Question

Prove root 7 is an irrational number

Answer 1

any value of odd no.... is irrational because.....rational no. wirrten in form of p/q...irratiinal no. is not written in form of P/Q

Answer 2

let's assume that root 7 is a rational number
then 7 can be written in the form of p by q

where p and q are co-prime]

√7 = p / q

=> √7 x q = p

squaring on both sides

=> 7q2= p2  ------> (1)

p2 is divisible by 7

p is divisible by 7

p = 7c  [c is a positive integer] [squaring on both sides ]

p2 = 49 c2 --------- > (2)

subsitute p2 in equ (1) we get

7q2 = 49 c2

q2 = 7c2

=> q is divisble by 7

thus q and p have a common factor 7.

there is a contradiction

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

so that √7 is an irrational.