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 $(1, 1)$ .

The unit square in the real plane

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 $I \times I$ , 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.[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 $(x,y)$ with $y\geq 1$ , suppose that $x^{2}+y^{2}={\frac {A^{2}}{B^{2}}}$ .

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

References

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

External links

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

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