Question

Prove that √5 is irrational.​

Answer 1

Answer:

Say, √5 is a rational number. ∴ It can be expressed in the form p/q where p,q are co-prime integers. Hence, p,q have a common factor 5. ... Therefore 2-√5 is also irrational because difference of a rational and an irrational number is always an irrational number.

Answer 2

Answer:

Step-by-step explanation:

Let root 5 be rational

Then it must in the form of pq [q is not equal to 0][p,q are co-prime]

5√=pq

5√q=p

squaring on both sides

5q2=p2  ------ (1)

q2=p25

p2 is divisible by 5

p is divisible by 5 ------{If p is a prime no. and p divides a2, then p divides a also, where a is a positive integer}

p=5c  [c is a positive integer]

Squaring on both sides,

p2=25c2  --------- (2)

substitute for p2 in (1)

5q2=25c2

q2=5c2

q is divisible by 5

Thus q and p have a common factor 5

There is a contradiction.

Therefore, p and q are not co-prime.

So 5√ is an irrational