Question

Prove that a rational number + an irrational number is an irrational number

Answer 1

I will prove this by doing a proof by contradiction:
Assume rational + irrational = rational.
Let a,c be rational, b be irrational.
Then a + b = c
Since a and c are rational, we can pick integers p,q,r,s such that
a = p/q, c = r/s
Thus
p/q + b = r/s
Then b = r/s - p/q
Then b = (rq - sp) / sq
Then b = rational since (rq-sp)/sq is integer/integer which is rational.
But b is defined to be irrational so this is a contradiction.
Therefore rational + irrational is not rational
Therefore rational + irrational is irrational.

Answer 2

All you had to do was provide an example, you could have added an irrational square root to a rational number like 2. It would have been simpler.

Answer 3

But that is a good method.

Answer 4

@Gdeinward I wish it was that easy, but I'd lose points of I did that. I'm in Mathematical Analysis is a college-level course.

Answer 5

Ah, thats understood.