Answer to Question #155090 in Real Analysis for Komal

Question

find rational number r=m/n . Such that √5 < r < √6

Answer 1

The distance between the numbers $\sqrt{5}$ and $\sqrt{6}$ is equal to

$\sqrt{6}-\sqrt{5} = \frac{(\sqrt{6}-\sqrt{5})(\sqrt{6}+\sqrt{5})}{\sqrt{6}+\sqrt{5}} = \frac{1}{\sqrt{6}+\sqrt{5}}>\frac15$

Therefore, if $n\geq 5$ , then the length of the interval ($\sqrt{5}n,\sqrt{6}n$ ) is equal to $\sqrt{6}n-\sqrt{5}n >1$ and this interval must contain an integer m.

We have $\sqrt{5}n < m <\sqrt{6}n$ and, therefore, the ratio m/n belongs to the interval $(\sqrt{5},\sqrt{6})$.

To find the ratio m/n in the interval $(\sqrt{5},\sqrt{6})$ we should seek a full-squared integer m² in the interval (5n², 6n²).

Let's take n=5.

$5n^2 = 5\cdot 5^2 = 125$

$6n^2 = 6\cdot 5^2 = 150$

The only full-squared integer between 125 and 150 is 144=12²=m².

Therefore, r=m/n = 12/5 = 2.4 satisfies the original condition.