Prove that 3+ 2√5 is irrational

Question – Prove that 3+ 2√5 is irrational.

Solution –

Proof: Let us assume the contrary that 3+2√5 is not irrational.

This means that 3+2√5 is a rational number

Then, we can find two co prime integers a and b, where (b ≠ 0) such that

3+2√5 = a/b

⇒ a/b – 3 = 2√5

⇒ (a-3b)/b = 2√5

⇒ (a-3b)/2b = √5

Since a, b, 3 and 2 are integers, (a-3b)/2b is rational.

Therefore, √5 is rational, which is a contradiction to the fact that √5 is irrational.

This contradiction has arisen because of our incorrect assumption that 3+ 2√5 is rational.

Therefore, we concluded that 3+ 2√5 is irrational.

Related Questions –

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