Year 10 CAIE Computer Science: Formula & Theorem Quick Reference | 公式定理速查手册

📚 Year 10 CAIE Computer Science: Formula & Theorem Quick Reference | 公式定理速查手册

Welcome to the Year 10 CAIE Computer Science Formula & Theorem Quick Reference Handbook. This guide condenses the essential formulas, conversions, and Boolean logic theorems you need to master IGCSE-style problem solving. Keep it handy for revision, homework, or quick checks during your study sessions.

欢迎使用 Year 10 CAIE 计算机科学公式定理速查手册。本手册浓缩了必备的公式、进制转换方法和布尔逻辑定理，帮助你掌握 IGCSE 风格的问题求解。保存下来，用于复习、作业或学习中的快速查阅。

1. Data Storage Unit Conversions | 数据存储单位换算

In computer science, all data is represented as bits (binary digits). A group of 8 bits makes a byte. When calculating file sizes and memory capacities, you must use the binary scaling factors rather than decimal: 1 KB = 1024 bytes (2¹⁰), 1 MB = 1024 KB, 1 GB = 1024 MB, and 1 TB = 1024 GB. For network speeds or some storage manufacturer specifications, decimal definitions (1 kbps = 1000 bps, 1 MB = 1 000 000 bytes) may apply, so always check the context given in the exam question.

在计算机科学中，所有数据都以位（二进制数字）表示。8 位组成 1 字节。计算文件大小和内存容量时，必须使用二进制换算因子，而非十进制：1 KB = 1024 字节（2¹⁰），1 MB = 1024 KB，1 GB = 1024 MB，1 TB = 1024 GB。对于网络速度或某些存储厂商规格，可能使用十进制定义（1 kbps = 1000 bps，1 MB = 1 000 000 字节），因此务必仔细审题。

Below is a quick reference table for the most common binary data units:

以下是最常见的二进制数据单位速查表：

The general formula to convert from any larger unit to bytes is to multiply by 1024 for each step down the scale. For instance, to convert gigabytes directly to bytes:

将较大单位转换为字节的一般公式是每向下跳一级乘以 1024。例如，直接将 GB 转换为字节的公式为：

Size (bytes) = Size (GB) × 1024 × 1024 × 1024

Example: Convert 2.5 GB to bytes. 2.5 × 1024 × 1024 × 1024 = 2,684,354,560 bytes. When a question involves transmission, always express the file size in bits first by multiplying the byte size by 8.

示例：将 2.5 GB 转换为字节。2.5 × 1024 × 1024 × 1024 = 2,684,354,560 字节。当题目涉及传输时，始终先将文件大小表示为位，即字节数乘以 8。

2. Image File Size | 图像文件大小

An uncompressed bitmap image’s storage requirement depends on its pixel dimensions and the colour depth (number of bits used to encode each pixel’s colour). The basic formula in bits is straightforward:

未压缩的位图图像存储需求取决于其像素尺寸和颜色深度（用于编码每个像素颜色的位数）。以位为单位的基本公式非常直观：

Image size (bits) = width (pixels) × height (pixels) × colour depth (bpp)

To obtain the size in bytes, divide the result by 8. Colour depth values you must know: 1 bit for black and white (monochrome), 8 bits for 256 colours, and 24 bits for true colour (about 16.7 million colours). If an image uses a palette, the colour depth is often smaller.

要得到以字节为单位的大小，将结果除以 8。必须掌握的颜色深度值：1 位用于黑白（单色），8 位用于 256 色，24 位用于真彩色（约 1670 万色）。如果图像使用调色板，颜色深度通常更小。

Example: A 1920×1080 Full HD photograph stored with 24-bit colour depth. Bits = 1920 × 1080 × 24 = 49,766,400 bits. Dividing by 8 gives 6,220,800 bytes. To express in megabytes, divide by 1024 twice: 6,220,800 ÷ 1,048,576 ≈ 5.93 MB.

示例：一张 1920×1080 全高清照片，以 24 位颜色深度存储。位数 = 1920 × 1080 × 24 = 49,766,400 位。除以 8 得到 6,220,800 字节。转换为兆字节需除以 1024²：6,220,800 ÷ 1,048,576 ≈ 5.93 MB。

If you need to calculate the number of colours available from a given bit depth, use 2ⁿ. For example, 8-bit colour depth provides 2⁸ = 256 possible colours.

如果需要从给定位深度计算可用颜色数，使用 2ⁿ。例如，8 位颜色深度提供 2⁸ = 256 种可能的颜色。

3. Sound File Size | 声音文件大小

Digital sound is stored by taking samples of the audio wave at regular intervals. The file size of an uncompressed audio clip depends on the sample rate, sample resolution (bit depth), number of channels, and duration:

数字声音通过在固定的时间间隔对音频波形采样来存储。未经压缩的音频剪辑的文件大小取决于采样率、采样精度（位深度）、声道数和时长：

Sound size (bits) = sample rate (Hz) × sample resolution (bits) × channels × duration (seconds)

Sample rate is the number of samples taken per second, typically 44,100 Hz for CD quality. Sample resolution is the number of bits per sample, usually 16 bits. Channels: 1 for mono, 2 for stereo. Convert the final bit value to bytes by dividing by 8, and then to larger units if required.

采样率是每秒采集的样本数，CD 质量通常为 44,100 Hz。采样精度是每个样本的位数，通常为 16 位。声道数：单声道为 1，立体声为

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