Find two rationals, one greater and one smaller than a given irrational number.

Question

Given an irrational number 0 < i < 1. Find two rational numbers a and b such that 0 < a < i < b < 1.

Answer 1

Since $0<i<1,$ then there exists some least positive integer $n$ such that $$\frac1n<i<1-\frac1n.$$ To see why, apply the Archimedean property to $$\frac1{\min(i,1-i)}.$$

Answer 2

Find the first non-zero digit in the decimal explasion, set everything after to 0 this gives a. Now find the first non 9 digit set add one to this digit and set everything after to 0 this gives b.

Answer 3

It is easy to do. Note that $\Bbb Q$ is dense in $\Bbb R$. Therefore $(0,i) \cap \Bbb Q \not=\emptyset$ and $(i,1)\cap \Bbb Q \not= \emptyset$ since $(0,i)$ and $(i,1)$ are two open sets. Then there exist $a$ and $b$ in $\Bbb Q$ such that $0<a<i<b<1$.