If the answer is always yes, why? If sometimes yes and sometimes no can you provide example pairs of x and y to show that sometimes x + y is rational and sometimes it isn't. If the answer is always no, can you prove it?

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I think Tussin's guess is correct. Say x/y = q is a rational number. Then 1 + x/y is rational as well, and we can rewrite the expression as

1 + x/y = (y+x)/y = q + 1

If we multiply both sides by y, we get

x + y = y(q + 1)

The right-hand side is an irrational number times a rational number. This will only equal a rational number if q = -1, making the product 0. When q = -1 it means x and y are opposites. Otherwise, for any other value of q the product is irrational. Therefore, x + y is irrational unless x = -y.

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Great explanation, thanks!

If x = sqrt(2) and y = -sqrt(2), then the ratio and product are rational, and since x+y = 0, the sum is rational as well.

But if x = y = sqrt(2), the ratio and product are rational but the sum is not.

I'm not sure if making x and y be opposite square roots is the only way to get a rational sum (0), but in the short amount of time I thought about the problem I couldn't see another way.

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