page_title: Square root of 2
section_title: Properties of the square root of two
hierarchical_section_title: Square root of 2 / Properties of the square root of two
context_page_description: The square root of 2, or the th power of 2, written in mathematics as √2 or 2​¹⁄², is the positive algebraic number that, when multiplied by itself, equals the number 2. Technically, it is called the principal square root of 2, to distinguish it from the negative number with the same property.
  Geometrically, the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational.
  As a good rational approximation for the square root of two, with a reasonably small denominator, the fraction is sometimes used.
  The sequence A002193 in the OEIS consists of the digits in the decimal expansion of the square root of 2, here truncated to 65 decimal places:
  1.41421356237309504880168872420969807856967187537694807317667973799...
context_section_description: One-half of √2, also the reciprocal of √2, approximately 0.707106781186548, is a common quantity in geometry and trigonometry because the unit vector that makes a 45° angle with the axes in a plane has the coordinates
  This number satisfies
  One interesting property of √2 is as follows:
  since
  This is related to the property of silver ratios.
  √2 can also be expressed in terms of the copies of the imaginary unit i using only the square root and arithmetic operations:
  if the square root symbol is interpreted suitably for the complex numbers i and −i.
  √2 is also the only real number other than 1 whose infinite tetrate (i.e., infinite exponential tower) is equal to its square. In other words: if for c > 1, x₁ = c and xₙ₊₁ = cˣⁿ for n > 1, the limit of xₙ will be called as n → ∞ (if this limit exists) f(c). Then √2 is the only number c > 1 for which f(c) = c². Or symbolically:
  √2 appears in Viète's formula for π:
  for m square roots and only one minus sign.
  Similar in appearance but with a finite number of terms, √2 appears in various trigonometric constants:
  It is not known whether √2 is a normal number, a stronger property than irrationality, but statistical analyses of its binary expansion are consistent with the hypothesis that it is normal to base two.
caption: Angle size and sector area are the same when the conic radius is √2. This diagram illustrates the circular and hyperbolic functions based on sector areas u.