📚 Year 7 SQA Computing: Quick Reference Guide to Formulas and Theorems | Year 7 SQA 计算机：公式定理速查手册
Welcome to your quick reference guide for the essential formulas and theorems studied in Year 7 SQA Computing. This handbook brings together the key calculations for data representation, file sizes, transfer timing, logic gates, and binary arithmetic. Each section presents a clear explanation with worked examples you can follow step by step. Keep this guide close as you revise or tackle homework problems.
欢迎使用 Year 7 SQA 计算机关键公式定理速查手册。本手册汇集了数据表示、文件大小、传输时间、逻辑门和二进制算术的核心计算。每个部分都提供了清晰的解释和逐步的示例，你可以随时仿照练习。在复习或做作业时，不妨随时翻阅这本手册。
1. Binary to Decimal Conversion | 二进制转十进制
Binary is a base‑2 number system that uses only the digits 0 and 1. Each position in a binary number corresponds to a power of 2, starting from 2⁰ on the right. To convert a binary value to decimal, multiply each binary digit by its place value (the corresponding power of 2) and add the results together.
二进制是一种以 2 为底的记数系统，只使用数字 0 和 1。二进制数中每个位置都对应一个 2 的幂，从最右边的 2⁰ 开始。要将二进制值转换为十进制，只需将每个二进制数字乘以它的位值（对应的 2 的幂），然后将结果相加。
Example: Convert 1101₂ to decimal. Write the powers: 2³ 2² 2¹ 2⁰, so 1×2³ + 1×2² + 0×2¹ + 1×2⁰ = 8 + 4 + 0 + 1 = 13. The subscript ₂ indicates that 1101 is a binary number.
示例：将 1101₂ 转换为十进制。写出幂次：2³ 2² 2¹ 2⁰，因此 1×2³ + 1×2² + 0×2¹ + 1×2⁰ = 8 + 4 + 0 + 1 = 13。下标 ₂ 表示 1101 是一个二进制数。
For longer binary numbers, simply extend the powers of 2 to the left. The leftmost bit is called the most significant bit (MSB), and the rightmost is the least significant bit (LSB).
对于更长的二进制数，只需将 2 的幂向左延伸。最左边的位称为最高有效位 (MSB)，最右边的位称为最低有效位 (LSB)。
2. Decimal to Binary Conversion | 十进制转二进制
To convert a decimal integer to binary, repeatedly divide the number by 2. Keep track of the quotients and remainders. Read the sequence of remainders from bottom to top to build the binary equivalent.
要将十进制整数转换为二进制，请反复将该数除以 2，并记录每次的商和余数。将余数从下往上排列，就得到了二进制数。
Example: Convert 13 to binary. 13 ÷ 2 = 6 remainder 1; 6 ÷ 2 = 3 remainder 0; 3 ÷ 2 = 1 remainder 1; 1 ÷ 2 = 0 remainder 1. Reading the remainders upwards gives 1101₂.
示例：将 13 转换为二进制。13 ÷ 2 = 6 余 1；6 ÷ 2 = 3 余 0；3 ÷ 2 = 1 余 1；1 ÷ 2 = 0 余 1。从下往上读取余数，得到 1101₂。
Always continue until the quotient becomes 0. This method works for any positive integer and is a useful skill for topics like character encoding and subnet masks later.
一定要继续除到商为 0 为止。该方法适用于任何正整数，并且是学习字符编码和子网掩码等后续主题的重要基本功。
3. Units of Data Storage | 数据存储单位
Digital information is measured in bits and bytes. A bit (binary digit) holds a single 0 or 1. A group of 8 bits forms one byte. Larger amounts are described with metric prefixes, though in computing the precise values are often powers of 2.
