Question

Prove that √5 is irrational, if √5 is multiplied with 2 and rational number 3 is added to it then it will be rational on irrational? Explain.​

Answer 1

Answer:

Lets assume √5 is a rational number

√5=p/q (q is not equal to 0, p and q are co

primes)

Squaring both the sides

(√5)^2=p^2/q^2

5=p^2/q^2

Answer 2

Step-by-step explanation:

• 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.