Binary to Decimal, Hex & Octal Converter

Each binary digit (bit) represents a power of 2, starting from the rightmost bit (position 0). Multiply each bit by its positional value and sum them. For 1011: (1×8) + (0×4) + (1×2) + (1×1) = 8+0+2+1 = 11. The step-by-step view shows the reverse process: repeated division by 2 with remainders read bottom-to-top.

Two's complement is the standard way computers represent negative integers in binary. To negate a number: flip all bits, then add 1. For example, -5 in 8-bit two's complement is 11111011. This system has one representation of zero and makes arithmetic consistent — adding a negative number works the same as subtracting, so CPUs only need an adder circuit.

One hex digit maps exactly to 4 binary bits (a nibble), and two hex digits map to one byte. This makes hex a compact, human-readable representation of binary data. Memory addresses, color codes, file format magic numbers, and byte values are all naturally expressed in hex. It's much easier to read 0xFF than 11111111 or 255 depending on context.

Unix permissions have three groups (owner, group, others), each with three flags (read=4, write=2, execute=1). Summing the flags gives a digit 0-7, and three digits give the full permission. chmod 755 means owner has read+write+execute (7), group has read+execute (5), others have read+execute (5). Octal maps cleanly to these 3-bit groups.

An unsigned 8-bit integer holds values 0–255 (2^8 - 1). A signed 8-bit integer holds -128 to +127. An unsigned 16-bit integer holds 0–65,535 (2^16 - 1), and signed holds -32,768 to +32,767. These limits explain common errors like integer overflow — for example, a counter stored in a byte will wrap from 255 back to 0 when incremented.