Question

18) Prove that
$\sqrt{2}$
is an irrational number. Hence show that
$3 \div \sqrt{2}$

is also an irrational number.​

Answer 1

Answer:

let √2 be a rational number in the form p/q where p&q are co-prime integers and q is not equal to zero.

√2=p/q

squaring both sides

2=p²/q²

2q²=p²

therefore 2 is a factor of p² as well as p - - -1

as 2 is a factor of p

p=2r

p²=4r²

but p²=2q²

so 2q²=4r²

therefore 2 is a factor of q² as well as q - - -2

from 1& 2

2 is a factor of both p& q

so p&q are not co-prime integers

p/q is not rational.

so √2 is an irrational no.

since√2 is irrational

therefore 3/√2 is irrational

rational /irrational=irrational