I have proved in earlier exercises of this book that √2 and √3 are irrational. Then, the sum of two irrational numbers is an irrational number. Thus, √2+√3 is irrational. My first question is, is this reasoning correct?
Secondly, the book wants me to use the fact that if n is an integer that is not a perfect square, then √n is irrational. This means that √6 is irrational. How are we to use this fact? Can we reason as follows:
√6 is irrational
⇒√2⋅3 is irrational.
⇒√2⋅√3 is irrational
⇒√2 or √3 or both are irrational.
⇒√2+√3 is irrational.
Is this way of reasoning correct?
Answer
If √2+√3 is rational, then so is (√2+√3)2=5+2√6. But this is absurd since √6 is irrational.
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