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I encountered a few questions on irrational numbers.

1. Prove that [tex] \sqrt{3} [/tex] is irrational [/tex]. So let [tex] l = \sqrt{3} [/tex]. Then if [tex] l [/tex] were a rational number and equal to [tex] \frac{p}{q} [/tex] where [tex] p, q [/tex] are integers different from zero then we have [tex] p^{2} = 3q^{2} [/tex]. We can assume that [tex] p, q [/tex] have no common factors, because they would be cancelled out in the beginning. Now [tex] p^2 [/tex] is divisible by 3. So let [tex] p = 3p' [/tex]. We have [tex] 9p'^2 = 3q^2 [/tex] or [tex] q^2 = 3p'^{2} [/tex]. So both [tex] p , q [/tex] are divisible by 3. But this contradicts the fact that common factors of [tex] p, q [/tex] were cancelled out. Hence [tex] \sqrt{3} [/tex] is irrational.

2. If we had to prove that [tex] \sqrt{n} [/tex] was an irrational number where [tex] n [/tex] is not a perfect square would be do basically the same thing as we did above?