Chapter 1: Real Numbers

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Q
Real Numbers Solutions

Question:

Prove that 3 + 2โˆš5 is irrational.

Answer:

Proof: To prove that 3 + 2√5 is an irrational number 

Let us assume that 3 + 25  is rational. Then 3 + 2√5  =   frac{a}{b}

Since a and b are co primes with only 1 common factor and b≠0.  

⇒ 3 + 2√5  =  frac{a}{b}   

⇒  2√5  = frac{a}{b}   – 3

Now divide by 2 into both sides,  we get

 √5  =  frac{a-3b}{2b}  

Here a and b are integer so frac{a-3b}{2b}  is a rational number, so √5 โ€‹ should be a rational number.


But √5 is an irrational number, so it is contradicted.

Therefore, 3 + 2√5
is an irrational number.

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