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Prove the root 5 is irrational |
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Answer» Let root 5 is a rational no. Root 5=p/q (where p and q are co prime)P=root5 qSquaring both sideP^2=5q^2 (let eqn. I)5 is a factor of q^25 is a factor of q also ,P= 5c for some integer cPut the value of eqn. ( 1 )5p^2=(5c)^25p^2=25c^2P^2=5c^25 is a factor of p^25 is a factor of p also,(Where p and q are integers )Therefore , our assumption is wrong Root 5 is irrational no. let root 5 be rationalthen it must in the form of p/q [q is not equal to 0][p and q are co-prime]root 5=p/q=> root 5 × q = psquaring on both sides=> 5 ×q ×q = p ×p ------> 1p ×p is divisible by 5p is divisible by 5p = 5c [c is a positive integer] [squaring on both sides ]p ×p = 25c ×c --------- > 2sub p ×p in 15 ×q ×q = 25 ×c ×cq ×q = 5 ×c ×c=> q is divisble by 5thus q and p have a common factor 5there is a contradictionas our assumsion p &q are co prime but it has a common factorso\xa0√5 is an irrational |
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