sqrt2s

The square root of two, often denoted by √2 or sqrt(2), is the positive real number that squared equals 2. It is approximately 1.41421356 and plays a foundational role in geometry and number theory. As a real number, sqrt(2) is irrational: it cannot be expressed as a ratio of two integers. A standard proof by contradiction uses parity to show that no integer solution to a^2 = 2b^2 exists in reduced form, establishing the irrationality.

Algebraically, √2 is a root of the polynomial x^2 − 2 and is an algebraic number of degree 2. Its minimal polynomial over the rational numbers is x^2 − 2. In radical form, it is simply √2, and it also has a simple continued fraction expansion, [1; 2, 2, 2, …]. The convergents 1, 3/2, 7/5, 17/12, 41/29, and so on, provide excellent rational approximations, closely tied to solutions of the Pell equation x^2 − 2y^2 = ±1.

Geometrically and historically, √2 equals the length of the diagonal of a unit square by the Pythagorean theorem. It is constructible with straightedge and compass, a fact tied to its geometric origin. The discovery of the irrationality of √2 is traditionally attributed to the early Pythagoreans, with Hippasus associated in historical accounts.

Applications of sqrt(2) appear across mathematics, computer science, and physics as a standard benchmark value and a basic irrational number. Its properties make it a central example in discussions of irrationality, continued fractions, and Diophantine approximation.