📚 IGCSE Edexcel Computer Science: Binary – Key Points | IGCSE Edexcel 计算机：二进制 考点精讲
In IGCSE Edexcel Computer Science, the binary system forms the fundamental language of all digital devices. Understanding how numbers, text, images and sound are represented in binary is essential for mastering data representation, calculation of file sizes, and logical operations. This article breaks down each key concept with clear explanations and examples to help you score top marks in your exam.
在IGCSE Edexcel计算机科学中，二进制是所有数字设备的基础语言。理解数字、文字、图像和声音是如何以二进制表示，对于掌握数据表示、文件大小计算以及逻辑运算至关重要。本文逐一讲解每个核心概念，配以清晰的解释与示例，帮助你在考试中取得高分。
1. Why Binary? | 为什么使用二进制？
Computers use binary because they are built from billions of tiny switches called transistors, which can only be in one of two states: ON (1) or OFF (0). Using a base‑2 number system makes it simple to store and process data reliably, even with electrical noise. Any form of data – numbers, characters, colours, sounds – is ultimately represented as sequences of 1s and 0s.
计算机之所以使用二进制，是因为其内部由数十亿个称为晶体管的小开关组成，它们只有两种状态：开（1）或关（0）。使用基数为2的数字系统可以确保数据在存储和处理时即便有电噪声也能保持可靠。任何形式的数据——数字、字符、颜色、声音——最终都被表示为一串串1和0。
In binary, each digit is called a bit (binary digit). A group of 8 bits is a byte. The value of a bit depends on its position, with each position representing a power of 2. This positional notation is identical in concept to the decimal system, except the base is 2 instead of 10.
在二进制中，每个数字称为一个比特（bit，即二进制位）。8个比特组成一个字节（byte）。一个比特的值取决于其所在的位置，每个位置代表2的幂。这种位值记数法与十进制在概念上完全相同，只是基数为2而非10。
2. Binary to Decimal Conversion | 二进制转十进制
To convert a binary number to decimal, write down the powers of 2 from right to left (starting at 2⁰ = 1). Then multiply each bit by its place value and sum the results. For example, the binary number 10110₂ (the subscript ₂ indicates base 2) has place values: 16, 8, 4, 2, 1. The bits give: 1×16 + 0×8 + 1×4 + 1×2 + 0×1 = 16 + 4 + 2 = 22 in decimal (22₁₀).
将二进制数转换为十进制数时，从右向左写出2的幂（从2⁰=1开始）。然后将每个比特乘以其位值，再把所有结果相加。例如，二进制数10110₂（下标₂表示基数为2）的位值为：16、8、4、2、1。比特给出：1×16 + 0×8 + 1×4 + 1×2 + 0×1 = 16 + 4 + 2 = 22（十进制22₁₀）。
Practice is key. Always check your answer by adding the place values where a 1 appears. A common exam question asks you to convert an 8‑bit binary number (maximum value 255) to decimal, so be comfortable with place values up to 128 (2⁷).
练习是关键。只需将出现1的位值相加便可验算。常见的考试题会让你把8位二进制数（最大值255）转换为十进制，因此要熟练掌握到128（2⁷）的位值。
3. Decimal to Binary Conversion | 十进制转二进制
There are two main methods. The division method repeatedly divides the decimal number by 2 and records the remainder. Read the remainders from bottom to top to get the binary equivalent. For 22: 22 ÷ 2 = 11 remainder 0; 11 ÷ 2 = 5 remainder 1; 5 ÷ 2 = 2 remainder 1; 2 ÷ 2 = 1 remainder 0; 1 ÷ 2 = 0 remainder 1. Reading upwards gives 10110₂.
主要有两种方法。除2取余法：将十进制数不断除以2，记录余数。从下往上读取余数即得二进制。以22为例：22÷2=11余0；11÷2=5余1；5÷2=2余1；2÷2=1余0；1÷2=0余1。向上读取得10110₂。
The subtraction method compares the decimal number with descending powers of 2. If the power can be subtracted, write a 1 and subtract it; otherwise write a 0. For 22: can subtract 16 → 1, remainder 6; 8 cannot → 0; subtract 4 → 1, remainder 2; subtract 2 → 1, remainder 0; subtract 1 → 0, final bit 0. Result: 10110₂. Both methods are equally valid; choose the one you find most reliable.
