page_title: Mathematical constant
section_title: Pythagoras' constant √2
hierarchical_section_title: Mathematical constant / Basic mathematical constants / Pythagoras' constant √2
caption_attribution_description: English: Dedekind cut defining √2, an irrational number, using two sets of positive rational numbers. All those whose square is less than two (red), and those whose square is equal to or greater than two (blue). ... 1/1 = 1 7/5 = 1.4 41/29 = 1.413793103448276... 239/169 = 1.414201183431953... 1393/985 = 1.414213197969543... √2 = 1.414213562373095... 3363/2378 = 1.41421362489487... 577/408 = 1.41421568627451... 99/70 = 1.414285714285714... 17/12 = 1.416 3/2 = 1.5 2/1 = 2 ... After: File:Dedekind cut sqrt 2.svg.
context_page_description: A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol, or by mathematicians' names to facilitate using it across multiple mathematical problems. Constants arise in many areas of mathematics, with constants such as e and π occurring in such diverse contexts as geometry, number theory, and calculus.
  What it means for a constant to arise "naturally", and what makes a constant "interesting", is ultimately a matter of taste, just as some mathematical constants are notable more for historical reasons—than for their intrinsic mathematical interest. The more popular constants have been studied throughout the ages and computed to many decimal places.
  All named mathematical constants are definable numbers, and usually are also computable numbers.
context_section_description: The square root of 2, often known as root 2, radical 2, or Pythagoras' constant, and written as √2, is the positive algebraic number that, when multiplied by itself, gives the number 2. It is more precisely 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. Its numerical value truncated to 65 decimal places is:
  1.41421356237309504880168872420969807856967187537694807317667973799... (sequence A002193 in the OEIS).
  Alternatively, the quick approximation 99/70 (≈ 1.41429) for the square root of two is frequently used. Despite having a denominator of only 70, it differs from the correct value by less than 1/10,000 (approx. 7.2 × 10 ⁻⁵).
caption: The square root of 2.
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