### Chapter 1: Real Numbers

Q
##### Real Numbers Solutions

Question:

Prove that 3 + 2√5 is irrational.

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 ​ 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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