凑数法则是将十进制数与从大到小的2的幂比较。若能减去该幂，则写1并作减法；否则写0。以22为例：可减去16→1，余6；8不能→0；减4→1，余2；减2→1，余0；减1→0，末位0。结果为10110₂。两种方法同样有效，选择你觉得最可靠的一种即可。
4. Hexadecimal System | 十六进制系统
Hexadecimal (base 16) is widely used in computing because it provides a shorter, more human‑readable way to represent binary – especially for memory addresses and colour codes. Each hex digit represents exactly 4 bits (a nibble). Digits 0–9 are used, plus letters A–F for decimal values 10–15.
十六进制（基数为16）在计算中应用广泛，因为它能以更短、更易读的方式表示二进制——尤其在内存地址和颜色代码中。每1个十六进制位恰好代表4个比特（一个半字节）。使用数字0–9，字母A–F分别代表十进制值10–15。
To convert binary to hex, group the bits into nibbles from the right, then convert each group. For 10110110₂: 1011₂ = 11 (B) and 0110₂ = 6, so the hex is B6₁₆. To convert hex to binary, replace each hex digit with its 4‑bit binary equivalent: 3F₁₆ becomes 0011 1111₂. You must be able to convert between binary, denary and hex quickly in the exam.
二进制转十六进制时，从右向左每4位一组，再逐组转换。如10110110₂：1011₂=11（B），0110₂=6，因此十六进制为B6₁₆。十六进制转二进制时，将每个十六进制位替换为4位二进制：3F₁₆变为0011 1111₂。考试中必须能快速地在二进制、十进制和十六进制之间转换。
5. Binary Addition and Overflow | 二进制加法与溢出
Binary addition follows four simple rules: 0+0=0, 0+1=1, 1+0=1, 1+1=0 carry 1, and 1+1+1=1 carry 1. Add two binary numbers column by column from the right, carrying over to the next column as needed. For example, 0110₂ (6) + 0111₂ (7) gives 1101₂ (13).
二进制加法遵循四条简单规则：0+0=0，0+1=1，1+0=1，1+1=0进位1，以及1+1+1=1进位1。从右向左逐列相加，必要时向下一列进位。例如，0110₂ (6) + 0111₂ (7) 得到 1101₂ (13)。
Computers store numbers in a fixed number of bits (e.g. 8 bits). When the result of an addition requires more bits than are available, an overflow error occurs. For example, adding 11111111₂ (255) and 00000001₂ (1) in an 8‑bit register produces 00000000₂ with a carry out, which is lost. The CPU usually has an overflow flag to detect such errors. Understanding overflow is vital for explaining why integer limits exist.
计算机用固定位数（例如8位）存储数字。当加法结果需要超过现有位数时，就会发生溢出错误。例如，在8位寄存器中将11111111₂（255）与00000001₂（1）相加，得到00000000₂并产生一个进位，该进位丢失。CPU通常设有溢出标志以检测此类错误。理解溢出对于解释为什么整数存在上限至关重要。
6. Binary Shifts (Multiplication and Division) | 二进制移位（乘法与除法）
Shifting a binary number left by one place multiplies it by 2. Shifting right by one place divides by 2 (integer division, discarding any remainder). For example, 00001100₂ (12) shifted left once becomes 00011000₂ (24); shifted right once becomes 00000110₂ (6).
将一个二进制数左移一位相当于乘以2；右移一位相当于除以2（整数除法，舍去余数）。例如，00001100₂（12）左移一位变为00011000₂（24）；右移一位变为00000110₂（6）。
A left shift of n places multiplies by 2ⁿ. A right shift of n places divides by 2ⁿ. In many processors, logical shifts fill vacant bits with 0, while arithmetic right shifts preserve the sign bit (for negative numbers in two’s complement). At IGCSE level, you mainly deal with logical shifts on positive integers. Shift operations are much faster than using the multiplication circuitry, so they are often used for optimisation.
