Show That the Following Numbers Are Irrational
Video Explanation
Watch the video below for a detailed NCERT-based explanation:
Solution
Question: Show that the following numbers are irrational:
(i) 1/√2 (ii) 7√5 (iii) 6 + √2 (iv) 3 − √5
(i) Prove that 1/√2 is irrational
We know that √2 is an irrational number.
Let us assume that 1/√2 is rational.
Then its reciprocal √2 must also be rational, because the reciprocal of a non-zero rational number is rational.
But this contradicts the fact that √2 is irrational.
∴ Our assumption is wrong.
Hence, 1/√2 is irrational.
(ii) Prove that 7√5 is irrational
We know that √5 is an irrational number.
Let us assume that 7√5 is rational.
Since 7 is a non-zero rational number, dividing both sides by 7 gives:
√5 = (7√5) / 7
This implies that √5 is rational, which is a contradiction.
∴ Our assumption is wrong.
Hence, 7√5 is irrational.
(iii) Prove that 6 + √2 is irrational
Let us assume that 6 + √2 is rational.
Since 6 is a rational number, subtracting 6 from both sides gives:
√2 = (6 + √2) − 6
This implies that √2 is rational, which is a contradiction.
∴ Our assumption is wrong.
Hence, 6 + √2 is irrational.
(iv) Prove that 3 − √5 is irrational
Let us assume that 3 − √5 is rational.
Since 3 is a rational number, subtracting both sides from 3 gives:
√5 = 3 − (3 − √5)
This implies that √5 is rational, which is a contradiction.
∴ Our assumption is wrong.
Hence, 3 − √5 is irrational.
Final Answer
(i) 1/√2 is irrational
(ii) 7√5 is irrational
(iii) 6 + √2 is irrational
(iv) 3 − √5 is irrational
Conclusion
Thus, all the given numbers are proved to be irrational using the standard NCERT Class 10 method of contradiction.