Question

Prove that √5 is irrational​

Answer 1

Answer:

Step-by-step explanation:

To prove that √5 is irrational number  

Let us assume that √5 is rational  

Then √5 =  

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

⇒ √5 =  

(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

Answer 2

Answer:

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Step-by-step explanation:

Let assume that √5is rational number

√5 =a/b (where a and b are coprimes)

5=a^2/b^2

a^2 =5b^2

b^2=a^2/5

a^2 is divisible by 5 so a is also divisible by 5

Let's take c

a/5=c

a=5c

b^2=a^2/5

c=a/5

a= 5c/5

=25c/5

b^2=5c^2

b^2/5=c