Question

prove that underroot 5 i​

Answer 1

To prove :

√5 is an irrational number .

Proof :

Let us assume that , √5 is a rational number .

Thus ,

Let √5 = p/q , ( p ≠ 0 ) where p and q are co-primes (ie. p/q is in its simplest form) .

Now ,

Squaring both the sides , we get ;

=> (√5)² = (p/q)²

=> 5 = p²/q²

=> 5q² = p² ------(1)

Clearly ,

5q² is divisible by 5 .

=> p² is divisible by 5 .

=> p is divisible by 5 . --------(2)

Since p is divisible by 5 , thus let

p = 5m , m € N

Now ,

Putting p = 5m in eq-(1) , we get ;

=> 5q² = (5m)²

=> 5q² = 25m²

=> q² = 5m²

Clearly ,

5m² is divisible by 5 .

=> q² is divisible by 5 .

=> q is divisible by 5 . ----------(3)

From statements (2) and (3) , we get that 5 is a common divisor of p and q which contradicts our assumption that p and q are co-primes .

Thus , √5 is an irrational number .

Hence proved .