NCERT Solutions for Class 9 Maths Exercise 1.2 Question 2
Understanding the Question 🧐
This question asks us to investigate a claim about the square roots of positive integers. A positive integer is just a whole number greater than zero, like &&1, 2, 3, 10, 100$$, and so on. We need to determine if taking the square root of *every* positive integer always results in an irrational number. If it doesn’t, we need to provide an example that proves it wrong.
Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.
Step-by-Step Solution 📝
Answer: No, the square roots of all positive integers are not irrational.
Justification: To prove the statement wrong, we only need to find one example (a “counterexample”) of a positive integer whose square root is a rational number.
-
Let’s test some positive integers:
- The square root of &&2&& is &&\sqrt{2} \approx 1.41421…&&, which is an irrational number.
- The square root of &&3&& is &&\sqrt{3} \approx 1.73205…&&, which is also an irrational number.
- The square root of &&4&& is &&\sqrt{4} = 2&&. Is &&2&& irrational? No!
-
Analyze the counterexample:
- The number &&2&& is a rational number because it can be written as a fraction &&\frac{2}{1}&&.
- Since we found a positive integer (&&4&&) whose square root (&&2&&) is rational, the original statement that the square roots of *all* positive integers are irrational must be false.
-
Provide the required example:
The question asks for an example. We can use the one we just found or any other perfect square.
Example: Consider the positive integer &&9&&. Its square root is:
&&\sqrt{9} = 3&&
The number &&3&& is a rational number.
Conclusion and Key Points ✅
The statement “the square roots of all positive integers are irrational” is false. The key takeaway from these ncert solutions is:
- The square root of a positive integer is rational if the integer is a perfect square (e.g., &&1, 4, 9, 16, 25, \dots&&).
- The square root of a positive integer is irrational if the integer is not a perfect square (e.g., &&2, 3, 5, 7, 8, \dots&&).
- Positive integers are also known as natural numbers (&&1, 2, 3, \dots&&).
- A perfect square is an integer that is the square of another integer (e.g., &&16 = 4^2&&).
- &&\sqrt{\text{perfect square}} = \text{rational number}&&.
- &&\sqrt{\text{non-perfect square}} = \text{irrational number}&&.
FAQ
Q: What is a positive integer?
A: Positive integers are the whole numbers greater than zero: &&1, 2, 3, 4$$, and so on. They are also known as natural numbers.
Q: Are all square roots irrational?
A: No. The square roots of positive integers that are perfect squares (like &&4, 9, 16&&) are rational integers. The square roots of non-perfect squares (like &&2, 3, 5&&) are irrational.
Q: What is a perfect square?
A: A perfect square is a number that is the result of multiplying an integer by itself. For example, &&25&& is a perfect square because it is &&5 \times 5&& or &&5^2&&.
Q: Give another example of a positive integer whose square root is rational.
A: The positive integer &&36&& is a perfect square. Its square root is &&\sqrt{36} = 6&&, which is a rational number because it can be written as &&\frac{6}{1}&&.
Q: Is &&\sqrt{8}&& rational or irrational?
A: The number &&8&& is not a perfect square. Therefore, its square root, &&\sqrt{8}&&, is an irrational number because its decimal value is non-terminating and non-repeating.
Further Reading
For a deeper understanding of number systems, you can refer to the official NCERT textbooks, which provide the foundational concepts for these exercises.
