# Show that there is a rational number and an irrational

Rapsinincke 2022-07-01 Answered
Show that there is a rational number and an irrational number between any two real numbers.
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## Answers (1)

fugprurgeil
Answered 2022-07-02 Author has 12 answers
We know that $\mathbb{Q}$ is dense in $\mathbb{R}$. This means for any $x\in \mathbb{R}$ and for any $r>0$ , $\mathbb{Q}\cap B\left(x;r\right)\ne \varphi$ where $B\left(x;r\right)=\left\{y\in \mathbb{R}:|x-y|.
Let $a,b\in \mathbb{R}$ be such that $a. Then we have $\mathbb{Q}\cap B\left(\frac{a+b}{2};\frac{b-a}{3}\right)\ne \varphi$ by above statement. This simply means there is at least one rational number between $a$ and $b$.
Similarly, $\mathbb{R}\mathbb{-}\mathbb{Q}$ is also dense in $\mathbb{R}$. By the similar process as above , one can easily derive that there is at least one irrational number between any two distinct real numbers.

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