The square root of 2, written √2, is the positive real number that when multiplied by itself yields 2. This specific value is known as the principal square root; the equation x2 = 2 also has the negative solution −√2. As a classification, √2 is an irrational number, meaning it cannot be expressed exactly as a ratio of two integers.

Definition and basic properties

Formally, √2 is the unique nonnegative real x satisfying x·x = 2. It is algebraic of degree two (a root of the polynomial x2 − 2 = 0) but not rational. Its decimal expansion begins 1.41421356... and continues without repeating. The continued fraction for √2 is simple and periodic, typically written as [1; 2, 2, 2, …], a fact that reflects its close connection to simple quadratic surds.

Geometric interpretation

Geometrically, √2 is the length of the diagonal of a unit square. If the sides of a square measure 1, the Pythagorean theorem gives the diagonal length as √(1² + 1²) = √2. This concrete appearance in Euclidean geometry is why the number is sometimes called Pythagoras's constant. The geometric view also motivates many constructions and approximations.

History and proofs

The irrationality of √2 is one of the earliest known results about incommensurable magnitudes. Classical proofs use contradiction and parity arguments to show that no pair of integers in lowest terms can square to 2. Over centuries, mathematicians have explored alternative demonstrations and generalized the reasoning to other non-square integers.

Representations and approximations

Several practical representations are used depending on context: decimals (≈1.41421356), continued fractions (periodic [1;2,2,2,...]), and convergent rational approximations such as 1, 3/2, 7/5, 17/12, which arise from Pell-type recurrences. For symbolic work one writes √2 or 2^(1/2). Numerical methods supply arbitrarily accurate approximations when required.

Uses, importance, and distinctions

√2 appears in design, engineering, and number theory: from aspect ratios (the ISO paper sizes are based on √2) to solutions of quadratic equations, lattice geometry, and continued-fraction studies. It should be distinguished from the negative root and from approximations used in computation. For further mathematical background see general references on irrational numbers and algebraic integers, or consult introductory materials linked here and here.

  • Principal value: √2 (positive)
  • Equation: x² = 2
  • Decimal: ≈1.41421356
  • Continued fraction: [1; 2, 2, 2, ...]