左移n位相当于乘以2ⁿ；右移n位相当于除以2ⁿ。许多处理器中，逻辑移位用0填充空位，而算术右移则保留符号位（用于补码表示负数）。IGCSE阶段主要处理正整数的逻辑移位。移位操作比使用乘法电路快得多，因此常被用于优化。
7. Representing Text – ASCII and Unicode | 文本表示 – ASCII与Unicode
Characters are represented by assigning each one a unique binary code. ASCII (American Standard Code for Information Interchange) uses 7 bits, providing 128 codes for English letters, digits, punctuation and control characters. For example, ‘A’ = 65 (01000001₂). Extended ASCII uses 8 bits for 256 characters, including additional symbols and accented letters.
字符通过为每个字符分配唯一的二进制代码来表示。ASCII（美国信息交换标准代码）使用7位，提供128个代码用于英文字母、数字、标点和控制字符。例如，’A’ = 65（01000001₂）。扩展ASCII使用8位，支持256个字符，包括其他符号和带重音字母。
Unicode was developed to cover all writing systems of the world (e.g. Chinese, Arabic, emojis). The most common form, UTF‑8, uses between 1 and 4 bytes per character, remaining backward‑compatible with ASCII. In exams, you should state that Unicode allows a far wider range of characters, which is essential for global communication, but it can require more storage space per character than plain ASCII.
Unicode被开发出来以涵盖世界上所有的书写系统（如中文、阿拉伯文、表情符号）。最常用的形式UTF‑8每字符使用1到4字节，保持与ASCII向后兼容。考试中应指出Unicode允许更广泛的字符范围，这对全球交流至关重要，但与纯ASCII相比每个字符可能需要更多存储空间。
8. Representing Images – Pixels and Colour Depth | 图像表示 – 像素与颜色深度
Images on a computer are made of a grid of tiny dots called pixels (picture elements). Each pixel’s colour is stored as a binary number. The colour depth (bit depth) is the number of bits used for each pixel. A 1‑bit image can show only two colours (e.g. black and white), while an 8‑bit image can represent 2⁸ = 256 colours, and a 24‑bit image gives about 16.7 million colours (True Colour).
计算机中的图像由被称为像素（图像元素）的微小点阵组成。每个像素的颜色用二进制数存储。颜色深度（位深度）是每个像素使用的比特数。1位图像只能显示两种颜色（如黑与白），8位图像则可表示2⁸=256种颜色，24位图像可呈现约1670万色（真彩色）。
Metadata (such as width, height and colour depth) is stored alongside the pixel data. The resolution is the total number of pixels in the image, often given as width×height (e.g. 1920×1080). Increasing resolution or colour depth improves image quality but also increases file size. You may be asked to calculate the file size of an image: size in bits = width × height × colour depth. Always convert to appropriate units (bytes, kilobytes) in your answer.
元数据（如宽度、高度和颜色深度）与像素数据一同存储。分辨率是图像中的像素总数，通常表示为宽×高（如1920×1080）。提高分辨率或颜色深度可以改善画质，但也会增大文件大小。你可能需要计算图像的文件大小：大小（比特）= 宽度 × 高度 × 颜色深度。答题时务必转换为合适的单位（字节、千字节）。
9. Representing Sound – Sampling and Bit Rate | 声音表示 – 采样与比特率
Sound is an analogue signal; to store it digitally, the sound wave is measured (sampled) at regular intervals. The sample rate is the number of samples taken per second, measured in hertz (Hz). A common CD sample rate is 44.1 kHz (44,100 samples per second). The sample resolution (bit depth) is the number of bits used to record each sample, e.g. 16 bits.
声音是模拟信号；为以数字形式存储，需按固定间隔对声波进行测量（采样）。采样率是每秒采集的样本数，以赫兹（Hz）为单位。CD的常用采样率为44.1 kHz（每秒44,100个样本）。采样精度（位深度）是记录每个样本所使用的比特数，例如16位。
Higher sample rate captures higher frequencies more accurately, and higher bit depth records finer volume differences, but both increase file size. The bit rate (in bits per second) = sample rate × sample resolution × number of channels (e.g. 2 for stereo). File size for a sound clip = bit rate × duration in seconds. You must be able to perform these calculations and explain the trade‑off between quality and storage.
