prove that √11 is an irrational number.
Answers
Answer:
Step-by-step explanation:
Let us assume that square root 11 is rational. Now since it is a rational number, as we have assumed, we can write it in the form p/q, where p, q ∈ Z, and coprime numbers, i.e., GCD (p,q) = 1.
⇒ √11 = p/q
Rearranging the terms,
⇒ p = √11 q ------- (1)
On squaring both sides we get,
⇒ p2 = 11 q2
Again rearranging the terms,
⇒ p2/11 = q2 ------- (2)
As we know, 11 is a prime number. Using the theory, which says that, if a prime number is a factor of a number, it will also be the factor of the given number's square, and vice versa is also true. This implies that since 11 is a factor of p2 then it will also be the factor of p.
Thus we can write p = 11a (where a is some constant)
Substituting p = 11a in equation (2), we get
(11a)2/11 = q2
⇒ (121a2)/11 = q2
⇒ 11a2 = q2
On rearranging, we get,
⇒ a2 = q2/11 ------- (3)
This shows that 11 will also be the factor of q.
Now, according to our initial assumptions, p and q are the coprime numbers hence only 1 is the number that can evenly divide both of them. But here, we have 11 as the factor of both p and q, which is contradictory to our initial assumption. This proves that the assumption of root 11 as a rational number was incorrect.
Therefore, the square root of 11 is irrational.
Answer:
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