Question

How to prove that a number is IRRATIONAL

prove that ✓2 is irrational​

Answer 1

To Prove :-

• √2 is an irrational number.

SoluTion :-

Let's assume on the contrary that √2 is a rational number.

Then, there exists two rational numbers a and b

such that √2 = a/b where, a and b are co primes.

(√2)² = (a/b)²

→ 2 = a²/b²

→ 2b² = a²

2 divides a²

So, 2 divides a.

a = 2k , (for some integer)

a² = 4k²

2b² = 4k²

b² = 2k²

2 divides b²

2 divides b

Now, 2 divides both a and b but this contradicts that a and b are co primes.

It happened due to our wrong assumption.

Hence, √2 is irrational.

Answer 2

$\bold{To \: Prove :-}$

√2 is an irrational number

$\bold \red{Solution :-}$

Let us assume on the contrary that 2 is a rational number. Then, there exist positive integers a and b such that,

√2 = b/a

where, a and b, are co-prime i.e. their HCF is 1

⇒(2)² =(a/b)²

⇒2 = a²/b²

⇒2b² = a²

⇒2∣a² [∵2∣2b²and 2b² = a² ]

⇒2∣a...(i)

⇒a=2c for some integer c

⇒a² = 4c²

⇒2b² = 4c² [∵2b² = a² ]

⇒b² = 2c²

⇒2∣b² [∵2∣2c² ]

⇒2∣b...(ii)

From (i) and (ii), we obtain that 2 is a common factor of a and b. But, this contradicts the fact that a and b have no common factor other than This means that our supposition is wrong.

Hence, √2 is an irrational number.