Number systems are ways to represent numbers using different bases. The most common are:

• Decimal (Base 10): Uses digits 0-9. This is what we use in everyday life.
• Binary (Base 2): Uses only 0 and 1. This is how computers store data.
• Hexadecimal (Base 16): Uses 0-9 and A-F. Often used in programming for memory addresses and colors.
• Octal (Base 8): Uses digits 0-7. Less common but still used in some systems.

Conversion methods vary by system:

• Decimal to Binary: Repeatedly divide by 2 and collect remainders.
• Binary to Decimal: Multiply each bit by its power of 2 and sum.
• Decimal to Hex: Divide by 16 and map remainders (10=A, 11=B, etc.).
• Binary to Hex: Group bits in sets of 4 and convert each group.

Bit manipulation operates on individual bits:

• AND (&): Returns 1 only if both bits are 1.
• OR (|): Returns 1 if at least one bit is 1.
• XOR (^): Returns 1 if bits are different.
• NOT (~): Flips all bits (0→1, 1→0).
• Left Shift (<<): Moves bits left, multiplying by 2.
• Right Shift (>>): Moves bits right, dividing by 2.

Two's complement is how computers represent negative numbers in binary. To get the two's complement:

1. Invert all bits (NOT operation)
2. Add 1 to the result

For example, -5 in 8-bit two's complement: 5 = 00000101 → NOT = 11111010 → +1 = 11111011

Number system conversions have many practical applications:

• Programming: Memory addresses, bitwise operations, color codes (#FF0000)
• Networking: IP addresses, subnet masks, MAC addresses
• Electronics: Digital circuit design, microcontroller programming
• Cryptography: Encryption algorithms, hash functions
• File Systems: File permissions (octal), data storage