Let $a$ is an irrational number. Then prove or disprove that $$a+a^2$$ is irrational number.
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Are you looking to prove this or disprove this? – Element118 Nov 20 '15 at 06:10
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I could not prove it and also could not find an example for disprove it – A.G Nov 20 '15 at 06:12
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Why negative vote? – A.G Nov 20 '15 at 06:13
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What if we force $a+a^2$ to be some nice rational number? – Element118 Nov 20 '15 at 06:13
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$a^2+a = \frac{1}{2} \implies a \in \{ \frac{1}{2}(-1 \pm \sqrt{3}) \}$, both irrational.
Eric Towers
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1Yes. Any $q in \Bbb{Q}$ such that $1+4q$ is not the square of a rational number produces an irrational $a$. This is just the quadratic formula... (For a negative instance, $q = -1/4$ forces $a = -1/2$.) – Eric Towers Nov 20 '15 at 06:23