Prove that √3 irrational.
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Answer:
Answer: Let us assume to the contrary that √3 is a rational number. where p and q are co-primes and q≠ 0. ... This demonstrates that √3 is an irrational number.
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Answer:
Let us assume, to the contrary, that 3 is rational.That is, we can find integers a and b (≠ 0) such that 3 = ab⋅Suppose a and b have a common factor other than 1, then we can divide by thecommon factor, and assume that a and b are coprime.So, 3ba=⋅Squaring on both sides, and rearranging, we get 3b2 = a2.Therefore, a2 is divisible by 3, and by Theorem 1.3, it follows that a is also divisibleby 3.So, we can write a = 3c for some integer c.Substituting for a, we get 3b2 = 9c2, that is, b2 = 3c2.This means that b2 is divisible by 3, and so b is also divisible by 3 (using Theorem 1.3with p = 3).Therefore, a and b have at least 3 as a common factor.But this contradicts the fact that a and b are coprime.This contradiction has arisen because of our incorrect assumption that 3 is rational.So, we conclude that 3 is irrational
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