How Number Bases Work: Binary, Hex, Octal, and Decimal

Every base uses positional notation: each digit is multiplied by the base raised to the power of its position, counting from zero on the right. In decimal, 205 means 2 times 100 plus 0 times 10 plus 5 times 1. The only thing that changes between bases is the base number and how many distinct digits are available.

Binary (base 2) uses digits 0 and 1, octal (base 8) uses 0 through 7, decimal (base 10) uses 0 through 9, and hexadecimal (base 16) uses 0 through 9 then A through F for the values ten through fifteen. The same quantity thirteen is 1101 in binary, 15 in octal, 13 in decimal, and D in hex.

Hexadecimal is popular because one hex digit maps to exactly four binary digits, so a byte of eight bits is always two clean hex digits. This makes hex a compact, readable stand-in for binary, which is why colors like #FF8800 and memory addresses are written in hex.

Octal groups bits in threes and survives mainly in Unix file permissions, where read, write, and execute map neatly onto three bits per digit. Both bases exist because they compress binary without hiding its structure the way decimal does.

To convert a decimal number to another base, repeatedly divide by the target base and collect the remainders from last to first. To go the other way, multiply each digit by the base raised to its position and add the results. Converting between binary and hex is faster: split the binary into groups of four bits from the right and translate each group into one hex digit.

Fractional values follow the same logic on the other side of the point, using negative powers of the base, though many fractions that are exact in decimal, like 0.1, repeat forever in binary.

The Number Base Converter takes a value in one base and shows it in binary, octal, decimal, and hexadecimal at once, so you can check your hand work or skip it entirely.

Bases themselves have no sign; a raw binary string is just a magnitude. Computers store negative integers using two's complement, where the leftmost bit carries a negative weight. In an 8-bit two's-complement byte, 11111111 is not 255 but -1, because the value depends on the fixed bit width.

This is the caveat that trips people up: the same bit pattern means different things depending on whether it is read as unsigned or signed, and on how many bits the register holds. A plain base conversion assumes an unsigned magnitude unless you specify the width and signing convention.