Editing Unit square

{{Short description|Square with side length one}}
{{distinguish|Square (unit)}}
[[Image:Unit Square.svg|thumb|300px|The unit square in the [[Euclidean geometry|real plane]]]]
In [[mathematics]], a '''unit square''' is a [[square (geometry)|square]] whose sides have length {{math|1}}.  Often, ''the'' unit square refers specifically to the square in the [[Cartesian coordinate system#Cartesian coordinates in two dimensions|Cartesian plane]] with corners at the four points {{math|(0, 0}}), {{math|(1, 0)}}, {{math|(0, 1)}}, and {{math|(1, 1)}}.<ref>{{citation
 | last = Horn | first = Alastair N.
 | editor-last1 = Crilly | editor-first1 = A. J.
 | editor-last2 = Earnshow | editor-first2 =  R. A.
 | editor-last3 = Jones | editor-first3 = H.
 | chapter = IFSs and the Interactive Design of Tiling Structures
 | title = Fractals and Chaos
 | date = 1991
 | publisher = Springer-Verlag
 | page = 136
 | chapter-url = https://books.google.com/books?id=PZHfBwAAQBAJ&pg=PA136
 | doi = 10.1007/978-1-4612-3034-2
| isbn = 978-1-4612-7770-5
 }}</ref>

==Cartesian coordinates==
In a [[Cartesian coordinate system]] with coordinates {{math|(''x'', ''y'')}}, a unit square is defined as a [[square (geometry)|square]] consisting of the points where both {{mvar|x}} and {{mvar|y}} lie in a closed [[unit interval]] from {{math|0}} to {{math|1}}.

That is, a unit square is the [[Cartesian product]] {{math|''I'' × ''I''}}, where {{mvar|I}} denotes the closed unit interval.

==Complex coordinates==
The unit square can also be thought of as a subset of the [[complex plane]], the topological space formed by the [[complex number]]s.
In this view, the four corners of the unit square are at the four complex numbers {{math|0}}, {{math|1}}, {{mvar|i}}, and {{math|1 + ''i''}}.

==Rational distance problem==
{{unsolved|mathematics|Is there a point in the plane at a rational distance from all four corners of a unit square?}}
It is not known whether any point in the plane is a [[Rational number|rational]] distance from all four vertices of the unit square.<ref>{{citation
 | last = Guy | first = Richard K. |authorlink = Richard K. Guy
 | title = Unsolved Problems in Number Theory
 | series = Problem Books in Mathematics | volume = 1
 | publisher = Springer-Verlag
 | edition = 2nd
 | year = 1991
 | pages = 181–185
 | doi = 10.1007/978-1-4899-3585-4
| isbn = 978-1-4899-3587-8 }}</ref>

== See also ==
* [[Unit circle]]
* [[Unit cube]]
* [[Unit sphere]]

== References ==
{{reflist}}

==External links==
* {{mathworld | urlname = UnitSquare | title = Unit square}}


[[Category:1 (number)]]
[[Category:Types of quadrilaterals]]
[[Category:Squares in number theory]]