300px|The unit square in the real plane
In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and .

Cartesian coordinates

In a Cartesian coordinate system with coordinates , a unit square is defined as a square consisting of the points where both and lie in a closed unit interval from to . That is, a unit square is the Cartesian product , where 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 numbers. In this view, the four corners of the unit square are at the four complex numbers , , , and .

Rational distance problem

It is not known whether any point in the plane is a rational distance from all four vertices of the unit square.. However, according to Périat, the only points included in the square of rational distances of the four vertices are necessarily on the sides: With the point $(x,y)$ with $y\; \backslash geq\; 1$, suppose that $x^2\; +\; y^2\; =\; \backslash frac$. Then the distance: $x^2\; +\; (1-y)^2\; =\; x^2\; +\; y^2\; -2y\; +1\; =\; \backslash frac\; -\; 2y\; +\; 1\; =\; \backslash left(\backslash frac-1\backslash right)^2\; \backslash implies\; y\; =\; \backslash frac\; \backslash implies\; x\; =\; 0$.

** See also **

* Unit circle
* Unit cube
* Unit sphere

** References **

External links

* {{mathworld | urlname = UnitSquare | title = Unit square Category:1 (number) Category:Types of quadrilaterals Category:Squares in number theory

Cartesian coordinates

In a Cartesian coordinate system with coordinates , a unit square is defined as a square consisting of the points where both and lie in a closed unit interval from to . That is, a unit square is the Cartesian product , where 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 numbers. In this view, the four corners of the unit square are at the four complex numbers , , , and .

Rational distance problem

It is not known whether any point in the plane is a rational distance from all four vertices of the unit square.. However, according to Périat, the only points included in the square of rational distances of the four vertices are necessarily on the sides: With the point $(x,y)$ with $y\; \backslash geq\; 1$, suppose that $x^2\; +\; y^2\; =\; \backslash frac$. Then the distance: $x^2\; +\; (1-y)^2\; =\; x^2\; +\; y^2\; -2y\; +1\; =\; \backslash frac\; -\; 2y\; +\; 1\; =\; \backslash left(\backslash frac-1\backslash right)^2\; \backslash implies\; y\; =\; \backslash frac\; \backslash implies\; x\; =\; 0$.

External links

* {{mathworld | urlname = UnitSquare | title = Unit square Category:1 (number) Category:Types of quadrilaterals Category:Squares in number theory