Bitwise Operators

Two’s Complement

Two’s complement is a way to represent signed integers (positive and negative) in binary. It’s the most common method used in computers because it simplifies arithmetic operations, particularly addition and subtraction.

Key Concepts:

• Sign Bit: The most significant bit (MSB) represents the sign. 0 indicates positive, 1 indicates negative.

• Positive Numbers: Represented in standard binary.

• Negative Numbers: Calculated by inverting all bits (0s become 1s, 1s become 0s) and then adding 1.

Example (8-bit):

Let’s represent 5 and -5:

• 5: 00000101 (Positive, standard binary)

• -5:

  1. Invert bits: 11111010
  2. Add 1: 11111011

Python Implementation:

Python handles this internally, so you don’t usually need to worry about the specifics. However, you can simulate it:

def twos_complement(n, bits):
    """Represents an integer using two's complement in a specified number of bits."""
    if n >= 0:
        return bin(n)2:]].zfill(bits)  #Positive numbers
    else:
        positive_equivalent = (1 << bits) + n #add 2^bits to get the positive equivalent
        return bin(positive_equivalent)2:]].zfill(bits)

print(twos_complement(5, 8))  # Output: 00000101
print(twos_complement(-5, 8)) # Output: 11111011

Advantages:

• Simplified Arithmetic: Addition and subtraction can be performed using the same circuitry regardless of the signs of the numbers. This simplifies hardware design.
• Unique Representation: Each integer has a unique representation in two’s complement (except for the most negative number in a given number of bits).

Disadvantages:

• Range Limitation: The range of representable numbers is limited by the number of bits used. For example, with 8 bits, the range is -128 to 127.
• Bitwise Operations (Need separate explanation)
• Binary Representation (Need separate explanation)

Number Systems (A note on different number systems like decimal, binary, hexadecimal would be useful)