Square root of 5

The square root of 5, denoted ⁠

5

{\displaystyle {\sqrt {5}}}

⁠, is the positive real number that, when multiplied by itself, gives the natural number 5. Along with its conjugate ⁠

−

5

{\displaystyle -{\sqrt {5}}}

⁠, it solves the quadratic equation ⁠

x

2

−

5

=

0

{\displaystyle x^{2}-5=0}

⁠, making it a quadratic integer, a type of algebraic number. ⁠

5

{\displaystyle {\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:

A length of ⁠

5

{\displaystyle {\sqrt {5}}}

⁠ can be constructed as the diagonal of a ⁠

2

×

1

{\displaystyle 2\times 1}

⁠ unit rectangle. ⁠

5

{\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 ⁠

φ

=

1

2

(

1

+

5

)

{\displaystyle \varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}~\!{\bigr )}}

⁠.

Rational approximations

The square root of 5 is an irrational number, meaning it can not be exactly represented as a fraction ⁠

x

/

y

{\displaystyle x/y}

⁠ where ⁠

x

{\displaystyle x}

⁠ and ⁠

y

{\displaystyle y}

⁠ are integers. However, it can be approximated arbitrarily closely by such rational numbers.

Particularly good approximations are the integer solutions of Pell's equations,

x

2

−

5

y

2

=

1

and

x

2

−

5

y

2

=

−

1

,

{\displaystyle x^{2}-5y^{2}=1\quad {\text{and}}\quad x^{2}-5y^{2}=-1,}

which can be algebraically rearranged into the form

x

y

=

5

±

1

y

2

.

{\displaystyle {\frac {x}{y}}={\sqrt {5\pm {\frac {1}{y^{2}}}}}.}

For example, the approximation ⁠

2

=

5

−

1

{\displaystyle \textstyle 2={\sqrt {5-1}}}

⁠, which is accurate to about 10%, satisfies the negative Pell's equation, ⁠

2

2

−

5

⋅

1

2

=

4

−

5

=

−

1

{\displaystyle \textstyle 2^{2}-5\cdot 1^{2}=4-5=-1}

⁠; likewise, the approximation ⁠

9

4

=

5

+

1

16

=

2.25

{\displaystyle \textstyle {\tfrac {9}{4}}={\sqrt {5+{\tfrac {1}{16}}}}=2.25}

⁠, which is accurate within 1%, satisfies the positive equation, ⁠

9

2

−

5

⋅

4

2

=

81

−

80

=

1

{\displaystyle \textstyle 9^{2}-5\cdot 4^{2}=81-80=1}

⁠. These two approximations are the respective fundamental solutions of each Pell's equation, to which additional solutions are algebraically related.