数字信息以比特和字节度量。一个比特（二进制位）存储一个 0 或 1。8 个比特组成一个字节。更大的数据量用词冠表示，但在计算机领域，精确的数值往往是 2 的幂。
| Unit | Abbreviation | Approximate (decimal) | Binary (accurate) |
|---|---|---|---|
| Bit | b | 1 | 1 |
| Byte | B | 8 bits | 8 bits |
| Kilobyte | KB | 1,000 bytes | 1,024 bytes (2¹⁰) |
| Megabyte | MB | 1,000 KB | 1,048,576 bytes (2²⁰) |
| Gigabyte | GB | 1,000 MB | 1,073,741,824 bytes (2³⁰) |
| Terabyte | TB | 1,000 GB | 1,099,511,627,776 bytes (2⁴⁰) |
For simple calculations in class, we typically use the decimal definitions (1 KB = 1000 B, 1 MB = 1000 KB). Make sure you check which convention your teacher or exam paper expects.
在课堂上的简单计算中，我们通常使用十进制定义（1 KB = 1000 B，1 MB = 1000 KB）。请务必确认老师或考卷要求使用哪一种约定。
4. Calculating File Size for Text | 文本文件大小计算
Plain text is stored using character codes such as ASCII (American Standard Code for Information Interchange). In ASCII, each letter, digit, or symbol is represented by exactly 8 bits, i.e. 1 byte. Therefore, the file size in bytes is simply the number of characters.
纯文本使用 ASCII（美国信息交换标准代码）等字符码存储。在 ASCII 中，每个字母、数字或符号恰好用 8 个比特（即 1 个字节）表示。因此，以字节为单位的文件大小就等于字符的个数。
Formula: Text file size (bytes) = number of characters × 1
公式：文本文件大小（字节） = 字符数 × 1
Example: A short essay contains 2500 characters, including spaces and punctuation. File size = 2500 bytes. To express in kilobytes: 2500 ÷ 1000 = 2.5 KB.
示例：一篇短文包含 2500 个字符（包括空格和标点）。文件大小 = 2500 字节。用千字节表示：2500 ÷ 1000 = 2.5 KB。
If a different encoding such as Unicode UTF‑16 is used, a character may require 2 bytes. In Year 7, we almost always assume 1 byte per character for simplicity.
如果使用 Unicode UTF‑16 等不同编码，一个字符可能需要 2 字节。在七年级，为了简化，我们几乎总是假定每个字符占用 1 字节。
5. Calculating Image File Size | 图像文件大小计算
Digital images are composed of tiny dots called pixels. Each pixel stores a colour value. The number of bits used to represent each pixel is called the colour depth or bit depth. The total size of an uncompressed image is found by multiplying width, height, and colour depth.
数字图像由称为像素的微小点阵组成。每个像素存储一个颜色值。用来表示每个像素的比特数称为颜色深度或位深度。未压缩图像的总大小可以通过宽度、高度和颜色深度的乘积求得。
Image file size (bits) = width × height × colour depth (bits per pixel)
图像文件大小（比特） = 宽度 × 高度 × 颜色深度（每像素比特数）
Example: A photograph with dimensions 800 × 600 pixels uses 24‑bit colour (8 bits each for red, green, and blue). Bits = 800 × 600 × 24 = 11,520,000 bits. Convert to bytes: 11,520,000 ÷ 8 = 1,440,000 bytes. Then to KB: 1,440,000 ÷ 1000 = 1,440 KB. Finally to MB: 1,440 ÷ 1000 = 1.44 MB.
示例：一张尺寸为 800 × 600 像素的照片使用 24 位颜色（红、绿、蓝各 8 位）。比特数 = 800 × 600 × 24 = 11,520,000 比特。转换为字节：11,520,000 ÷ 8 = 1,440,000 字节。再转换为 KB：1,440,000 ÷ 1000 = 1,440 KB。最后转换为 MB：1,440 ÷ 1000 = 1.44 MB。
If the colour depth is given as a number of possible colours, you first need to work out how many bits are needed. For example, 256 colours require 8 bits because 2⁸ = 256.
