Binäärien

Binäärien (binary numbers) are numbers expressed in the base-2 numeral system, which uses only two digits: 0 and 1. In digital devices, each binary digit (bit) represents an on or off state, enabling compact data encoding and reliable arithmetic with simple logic circuits.

Unsigned binary encodes nonnegative integers. For signed numbers, several schemes exist; two's complement is the most widely used in modern computers, providing a straightforward method for representing negative values and performing subtraction via addition. Other schemes include sign-magnitude and one's complement, though they are less common in contemporary hardware. Binary also supports real numbers through floating-point representations, most prominently according to the IEEE 754 standard, which uses a sign bit, an exponent, and a significand.

Conversions and arithmetic: to convert a decimal integer to binary, repeatedly divide by two and record remainders; binary to decimal sums powers of two for each set bit. Basic binary addition follows the same carry rules as decimal addition, with carries propagating through positions. Bitwise operations such as AND, OR, XOR, and NOT underpin digital logic, while shifting bits serves as multiplication or division by powers of two.

Storage and encoding: sequences of bits form bytes (typically eight bits) and larger memory units. Data and instructions in computers are stored as binary sequences, and text characters are encoded using schemes such as ASCII or Unicode in binary form. History: the binary numeral system was popularized by Gottfried Wilhelm Leibniz, and it later became central to the development of electronic computing in the 20th century, where binary logic governs processors and digital communication.