**Extra Questions On Irrationality of Numbers**

** Question:** Prove that âˆš5 is an irrational number.

** Solution: **Let âˆš5 is a rational number then we have

âˆš5=p/q, where p and q are co-primes.

â‡’ p =âˆš5q

Squaring both sides, we get

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

â‡’ p^{2} is divisible by 5

â‡’ 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}

â‡’ q^{2} = 5m^{2}

â‡’ q^{2} is divisible by 5

â‡’ 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, âˆš5 is an irrational number.

**Q.1. **Prove that âˆš2 is an irrational number.

**Q.2. **Prove that âˆš3 is an irrational number.

**Q.3.** Prove that 2 + 5âˆš3 is an irrational number.

**Q.4.** Prove that 3- 2âˆš5 is an irrational number.

**Q.5. **Prove that âˆš2 + âˆš3 is an irrational number.

**Q.6. **Prove that âˆš3 + âˆš5 is an irrational number.

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