The following result shows that in every little interval of real numbers there is at least one rational number. Consequently, in every little interval of real numbers, there are infinitely many rational numbers! This is what is meant by rational numbers lie "densely" in real numbers.

Moreover, the result demonstrates in theory that every real number can be arbitrarily well approximated by rational numbers. As an practical example, take the decimal representation of the irrational number $$\pi=3.1415\ldots$$ which can be also interpreted as an approximation using the rational numbers $3,$ $3.1,$ $3.14,$ $3.141,$ $3.1415,$ ...

applicability: $\mathbb {Q}$

Proposition: Rational Numbers are Dense in Real Numbers

Let $x\in\mathbb R$ be any real number and let $\epsilon > 0$ be (an arbitrarily small) real number. Let $]x-\epsilon, x+\epsilon[$ be the open real interval containing the real number $x$. Then there is at least one rational number $q\in\mathbb Q$ with $q\in]x-\epsilon, x+\epsilon[$.

Because of this property, the rational numbers $\mathbb Q$ are set to be dense in the set of the real numbers $\mathbb R$.

Proofs: 1

Chapters: 1
Explanations: 2

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983