Question

27. Prove that √5 is an irrational number,​

Answer 1

Answer:

proving √5 is irrational by contradiction method

Step-by-step explanation:

firstly, assume √5 is rational

⇒ √5 = p/q (where q ≠ 0 and p and q are co-primes)

⇒ √5q = p

now, squaring both sides, we get

5q² = p²                            (--- 1)

so, 5 is a factor of p² ⇒ 5 is also a factor of p

we can write this as:

p = 5x; where x is any natural number      (----2)

now putting (2) in (1) we get

5q² = (5x)²

⇒ 5q² = 25x²

⇒ q² = 5x²

so, 5 is a factor of q²⇒ 5 is a factor of q

this means that both p and q have a common factor, 5

however this contradicts the fact that p and q are co-primes

hence, our assumption was wrong

∴ √5 is irrational