Binary Converter

The binary number system (base 2) uses only two digits: 0 and 1. It is the fundamental language of computers and digital electronics because electronic circuits can easily represent two states (on/off, high/low voltage).

• Each digit in binary is called a "bit" (binary digit)
• 8 bits make up 1 byte
• Binary 1010 equals decimal 10 (8 + 0 + 2 + 0)
• Place values double from right to left: 1, 2, 4, 8, 16, 32...

To convert binary to decimal, multiply each bit by its position value (power of 2) and add the results:

• Write down the binary number (e.g., 1101)
• Assign powers of 2 from right to left: 2^0, 2^1, 2^2, 2^3...
• Multiply each bit by its power: (1x8) + (1x4) + (0x2) + (1x1)
• Add the results: 8 + 4 + 0 + 1 = 13
• So binary 1101 = decimal 13

Hexadecimal (base 16) uses 16 symbols: 0-9 and A-F (where A=10, B=11, C=12, D=13, E=14, F=15). It's popular in computing because:

• One hex digit represents exactly 4 binary bits
• Easier to read than long binary strings (FF vs 11111111)
• Commonly used for memory addresses, colors (e.g., #FF5733), and MAC addresses
• Two hex digits can represent a byte (00 to FF = 0 to 255)

Octal (base 8) uses digits 0-7. While less common than hexadecimal today, it still has specific uses:

• Unix/Linux file permissions (e.g., chmod 755)
• Each octal digit represents exactly 3 binary bits
• Historically used in early computing systems
• Octal 777 = binary 111111111 = decimal 511

Each number system has specific valid characters:

• Binary (Base 2): Only 0 and 1
• Octal (Base 8): 0, 1, 2, 3, 4, 5, 6, 7
• Decimal (Base 10): 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• Hexadecimal (Base 16): 0-9 and A-F (case insensitive)

Using invalid digits for a number system will result in an error. For example, "2" is not valid in binary, and "G" is not valid in hexadecimal.