How can I prove that
nonzero integer linear combination of two rational independent irrational numbers is still a irrational number?That is to say, given two irrational numbers a and b, if a/b is a irrational number too, then for any m,n is nonzero integer, we have that the number ma+nb is a irrational number, why?
Answer
That's not true: Take $a=\sqrt{2} -1$, $b=\sqrt{2}$. Then $\frac{a}{b} = \frac{1}{\sqrt{2}} - 1 $ isn't rational, but $a-b=1$
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