Euclidean

Euclidean refers to ideas and methods associated with the ancient Greek mathematician Euclid of Alexandria, active around 300 BCE. In mathematics, the term is used for theories, spaces, and algorithms that embody the flat, standard geometry of ordinary space. The label contrasts with non-Euclidean geometries that arise when the parallel postulate is altered.

Euclidean geometry is the branch based on Euclid's Elements, using a concise set of axioms about points, lines, and planes. It studies figures in the plane and in three-dimensional space, deriving theorems such as the Pythagorean theorem and criteria for congruence. The parallel postulate, which concerns the uniqueness of parallels, is a defining feature; in the 19th century, alternative geometries showed that this postulate can be replaced, leading to hyperbolic and elliptic geometries.

In modern mathematics, Euclidean space, denoted R^n, is the standard model of flat n-dimensional space. It is equipped with the Euclidean metric, distance between x and y being sqrt(sum (xi - yi)^2), which leads to familiar results about lengths, angles, and volumes. The space has an inner product that induces distances and angles, and is preserved by the usual isometries such as translations and rotations.

The Euclidean algorithm is a method for computing the greatest common divisor of two integers. Attributed to Euclid in Elements, Book VII, it repeatedly replaces the pair (a,b) with (b, a mod b) until the remainder is zero; the last nonzero remainder is the gcd. The algorithm underpins many number-theoretic procedures and is efficient for large integers.