Question

prove that √5 is irrational​

Answer 1

Answer:

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. Therefore 2-√5 is also irrational because difference of a rational and an irrational number is always an irrational number.

Answer 2

♡♥JAY SHREE KRISHNA..

♡YOUR♡ANSWER♡IS♡AS♡FOLLOW♡

Suppose √5 is a rational number. Then, √5 can be expressed in the form α/β, where a and b are integers, which have no common factor other than one and b is not equal to 0

therfore, √5 = a

——

b.

on squaring both sides , we get

5= a²/b² ➡ a²= 5b² _______(i)

here , 5 divides a² and a both __(by theoram 1)

So, we can take a=5m

a²= 25m² [on squaring both side]

On putting the value of a² in equation (i) we get

25m²= 5b² ➡ 5m² = b²

➡5 divides b²

5divides b also -------(by theoram 1)---(ii)

Thus, from equation (ii) 5 divides a and from equation (i) 5 divides b . it means 5 is a common factor of a and b.

this contradiction arises by our own assumption

Hence, √5 is irration number...

♡♥HOPE THIS HELPS

♡♥PLS MARK AS BRAINLIST AND LIKE

♡♥JAY SHREE KRISHNA