Question

x. Prove that 5 is irrational.​

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

Answer 2

Answer:

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To prove that √5 is a irrational no.

Let us assume that √5 is rational

Then √5 = $\frac{a}{b}$

(cross multiply)  

⇒ a = √5b  

⇒ a² = 5b² -------> α

⇒ 5/a²

(by theorem if p divides q then p can also divide q²)  

⇒ 5/a ----> 1  

⇒ a = 5c  

(squaring on both sides)  

⇒ a² = 25c² ----> β  

From equations α and β  

⇒ 5b² = 25c²

⇒ b² = 5c²  

⇒ 5/b²  

(again by theorem)  

⇒ 5/b-------> 2  

we know that a and b are co-primes having only 1 common factor but from 1 and 2 we can that it is wrong.  

This contradiction arises because we assumed that √5 is a rational number  

∴ our assumption is wrong  

∴ √5 is irrational number  

Step-by-step explanation:

⇒ hope it helps you dear :)