Question

prove that _/5 is an irrational number​

Answer 1

Answer:

Step-by-step explanation:

To prove that √5 is irrational number,

Let us assume that √5 is rational,

Then √5 = a/b,

(a and b are co primes, with only 1 common factor and b≠0),

 

⇒ √5 =  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.

Hope it helps you.

Answer 2

Hey mate !

Here's your answer :

$\sqrt{5} \: is \: irrational$

Kindly refer to the Attachment.

Step-by-step explanation:

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