Answer to Question #32002 in Combinatorics | Number Theory for RAJESH

Question

rational number between root 3 and root 5

Answer 1

To find such number x we can use the fact that if 0<a<b<c then root( a )< root(b)<root(c) and vice versa.
If we want to find b - rational then x=m/n for some integers m,n then in our case a=3, c=5, b=x^2
root( 3 )< root( m^2/n^2 )<root( 5 )
then 3<m^2/n^2<5 so m^2>3n^2 and 5n^2 > m^2
let n=1 then m^2<5 and m^2 > 3
m=2
4<5, 4>3
so x=2/1=2