Square root of 5

Square root of 5

The diagonal of a 2 × 1 rectangle has length √5.
Rationality	Irrational
Representations
Decimal	2.23606797749978969...
Algebraic form	${\sqrt {5}}$
Continued fraction	$2+{\cfrac {1}{4+{\cfrac {1}{4+{\cfrac {1}{4+{{\vphantom {x}} \atop \displaystyle \ddots }}}}}}}$

The square root of 5, denoted ⁠${\sqrt {5}}$⁠, is the positive real number that, when multiplied by itself, gives the natural number 5. Along with its conjugate ⁠$-{\sqrt {5}}$⁠, it solves the quadratic equation ⁠$x^{2}-5=0$⁠, making it a quadratic integer, a type of algebraic number. ⁠${\sqrt {5}}$⁠ is an irrational number, meaning it cannot be written as a fraction of integers. The first thirty significant digits of its decimal expansion are:

2.23606797749978969640917366873...

A length of ⁠${\sqrt {5}}$⁠ can be constructed as the diagonal of a ⁠$2\times 1$⁠ unit rectangle. ⁠${\sqrt {5}}$⁠ also appears throughout in the metrical geometry of shapes with fivefold symmetry; the ratio between diagonal and side of a regular pentagon is the golden ratio ⁠$\varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}~\!{\bigr )}$⁠.