I m giving u some simple steps ok first of all u just assume √5 as a rational no. √5=a/b whose a is an integer and b is not equal to zero u know only.√5b=a then u sqaure the both side b2 ×5 =a2 =5b2 =a2. so we can write as a =5c for some integer c.now u just substitute for a then we get 5b2 =(5c2) =b2= 25/5 so the ans is 5c2 so b2 is divisible by 5 hence we get √5 as a irrational no.

**Proof that √5 is irrational**

To prove that √5 is irrational, we will assume the opposite, that √5 is rational. Then we will derive a contradiction, which will prove our assumption false and establish that √5 is indeed irrational.

**Assumption: √5 is rational**

Let's assume that √5 is rational. By definition, a rational number can be written as a fraction a/b, where a and b are integers and b is not equal to 0. So, we can express √5 as √5 = a/b.

**Deriving a contradiction**

To derive a contradiction, we will square both sides of the equation √5 = a/b.

(√5)^2 = (a/b)^2

5 = (a^2) / (b^2) [since (√x)^2 = x]

Multiplying both sides of the equation by (b^2), we get:

5 * (b^2) = a^2

Therefore, a^2 is divisible by 5.

**Proof by contradiction**

Now, we will prove by contradiction that if a^2 is divisible by 5, then a is also divisible by 5.

Assume that a is not divisible by 5. This implies that a can be written as a = 5k + r, where k is an integer and r is the remainder when a is divided by 5. Since a is not divisible by 5, the remainder r must be a non-zero integer between 1 and 4.

Now, substitute the value of a in the equation a^2 = 5 * (b^2):

(5k + r)^2 = 5 * (b^2)

Expanding the equation:

25k^2 + 10kr + r^2 = 5 * (b^2)

Rearranging the terms:

5(5k^2 + 2kr) + r^2 = 5 * (b^2)

This implies that r^2 is divisible by 5.

**Contradiction**

We have derived a contradiction because if r^2 is divisible by 5, then r must also be divisible by 5. However, we assumed earlier that r is a non-zero integer between 1 and 4. This contradiction proves that our initial assumption, that √5 is rational, is false.

Therefore, we conclude that √5 is irrational.