Square root of two

List of numbers Irrational and suspected irrational numbers $γ$ $ζ$ (3) √2 √3 √5 $φ$ $ρ$ $δ$ _$S$ $e$ $π$ $δ$
Binary	1.01101010000010011110…
Decimal	1.4142135623730950488…
Hexadecimal	1.6A09E667F3BCC908B2F…
Continued fraction	$1+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+\ddots }}}}}}}}$

The square root of 2, or the (1/2)th power of 2, written in mathematics as $\sqrt 2$ or $2 1 ⁄ 2$ , 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.

The rational approximation of the square root of two, 665,857/470,832, derived from the fourth step in the Babylonian algorithm starting with $a 0 = 1$ , is too large by approx. 6988160000000000000♠1.6×10⁻¹²: its square is 7000200000000000450♠2.0000000000045…

The rational approximation 99/70 (≈ 1.41429) is frequently used. Despite having a denominator of only 70, it differs from the correct value by less than 1/10,000 (approx. 6995720000000000000♠+0.72×10⁻⁴). Since it is a convergent of the continued fraction representation of the square root of two, any better rational approximation has a denominator not less than 169, since 239/169 is the next convergent with an error of approx. 3004880000000000000♠−0.12×10⁻⁴.

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