性能方程，涵盖布尔代数、数据表示、多媒体计算、网络延迟、算法复杂度、硬件性能指标及存储换算。每个小节均以清晰且贴近考试的形式呈现核心关系，并附有简要说明。
1. Boolean Algebra Laws | 布尔代数定律
The fundamental laws of Boolean algebra allow simplification of logic circuits and expressions. They include commutative, associative, distributive, identity, annihilator, complement, idempotent, absorption, and De Morgan's laws. Mastery of these theorems is critical for logic minimisation in Paper 1.
布尔代数的基本定律用于化简逻辑电路与表达式，主要包括交换律、结合律、分配律、同一律、归零律/归一律、互补律、幂等律、吸收律以及德摩根定律。掌握这些定理对 Paper 1 中的逻辑化简至关重要。
| Law | Expression (AND form) | Expression (OR form) |
|---|---|---|
| Commutative | A · B = B · A | A + B = B + A |
| Associative | (A · B) · C = A · (B · C) | (A + B) + C = A + (B + C) |
| Distributive | A · (B + C) = (A · B) + (A · C) | A + (B · C) = (A + B) · (A + C) |
| Identity | A · 1 = A | A + 0 = A |
| Annihilator | A · 0 = 0 | A + 1 = 1 |
| Complement | A · A' = 0 | A + A' = 1 |
| Idempotent | A · A = A | A + A = A |
| Absorption | A · (A + B) = A | A + (A · B) = A |
| De Morgan | (A · B)' = A' + B' | (A + B)' = A' · B' |
De Morgan’s laws are frequently used to convert between sum-of-products and product-of-sums forms, and to push inversions through gates in circuit diagrams.
德摩根定律常用于在积之和与和之积形式之间相互转换，以及在电路图中将反相传递至各输入端。
2. Logic Gate Symbols and Boolean Expressions | 逻辑门符号与布尔表达式
Each basic logic gate implements a Boolean function. The table below summarises the symbols, Boolean expressions, and truth table outputs used in AQA examinations.
每种基本逻辑门都实现一个布尔函数。下表汇总了 AQA 考试中使用的符号、布尔表达式和真值表输出。
| Gate | Boolean Expression | Output (for inputs A,B) |
|---|---|---|
| AND | Q = A · B | 1 only when A=1 and B=1 |
| OR | Q = A + B | 1 when at least one input is 1 |
| NOT | Q = A' | Inverts the input |
| NAND | Q = (A · B)' | 0 only when A=1 and B=1 |
| NOR | Q = (A + B)' | 1 only when A=0 and B=0 |
| XOR | Q = A ⊕ B = A'·B + A·B' | 1 when A and B differ |
| XNOR | Q = (A ⊕ B)' = A·B + A'·B' | 1 when A and B are equal |
The NAND and NOR gates are particularly important because they are functionally complete; any Boolean function can be implemented using only NAND gates or only NOR gates.
NAND 和 NOR 门尤为重要，因为它们具有功能完备性：任何布尔函数都可以仅用 NAND 门或仅用 NOR 门实现。
3. Binary Two’s Complement Representation | 二进制补码表示
egister, the most significant bit (MSB) acts as the sign bit (0 = positive, 1 = negative).
补码是表示有符号整数的标准二进制方法。对于 n 位寄存器，最高有效位 (MSB) 作为符号位（0 表示正，1 表示负）。
To obtain the negative of a number: invert all bits and add 1 to the least significant bit (LSB). The range of values for an n-bit two’s complement integer is:
求负数的方法：将所有位取反并在最低有效位加 1。n 位补码整数的取值范围为：
Range = -2n-1 to 2n-1 - 1
For example, with 8 bits the range is -128 to 127. When adding two’s complement numbers, any carry beyond the MSB is ignored, and overflow occurs if the sign bits of the two operands are identical but the sign of the result differs.
例如，对于 8 位，范围为 -128 至 127。两补数相加时，最高有效位之外的进位被忽略；若两个操作数的符号位相同而结果的符号位不同，则发生溢出。
4. Floating-Point Binary Representation | 浮点数二进制表示
A floating-point binary number consists of a mantissa (fractional part) and an exponent. In normalised form, the mantissa’s MSB is the sign bit, and the binary point immediately follows it. The format used in AQA a-level typically stores both mantissa and exponent as two’s complement numbers.
浮点二进制数由尾数（小数部分）和指数组成。在规格化形式中，尾数的最高有效位为符号位，其后紧跟二进制小数点。AQA 高级考试中使用的格式通常将尾数和指数均保存为补码数。
The decimal value is given by:
Value = Mantissa × 2Exponent
To normalise a binary floating-point number, shift the mantissa left or right and adjust the exponent so that the first two bits of the mantissa are different (01… for positive, 10… for negative). Normalisation maximises precision.
