Question

1. Prove that V5 is irrational.​

Answer 1

Answer:

Hence Proved!!

Step-by-step explanation:

let root 5 be rational

then 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 = p

squaring on both sides

=> 5*q*q = p*p  ------> 1

p*p is divisible by 5

p is divisible by 5

p = 5c  [c is a positive integer] [squaring on both sides ]

p*p = 25c*c  --------- > 2

sub p*p in 1

5*q*q = 25*c*c

q*q = 5*c*c

=> q is divisble by 5

thus q and p have a common factor 5

there is a contradiction

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

so √5 is an irrational

Answer 2

let √5 is rational

√5 =a/b here aand b have no common factor

5= a²/b²

5b²= a²

here a² divides 5

a divides 5 also

let a= 5c

5b² = (5c)²

5b² = 25c²

b² = 5c²

here b² divides 5

b divides 5 also

now we get a and b have common factor 5

hence our assumption is wrong

this wrong assumption is ariese be supposing

that √5 is rational

hence √5 is not rational

hence √5 is rational