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

Last Updated : 19 Mar, 2024
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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.

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!

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