Question

if√ab be an irrational number prove that √a+√b is a irrational number

Answer 1

Instead of working with inequalities, we could perhaps prove the contrapositive of the statement.

Answer 2

Which would be: if \[\sqrt{a} + \sqrt{b}\] is rational, then \[\sqrt{ab}\] is rational. Or in symbolic form, \[\forall m,n \in \mathbb{Z}, \sqrt{a} + \sqrt{b} = \frac{m}{n}\] and \[\forall x,y \in \mathbb{Z}, \sqrt{ab} = \frac{x}{y}\]

Answer 3

Correction, I used a universal quantifier instead of a "there exists" one.

Answer 4

This relation may perhaps help:\[\left(\sqrt a + \sqrt b ~\right)^2 = a + b + 2\sqrt{ab} \]\[\Rightarrow \sqrt{ab} = \dfrac{1}{2}\left( (\sqrt a + \sqrt b ) ^2 - a - b\right)\]I'm not entirely sure how to proceed further with this information. Is it given that a, b are rational?

Answer 5

Actually, a counter example to this would be the following.
\[\pi \cdot -\pi = -\pi^{2}\] which is irrational. Then \[\pi + (-\pi) = 0\] which is rational.

Answer 6

*

Answer 7

well it should say a,b \(\in\) Z

Answer 8

Of course, my bad.

Answer 9

@ganeshie8 what do u think mm ?

Answer 10

ok i might have this solution :-
as @dinnertable startted
ok for a,b ∈ Z
let √a+ √b be rational, then
√a+ √b = m/n such that gcd (m,n )=1
then both √a, √b are rational

√b = m/n -√a *

from
√ab =√a . √b =√a(m/n -√a)=( √a . m/n) -a

but √a is rational so , √a m/n is rational
hence
( √a . m/n) -a is rational :)