更高的采样率能更准确地捕捉高频，更高的位深度能记录更精细的音量差异，但两者都会增大文件大小。比特率（每秒比特数）= 采样率 × 采样精度 × 声道数（如立体声为2）。声音片段文件大小 = 比特率 × 时长（秒）。必须能完成这些计算并解释质量与存储之间的权衡。
10. Storage Units and File Size Calculations | 存储单位与文件大小计算
Data quantities are measured using the following units: bit (b), nibble (4 bits), byte (B = 8 bits), kilobyte (KB = 1000 B in decimal and 1024 B in binary, but for IGCSE exam purposes the binary definition is often expected: 1 KB = 1024 B), megabyte (MB = 1024 KB), gigabyte (GB = 1024 MB), terabyte (TB = 1024 GB).
数据量用以下单位衡量：位（bit）、半字节（nibble，4位）、字节（B=8位）、千字节（KB，十进制中1 KB=1000 B，二进制中1 KB=1024 B；IGCSE考试中通常要求使用二进制定义：1 KB=1024 B）、兆字节（MB=1024 KB）、吉字节（GB=1024 MB）、太字节（TB=1024 GB）。
Carefully read the question to determine whether it expects decimal (1000) or binary (1024) conversions. Typical exam problems ask you to calculate the storage required for a set of images, a sound recording or a text file. Always show your working, include the correct units, and check the final answer makes sense (e.g. an uncompressed image file should not be ridiculously small or large).
仔细读题，判断是使用十进制（1000）还是二进制（1024）换算。常见考题要求计算一组图像、一段录音或文本文件所需的存储空间。务必展示计算过程，使用正确单位，并检查最终答案是否合理（如未压缩的图像文件不应过小或过大）。
11. Common Pitfalls and Exam Tips | 常见误区与答题技巧
Many students lose marks by miscounting bit positions (remember the rightmost bit is 2⁰, not 2¹). In conversions, always annotate your working; even if the final answer is wrong, method marks are awarded. When adding binary numbers, double‑check carries – a small slip can cascade through the whole sum.
许多学生因为数错比特位置而丢分（记住最右边的比特是2⁰，不是2¹）。转换时多在草稿上做标注；即使最终答案错误，步骤正确也能得分。二进制加法时，仔细复核进位——一个小失误可能影响整个计算结果。
In file size calculations, ensure all values are in consistent units before multiplying. For example, if length is given in minutes, convert to seconds. Also, image dimensions must be in pixels; if you are given a size in inches and a DPI (dots per inch), multiply them to get pixel dimensions before computing. Practise past paper questions to become familiar with the exact phrasing and expected level of detail.
计算文件大小时，确保所有值在相乘前单位一致。例如，时长以分钟为单位时需转换为秒。另外，图像尺寸必须使用像素；如果给出以英寸为单位的尺寸和DPI（每英寸点数），需先相乘得到像素尺寸再计算。通过练习往年真题熟悉出题措辞与要求的详细程度。
12. Summary | 总结
Mastering binary is not just about memorising conversion steps – it is about understanding why data representation works as it does. Binary underpins every operation in a computer. From simple arithmetic to storing a video, the principles of bits, bytes, hex, and sampling let you predict storage needs and appreciate hardware limitations. Revise each section actively by solving problems and explaining the concepts to a friend – this will reinforce your knowledge and prepare you perfectly for the IGCSE Edexcel Computer Science exam.
掌握二进制不仅仅是记住转换步骤——更重要的是理解数据表示为何如此运作。二进制是计算机一切运算的基础。从简单的算术到存储视频，比特、字节、十六进制和采样的原理使你能够预测存储需求并理解硬件局限。通过积极做题和向同学讲解概念来复习每一部分——这将巩固你的知识，为IGCSE Edexcel计算机科学考试做好完美准备。
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