**Extra Questions On Irrationality of Numbers**

** Question:** Prove that √5 is an irrational number.

** Solution: **Let √5 is a rational number then we have

√5=p/q, where p and q are co-primes.

⇒ p =√5q

Squaring both sides, we get

p^{2}=5q^{2}

⇒ p^{2} is divisible by 5

⇒ p is also divisible by 5

So, assume p = 5m where m is any integer.

Squaring both sides, we get p^{2} = 25m^{2}

But p^{2} = 5q^{2}

Therefore, 5q^{2} = 25m^{2}

⇒ q^{2} = 5m^{2}

⇒ q^{2} is divisible by 5

⇒ q is also divisible by 5

From above we conclude that p and q has one common factor i.e. 5 which contradicts that p and q are co-primes.

Therefore our assumption is wrong.

Hence, √5 is an irrational number.

**Q.1. **Prove that √2 is an irrational number.

**Q.2. **Prove that √3 is an irrational number.

**Q.3.** Prove that 2 + 5√3 is an irrational number.

**Q.4.** Prove that 3- 2√5 is an irrational number.

**Q.5. **Prove that √2 + √3 is an irrational number.

**Q.6. **Prove that √3 + √5 is an irrational number.