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Sum
Prove that `sqrt2` is an irrational number.
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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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