Prove that root 2 is an irrational number.
Rational numbers are integers that are expressed in the form of p / q where p and q are both co-prime numbers and q is non zero.
Answer: Hence proved that √2 is an irrational number with 2 as a common multiple.
Let's find if √2 is irrational.
To prove that √2 is an irrational number, we will use the contradiction method.
⇒ √2 = p/q
On squaring both sides we get,
⇒ 2q2 = p2
⇒ p2 is an even number that divides q2. Therefore, p is an even number that divides q.
Let p = 2x where x is a whole number.
By substituting this value of p in 2q2 = p2, we get
⇒ 2q2 = (2x)2
⇒ 2q2 = 4x2
⇒ q2 = 2x2
⇒ q2 is an even number that divides x2. Therefore, q is an even number that divides x.
Since p and q both are even numbers with 2 as a common multiple which means that p and q are not co-prime numbers as their HCF is 2.
This leads to the contradiction that root 2 is a rational number in the form of p/q with p and q both co-prime numbers and q ≠ 0.