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	${\displaystyle {\sqrt {5}}}$
Continued fraction	${\displaystyle 2+{\cfrac {1}{4+{\cfrac {1}{4+{\cfrac {1}{4+{{\vphantom {x}} \atop \displaystyle \ddots }}}}}}}}$

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

2.236067977499789696409173668731276235440... (sequence A002163 in the OEIS).

A length of ⁠${\displaystyle {\sqrt {5}}}$⁠ can be constructed as the diagonal of a ⁠${\displaystyle 2\times 1}$⁠ unit rectangle. ⁠${\displaystyle {\sqrt {5}}}$⁠ 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 ⁠${\displaystyle \varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}~\!{\bigr )}}$⁠.