Question

prove that root 2 and root 5 is irrational

Answer 1

HI !

To prove :-

√2 is irrational

Proof :-
Let us assume that √2 is rational 

Let ,
√2 = p/q , where p and q are integers and q≠ 0 , p and q are co prime numbers.

Squaring both sides ,

2 = p²/q²

2q² = p²

2 divides p²

2 divides p             ----> [1]

p = 2m

2q² = (2m)²

2q² = 4m²

q² = 2m²

2 divides q²

2 divides q            ----> [2]

From 1 and 2 , 2 divides p and q .
2 is a common factor of p and q .
This is a contradiction. p and q are not co prime
Hence ,√2 is irrational.
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To prove :-

√5 is irrational

Proof :-
Let us assume that √5 is rational 

Let ,
√5 = p/q , where p and q are integers and q≠ 0 , p and q are co prime numbers.

Squaring both sides ,

5 = p²/q²

5q² = p²

5 divides p²

5 divides p             ----> [1]

p = 5m

5q² = (5m)²

5q² = 25m²

q² = 5m²

5 divides q²

5 divides q            ----> [2]

From 1 and 2 ,
 5 divides p and q .
5 is a common factor of p and q .
This is a contradiction. p and q are not co prime
Hence ,√5 is irrational.