Editing Square root (section)

==Square roots of positive integers==
A positive number has two square roots, one positive, and one negative, which are [[opposite (mathematics)|opposite]] to each other. When talking of ''the'' square root of a positive integer, it is usually the positive square root that is meant.

The square roots of an integer are [[algebraic integer]]s—more specifically [[quadratic integer]]s.

The square root of a positive integer is the product of the roots of its [[prime number|prime]] factors, because the square root of a product is the product of the square roots of the factors. Since  <math display="inline">\sqrt{p^{2k}} = p^k,</math> only roots of those primes having an odd power in the [[integer factorization|factorization]] are necessary. More precisely, the square root of a prime factorization is<math display="block">\sqrt{p_1^{2e_1+1} \cdots p_k^{2e_k+1}p_{k+1}^{2e_{k+1}} \dots p_n^{2e_n}} = p_1^{e_1} \dots p_n^{e_n} \sqrt{p_1\dots p_k}.</math> 

===As decimal expansions===
The square roots of the [[square number|perfect square]]s (e.g., 0, 1, 4, 9, 16) are [[integers]]. In all other cases, the square roots of positive integers are [[irrational number]]s, and hence have non-[[repeating decimal]]s in their [[decimal representation]]s. Decimal approximations of the square roots of the first few natural numbers are given in the following table.

{|class="wikitable"
! {{mvar|n}} !! <math>\sqrt{n},</math> truncated to 50 decimal places
|- 
|align="right" | 0 || 0
|- 
|align="right" | 1 || 1
|-
|align="right" | 2 || [[Square root of 2|{{gaps|1.4142135623|7309504880|1688724209|6980785696|7187537694}}]]
|-
|align="right" | 3 || [[Square root of 3|{{gaps|1.7320508075|6887729352|7446341505|8723669428|0525381038}}]]
|-
|align="right" | 4 || 2
|-
|align="right" | 5 || [[Square root of 5|{{gaps|2.2360679774|9978969640|9173668731|2762354406|1835961152}}]]
|-
|align="right" | 6 || [[Square root of 6|{{gaps|2.4494897427|8317809819|7284074705|8913919659|4748065667}}]]
|-
|align="right" | 7 || [[Square root of 7|{{gaps|2.6457513110|6459059050|1615753639|2604257102|5918308245}}]]
|-
|align="right" | 8 || {{gaps|2.8284271247|4619009760|3377448419|3961571393|4375075389}}
|-
|align="right" | 9 || 3
|-
|align="right" | 10 || {{gaps|3.1622776601|6837933199|8893544432|7185337195|5513932521}}
|}

===As expansions in other numeral systems===
As with before, the square roots of the [[square number|perfect square]]s (e.g., 0, 1, 4, 9, 16) are integers. In all other cases, the square roots of positive integers are [[irrational number]]s, and therefore have non-repeating digits in any standard [[positional notation]] system.

The square roots of small integers are used in both the [[SHA-1]] and [[SHA-2]] hash function designs to provide [[nothing up my sleeve number]]s.

===As periodic continued fractions===
A result from the study of [[irrational number]]s as [[simple continued fraction]]s was obtained by [[Joseph Louis Lagrange]] {{circa|1780}}. Lagrange found that the representation of the square root of any non-square positive integer as a continued fraction is [[Periodic continued fraction|periodic]]. That is, a certain pattern of partial denominators repeats indefinitely in the continued fraction. In a sense these square roots are the very simplest irrational numbers, because they can be represented with a simple repeating pattern of integers.

{|
|- 
|align="right"|<math>\sqrt{2}</math>|| = [1; 2, 2, ...]
|-
|align="right"|<math>\sqrt{3}</math>|| = [1; 1, 2, 1, 2, ...]
|-
|align="right"|<math>\sqrt{4}</math>|| = [2]
|-
|align="right"|<math>\sqrt{5}</math>|| = [2; 4, 4, ...]
|-
|align="right"|<math>\sqrt{6}</math>|| = [2; 2, 4, 2, 4, ...]
|-
|align="right"|<math>\sqrt{7}</math>|| = [2; 1, 1, 1, 4, 1, 1, 1, 4, ...]
|-
|align="right"|<math>\sqrt{8}</math>||= [2; 1, 4, 1, 4, ...]
|-
|align="right"|<math>\sqrt{9}</math>|| = [3]
|-
|align="right"|<math>\sqrt{10}</math>|| = [3; 6, 6, ...]
|- 
|align="right"|<math>\sqrt{11}</math>|| = [3; 3, 6, 3, 6, ...]
|-
|align="right"|<math>\sqrt{12}</math>|| = [3; 2, 6, 2, 6, ...]
|-
|align="right"|<math>\sqrt{13}</math>|| = [3; 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, ...]
|-
|align="right"|<math>\sqrt{14}</math>|| = [3; 1, 2, 1, 6, 1, 2, 1, 6, ...]
|-
|align="right"|<math>\sqrt{15}</math>|| = [3; 1, 6, 1, 6, ...]
|-
|align="right"|<math>\sqrt{16}</math>|| = [4]
|-
|align="right"|<math>\sqrt{17}</math>|| = [4; 8, 8, ...]
|-
|align="right"|<math>\sqrt{18}</math>|| = [4; 4, 8, 4, 8, ...]
|-
|align="right"|<math>\sqrt{19}</math>|| = [4; 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, ...]
|-
|align="right"|<math>\sqrt{20}</math>|| = [4; 2, 8, 2, 8, ...]
|}

The [[square bracket]] notation used above is a short form for a continued fraction. Written in the more suggestive algebraic form, the simple continued fraction for the square root of 11, [3; 3, 6, 3, 6, ...], looks like this:<math display="block">
\sqrt{11} = 3 + \cfrac{1}{3 + \cfrac{1}{6 + \cfrac{1}{3 + \cfrac{1}{6 + \cfrac{1}{3 + \ddots}}}}}
</math>

where the two-digit pattern {3, 6} repeats over and over again in the partial denominators. Since {{math|1=11 = 3<sup>2</sup> + 2}}, the above is also identical to the following [[generalized continued fraction#Roots of positive numbers|generalized continued fractions]]:

<math display="block">
\sqrt{11} = 3 + \cfrac{2}{6 + \cfrac{2}{6 + \cfrac{2}{6 + \cfrac{2}{6 + \cfrac{2}{6 + \ddots}}}}} = 3 + \cfrac{6}{20 - 1 - \cfrac{1}{20 - \cfrac{1}{20 - \cfrac{1}{20 - \cfrac{1}{20 - \ddots}}}}}.
</math>