Editing Square root of 2 (section)

{{Short description|Unique positive real number which when multiplied by itself gives 2}}
{{use dmy dates |cs1-dates=sy |date=October 2024}}
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{{redirect-distinguish|Pythagoras's constant|Pythagoras number}}
{{infobox non-integer number
|image              = Isosceles right triangle with legs length 1.svg
|image_caption      = The square root of 2 is equal to the length of the [[hypotenuse]] of an [[Isosceles triangle|isosceles]] [[right triangle]] with legs of length 1.
|decimal            = {{gaps|1.41421|35623|73095|0488...}}
|continued_fraction = <math>1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}}</math>
}}

The '''square root of 2''' (approximately 1.4142) is the positive [[real number]] that, when multiplied by itself or squared, equals the [[number 2]]. It may be written as <math>\sqrt{2}</math> or <math>2^{1/2}</math>. It is an [[algebraic number]], and therefore not a [[transcendental number]]. Technically, it should be 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 [[Unit square|square with sides of one unit of length]]; this follows from the [[Pythagorean theorem]]. It was probably the first number known to be [[irrational number|irrational]].<ref>{{citation |last=Fowler |first=David H. |editor-last1=Gavroglu |editor-first1=Kostas |editor-last2=Christianidis |editor-first2=Jean |editor-last3=Nicolaidis |editor-first3=Efthymios |date=1994 |chapter=The Story of the Discovery of Incommensurability, Revisited |title=Trends in the Historiography of Science |series=Boston Studies in the Philosophy of Science |volume=151 |location=Dortrecht |publisher=Springer |pages=221–236 |doi=10.1007/978-94-017-3596-4 |isbn=978-9048142644}}</ref> The fraction {{sfrac|99|70}} (≈ '''1.4142'''857) is sometimes used as a good [[Diophantine approximation|rational approximation]] with a reasonably small [[denominator]].

Sequence {{OEIS link|A002193}} in the [[On-Line Encyclopedia of Integer Sequences]] consists of the digits in the [[decimal expansion]] of the square root of 2, here truncated to 60 decimal places:<ref>{{cite OEIS |1=A002193 |2= Decimal expansion of square root of 2 |access-date=2020-08-10 }}</ref>

:{{gaps|1.41421|35623|73095|04880|16887|24209|69807|85696|71875|37694|80731|76679}}