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 $(x, y)$, 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
It is not known whether any point in the plane is a rational distance from all four vertices of the unit square.