Question

Prove that under root 11 is an irrational number

Answer 1

let√11 be rational.

then it must in the form of p / q [q is not equal to 0] [p and q are co-prime]

√11 = p / q

=> √11 x q = p

squaring on both sides

=> 11q2= p2  ------> (1)

p2 is divisible by 11

p is divisible by 11

p = 11c  [c is a positive integer] [squaring on both sides ]

p2 = 121 c2 --------- > (2)

subsitute p2 in equ (1) we get

11q2 = 121c2

q2 = 11c2

=> q is divisble by 11

thus q and p have a common factor 11

there is a contradiction

as our assumsion p & q are co prime but it has a common factor.

so √11 is an irrational.