Question

prove that ✓2 is irrational number

Answer 1

Hello !!!

-------------


Solution --->>>

Given √2 is irrational number.

Let √2 = a / b wher a,b are integers b ≠ 0 we also suppose that a / b is written in the simplest formNow √2 = a / b ⇒ 2 = a2/ b2 ⇒ 2b2= a2

∴ 2b2is divisible by 2⇒ a2is divisible by 2 ⇒ a is divisible by 2

∴ let a = 2ca2= 4c2⇒ 2b2= 4c2⇒ b2= 2c2∴ 2c2 is divisible by 2∴ b2 is divisible by 2∴ b is divisible by 2

∴a are b are divisible by 2 .

This contradicts our supposition that a/b is written in the simplest formHence our supposition is wrong

So,√2 is irrational number.✌ ☺

Hope i helped

Answer 2

Answer:

Let √2 be a rational number

Therefore, √2= p/q [ p and q are in their least terms i.e., HCF of (p,q)=1 and q ≠ 0

On squaring both sides, we get

p²= 2q² ...(1)

Clearly, 2 is a factor of 2q²

⇒ 2 is a factor of p² [since, 2q²=p²]

⇒ 2 is a factor of p

Let p =2 m for all m ( where m is a positive integer)

Squaring both sides, we get

p²= 4 m² ...(2)

From (1) and (2), we get

2q² = 4m² ⇒ q²= 2m²

Clearly, 2 is a factor of 2m²

⇒ 2 is a factor of q² [since, q² = 2m²]

⇒ 2 is a factor of q

Thus, we see that both p and q have common factor 2 which is a contradiction that H.C.F. of (p,q)= 1

Therefore, Our supposition is wrong

Hence √2 is not a rational number i.e., irrational number.