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The square root of 2, written in mathematics as or , is the positive algebraic number that, when multiplied by itself, gives 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. Its numerical value, truncated to 65 decimal places, is: : . The 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). The approximation 665857/470832 is valid to within 1.13 x 10−12: its square is 2.0000000000045.... }}} |} ==History== The Babylonian clay tablet YBC 7289 (c. 1800–1600 BC) gives an approximation of in four sexagesimal figures, 1 24 51 10, which is accurate to about six decimal digits,〔Fowler and Robson, p. 368. (Photograph, illustration, and description of the ''root(2)'' tablet from the Yale Babylonian Collection ) (High resolution photographs, descriptions, and analysis of the ''root(2)'' tablet (YBC 7289) from the Yale Babylonian Collection )〕 and is the closest possible three-place sexagesimal representation of : : Another early close approximation is given in ancient Indian mathematical texts, the Sulbasutras (c. 800–200 BC) as follows: ''Increase the length (the side ) by its third and this third by its own fourth less the thirty-fourth part of that fourth.''〔Henderson.〕 That is, : This ancient Indian approximation is the seventh in a sequence of increasingly accurate approximations based on the sequence of Pell numbers, that can be derived from the continued fraction expansion of . Despite having a smaller denominator, it is only slightly less accurate than the Babylonian approximation. Pythagoreans discovered that the diagonal of a square is incommensurable with its side, or in modern language, that the square root of two is irrational. Little is known with certainty about the time or circumstances of this discovery, but the name of Hippasus of Metapontum is often mentioned. For a while, the Pythagoreans treated as an official secret the discovery that the square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it.〔Stephanie J. Morris, ("The Pythagorean Theorem" ), Dept. of Math. Ed., University of Georgia.〕〔Brian Clegg, ("The Dangerous Ratio ..." ), Nrich.org, November 2004.〕〔Kurt von Fritz, ("The discovery of incommensurability by Hippasus of Metapontum" ), Annals of Mathematics, 1945.〕 The square root of two is occasionally called "Pythagoras' number" or "Pythagoras' Constant", for example by . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「square root of 2」の詳細全文を読む スポンサード リンク
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