Prove that √ 2 is an Irrational Number. - Mathematics

Sum

Prove that sqrt2 is an irrational number.

Solution

Let sqrt2 is rational.
∴ sqrt2 = "p"/"q" where p and q are co-prime integers and q ≠ 0.

⇒ sqrt2"q" = "p"
⇒ 2q2 = p2  ....(1)
⇒ 2 divides p2
⇒ 2 divides p  ......(A)

Let p = 2c where c is an integer
⇒ p2 = 4c2
⇒ 2q2 = 4c2  ...[from (1)]
⇒ q2 = 2c2
⇒ 2 divides q2
⇒ 2 divides q  .....(B)

From statements (A) and (B), 2 divides p and q both that means p and q are not co-prime which contradicts our assumption.
So, our assumption is wrong.

Hence sqrt2 is irrational.
Proved.

Concept: Proofs of Irrationality
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