Question

Prove that  √5  is an irrational number.​

Answer 1

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Let's prove this by the method of contradiction-

Say, √5 is a rational number.

∴ It can be expressed in the form p/q where p,q are co-prime integers.

⇒√5=p/q

⇒5=p²/q²  {Squaring both the sides}

⇒5q²=p² .....(1)

⇒p² is a multiple of 5. {Euclid's Division Lemma}

⇒p is also a multiple of 5. {Fundamental Theorm of arithmetic}

⇒p=5m

⇒p²=25m²  .... (2)

From equations (1) and (2), we get,

5q²=25m²

⇒q²=5m²

⇒q² is a multiple of 5. {Euclid's Division Lemma}

⇒q is a multiple of 5.{Fundamental Theorm of Arithmetic}

Hence, p,q have a common factor 5.

this contradicts that they are co-primes.

Therefore, p/q is not a rational number.

This proves that √5 is an irrational number. 

For the second query, as we've proved √5 irrational.

Therefore 2-√5 is also irrational because difference of a rational and an irrational number is always an irrational number.

Answer 2

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