Question

let x be a rational no and y be irrational . is x+y necessary an irrational No..

Answer 1

Yes. If x is rational and y is irrational in x+y then the answer will be irrational.
PROOF:
Let x+y be a rational number where y is irrational.
Then

$x + y = \frac{a}{b} \: where \: a \: and \\ \: b \: are \: co - primes$
Then again by transporting x from L. H. S to R. H. S. we have,
$y = \frac{a}{b} - x = \frac{a - bx}{b} \\ y= \frac{a}{b} - \frac{bx}{b}$
Now a/b and bx/b are rational numbers and also
Rational - Rational = Rational
But here
Irrational = Rational - Rational
= Rational
Now this contradicts our Supposition that x+y is rational. Hence x+y is an irrational number.
Hope it's clear to you And don't forget to mark this answer as the brainliest.