Question

prove that root 5 is irrational​

Answer 1

$\Huge\orange\bigstar$ $\Huge\mathfrak\red{Solution}$

On the contrary, let us assume that √5 is a rational number.

therefore √5 = P/q (q is not equals to 0)

If P, q have a common factor, on cancelling the common factor let it be reduces to a/b , where a, b are co-primes.

Now √5 = a/b , where HCF (a, b) = 1

Squaring on both sides we get

(√5)² = (a/b)²

5 = a²/b²

=> 5b² = a²

=> 5 divides a² and thereby 5 divides 5

Now, take a = 5c

then, a² = 25c²

I.e., 5b² = 25c²

=> b² = 5c²

=> 5 divides b² and thereby b.

=> 5 divides both b and a.

• This contradicts that a and b are co-primes.
• This contradiction arises due to our assumption that √5 is a rational number.
• Hence our assumption is wrong and the given statement is true.
• So, √5 is an irrational number.

$\rule{200}{5}$

Answer 2

let us assume that √5 be a rational number

then we can write in p/q form.

where q is does not equal to 0

P & q = co-prime

√5 = p/q

squaring on the two sides

5 = p²/q²

5q² = p²

now here P OS divisible by 5

p=5k where k is positive integer

Here, squaring on two side

p² = 25k²

substituting 5q² = p²

5q²=25k²

q²=5k²

q is divisible by 5

From this we can say that P and q have th e common factor called 5.

such that it contradicts to our assumption P&q are co-primes

so, √5 is an irrational number