Sum and product of irrational numbers

Question

So, let's say I have two irrational numbers a, and b. Is it possible to have a + b to be rational, and ab to be rational?

Answer 1

Well, sure. Take $a = \sqrt{2}$ and $b = -\sqrt{2}$.

Answer 2

If $a$ is irrational and $c$ is rational. Let $b=c-a$. Then $b$ is irrational and $a+b=c$ is rational.

If $a$ is irrational and $d$ is rational. Let $b=\frac{d}{a}$. Then $b$ is irrational and $ab=d$ is rational.

Answer 3

Take $a,b$ to be the roots of any equation $x^2 - m x + n =0$ where $m,n$ are integers and the quadratic does not have integer roots i.e. the discriminant $m^2-4n$ is not a perfect square. Then $a,b$ will necessarily be irrational (by the rational root theorem), while $a+b=m$ and $ab = n$ are integers (by Vieta's relations).

Answer 4

The hypothesis $a,b$ irrational is not strong enough to conclude that $a+b$ or $ab$ must be irrational, but $a,b$ transcendental implies that one of $a+b$ or $ab$ must be irrational. Otherwise the the polynomial $x^2-(a+b)x+ab$ would have rational coefficients and hence roots which are algebraic which contradicts $a,b$ transcendental.