Number systems and codes

Digital systems represent every quantity as a pattern of bits, so the way numbers and symbols are encoded is fundamental to their design. The binary system uses base two, while octal (base eight) and hexadecimal (base sixteen) are compact shorthands that group bits for easier reading. In a weighted binary system each bit position carries a fixed value; the ordinary 8421 weighting is the most common, and it forms the basis of binary-coded decimal (BCD), where each decimal digit is stored in its own group of four bits.

Beyond simple weighted codes there are codes chosen for special properties. The Excess-3 code is a self-complementing variant of BCD that simplifies decimal arithmetic. The Gray code is arranged so that successive values differ by only one bit, which reduces errors in position encoders and switching circuits where several bits might otherwise change at once. Text is handled by character codes: ASCII assigns numbers to letters, digits and control characters, and the older EBCDIC code served the same purpose on some mainframes.

Because storage and transmission can introduce errors, codes are often given redundancy. A simple parity bit detects single-bit errors, while the Hamming code adds several check bits placed so that the position of a faulty bit can be identified and corrected. Real numbers with very large or very small magnitudes are stored in floating-point form, separating a number into a sign, a fractional part and an exponent so that a wide range of values fits in a fixed number of bits.

Frequently asked questions

What makes Gray code useful?

Only one bit changes between consecutive values, so devices such as rotary encoders avoid the large errors that can occur when several bits switch at the same instant.

How does BCD differ from plain binary?

BCD stores each decimal digit separately in four bits, which makes decimal display and arithmetic straightforward but uses more bits than a pure binary number.

What does a Hamming code add over a parity bit?

A single parity bit only detects an error; a Hamming code uses several check bits so the exact position of a wrong bit can be found and corrected.