KS3 CAIE Computer Science: Formula & Theorem Quick Reference | KS3 CAIE计算机：公式定理速查手册

📚 KS3 CAIE Computer Science: Formula & Theorem Quick Reference | KS3 CAIE计算机：公式定理速查手册

This quick reference handbook compiles the most important formulas, theorems, definitions and calculation methods you need for the KS3 CAIE Computer Science course, from data representation and Boolean logic to CPU performance, network transfer times and algorithm complexity. Every formula is paired with a short explanation, a worked example where useful, and a note on the mistakes examiners see most often, so you can use it to refresh your memory before a test, while completing homework, or as a final checklist the night before an assessment.

这本速查手册汇集了KS3 CAIE计算机科学课程中最重要的公式、定理、定义和计算方法，涵盖数据表示、布尔逻辑、CPU性能、网络传输时间以及算法复杂度等核心内容。每一条公式都配有简明解释、必要的例题演示，以及考官最常看到的易错提示。你可以在考试前快速复习、完成作业时随手查阅，也可以在测评前一晚把它当作最终检查清单使用。

1. Data Storage Units | 数据存储单位

Every quantitative question in computer science starts with the bit. A bit is a single binary digit, 0 or 1. Four bits make a nibble, which is exactly one hexadecimal digit, and eight bits make a byte, which is the standard unit for storing one ASCII character. Larger units are built from the byte using decimal prefixes in the CAIE specification, so 1 kilobyte is 1 000 bytes, not 1 024.

计算机科学中所有定量问题都从位（bit）开始。一个位就是一个二进制数字，取值为0或1。4个位组成一个半字节（nibble），正好等于一个十六进制数字；8个位组成一个字节（byte），是存储一个ASCII字符的标准单位。在CAIE课程规范中，更大的单位以字节为基础、使用十进制前缀，因此1千字节等于1000字节，而不是1024字节。

Converting between units only ever means multiplying or dividing by a power of ten, or by 8 when you move between bits and bytes. Always write the unit next to every number in your working, because a correct number with the wrong unit still loses marks. Some textbooks use 1 024 bytes for a kilobyte, which is properly called a kibibyte; if a question specifies 1 024, follow the question, not the general rule.

单位换算本质上只是乘以或除以10的幂，而在位与字节之间转换时则要乘以或除以8。计算过程中一定要在每个数字后写出单位，因为数字正确但单位错误依然会失分。有些教材把1千字节写作1024字节，那其实是"千二进制字节"（kibibyte）；如果题目明确给出1024，就按题目要求做，而不要套用通用规则。

Bits = Bytes × 8   |   Bytes = Bits ÷ 8

2. Number Systems Overview | 数制概览

ps directly onto the on and off states of a transistor. Hexadecimal is base 16 and uses 0 to 9 plus A to F, where A = 10 and F = 15; it is popular with programmers because one hex digit represents exactly four bits, so a byte can be written with only two characters.

KS3阶段的计算机科学涉及三种数制。十进制（denary）是基数为10的数制，使用数字0到9。二进制是基数为2的数制，只使用0和1，正好对应晶体管的"开"与"关"两种状态。十六进制是基数为16的数制，使用0到9以及A到F，其中A代表10，F代表15；程序员偏爱十六进制，因为一个十六进制数字正好代表4个位，所以一个字节只需两个字符就能表示。

The place values of an 8-bit binary number, read from left to right, are 128, 64, 32, 16, 8, 4, 2 and 1. With n bits you can represent 2ⁿ different values, numbered from 0 up to 2ⁿ − 1. That is why an 8-bit byte holds values from 0 to 255, a 4-bit nibble holds 0 to 15, and a 16-bit value holds 0 to 65 535.

一个8位二进制数的位值从左到右依次是128、64、32、16、8、4、2、1。用n个位可以表示2ⁿ个不同的值，编号从0到2ⁿ − 1。因此一个8位字节的取值是0到255，一个4位半字节的取值是0到15，而一个16位数值的取值是0到65535。

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Signed numbers use two's complement. In an n-bit two's complement system the range becomes −2ⁿ⁻¹ up to 2ⁿ⁻¹ − 1, so an 8-bit signed byte runs from −128 to +127. Notice that the total number of values is unchanged; only the boundary has moved.

有符号数使用二进制补码（two's complement）表示。在n位补码系统中，取值范围变为 −2ⁿ⁻¹ 到 2ⁿ⁻¹ − 1，因此8位有符号字节的范围是 −128 到 +127。注意，可表示值的总数没有改变，只是边界发生了移动。

Number of values = 2ⁿ  |  Unsigned range = 0 to 2ⁿ − 1  |  Signed range = −2ⁿ⁻¹ to 2ⁿ⁻¹ − 1

3. Conversion Methods and Full Lookup Table | 转换方法与完整对照表

0 × 128 + 0 × 64 + 1 × 32 + 0 × 16 + 1 × 8 + 1 × 4 + 0 × 2 + 1 × 1 = 32 + 8 + 4 + 1 = 45.

把二进制转换为十进制，就是用每一位乘以其位值，再把结果相加。例如 = 0 × 128 + 0 × 64 + 1 × 32 + 0 × 16 + 1 × 8 + 1 × 4 + 0 × 2 + 1 × 1 = 32 + 8 + 4 + 1 = 45。

gives remainders 0, 0, 1, 1, 1, 0, 0, 1 read upwards as 1001 1100, which checks out as 128 + 16 + 8 + 4 = 156. A faster method is to subtract the largest place values that fit.

把十进制转换为二进制，可以反复除以2并记录每次的余数，然后从下往上读取余数。例如156连续除以2得到的余数从下往上读作 1001 1100，验算得 128 + 16 + 8 + 4 = 156。更快捷的方法是不断减去能够容纳的最大位值。

Binary to hexadecimal is done by splitting the bits into nibbles from the right and replacing each nibble with one hex digit: 1101 0111 becomes D7. Hexadecimal to binary is the reverse: expand each hex digit into exactly four bits. Denary to hexadecimal is easiest via binary, or by repeated division by 16. Keep leading zeros in each nibble, because 1101 011 would be misread if the missing zero were dropped.

二进制转十六进制的方法是从右往左每4位分成一组，再用一个十六进制数字替换每一组：1101 0111 变为 D7。十六进制转二进制则相反：把每个十六进制数字展开成恰好4个位。十进制转十六进制最简便的路径是先转二进制，或者反复除以16。每组都要保留前导零，因为若把 1101 011 中缺少的零丢掉就会被误读。

Binary → Denary: Σ(bit × 2^position)  |  4 bits = 1 hex digit  |  8 bits = 1 byte = 2 hex digits

Denary	Hexadecimal	4-bit Binary
0	0	0000
1	1	0001
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