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Prove that 2 root 3 divided by 5 is irrational

 

Asked by anirudhbanga2017 18th February 2018, 8:27 AM
Answered by Expert
Answer:
begin mathsize 16px style let space fraction numerator 2 square root of 3 over denominator 5 end fraction be space rational
fraction numerator begin display style 2 square root of 3 end style over denominator begin display style 5 end style end fraction equals straight p over straight q...... straight p space and space straight q space are space co space prime
square
12 over 25 equals fraction numerator begin display style straight p squared end style over denominator begin display style straight q squared end style end fraction
straight p squared equals fraction numerator begin display style 12 end style over denominator begin display style 25 end style end fraction straight q squared
straight p squared equals straight c space cross times space straight q squared
so space straight c space divides space straight p squared
ie. space straight c space divides space straight p
straight p equals straight c cross times straight k....... say
fraction numerator begin display style 2 square root of 3 end style over denominator begin display style 5 end style end fraction equals straight p over straight q
straight q equals fraction numerator begin display style 5 end style over denominator begin display style 2 square root of 3 end style end fraction cross times straight p
straight q equals fraction numerator begin display style 5 end style over denominator begin display style 2 square root of 3 end style end fraction cross times straight c cross times straight k
straight q equals space straight t space cross times space straight c......... straight c space divides space straight q
now space straight c space divides space both space straight p space and space straight q space
so space contradiction space since space they space are space co space prime
so space our space assumption space is space wrong end style
Answered by Expert 18th February 2018, 3:27 PM
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Tags: real-numbers
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