Unit square

Not to be confused with Square (unit).

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 the four points (0, 0), (1, 0), (0, 1), and (1, 1).

Cartesian coordinates

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

That is, a unit square is the Cartesian product I × I, where 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 numbers. In this view, the four corners of the unit square are at the four complex numbers 0, 1, i, and 1 + i.

Rational distance problem

Unsolved problem in mathematics:

Is there a point in the plane at a rational distance from all four corners of a unit square?

(more unsolved problems in mathematics)

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 $y\geq 1$, suppose that $x^{2}+y^{2}={\frac {A^{2}}{B^{2}}}$.

Then the distance: $x^{2}+(1-y)^{2}=x^{2}+y^{2}-2y+1={\frac {A^{2}}{B^{2}}}-2y+1=\left({\frac {A}{B}}-1\right)^{2}\implies y={\frac {A}{B}}\implies x=0$.

See also

• Unit circle
• Unit cube
• Unit sphere

External links

• Weisstein, Eric W. "Unit square". MathWorld.