#
Unit square

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*) 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.

That is, the 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

It is not known whether any point in the plane is a rational distance from all four vertices of the unit square.^{[1]} 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: $(x,y)$, suppose that $x^{2}+y^{2}={\frac {A^{2}}{B^{2}}}$. Then the distance: $x^{2}+(1-y)^{2}=x^{2}+y^{2}-2y+1={\frac {A^{2}}{B^{2}}}-2y+1=({\frac {A}{B}}-1)^{2}\implies y={\frac {A}{B}}\implies x=0$.

What is becoming general for the plan; let the point $(x,y)$, be in the first dial $y\geq 1$ if we have the distance: $x^{2}+y^{2}={\frac {A^{2}}{B^{2}}}$ then the distance: $x^{2}+(y-1)^{2}=x^{2}+y^{2}-2y+1={\frac {A^{2}}{B^{2}}}-2y+1=({\frac {A}{B}}-1)^{2}\implies y={\frac {A}{B}}\implies x=0$

## See also

## References

**^** Guy, Richard K. (1991), *Unsolved Problems in Number Theory, Vol. 1* (2nd ed.), Springer-Verlag, pp. 181–185.

## External links

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