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Demonstrate that [itex]\sqrt{2}+\sqrt{3}[/itex] is irrational.
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This feels like homework. However, I will give a proof just to make sure I still can:Demonstrate that [itex]\sqrt{2}+\sqrt{3}[/itex] is irrational.
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It is a homework-type question. It still belongs in homework and the homework rules apply.It is not to homwork, is to pass an exam.
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To avoid misinforming the person asking the question, I would like to note that a conjugate is not an inverse. A conjugate of (a+b) is (a-b) whereas the inverse of (a+b) is (a-b)/(a^{2}-b^{2}). Applying this:Yes, probably homework, but we should not give misleading advice, so:
If Eval had done the first part correctly, he/she would have arrived at the otherwise immediate
√3 - √2 = m/n. Adding this equation to √3 + √2 = n/m, you obtain that √3 is rational, which is a contradiction.
Another way is just squaring both sides of √3 + √2 = n/m, giving √6 rational, and again a contradiction.
Hi Eval,If √3 + √2 = n/m, then 1/(√3 + √2) = m/n. Then, multiplying the top and bottom of the lefthand side by its conjugate, we get (√3 - √2)/(9^{2}+2^{2}) = (√3 - √2)/13 = m/n, not √3 - √2 = m/n.