Mathematics
Prove that the following numbers are irrational:
(i)
(ii)
(iii)
Rational Irrational Nos
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Answer
(i) can be written as , now we are going to show that is an irrational number.
Let be a rational number, then
where p, q are integers, q ≠ 0 and p, q have no common factors (except 1)
As 2 divides 2q2, so 2 divides p2 but 2 is prime
Let p = 2m, where m is an integer.
Substituting this value of p in (i), we get
As 2 divides 2m2, so 2 divides q2 but 2 is prime
Thus, p and q have a common factor 2. This contradicts that p and q have no common factors (except 1).
Hence, is not a rational number. So, we conclude that is an irrational number.
Since, product of non-zero rational number and an irrational number is an irrational number.
And is an irrational number this implies that = is an irrational number.
(ii) Suppose that is a rational number, then
where p, q are integers, q ≠ 0 and p, q have no common factors (except 1)
As 2 divides 14q2, so 2 divides p2 but 2 is prime
Let p = 2k, where k is some integer.
Substituting this value of p in (i), we get
As 2 divides 2k2, so 2 divides 7q2
2 divides 7 or 2 divides q2
But 2 does not divide 7, therefore, 2 divides q2
2 divides q (Theorem 1)
Thus, p and q have a common factor 2. This contradicts that p and q have no common factors (except 1).
Hence, our supposition is wrong. Therefore, is not a rational number. So, we conclude that is an irrational number.
(iii) Suppose that = , where p, q are integers , q ≠ 0 , p and q have no common factors (except 1)
As 2 divides 2q3 2 divides p3
2 divides p (using generalisation of theorem 1)
Let p = 2k , where k is an integer.
Substituting this value of p in (i), we get
(2k)3 = 2q3
8k3 = 2q3
4k3 = q3
As 2 divides 4k3 2 divides q3
2 divides q (using generalisation of theorem 1)
Thus, p and q have a common factor 2. This contradicts that p and q have no common factors (except 1).
Hence, our supposition is wrong. It follows that cannot be expressed as , where p, q are integers, q > 0, p and q have no common factors (except 1).
∴ is an irrational number.
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