Question

given that root 2 is irrational prove that 5 + 3 root 2 an irational number

Answer 1

let 5+3√2=a\bπs.b.s
25+18+30√2=a^2\b^2
43+30√2=a^2\b^2
30√2=a^2-43b^2\b^2
√2=a^2-43b^2\b^2*30
therefore,5+3√2is irrational

Answer 2

Answer:

Let √2 be a Rational Number.

⇒√2= p/q            ( p & q are integers and q ∦ 0 )

Let p & q have some common factor; so by cancelling them, we get a/ b

⇒√2 = a/b       ( a & b are co- prime numbers )

Squaring on both sides :-

⇒(√2)² = (a/b)²

⇒ 2 =a²/b²

⇒2b² = a²

⇒b² = a²/2     .........(i)

∵2 divides a²

∴2 divids a also.

⇒ a/2 = c       (say)

Squaring on both sides:-

⇒(a/2)² = c²

⇒a²/4 = c²

⇒a² = 4c²

Putting the value of a² in equation ...(i)

⇒b² = 4c²/2

⇒ b² = 2c²

⇒ c² = b²/2

∵ 2 divides b²

∴2 divides b also.

∴2 divides a& b both.

It means that a& b have some common factor 2. But it contradict the fact that a & b are co- prime numbers.

And this contradiction is arisen due to wrong assumption that √2 is a Rational Number.

Haence, √2 is an Irrational Number.