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Prove that√p+√q where these are irrattional numbers |
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Answer» Answer: LETS ASSUME that √p is rational, ⇒ √p = a/b ( where 'a' and 'b' are co primes, meaning they don't have any COMMON factors EXCEPT for 1) From squaring both sides, p = a²/b² ⇒pb² = a² ⇒ b² = a²/p Since 'p' divides a², it also divides 'a' meaning 'a' has a factor of p Let 'a' = pm (where m is a positive integer) ⇒ a² = p²m² Now, pb² = a² pb² = p²m² pb²/p²= m² b²/p =m² ∴ 'p' divides 'b' ⇒ 'b' also has a factor 'p' ∴ 'a' and 'b' are not co primes and our assumption was wrong ⇒ √p is irrational Similarly √q is irrational ∴⇒ √p + √q is irrational |
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