I proved something wrong. If a and b are irrational proof that a + b is irrational or rational.

Question

I'm practicing and Found this question.

If $ a $ and $b$ are irrational, either prove or disprove that $a + b$ is irrational.

So I tried contradiction (to a + b is irrational).

Let $a$ and $b$ be arbitrary irrational numbers. Assume that$ a + b $is rational.

Then $ a + b = x/y$ for some integers $x$ and $y$.

then $y(a + b) = x$

and $ay + by = x$

Because $x$ was an integer $ay$ is an integer and $by$ is an integer.

then $a$ divides $ay$ and $b$ divides $by$. But that's impossible because a is irrational and b is irrational and y is an integer.

So $a+b$ must be irrational as well.

Now I know this is wrong. Because I found a counterexample as the solution.

$sqrt(2)$ + $-sqrt(2)$ = 0.

Can someone point out my logic mistake? Thank you very much in advance!

Answer 1

$ay$ and $by$ need not be integers in your proof.

$0=\sqrt 2 +(-\sqrt 2)$. If sum of two numbers is an integer you cannot say that both numbers are integers.

Answer 2

The mistake is in the step when you say "Because $x$ was an integer $ay$ is an integer and $by$ is an integer."

As your counterexample shows, the sum of two non-integer real numbers may be an integer.

Answer 3

The mistake is that $\ ay\ $ and $\ by\ $ cannot be integers since $\ a\ $ and $\ b\ $ are irrational and $\ y\ $ a non-zero integer.