Question

prove that √5 is irrational number

and

also prove that 5 - √3 is irrational number

Answer 1

let us suppose that 5 - √3 is rational number.Then it will

5 - √3 = a / b

where, a and b are co-prime number and b is not equal to 0.

- √3 = (a / b) - 5

Here, L.H.S is irrational number where as R.H.S is rational number.

Therefore, it is condradictory.

Our assumption was wrong.

Hence, 5 - √3 is an irrational number.

Answer 2

Answer:

Let's prove this by the method of contradiction-

Say, √5 is a rational number. ∴ It can be expressed in the form p/q where p,q are co-prime integers.

⇒√5=p/q

⇒5=p²/q² {Squaring both the sides}

⇒5q²=p² (1)

⇒p² is a multiple of 5. {Euclid's Division Lemma}

⇒p is also a multiple of 5. {Fundamental Theorm of arithmetic}

⇒p=5m

⇒p²=25m² (2)

From equations (1) and (2), we get,

5q²=25m²

⇒q²=5m²

⇒q² is a multiple of 5. {Euclid's Division Lemma}

⇒q is a multiple of 5.{Fundamental Theorm of Arithmetic}

Hence, p,q have a common factor 5. this contradicts that they are co-primes. Therefore, p/q is not a rational number. This proves that √5 is an irrational number.

For the second query, as we've proved √5 irrational.