find 2 rational and irrational numbers between 3root 8 and root 9 - Maths - Number Systems

Question

find 2 rational and irrational numbers between 3root 8 and root
9

Ajanta Trivedi · Accepted Answer

let $a = 3 \sqrt{8} = 3 * 2 \sqrt{2} = 6 \sqrt{2}$ and $b = \sqrt{9} = 3$

therefore $a > 6 * 1.414 \Rightarrow a > 8.484$ and b =3

thus the rational numbers between b and a are 4 , 5, 6, 7, 4.1 , 4.2 . any terminating or recurring decimal between 3 and 8.484.

to write the irrational number :

we can write it by the mean method:

i.e. if $a_{1} a n d a_{2}$ are two irrational numbers between a and b.

$a_{1} = \frac{a + b}{2} a n d a_{2} = \frac{a_{1} + b}{2}$

therefore two irrational numbers between $3 \sqrt{8} a n d \sqrt{9}$ are:

$a_{1} = \frac{6 \sqrt{2} + 3}{2} a n d a_{2} = \frac{\frac{6 \sqrt{2} + 3}{2} + 3}{2} = \frac{6 \sqrt{2} + 9}{4}$

hope this helps you.

cheers!!