数据表示 | Data Representation
整数的表示和运算方式 | Integer Representation and Arithmetic
表示方式 | Representation
二进制 | Binary
整数在计算机中以二进制表示，每个位只包含 0 或 1。
Integers are represented in computers using binary, where each bit can be either 0 or 1.
例如：10 进制的 10 在二进制中表示为 1010
For example: Decimal 10 is represented as 1010 in binary.
补码 | Two’s Complement
补码用于表示有符号整数，最高位表示符号位，0 表示正数，1 表示负数。
Two’s complement is used to represent signed integers, where the most significant bit represents the sign, 0 for positive and 1 for negative.
例如：8 位的-5 在补码中表示为 11111011
For example: -5 in 8-bit two’s complement is represented as 11111011.
运算方式 | Arithmetic
加法 | Addition
将两个二进制数相加，处理进位。
Add two binary numbers and handle the carry.
# Example in Python
a = 0b1010 # 10 in decimal
b = 0b0110 # 6 in decimal
result = a + b # 0b10000, which is 16 in decimal
print(bin(result)) # Output: 0b10000减法 | Subtraction
将被减数加上减数的补码。
Subtract by adding the minuend to the two’s complement of the subtrahend.
# Example in Python
a = 0b1010 # 10 in decimal
b = 0b0011 # 3 in decimal
result = a + (~b + 1) # 0b0111, which is 7 in decimal
print(bin(result)) # Output: 0b111乘法和除法 | Multiplication and Division
通过移位和加法/减法实现。
Implemented via shifts and additions/subtractions.
# Example of multiplication in Python
a = 0b101 # 5 in decimal
b = 0b11 # 3 in decimal
result = a * b # 0b1111, which is 15 in decimal
print(bin(result)) # Output: 0b1111浮点数的表示和运算方式 | Floating Point Representation and Arithmetic
表示方式 | Representation
IEEE 754 标准 | IEEE 754 Standard
浮点数使用 IEEE 754 标准表示，包含三个部分：符号位、指数位和尾数位。
Floating-point numbers are represented using the IEEE 754 standard, consisting of three parts: sign bit, exponent, and mantissa.
例如：32 位浮点数
For example: 32-bit floating point number
符号位 | Sign Bit: 1 位 (bit)
指数位 | Exponent: 8 位 (bits)
尾数位 | Mantissa: 23 位 (bits)
规格化 | Normalization
浮点数表示为
Floating-point numbers are represented as
运算方式 | Arithmetic
加法和减法 | Addition and Subtraction
对齐指数后相加或相减。
Align exponents and then add or subtract.
# Example in Python
import numpy as np
a = np.float32(1.5)
b = np.float32(2.25)
result = a + b # 3.75 in decimal
print(result) # Output: 3.75乘法和除法 | Multiplication and Division
分别乘或除尾数，并加或减指数。
Multiply or divide mantissas and add or subtract exponents.
# Example of multiplication in Python
a = np.float32(1.5)
b = np.float32(2.0)
result = a * b # 3.0 in decimal
print(result) # Output: 3.0精度问题 | Precision Issues
浮点数运算可能产生精度误差，特别是累加和大数相减。
Floating-point arithmetic can introduce precision errors, especially in summation and subtraction of large numbers.
# Precision issue example
a = np.float32(1e10)
b = np.float32(1)
result = a + b - a # Expected: 1, Actual: 0
print(result) # Output: 0.0