Editing Square number (section)

{{short description|Product of an integer with itself}}
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[[File:Square number 16 as sum of gnomons.svg|thumb|Square number 16 as sum of [[Gnomon (figure)|gnomons]].]]
In [[mathematics]], a '''square number''' or '''perfect square''' is an [[integer]] that is the [[square (algebra)|square]] of an integer;<ref>Some authors also call squares of [[rational numbers]] perfect squares.</ref> in other words, it is the [[multiplication|product]] of some integer with itself. For example, 9 is a square number, since it equals {{math|3<sup>2</sup>}} and can be written as {{math|3 × 3}}.

The usual notation for the square of a number {{mvar|n}} is not the product {{math|''n'' × ''n''}}, but the equivalent [[exponentiation]] {{math|''n''<sup>2</sup>}}, usually pronounced as "{{mvar|n}} squared". The name ''square'' number comes from the name of the shape. The unit of [[area]] is defined as the area of a [[unit square]] ({{math|1 × 1}}). Hence, a square with side length {{mvar|n}} has area {{math|''n''<sup>2</sup>}}. If a square number is represented by ''n'' points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of ''n''; thus, square numbers are a type of [[Figurate number|figurate numbers]] (other examples being [[Cube (algebra)|cube numbers]] and [[triangular numbers]]).

In the [[Real number|real number system]], square numbers are [[non-negative]]. A non-negative integer is a square number when its [[square root]] is again an integer. For example, <math>\sqrt{9} = 3,</math> so 9 is a square number.

A positive integer that has no square [[divisor]]s except 1 is called [[Square-free integer|square-free]].

For a non-negative integer {{mvar|n}}, the {{mvar|n}}th square number is {{math|''n''<sup>2</sup>}}, with {{math|1=0<sup>2</sup> = 0}} being the [[0 (number)|zeroth]] one. The concept of square can be extended to some other number systems. If [[rational number|rational]] numbers are included, then a square is the ratio of two square integers, and, conversely, the ratio of two square integers is a square, for example, 
<math>\textstyle \frac{4}{9} = \left(\frac{2}{3}\right)^2</math>.

Starting with 1, there are <math>\lfloor \sqrt{m} \rfloor</math> square numbers up to and including {{mvar|m}}, where the expression <math>\lfloor x \rfloor</math> represents the [[floor function|floor]] of the number&nbsp;{{mvar|x}}.