Prove that 2+(√3) , (((√3)−2)/3) and (1/( (√3)−5)) are the number irrational


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Question Number 159079 by LEKOUMA last updated on 12/Nov/21

$P r o v e t h a t 2 + \sqrt{3}, \frac{\sqrt{3} - 2}{3} a n d \frac{1}{\sqrt{3} - 5}$ $a r e t h e n u m b e r i r r a t i o n a l$
	Answered by mindispower last updated on 13/Nov/21

	$(2 + \sqrt{3}) \in Q \Leftrightarrow \sqrt{3} \in Q$ $\sqrt{3} = \frac{a}{b}, a \land b = 1$ $\Rightarrow 3 b^{2} = a^{2} . . . . 1 \Rightarrow 3 ∣ a \Rightarrow a = 3 s$ $1 \Leftrightarrow b^{2} = 3 s^{2} \Rightarrow 3 ∣ b . . . 2$ $(1) & (2) \Rightarrow 3 ∣ a \land b = 1 a b s u r d \Rightarrow \sqrt{3} \notin Q$ $(2) \frac{\sqrt{3} - 2}{3} = \frac{a}{b} \Rightarrow \sqrt{3} = \frac{3 a + 2 b}{b} \in Q a b s u r d p r e v i o u s p r o o f$ $\frac{1}{\sqrt{3} - 5} = \frac{\sqrt{3} + 5}{- 21} \in Q \Rightarrow \sqrt{3} \in Q a b s u r d$
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