Proof: Why There Is No Rational Number Whose Square is 2?

Problem Statement : Why There Is No Rational Number Whose Square is 2.

Rational Numbers: 
A number that can be expressed in the form of p/q, where p and q are integers and q ≠ 0, is known as a rational number. Examples are 0, 1, -1, 5/2, etc.

Solution:
Let’s get started with the proof of the above problem statement. Proof in this article will be done using a mathematical technique called Proof By Contradiction.

Proof: Suppose to the contrary that P/q is a rational number in the lowest terms, whose square is 2. This means

(P/q)² = 2. Then p²/q² = 2,

So, p² = 2q² thus

P² is even and hence is p even. If it was not even, then p is odd, but the P² is odd, contradicting that it is even.

Since p is even, we have p = 2k for some integer k.

Then (2k)²=2q² so 4k²=2q², or q² = 2k², so q² is even, hence q is also even.

Thus p and q are both even, and so they have a common factor of 2. This contradicts the fact that they have no common factors. Thus there are no rational numbers whose square is 2.

Hence Proved!