📚 Year 7 CCEA Computer Science: Formula and Theorem Quick Reference Guide | Year 7 CCEA 计算机：公式定理速查手册
This quick reference guide brings together the essential formulas, number systems, Boolean laws, and calculation methods you will meet in the Year 7 CCEA Computer Science course. Keep it handy while working through binary conversions, logic circuits, and data representation tasks.
本速查手册汇集了 Year 7 CCEA 计算机课程中会遇到的基本公式、数制、布尔定律和计算方法。在做二进制转换、逻辑电路和数据表示练习时，随手查阅非常方便。
1. Binary and Decimal Number Systems | 二进制与十进制数制
All digital computers store and process data using the binary number system. Binary uses only two digits: 0 and 1. Each position in a binary number represents a power of 2, just as each position in decimal represents a power of 10.
所有数字计算机都使用二进制数制存储和处理数据。二进制只使用两个数字：0 和 1。二进制数中每个位置代表 2 的幂，就像十进制中每个位置代表 10 的幂一样。
A binary number is often written with a subscript 2 (e.g. 1011₂) and a decimal number with a subscript 10 (e.g. 11₁₀). The right‑most bit is the least significant bit (LSB) and the left‑most is the most significant bit (MSB).
二进制数常写作下标 2（如 1011₂），十进制数写作下标 10（如 11₁₀）。最右边的位是最低有效位（LSB），最左边的是最高有效位（MSB）。
Place values: … 2⁴ 2³ 2² 2¹ 2⁰ (16, 8, 4, 2, 1)
位权：… 2⁴ 2³ 2² 2¹ 2⁰ （16、8、4、2、1）
2. Converting Binary to Decimal | 二进制转十进制
To convert a binary number to decimal, multiply each bit by its place value (power of 2) and add the results.
要将二进制数转换为十进制，将每一位乘以其位权（2 的幂），再将结果相加。
Decimal value = Σ (bitᵢ × 2ⁱ)
十进制值 = Σ (位ᵢ × 2ⁱ)
Example: 1101₂ = (1 × 8) + (1 × 4) + (0 × 2) + (1 × 1) = 8 + 4 + 0 + 1 = 13₁₀.
示例：1101₂ = (1 × 8) + (1 × 4) + (0 × 2) + (1 × 1) = 8 + 4 + 0 + 1 = 13₁₀。
3. Converting Decimal to Binary | 十进制转二进制
To convert a decimal integer to binary, repeatedly divide the number by 2 and record the remainders. Read the remainders from the bottom upwards to get the binary equivalent.
要将十进制整数转换为二进制，反复将数字除以 2 并记录余数。从下往上读取余数，就得到对应的二进制数。
Example: Convert 19₁₀ to binary:
示例：将 19₁₀ 转换为二进制：
- 19 ÷ 2 = 9 remainder 1
- 9 ÷ 2 = 4 remainder 1
- 4 ÷ 2 = 2 remainder 0
- 2 ÷ 2 = 1 remainder 0
- 1 ÷ 2 = 0 remainder 1
Reading upwards: 10011₂. So 19₁₀ = 10011₂.
从下往上读：10011₂。所以 19₁₀ = 10011₂。
4. Binary Arithmetic: Addition | 二进制算术：加法
Binary addition follows simple rules, very similar to decimal addition but only using 0s and 1s.
二进制加法遵循简单的规则，与十进制加法非常相似，但只用 0 和 1。
| 0 + 0 = 0 |
| 0 + 1 = 1 |
| 1 + 0 = 1 |
| 1 + 1 = 0, carry 1 to the next column |
| 1 + 1 + 1 = 1, carry 1 |
When a column adds up to 2, write 0 and carry 1. When a column adds up to 3, write 1 and carry 1.
当一列相加为 2 时，写下 0 并进位 1。当一列相加为 3 时，写下 1 并进位 1。
Example: 1011₂ + 0110₂ = 10001₂.
示例：1011₂ + 0110₂ = 10001₂。
5. Boolean Logic: AND, OR, NOT | 布尔逻辑：与、或、非
Boolean logic operates on true/false values, often represented as 1 (true) and 0 (false). The three fundamental operations are AND, OR, and NOT.
布尔逻辑处理真/假值，通常用 1（真）和 0（假）表示。三种基本运算是与、或、非。
- AND (∧): output is 1 only if all inputs are 1.
- AND (∧)：只有所有输入都为 1 时，输出才为 1。
- OR (∨): output is 1 if at least one input is 1.
- OR (∨)：只要至少有一个输入为 1，输出就为 1。
- NOT (¬): flips the input; 1 becomes 0, 0 becomes 1.
- NOT (¬)：将输入取反；1 变成 0，0 变成 1。
These operations can be expressed using Boolean expressions like A AND B, A OR B, NOT A.
这些运算可以用布尔表达式表示，如 A AND B、A OR B、NOT A。
6. Truth Tables for Logic Gates | 逻辑门真值表
A truth table shows every possible input combination and the resulting output for a logic gate.
真值表显示逻辑门所有可能的输入组合及对应的输出。
AND gate (A ∧ B):
| A | B | Output |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
OR gate (A ∨ B):
| A | B | Output |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
NOT gate (¬A):
| A | Output |
|---|---|
| 0 | 1 |
| 1 | 0 |
7. Boolean Algebra Laws | 布尔代数定律
Boolean algebra has a set of laws that allow us to simplify expressions and design efficient logic circuits.
