Question

prove that root 11 is irrational

Answer 1

Sol:
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.

Answer 2

let √11 be rational no.
so ,√11= p/q where p and q are coprimes.
squaring both sides,
11=p^2/q^2
p^2=11q^2-(eq i)
so 11 is a factor of p^2.
therefore 11 is a factor of p.
=>p=11m put the value of p in eq i
121m^2=11q^2
121/11m^2=q^2
11m^2=q^2
so 11 is a factor of q^2.
therefore 11 is a factor of q.
but this is a contradiction. since we supposed that p and q are coprimes. therefore our supposition is wrong. hence proves that √11 is irrational no.