Prove that Root 2 is an Irrational Number

Q.  Prove that root 2 is an irrational number

Answer: The method of showing numbers by writing and representing them as a set of numbers by using symbols or arithmetic forms is called the number system.

The number that has no pattern in digits after the decimal and is actually non-terminating are called irrational numbers.

Solution- 

To prove- Given √2 is an irrational number

Proof-

For instance, let us consider, that √2 is a rational number

So, it can be expressed in the form of p/q 

Let us consider that p and q are co-prime integers where q ≠ 0

√2 = p/q

So, p and q are coprime numbers and also q ≠ 0

Therefore,

√2 = p/q

By squaring both the sides we get,

=>2 = (p/q)2

=> 2q2 = p2…………………………  (1)

p2/2 = q2

So, 2 divides p and p is a multiple of 2.

⇒ p = 2m

⇒ p² = 4m² ……………………………  (2)

From equations (1) and (2), we get,

2q² = 4m²

⇒ q² = 2m²

⇒ q² is a multiple of 2

⇒ q is a multiple of 2

Hence, a common factor is 2 between p and q.

This contradicts the assumption that we have considered that they are co-primes. 

Therefore p/q is not a rational number

Then, √2 is an irrational number.