布尔代数有一组定律，可以让我们简化表达式并设计高效逻辑电路。
Commutative Law | 交换律
A ∧ B = B ∧ A A ∨ B = B ∨ A
Associative Law | 结合律
(A ∧ B) ∧ C = A ∧ (B ∧ C) (A ∨ B) ∨ C = A ∨ (B ∨ C)
Distributive Law | 分配律
A ∧ (B ∨ C) = (A ∧ B) ∨ (A ∧ C)
A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C)
De Morgan’s Theorems | 德摩根定理
¬(A ∧ B) = ¬A ∨ ¬B
¬(A ∨ B) = ¬A ∧ ¬B
These laws help turn complex AND/OR combinations into simpler forms using only NAND or NOR gates.
这些定律有助于将复杂的 AND/OR 组合转换为仅使用 NAND 或 NOR 门的更简单形式。
8. Data Storage Units | 数据存储单位
Computer storage is measured using a hierarchy of units based on the byte. A byte consists of 8 bits.
计算机存储使用基于字节的单位层级来衡量。一个字节由 8 个比特组成。
| Unit | Abbreviation | Size in bytes |
|---|---|---|
| 1 bit | b | 1/8 byte |
| 1 byte | B | 1 byte |
| 1 kilobyte | KB | 1,024 bytes |
| 1 megabyte | MB | 1,024 KB = 1,048,576 bytes |
| 1 gigabyte | GB | 1,024 MB |
| 1 terabyte | TB | 1,024 GB |
Remember: when converting between units, multiply or divide by 1,024, not 1,000.
请记住：在单位之间转换时，要乘以或除以 1,024，而不是 1,000。
9. Calculating Image File Size | 图像文件大小计算
An image file size depends on its dimensions (width and height in pixels) and the colour depth (bits per pixel).
图像文件的大小取决于其尺寸（以像素为单位的宽度和高度）以及颜色深度（每像素的比特数）。
Image file size (bytes) = (width in pixels × height in pixels × colour depth in bits) ÷ 8
图像文件大小（字节）= （宽（像素）× 高（像素）× 颜色深度（比特））÷ 8
Example: A 200 × 100 pixel image with a colour depth of 24 bits. Size = (200 × 100 × 24) ÷ 8 = 480,000 ÷ 8 = 60,000 bytes = 60 KB.
示例：一张 200×100 像素、颜色深度为 24 比特的图像。大小 = (200 × 100 × 24) ÷ 8 = 480,000 ÷ 8 = 60,000 字节 = 60 KB。
10. Sound File Size Estimation | 声音文件大小估算
Uncompressed sound file size is calculated from the sample rate, sample resolution (bit depth), number of channels, and duration.
未压缩声音文件的大小是根据采样率、采样精度（位深度）、声道数和持续时间计算的。
Sound file size (bytes) = sample rate (Hz) × bit depth × channels × time (seconds) ÷ 8
声音文件大小（字节）= 采样率（Hz）× 位深度 × 声道数 × 时间（秒）÷ 8
Example: 10 seconds of CD‑quality audio (44,100 Hz, 16‑bit, stereo): size = 44,100 × 16 × 2 × 10 ÷ 8 = 1,764,000 bytes ≈ 1.76 MB.
示例：10 秒 CD 质量音频（44,100 Hz、16 位、立体声）：大小 = 44,100 × 16 × 2 × 10 ÷ 8 = 1,764,000 字节 ≈ 1.76 MB。
11. Data Transfer Time | 数据传输时间
To find how long it takes to transfer a file, divide the data size by the transfer rate. Keep units consistent (both in bits or both in bytes).
要计算传输文件所需的时间，用数据大小除以传输速率。保持单位一致（都用比特或都用字节）。
Transfer time (seconds) = file size (bits) ÷ transfer rate (bits per second)
传输时间（秒）= 文件大小（比特）÷ 传输速率（比特/秒）
If file size is given in bytes, multiply by 8 to convert to bits. Transfer rates are often expressed in bps (bits per second) or Mbps.
如果文件大小以字节给出，需乘以 8 转换为比特。传输速率通常以 bps（比特每秒）或 Mbps 表示。
12. Key Programming Operators | 关键编程运算符
Programming often uses mathematical, comparison, and logical operators. Understanding their symbols is essential for reading and writing code.
编程中经常使用算术、比较和逻辑运算符。理解它们的符号对读写代码至关重要。
Arithmetic operators | 算术运算符
- + (addition / 加), – (subtraction / 减), * (multiplication / 乘), / (division / 除), MOD (remainder / 取余)
- +（加）、-（减）、*（乘）、/（除）、MOD（取余数）
Comparison operators | 比较运算符
- == (equal to / 等于), != or <> (not equal / 不等于), > (greater than / 大于), < (less than / 小于), >= (greater than or equal / 大于等于), <= (less than or equal / 小于等于)
- ==（等于）、!= 或 <>（不等于）、>（大于）、<（小于）、>=（大于等于）、<=（小于等于）
Logical operators | 逻辑运算符
- AND, OR, NOT – often used in conditions.
- AND、OR、NOT – 常用于条件判断。
These operators form the basis of decisions and loops in programming.
这些运算符构成了编程中判断和循环的基础。
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