📚 Year 7 CAIE Computer Science: Formulae & Theorems Quick Reference Handbook | 公式定理速查手册
This quick reference handbook provides the essential formulae, conversion rules, and Boolean theorems for the Year 7 CAIE Computer Science curriculum. Each section presents a key computational concept with its formula or theorem clearly stated, followed by examples where helpful. Use this as a handy revision tool before tests.
本速查手册针对 Year 7 CAIE 计算机科学课程，提供核心公式、转换规则与布尔定理。每个小节讲解一个关键计算概念，清晰给出其公式或定理，并附上示例以便理解。可在考前用作便捷的复习工具。
1. Binary Place Value & Decimal Conversion | 二进制位值与十进制转换
In an n-bit binary number, each digit (bit) holds a place value which is a power of 2. The rightmost bit (least significant) has the weight 2⁰, the next 2¹, and so on up to 2ⁿ⁻¹ for the most significant bit. To obtain the decimal equivalent, multiply each bit by its place value and add all the products.
在 n 位二进制数中，每一位（比特）的位值是 2 的幂。最右边的位（最低有效位）权重为 2⁰，下一位为 2¹，依次增加，最高有效位为 2ⁿ⁻¹。要得到十进制值，只需将每一位乘以其位值，再将所有乘积相加。
Decimal = bₙ₋₁×2ⁿ⁻¹ + bₙ₋₂×2ⁿ⁻² + … + b₀×2⁰
Example: Convert the 8-bit binary number 01101001₂ to decimal.
0×2⁷ + 1×2⁶ + 1×2⁵ + 0×2⁴ + 1×2³ + 0×2² + 0×2¹ + 1×2⁰ = 0 + 64 + 32 + 0 + 8 + 0 + 0 + 1 = 105₁₀.
示例：将 8 位二进制数 01101001₂ 转换为十进制。
0×2⁷ + 1×2⁶ + 1×2⁵ + 0×2⁴ + 1×2³ + 0×2² + 0×2¹ + 1×2⁰ = 0 + 64 + 32 + 0 + 8 + 0 + 0 + 1 = 105₁₀。
2. Decimal to Binary: Repeated Division by 2 | 十进制转二进制：重复除以 2
To convert a decimal integer to binary, repeatedly divide the number by 2, recording the remainder each time. Continue until the quotient becomes 0. The binary representation is the sequence of remainders read from the last remainder obtained (most significant) to the first (least significant).
要将十进制整数转换为二进制，反复用 2 除该数，记录每次的余数。当商变为 0 时停止。从最后得到的余数（最高位）读至第一个余数（最低位），即为二进制表示。
Decimal → Binary: divide by 2, stack remainders, read upwards
÷ 2 = 6 r 1; 6 ÷ 2 = 3 r 0; 3 ÷ 2 = 1 r 1; 1 ÷ 2 = 0 r 1. Reading remainders upwards gives 1101001₂ (7 bits, often padded to 8 bits as 01101001₂).
示例：将 105 转换为二进制。105 ÷ 2 = 52 余 1; 52 ÷ 2 = 26 余 0; 26 ÷ 2 = 13 余 0; 13 ÷ 2 = 6 余 1; 6 ÷ 2 = 3 余 0; 3 ÷ 2 = 1 余 1; 1 ÷ 2 = 0 余 1。从下往上读余数得 1101001₂（7 位，常补为 8 位 01101001₂）。
3. Binary Addition Rules | 二进制加法规则
Binary addition follows these fundamental rules. When the sum of two bits is 2 (binary 10), write down 0 and carry 1 to the next higher column.
二进制加法遵循以下基本规则。当两个比特之和为 2（二进制 10），本位列写 0，并向高一位进 1。
| A | B | Sum | Carry-out |
| 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 1 |
When adding more than two bits, include the carry-in. Overflow occurs when the result exceeds the available bit width (e.g., two 8-bit numbers sum to a 9-bit value).
当相加的比特超过两位时，要包括进位输入。若结果超出了规定的位宽（如两个 8 位数之和产生第 9 位），则发生溢出。
4. Left Shift for Multiplying by Powers of 2 | 左移实现乘幂
Shifting a binary number n places to the left is equivalent to multiplying it by 2ⁿ. Zeros fill the vacated rightmost bits. This is useful for fast multiplication by powers of two.
将二进制数左移 n 位，等价于乘以 2ⁿ。腾空的最右位补 0。这种方法可以快速实现 2 的幂次乘法。
Number << n = Number × 2ⁿ
Example: 00000101₂ (5₁₀) left-shifted by 2 places → 00010100₂ = 20₁₀ = 5 × 2².
示例：00000101₂（5₁₀）左移 2 位 → 00010100₂ = 20₁₀ = 5 × 2²。
5. Hexadecimal, Binary and Decimal Interconversion | 十六进制、二进制与十进制互转
Each hexadecimal digit represents exactly 4 binary digits (bits). The conversion is direct: split the binary number into groups of four bits from the right, then replace each group with its hex equivalent (0–9, A–F).
每个十六进制数字恰好代表 4 位二进制数。转换直接：从右往左将二进制数每 4 位分为一组，每组替换为对应的十六进制符号（0–9, A–F）。
1 hex digit = 4 binary bits (nibble)
al, first go through binary or use place values: D2₁₆ = D×16¹ + 2×16⁰ = 13×16 + 2 = 210₁₀.
示例：二进制 11010010₂ → 1101 0010 → D2₁₆（1101₂ = D，0010₂ = 2）。要将十六进制转为十进制，可先经二进制或使用权值：D2₁₆ = D×16¹ + 2×16⁰ = 13×16 + 2 = 210₁₀。
6. Data Storage Unit Conversions | 数据存储单位换算
Computer storage is measured in bytes, and larger units are powers of 2 (binary prefixes). The key relationships for Year 7 CAIE are:
计算机存储以字节为单位，更大单位基于 2 的幂（二进制前缀）。Year 7 CAIE 的核心换算关系如下：
1 KB = 2¹⁰ bytes = 1024 B
1 MB = 2²⁰ bytes = 1024 KB
1 GB = 2³⁰ bytes = 1024 MB
To convert from a larger unit to bytes, multiply by the appropriate power of 2; to go from bytes to a larger unit, divide.
从大单位转换为字节，乘以相应的 2 的幂；从字节转换为大单位，则做除法。
7. Image File Size Formula | 图像文件大小公式
A bitmap image's file size depends on its resolution and colour depth. The formula multiplies width in pixels, height in pixels, and the number of bits used to represent each pixel's colour.
位图图像的文件大小取决于分辨率和颜色深度。公式将图像的像素宽、像素高以及每个像素颜色所用的比特数相乘。
size (bits) = Image width (pixels) × Image height (pixels) × Colour depth (bits per pixel)
0 bits. Divide by 8 to get bytes (1,440,000 B) or by 8×1024 to get KiB (approx. 1406 KiB).
示例：
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