Question

prove that Root over √13 is irrational number​

Answer 1

let us assume that /13 is a rational number.

so it can be written in the form p/q where q is not equal to 0.

/13=p/q

let p and q have common factor and a and b be their co primes. so

/13=a/b

/13b=a

squaring both the sides

13b2= a2----------(i)

13 divides a2

so 13 divides a---------(ii)

let c be any multiple of 12 through which we get a

13c= a

squaring both sides

169c2= a2

from (i)

169c2= 13b2

13c2= b2

13 divides b2

so 13 divides b-----------(iii)

from (ii) and(iii)

we get that 13 is factor of boty a and b

but we meant a and b as co primes

this contradiction arise as we meant /13 as rational number

so /13 is an irrational number

hence proved