Question

Prove that V5 is an irrational number.​

Answer 1

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

Answer:

Let us assume

√5 = rational

√5= a/b where a , b are co primes

√5b=a

Squaring on both sides

( √5b)²=a²

5b² = a².........equation 1

b²= a²/5

Here 5 divides a² , so it also divides a .

Now let a= 5c for some integer c.

Squaring on both sides.

a²= (5c)²

a²=25c²

15b²=25c² ( from equation1 )

b²= 5c²

b²/5=c²

5 divides b² , so it can also divide b.

It isn't possible because a,b are co primes.

Therefore, our assumption is wrong.

√5 is irrational.

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