要将二进制浮点数规格化，需对尾数进行左移或右移并相应调整指数，使得尾数的最高两位互异（正数为 01…，负数为 10…）。规格化可最大限度提高精度。
5. Sampling Theorem and Audio File Size | 采样定理与音频文件大小
According to the Nyquist-Shannon sampling theorem, the sampling frequency must be at least twice the highest frequency component in the analogue signal to allow faithful reconstruction.
根据奈奎斯特-香农采样定理，采样频率必须至少是模拟信号中最高频率分量的两倍，才能实现忠实重建。
fsample ≥ 2 × fmax
The uncompressed file size of a digital audio recording is calculated with:
未压缩数字音频文件的大小计算公式为：
File Size (bits) = Sample Rate (Hz) × Bit Depth × Duration (s) × Number of Channels
Divide by 8 to convert to bytes, and further divide by 1024 for KiB, etc. For exam questions, always pay attention to the units requested.
除以 8 得到字节数，再除以 1024 可得到 KiB 等。考试答题时务必注意所需单位。
6. Image File Size and Colour Depth | 图像文件大小与色彩深度
olution (width × height in pixels) and colour depth (bits per pixel). The relationship is:
位图图像所需的存储空间取决于其分辨率（以像素为单位的宽度 × 高度）和色彩深度（每像素位数）。公式如下：
Image File Size (bits) = Width × Height × Colour Depth
The number of distinct colours that can be represented is 2colour depth. For example, a colour depth of 8 bits allows 256 colours, while 24 bits allows approximately 16.7 million colours (True Colour). Metadata and compression may alter the actual file size.
可表示的不同颜色数量为 2色彩深度。例如，8 位色彩深度可表示 256 种颜色，而 24 位可表示约 1670 万种颜色（真彩色）。元数据和压缩可能会改变实际文件大小。
7. Network Transmission Time and Delay | 网络传输时间与延迟
The time taken to send a file across a network is the sum of transmission time and propagation delay, plus any queuing and processing delays. The core formula for transmission time is:
通过网络发送文件所需的时间由传输时间、传播延迟以及可能的排队和处理延迟组成。传输时间的核心公式为：
Transmission Time (s) = Data Size (bits) / Bandwidth (bps)
Propagation delay depends on the physical distance and the speed of the signal in the medium:
传播延迟取决于物理距离和信号在介质中的速度：
Propagation Delay (s) = Distance (m) / Propagation Speed (m/s)
In fibre optic cables the speed is approximately 2 × 108 m/s. When calculating total end-to-end delay, always convert all components to the same time unit.
在光缆中，信号速度约为 2 × 108 米/秒。在计算端到端总延迟时，务必将所有分量统一为相同时间单位。
8. Algorithmic Complexity – Big-O Notation | 算法时间复杂度 – 大 O 表示法
Big-O notation expresses the upper bound of an algorithm's growth rate as the input size n increases. The table below lists the most common complexity classes seen in Year 12.
大 O 表示法表示随着输入规模 n 增大，算法增长率的上界。下表列出了 Year 12 中常见的复杂度类别。
| Complexity | Name | Example Algorithm |
|---|---|---|
| O(1) | Constant | Indexing an array |
| O(log n) | Logarithmic | Binary search |
| O(n) | Linear | Linear search |
| O(n log n) | Linearithmic | Merge sort, quicksort (average) |
| O(n²) | Quadratic | Bubble sort, insertion sort |
| O(2ⁿ) | Exponential | Recursive calculation of all subsets |
When discussing complexity in exams, focus on the dominant term and ignore constant factors. A repeated linear search inside a linear loop yields O(n²), not O(2n).
考试中讨论复杂度时，只需关注主导项，忽略常数因子。在线性循环内重复执行线性搜索会产生 O(n²)，而非 O(2n)。
9. Complexity of Common Data Structure Operations | 常用数据结构操作复杂度
The efficiency of fundamental operations (access, search, insertion, deletion) varies across different data structures. The following table summarises average-case time complexities examined in the AQA specification.
基本操作（访问、查找、插入、删除）的效率因数据结构而异。下表汇总了 AQA 规范中考核的平均情况时间复杂度。
| Data Structure | Access | Search | Insertion | Deletion |
|---|---|---|---|---|
| Array | O(1) | O(n) | O(n) | O(n) |
| Linked List | O(n) | O(n) | O(1) at head | O(1) at head |
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