Using a specific Lemma, how to prove that between any two distinct rational numbers there exists an irrational number? Lemma: If m/n and r/s are rational, with r/s ≠ 0, then m/n + r/s xx sqrt 2 is irrational.

Aleah Avery 2022-11-21 Answered
Using a specific Lemma, how to prove that between any two distinct rational numbers there exists an irrational number?
Lemma: If m / n and r / s are rational, with r / s 0, then r / s 0 is irrational.
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Answers (1)

martinmommy26nv8
Answered 2022-11-22 Author has 16 answers
The lemma simply says that if p and q are rational, and q 0, then p + q 2 is irrational; it doesn’t matter which rational numbers p and q are (as long as q 0). Now take p = m n and q = r s m n ; these are rational, and by hypothesis m n < r s , so q 0, so the lemma applies to let us conclude that p + q 2 = m n + 2 ( r s m n ) is irrational.
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