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
p2=5q2
⇒ p2 is divisible by 5
⇒ p is also divisible by 5
So, assume p = 5m where m is any integer.
Squaring both sides, we get p2 = 25m2
But p2 = 5q2
Therefore, 5q2 = 25m2
⇒ q2 = 5m2
⇒ q2 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.
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