In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a unit square is a
square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
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
In geometry, a Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of real numbers called ''coordinates'', which are the positive and negative number ...
with coordinates , a unit square is defined as a
square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
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.
See also
*
Unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
*
Unit cube
*
Unit sphere
References
External links
* {{mathworld , urlname = UnitSquare , title = Unit square
1 (number)
Types of quadrilaterals
Squares in number theory