Binary Converter

Binary is a base-2 number system that uses only two digits: 0 and 1. Each digit (bit) represents a power of 2. For example, 1011 in binary equals 1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 8 + 0 + 2 + 1 = 11 in decimal. Computers use binary because electronic circuits can easily distinguish between two states (on/off).

To convert decimal to binary, repeatedly divide the number by 2 and keep track of the remainders. Read the remainders from bottom to top. For example, to convert 13 to binary: 13÷2=6 remainder 1, 6÷2=3 remainder 0, 3÷2=1 remainder 1, 1÷2=0 remainder 1. Reading from bottom to top: 1101.

Hexadecimal (hex) is a base-16 number system using digits 0-9 and letters A-F (where A=10, B=11, C=12, D=13, E=14, F=15). It's widely used in computing because it's more compact than binary and each hex digit represents exactly 4 binary digits. For example, hex FF equals binary 11111111 equals decimal 255.

Each hexadecimal digit converts to exactly 4 binary digits. For example: 0=0000, 1=0001, 2=0010, ..., 9=1001, A=1010, B=1011, C=1100, D=1101, E=1110, F=1111. To convert hex to binary, replace each hex digit with its 4-bit binary equivalent. To convert binary to hex, group the binary digits into sets of 4 (from right to left) and convert each group.

Octal is a base-8 number system using digits 0-7. Each octal digit represents exactly 3 binary digits. It was commonly used in older computing systems and is still used in Unix/Linux file permissions. For example, octal 755 in file permissions means rwxr-xr-x (read, write, execute for owner; read and execute for group and others).

Unsigned integers can only represent positive numbers (and zero), using all bits for the magnitude. Signed integers can represent both positive and negative numbers, typically using two's complement representation where the most significant bit indicates the sign (0 for positive, 1 for negative). For example, in 8-bit: unsigned range is 0-255, signed range is -128 to 127.

Two's complement is the standard method for representing negative numbers in binary. To find the two's complement: 1) Invert all bits (one's complement), 2) Add 1 to the result. For example, to represent -5 in 8-bit: 5 = 00000101, invert to get 11111010, add 1 to get 11111011. This equals -5 in two's complement.

The maximum depends on the number of bits. For n bits: Binary can represent 2^n different values. For unsigned integers: 0 to 2^n-1. For signed integers: -2^(n-1) to 2^(n-1)-1. Common sizes: 8-bit unsigned (0-255), 16-bit unsigned (0-65,535), 32-bit unsigned (0-4,294,967,295), 64-bit unsigned (0-18,446,744,073,709,551,615).