📚 A-Level Edexcel Computer Science: Formula Handbook | A-Level Edexcel 计算机：公式汇总手册
This handbook brings together all the essential formulas and computational rules required for the A-Level Edexcel Computer Science specification. From number system conversions and Boolean algebra to data storage calculations and encryption steps, each section presents the key relationships in a clear, dual-language format. Use it as a quick reference for revision and problem-solving.
本手册汇总了A-Level Edexcel计算机科学课程中所有必备的公式与计算规则。从数制转换、布尔代数到数据存储计算与加密步骤，每一节都以清晰的双语格式呈现关键关系，可用作快速复习和解题参考。
1. Number Systems and Conversion | 数制与转换
The value of a number represented in base b can be expanded using positional weights. For a binary number with bits bn-1...b0.b-1..., the decimal equivalent is given by:
以基数 b 表示的数字可以使用位权展开求值。对于二进制数 bn-1...b0.b-1...，十进制等价值由下式给出：
Decimal Value = ∑ (bi × 2i)
十进制值 = ∑ (bi × 2i)
where i ranges from the most significant to the least significant digit, including fractional parts with negative exponents. To convert a decimal integer to binary, repeatedly divide by 2 and record the remainders; the binary representation is the sequence of remainders read from bottom to top. For hexadecimal, group binary digits in sets of four from the right, then convert each 4-bit group to a hex digit (0–F).
其中 i 范围从最高有效位到最低有效位，包括具有负指数的小数部分。将十进制整数转换为二进制时，重复除以2并记录余数；二进制表示即为从下往上读取的余数序列。对于十六进制，从右侧起每4个二进制位一组，然后将每个4位组转换为一个十六进制数字（0–F）。
2. Binary Arithmetic | 二进制算术
le rules: 0+0=0, 0+1=1, 1+0=1, 1+1=10 (sum 0, carry 1). Subtraction is performed using two's complement addition: A − B = A + (two's complement of B). The two's complement of a binary number is obtained by inverting all bits and adding 1.
二进制加法遵循简单规则：0+0=0，0+1=1，1+0=1，1+1=10（和为0，进位1）。减法通过补码加法实现：A − B = A + (B的补码)。一个二进制数的补码通过将所有位取反后加1得到。
Overflow occurs in signed two' A · B. A full adder extends this with a carry-in input: Sum = A ⊕ B ⊕ Cin, Carryout = (A · B) + (Cin · (A ⊕ B)).
有符号补码运算中，当结果超出可表示范围时发生溢出。一种实用的检测规则是：进入符号位的进位 ≠ 离开符号位的进位。对于半加器设计，和 = A ⊕ B，进位 = A · B。全加器加入进位输入后扩展为：和 = A ⊕ B ⊕ Cin，进位出 = (A · B) + (Cin · (A ⊕ B))。
3. Two's Complement Representation | 补码表示
An n-bit two's complement integer can represent values in the range:
n 位补码整数可表示的数值范围为：
−2n−1 to 2n−1 − 1
−2n−1 到 2n−1 − 1
ger. If the sign bit is 1, the value is negative and can be evaluated as: Value = −2n−1 + (sum of remaining bits interpreted as unsigned). The quick method to negate a number is to flip all bits and add 1.
如果最高有效位（符号位）为0，该值即为正的二进制整数。如果符号位为1，该值为负，可按下式求值：值 = −2n−1 +（剩余位按无符号数求和）。快速求负数的方法是所有位取反后加1。
4. Floating Point Representation | 浮点数表示
A binary floating point number is expressed in the form ±1.M × 2E−bias. The stored fields are: sign S, mantissa (fraction) M, and exponent E encoded with a bias. The value is reconstructed as:
二进制浮点数表示为形式 ±1.M × 2E−bias。存储的字段有：符号 S、尾数（小数部分）M 以及用偏置编码的指数 E。数值重构公式为：
Value = (−1)S × (1 + M) × 2E − bias
值 = (−1)S × (1 +
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