In mathematics, a partial order ≤ on a set X is said to be dense (or dense-in-itself) if, for all x and y in X for which x < y, there is a z in X such that x < z < y. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ...
The rational numbers with the ordinary ordering are a densely ordered set in this sense, as are the real numbers. On the other hand, the ordinary ordering on the integers is not dense. In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, the real numbers may be described informally in several different ways. ... The integers are commonly denoted by the above symbol. ...
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