A-Level Edexcel Computer Science: Formula Handbook | A-Level Edexcel 计算机：公式汇总手册

📚 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 b_n-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 b_n-1…b₀.b_-1…，十进制等价值由下式给出：

Decimal Value = ∑ (b_i × 2ⁱ)

十进制值 = ∑ (b_i × 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

Binary addition follows simple 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’s complement arithmetic when the result exceeds the representable range. A practical detection rule is: carry into the sign bit ≠ carry out of the sign bit. For half-adder design, Sum = A ⊕ B, Carry = A · B. A full adder extends this with a carry-in input: Sum = A ⊕ B ⊕ C_in, Carry_out = (A · B) + (C_in · (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 ⊕ C_in，进位_出 = (A · B) + (C_in · (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

If the most significant bit (sign bit) is 0, the value is simply the positive binary integer. 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 × 2^E−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 × 2^E−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) × 2^{E − bias}

值 = (−1)^S × (1 +

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