Number Base Converter

Why 2, 8, 16 and not 7 or 13

Hardware is binary, but binary is unreadable at length. Because 8 and 16 are both powers of two, converting between them and binary needs no arithmetic at all — every hex digit is exactly four bits, every octal digit exactly three. 0xFF is 1111 1111 by inspection. That property is the whole reason those bases became standard, and it is why base 10 is the awkward one in computing.

Two's complement: how negatives are stored

Flip every bit and add one. In 8 bits, −1 becomes 11111111 and −128 becomes 10000000. The reason this representation won is that addition and subtraction need no special case — the same adder circuit handles signed and unsigned values identically, and the carry out of the top bit is simply discarded.

It also explains the asymmetry every programmer meets eventually: an 8-bit signed value ranges from −128 to +127, not −127 to +127, because zero takes up one of the positive slots.

Reading the place-value breakdown

Every base works identically: each digit is multiplied by the base raised to its position, counting from zero on the right. 1011 in binary is 8 + 0 + 2 + 1 = 11. 2F in hex is 2×16 + 15 = 47. The breakdown above spells this out digit by digit, which is the fastest way to convert by hand and to spot a mis-typed digit.

Bits, bytes and why the count matters

The number of bits needed is ⌊log₂(n)⌋ + 1. It determines whether a value fits in a field, how wide a database column has to be, and how many distinct values an identifier can hold. 255 needs 8 bits and fills a byte exactly, which is why it is the boundary that shows up everywhere from colour channels to IPv4 octets.

Frequently Asked Questions