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{{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 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 (0,&nbsp;0), (1,&nbsp;0), (0,&nbsp;1), and (1,&nbsp;1).


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==In the real plane==
In a [[Coordinate system#Cartesian coordinate system|Cartesian coordinate system]] with coordinates (''x'', ''y'') the '''unit square''' is defined as the [[square (geometry)|square]] consisting of the points where both ''x'' and ''y'' lie in a closed [[unit interval]] from 0 to 1 on their respective axes.
 
That is, the unit square is the [[Cartesian product]] {{math|I}}&nbsp;&times;&nbsp;{{math|I}}, where {{math|I}} denotes the closed unit interval.
 
It is not known whether any point in the plane is a rational distance from all four vertices of the unit square.<ref>Guy, Richard K. ''Unsolved Problems in Number Theory'', Vol. 1, Springer-Verlag, 2nd ed. 1991, 181-185.</ref> However, no such point is on an edge of the square.<ref>Barbara, Roy. "The rational distance problem", ''[[Mathematical Gazette]]'' 95, March 2011, 59-61.</ref>
 
==In the complex plane==
In the [[complex plane]], the corners of the unit square are at 0, 1, <math>i</math>, and 1&nbsp;+&nbsp;<math>i</math>.
 
== See also ==
* [[Unit circle]]
* [[Unit sphere]]
* [[Unit cube]]
 
== References ==
{{reflist}}
* {{mathworld | urlname = UnitSquare | title = Unit square}}
 
[[Category:Quadrilaterals]]
[[Category:One]]
 
{{Elementary-geometry-stub}}

Revision as of 21:35, 24 March 2013

Template:Distinguish

The unit square in the real plane.

In mathematics, a unit square is a square whose sides have length 1. Often, "the" unit square refers specifically to the square in the Cartesian plane with corners at (0, 0), (1, 0), (0, 1), and (1, 1).

In the real plane

In a Cartesian coordinate system with coordinates (x, y) the unit square is defined as the square consisting of the points where both x and y lie in a closed unit interval from 0 to 1 on their respective axes.

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It is not known whether any point in the plane is a rational distance from all four vertices of the unit square.[1] However, no such point is on an edge of the square.[2]

In the complex plane

In the complex plane, the corners of the unit square are at 0, 1, i, and 1 + i.

See also

References

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Template:Elementary-geometry-stub

  1. Guy, Richard K. Unsolved Problems in Number Theory, Vol. 1, Springer-Verlag, 2nd ed. 1991, 181-185.
  2. Barbara, Roy. "The rational distance problem", Mathematical Gazette 95, March 2011, 59-61.