prove that root 3 is irrational
Answers
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ANSWER
ANSWERIf possible , let 3 be a rational number and its simplest form be
ANSWERIf possible , let 3 be a rational number and its simplest form be ba then, a and b are integers having no common factor
ANSWERIf possible , let 3 be a rational number and its simplest form be ba then, a and b are integers having no common factor other than 1 and b=0.
ANSWERIf possible , let 3 be a rational number and its simplest form be ba then, a and b are integers having no common factor other than 1 and b=0.Now, 3=ba⟹3=b2a2 (On squaring both sides )
ANSWERIf possible , let 3 be a rational number and its simplest form be ba then, a and b are integers having no common factor other than 1 and b=0.Now, 3=ba⟹3=b2a2 (On squaring both sides )or, 3b2=a2 .......(i)
⟹3 divides a2 (∵3 divides 3b2)
⟹3 divides a2 (∵3 divides 3b2)⟹3 divides a
⟹3 divides a2 (∵3 divides 3b2)⟹3 divides aLet a=3c for some integer c
⟹3 divides a2 (∵3 divides 3b2)⟹3 divides aLet a=3c for some integer cPutting a=3c in (i), we get
⟹3 divides a2 (∵3 divides 3b2)⟹3 divides aLet a=3c for some integer cPutting a=3c in (i), we getor, 3b2=9c2⟹b2=3c2
⟹3 divides a2 (∵3 divides 3b2)⟹3 divides aLet a=3c for some integer cPutting a=3c in (i), we getor, 3b2=9c2⟹b2=3c2⟹3 divides b2 (∵3 divides 3c2)
⟹3 divides a2 (∵3 divides 3b2)⟹3 divides aLet a=3c for some integer cPutting a=3c in (i), we getor, 3b2=9c2⟹b2=3c2⟹3 divides b2 (∵3 divides 3c2)⟹3 divides a
⟹3 divides a2 (∵3 divides 3b2)⟹3 divides aLet a=3c for some integer cPutting a=3c in (i), we getor, 3b2=9c2⟹b2=3c2⟹3 divides b2 (∵3 divides 3c2)⟹3 divides aThus 3 is a common factor of a and b
⟹3 divides a2 (∵3 divides 3b2)⟹3 divides aLet a=3c for some integer cPutting a=3c in (i), we getor, 3b2=9c2⟹b2=3c2⟹3 divides b2 (∵3 divides 3c2)⟹3 divides aThus 3 is a common factor of a and bThis contradicts the fact that a and b have no common factor other than 1.
The contradiction arises by assuming 3 is a rational.
The contradiction arises by assuming 3 is a rational.Hence, 3 is irrational.
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Answer:
proof is given by indirect method (method of contradictory)