如果给出的颜色深度是可用的颜色数量，你首先需要计算出需要多少比特。例如，256 种颜色需要 8 比特，因为 2⁸ = 256。
6. Calculating Sound File Size | 声音文件大小计算
Digital sound is created by sampling the amplitude of an analogue signal many times per second. Three factors determine the size of an uncompressed sound file: the sample rate, the bit depth (resolution), and the duration. For stereo recordings, you must also multiply by the number of channels.
数字声音是通过每秒多次对模拟信号的幅度进行采样而生成的。决定未压缩声音文件大小的三个因素是：采样率、位深度（分辨率）和持续时间。对于立体声录音，还必须乘以声道数。
Sound file size (bits) = sample rate (Hz) × bit depth × duration (s) × number of channels
声音文件大小（比特） = 采样率（Hz） × 位深度 × 持续时间（秒） × 声道数
Example: A one‑minute CD‑quality recording has a sample rate of 44,100 Hz, a bit depth of 16 bits, and is in stereo (2 channels). Bits = 44,100 × 16 × 60 × 2 = 84,672,000 bits. In bytes: 84,672,000 ÷ 8 = 10,584,000 bytes. In MB: 10,584,000 ÷ 1000 = 10,584 KB, and 10,584 ÷ 1000 ≈ 10.58 MB.
示例：一段时长一分钟的 CD 品质录音，采样率为 44,100 Hz，位深度为 16 比特，立体声（2 声道）。比特数 = 44,100 × 16 × 60 × 2 = 84,672,000 比特。以字节计：84,672,000 ÷ 8 = 10,584,000 字节。以 MB 计：10,584,000 ÷ 1000 = 10,584 KB，10,584 ÷ 1000 ≈ 10.58 MB。
If the file is mono (1 channel), simply omit the channel multiplier. Remember to keep your units consistent – always convert minutes to seconds first.
如果文件是单声道（1 声道），只需省略声道乘数。请记住保持单位一致——始终先把分钟转换为秒。
7. Data Transfer Time | 数据传输时间
When downloading or uploading a file, the time taken depends on the size of the file and the speed of the network connection. The basic formula uses bits for both data amount and transfer rate.
在下载或上传文件时，所花费的时间取决于文件的大小和网络连接的速度。基本公式在数据量和传输速率上都采用比特。
Time (seconds) = data size (bits) ÷ transfer rate (bits per second, bps)
时间（秒） = 数据量（比特） ÷ 传输速率（比特每秒，bps）
Example: You wish to download a 20 MB video file on a connection rated at 10 Mbps (megabits per second). First convert 20 MB to megabits. 20 MB = 20 × 1000 × 1000 bytes = 20,000,000 bytes. Since 1 byte = 8 bits, total bits = 20,000,000 × 8 = 160,000,000 bits = 160 megabits. Time = 160 Mb ÷ 10 Mbps = 16 seconds.
示例：你希望在一个标称 10 Mbps（兆比特每秒）的网络上 下载一个 20 MB 的视频文件。先将 20 MB 转换为兆比特。20 MB = 20 × 1000 × 1000 字节 = 20,000,000 字节。由于 1 字节 = 8 比特，总比特数 = 20,000,000 × 8 = 160,000,000 比特 = 160 兆比特。时间 = 160 Mb ÷ 10 Mbps = 16 秒。
Always check whether the given rate is in bits per second (bps) or bytes per second (Bps). A rate of 10 Mbps is ten million bits per second, not ten million bytes per second.
务必检查给出的速率是比特每秒 (bps) 还是字节每秒 (Bps)。10 Mbps 的速率表示每秒一千万比特，而不是每秒一千万字节。
8. Compression Ratio | 压缩比
Compression reduces the number of bits needed to store or transmit data. The compression ratio tells you how much a file has been shrunk. It compares the size of the original file to the size of the compressed file.
压缩减少了存储或传输数据所需的比特数。压缩比告诉你文件被缩小了多少。它比较了原始文件的大小和压缩后文件的大小。
Compression ratio = original size ÷ compressed size
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