“Unit square”的意思、由来-开放百科全书

词条

Unit square

释义

Cartesian coordinates
Complex coordinates
Rational distance problem
See also
References
External links

{{Merge from|Lexicographic order topology on the unit square|discuss=Talk:Lexicographic order topology on the unit square#Merge proposal|date=September 2017}}{{distinguish|Square (unit)}}

In mathematics, a unit square is a square whose sides have length {{math|1}}. Often, "the" unit square refers specifically to the square in the Cartesian plane with corners at the four points {{math|(0, 0}}), {{math|(1, 0)}}, {{math|(0, 1)}}, and {{math|(1, 1)}}.

Cartesian coordinates

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

That is, the unit square is the Cartesian product {{math|I × I}}, where {{mvar|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 {{math|0}}, {{math|1}}, {{mvar|i}}, and {{math|1 + i}}.

Rational distance problem

{{unsolved|mathematics|Is there a point in the plane at a rational distance from all four corners of a unit square?}}

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: , suppose that . Then the distance: .

What is becoming general for the plan; let the point , be in the first dial if we have the distance: then the distance:

References

1. ^{{citation|last=Guy|first=Richard K.|authorlink=Richard K. Guy|title=Unsolved Problems in Number Theory, Vol. 1|publisher=Springer-Verlag|edition=2nd|year=1991|pages=181–185}}.

External links

{{mathworld | urlname = UnitSquare | title = Unit square}}

3 : 1 (number)|Quadrilaterals|Squares in number theory

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Cartesian coordinates

Complex coordinates

Rational distance problem

See also

References

External links