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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

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    $\begingroup$ You are over-thinking this. All you have to do is give an actual example of two irrational numbers whose sum is irrational. $\endgroup$ Commented Mar 22, 2022 at 14:23
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    $\begingroup$ To echo @MichaelCohen's comment, an "exists" statement simply requires an example. To prove that there exist irrationals $x,y,$ such that $x+y$ is (ir)rational, just give an example. $\endgroup$ Commented Mar 22, 2022 at 14:26
  • $\begingroup$ @MichaelCohen Thank you for the interest. But, wouldn't that just be a trivial answer? Thank you $\endgroup$
    – 74H54N3
    Commented Mar 22, 2022 at 14:28
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    $\begingroup$ The easiest example is $\sqrt{2}-1$ and $\sqrt{2}+1$. $\endgroup$
    – Peter
    Commented Mar 22, 2022 at 14:28
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    $\begingroup$ The exercise could be interpreted as that the numbers must be distinct, therefore I chose an example with distinct numbers to cover this case as well. $\endgroup$
    – Peter
    Commented Mar 22, 2022 at 14:45

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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.

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    $\begingroup$ Thank you for attempting to help. But, you are wrong correct me if I myself am wrong. But, "The sum of an irrational must be an irrational number" is a false statement. The sum can be rational as well which is mentioned in my post, I encourage you to read it. Here's an example, 5-√3 is a irrational number, √3 is also an irrational one and the sum of these two irrational numbers is 5 which is rational. Hope you understand. $\endgroup$
    – 74H54N3
    Commented Mar 22, 2022 at 14:39
  • $\begingroup$ @74H54N3 This answer is correct. If $x$ is irrational then so is $x+x$, which provides the needed example. $\endgroup$ Commented Mar 22, 2022 at 14:42
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    $\begingroup$ The statement is that the sum of two irrational numbers CAN be irrational, not that it MUST be irrational. Note the difference ! The answer does not claim either that this is ALWAYS the case. $\endgroup$
    – Peter
    Commented Mar 22, 2022 at 14:49
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    $\begingroup$ @74H54N3 This answer explicitly says "add to itself" and later "doubling an irrational number". It does answer your question, though you may not like the fact that the two irrationals being added are the same number. $\endgroup$ Commented Mar 22, 2022 at 15:23
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    $\begingroup$ @74H54N3 Nowhere have I said that the sum of any two irrational numbers must be irrational. I have shown a specific case that adding an irrational number to itself always yields an irrational number. This proves that it is possible to sum two irrational numbers and find an irrational result. There's no claim here that summing two irrational numbers always yields an irrational result. One specific case of the sum of irrationals being irrational is sufficient to prove "the sum of two irrational numbers can be irrational", and here I've presented an entire class of cases. $\endgroup$ Commented Mar 22, 2022 at 15:50

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