The question title itself is self explanatory. We need to prove that the sum of two irrational numbers can be irrational (it necessarily doesn't need to be always, rather i am trying to prove that, the sum of two irrational numbers can be irrational or rational)
Also, here's a "simple" proof that, the sum of two irrational numbers can be rational (aint saying it will always happen, rather saying, it can happen) :
if $z$ is a rational number and $x$ is an irrational one and $y = z-x$; then $y$ is an irrational number. [Difference between a rational and an irrational number is always irrational]
Here : $y = z-x$ or, $x+y = z$; where $z$ is the sum of two irrational numbers being a rational number itself... Proving that the sum of two irrational numbers can be rational.
Like wise, there should be a proof that shows that the sum of two irrational numbers can be irrational. But, i can't prove it, tried for two hours :(
Any hint or help would be much appreciated
An irrational number cannot be expressed as a ratio of integers A/B. Suppose there is an irrational number N that you can add to itself to get a rational number. If 2N can be expressed as a ratio of integers A/B, then N can be expressed as A/2B, where both A and 2B are integers. This is a contradiction, since we stipulated that N is not a rational number in the first place. Therefore, doubling an irrational number (adding an irrational number to itself) cannot yield a rational number - twice an irrational number must also be irrational. The sum of two irrational numbers can be irrational, and in fact, the sum of an irrational number and itself must be irrational.
$\sqrt{2}$ is irrational, therefore $\sqrt{2}$ +$\sqrt{2}$ is irrational. This is sufficient to prove that the sum of irrationals can be irrational.
You can extend this to recognize that any integer multiple of an irrational number must be irrational, and use that to reason about the sum of different irrationals.
$\sqrt{2}$ is irrational, therefore $\sqrt{2}$ +$\sqrt{2}$+$\sqrt{2}$, or $\sqrt{2}$ + $2\sqrt{2}$ is irrational. This is sufficient to prove that the sum of different irrationals ($\sqrt{2}$ and $2\sqrt{2}$) can be irrational.