Question

Proof that √5 is a irrational number

Answer 1

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Answer 2

Hello dear I am tanwi and your answer is in down


= Let√5 be an rational number
=> Then √5=p/q where p,q are integers, q is not equal to to 0 and p,q have no common factor ( except 1)
=> (√5)=(p/q) 2 (square) [squaring on both side]
= 5=P2/Q2 ( p square / q square)

P2= 5q2.......(I)

As 5 divides 5q , so 5 divides p but 5 is prime
5 divides p

Let's=5m, where m is an integer from equation(I)

P2(p square ) = 5q2( 5 q square )
(5m)2 (square) =5q2(square)
25m2 / 5 = 4q2 ( square )
(5 and 25 cancelled)

So remains 5m2 =q2
As 5 divides 5m2, so 5 divides q2

5 divides q

This , p and q have common factor 5

This contradicts that p and q have no common factor ( except 1 )

Hence,√5 is not a rational number

So it can be concluded that √5 is an irrational number.





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