Question

Find two irrational numbers between 2 and 2.5.

Answer 1

Solution:

Let $x$ be an irrational number between $2$ and $2.5$, therefore,

$$\begin{gathered} 2<x<2.5\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\left[}\mathbf{A}\mathbf{s}\mathbf{s}\mathbf{u}\mathbf{m}\mathbf{p}\mathbf{t}\mathbf{i}\mathbf{o}\mathbf{n}\mathbf{\right]} \\ \Rightarrow \sqrt{2^2}<x<\sqrt{2.5^2}\mathbf{}\mathbf{[}\mathbf{T}\mathbf{a}\mathbf{k}\mathbf{i}\mathbf{n}\mathbf{g}\mathbf{}\mathbf{s}\mathbf{quar}\mathbf{e}\mathbf{s}\mathbf{}\mathbf{a}\mathbf{n}\mathbf{d}\mathbf{}\mathbf{s}\mathbf{q}\mathbf{u}\mathbf{a}\mathbf{r}\mathbf{e}\mathbf{}\mathbf{r}\mathbf{o}\mathbf{o}\mathbf{t}\mathbf{s}\mathbf{]} \\ \Rightarrow \sqrt{4}<x<\sqrt{6.25} \\ \mathbf{}\mathbf{[}\mathbf{\because }\mathbf{4}\mathbf{}\mathbf{and}\mathbf{}\mathbf{6}\mathbf{.}\mathbf{25}\mathbf{}\mathbf{are}\mathbf{}\mathbf{two}\mathbf{}\mathbf{perfect}\mathbf{}\mathbf{squares}\mathbf{,}\mathbf{}\mathbf{thus}\mathbf{}\mathbf{the}\mathbf{}\mathbf{square}\mathbf{}\mathbf{root}\mathbf{}\mathbf{of}\mathbf{}\mathbf{all}\mathbf{}\mathbf{integres}\mathbf{}\mathbf{between}\mathbf{}\mathbf{them}\mathbf{}\mathbf{are}\mathbf{}\mathbf{irrational}\mathbf{]} \\ \therefore x=\sqrt{5},\sqrt{6} \end{gathered}$$

Final answer: Hence, any two irrational numbers between $2$ and $2.5$.are $\sqrt{\mathbf{5}}$ and $\sqrt{\mathbf{6}}$.