Question

root 5Prove that is an irrational number​

Answer 1

Step-by-step explanation:

Let us assume that √5 is a rational number.

Sp it t can be expressed in the form p/q where p,q are co-prime integers and q≠0

⇒√5=p/q

On squaring both the sides we get,

⇒5=p²/q²

⇒5q²=p² —————–(i)

p²/5= q²

So 5 divides p

p is a multiple of 5

⇒p=5m

⇒p²=25m² ————-(ii)

From equations (i) and (ii), we get,

5q²=25m²

⇒q²=5m²

⇒q² is a multiple of 5

⇒q is a multiple of 5

Hence, p,q have a common factor 5. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number

√5 is an irrational number

Hence proved

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Answer 2

Step-by-step explanation:

Let us assume, on the contrary, that √5 is a rational number

We know that,

A rational number is of the form p/q, where p and q are integers, and coprimes, and q not equal to 0

Coprimes are 2 numbers having a common factor of 1

Now,

p/q = √5

Squaring both sides we get,

p²/q² = 5

p² = 5q² ----- 1

Thus, 5 is a factor p², then 5 is also a factor of p

OR

5 divides p², then 5 divides p

Let p = 5m ----- 2

Now, Putting eq.2 in eq.1,

(5m)² = 5q²

5²m² = 5q²

25m² = 5q²

25m²/5 = q²

5m² = q²

Thus,

5 is a factor of q², then 5 is a factor of q

OR

5 divides q², thus, it divides q

But this contradicts the fact that p and q are coprimes and have only 1 as it's common factor

This contradiction has risen due to our incorrect assumption

Thus,

√5 is an irrational number

Hope it helped and you understood it........All the best