Hello good sir/mam, Can you help me understand the logic and significance behind

Question

Hello good sir/mam, Can you help me understand the logic and significance behind each step in this proof? Thank you very much. Theorem: If an integer N is not a square, the square root of N is irrational Proof: By contradiction, Assume, if possible that ?N = a/b, where a and b are positive integers b?N = a let b equare the samllest positive integer so that b ?N = an integer Since ?N is not an integer is it between two integers K< ?N < ?N-K < 1 let B` (B prime) = b(?N-K) = b?N - bk, so B` is an integer B`?N = b(?N-k)?N =(b?N - bk)?N =b?N(?N) - bk?N

Explanation / Answer

Suppose N is an integer and that it is not a square. Suppose, BWOC (by way of contradiction), that its square root is rational. Then N^.5=a/b for some positive integers a/b. (The equality comes from our assumption and the definition of a rational number.) Then bN^.5=a (This is by multiplication of b on both sides.) Let b be the smallest positive integer such that bN^.5 is an integer. Now, we know that N^.5 is not an integer (by definition of a square). However, we can see that it is between two integers K and K+1. Then K1, since if b=1, we have the contradiction that N^.5 is an integer and not an integer. Then bN^.5 is one of the integers in the set {bK+1, bK+2, ..., bK+b-1}, inclusive. Then it follows that N^.5 one of the members in the set {K+1/b, K+2/b, ..., K+1-1/b}, inclusive. Then it follows that (N^.5)^2=N is one of the members in the set {K^2+2K/b+1/b^2, K^2+4K/b+4/b^2, ..., K^2+2K-2K/b+1-2/b+1/b^2}. But since b>1, it must be an integer 2 or more. Thus, none of the members of this set are integers. But N is an integer. Thus, we have a contradiction. Thus, if N is an integer that is not a square, it must have an irrational square root.

Related Questions

Navigate

• Browse (All)
• Browse H
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.