Rational Numbers Set Is Dense. In the figure below, we illustrate the density property with a number line. Note that the set of irrational numbers is the complementary of the set of rational numbers.
Due to the fact that between any two rational numbers there is an infinite number of other rational numbers, it can easily lead to the wrong conclusion, that the set of rational numbers is so dense, that there is no need for further expanding of the rational numbers set. Points with rational coordinates, in the plane is dense in the plane. Theorem 1 (the density of the rational numbers):.
Let the ordered > pair (p_i, q_i) be an element of a function, as a set, from p to q.
The set of complex numbers includes all the other sets of numbers. The integers, for example, are not dense in the reals because one can find two reals with no integers between them. Real analysis grinshpan the set of rational numbers is not g by baire’s theorem, the interval [0; > else the rational numbers are not dense in the reals thus that between > any two irrational numbers there is a rational number.
