Knowlpedia

Density of ℝ \\ ℚ in ℝ

Between any two real numbers there is an irrational number

Density of R∖Q\mathbb{R}\setminus\mathbb{Q} in R\mathbb{R}: If a<ba<b are real numbers, then there exists an irrational number x∈R∖Qx\in\mathbb{R}\setminus\mathbb{Q} such that a<x<b.a<x<b.

Remarks

This shows that irrationals are just as ubiquitous as rationals and prevents "gaps" of irrationality; it is often used in constructing sequences with prescribed properties.