NCERT Solutions for Class 9 Maths Exercise 1.1 Question 5
Understanding the Question 🧐
In this ncert solutions guide, we will classify numbers as rational or irrational for Exercise 1.1 Question 5. A rational number can be written as &&\frac{p}{q}&&, where &&p&& and &&q&& are integers and &&q \neq 0&&. An irrational number cannot be expressed in this form, like &&\sqrt{2}&& or non-repeating, non-terminating decimals.
Part (i): &&\sqrt{23}&& 📝
Step 1: Check if 23 is a perfect square. A number is a perfect square if it can be written as &&n^2&& for some integer &&n&&. Let’s test: &&4^2 = 16&&, &&5^2 = 25&&. Since &&16 < 23 < 25&&, 23 is not a perfect square.
Step 2: Conclusion. Since &&\sqrt{23}&& is the square root of a non-perfect square, it cannot be written as &&\frac{p}{q}&&. Hence, &&\sqrt{23}&& is irrational.
Part (ii): &&\sqrt{225}&& 📝
Step 1: Check if 225 is a perfect square. Calculate: &&15^2 = 15 \times 15 = 225&&. So, &&\sqrt{225} = 15&&.
Step 2: Express as a fraction. Since &&15 = \frac{15}{1}&&, it is in the form &&\frac{p}{q}&&. Hence, &&\sqrt{225}&& is rational.
Part (iii): 0.3796 📝
Step 1: Analyze the decimal. The number 0.3796 has a finite number of digits after the decimal point, so it is a terminating decimal.
Step 2: Convert to fraction. Write &&0.3796 = \frac{3796}{10000}&&. Simplify: &&\frac{3796 \div 4}{10000 \div 4} = \frac{949}{2500}&&, which is &&\frac{p}{q}&& form. Hence, 0.3796 is rational.
Part (iv): 7.478478… 📝
Step 1: Identify the pattern. The decimal 7.478478… has “478” repeating, so it is a repeating decimal.
Step 2: Convert to fraction. Let &&x = 7.478478…&&. Then, &&1000x = 7478.478478…&&. Subtract: &&1000x – x = 7478.478478… – 7.478478…&&, so &&999x = 7471&&, and &&x = \frac{7471}{999}&&. Hence, 7.478478… is rational.
Caution: Ensure the repeating block is correctly identified.
Part (v): 1.1010010001… 📝
Step 1: Analyze the decimal. The number 1.1010010001… has gaps between 1’s increasing by one zero each time, so it is non-repeating and non-terminating.
Step 2: Conclusion. Since it cannot be written as &&\frac{p}{q}&&, 1.1010010001… is irrational.
Conclusion and Key Points ✅
In this ncert solutions guide, we classified numbers based on whether they can be expressed as &&\frac{p}{q}&&. Terminating or repeating decimals and perfect square roots are rational, while non-perfect square roots and non-repeating, non-terminating decimals are irrational.
Trick: To check if a square root is rational, verify if the number is a perfect square (e.g., &&\sqrt{225} = 15&&). For decimals, check if they terminate or repeat.
- Rational numbers are of the form &&\frac{p}{q}&&, where &&q \neq 0&&.
- &&\sqrt{n}&& is rational if &&n&& is a perfect square, else irrational.
- Terminating decimals (e.g., &&0.3796&&) are rational.
- Repeating decimals (e.g., &&7.478478…&&) are rational.
- Non-repeating, non-terminating decimals (e.g., &&1.1010010001…&&) are irrational.
FAQ
Q: What is a rational number?
A: A rational number can be expressed as &&\frac{p}{q}&&, where &&p&& and &&q&& are integers and &&q \neq 0&&, like &&\frac{3}{4}&& or &&15 = \frac{15}{1}&&.
Q: How to determine if &&\sqrt{23}&& is rational or irrational?
A: Since &&23&& is not a perfect square (&&4^2 = 16&&, &&5^2 = 25&&), &&\sqrt{23}&& cannot be written as &&\frac{p}{q}&&. Hence, it is irrational.
Q: Is &&\sqrt{225}&& rational or irrational?
A: Since &&225 = 15^2&&, we have &&\sqrt{225} = 15 = \frac{15}{1}&&, which is in &&\frac{p}{q}&& form. Hence, it is rational.
Q: How to classify 0.3796 as rational or irrational?
A: Since &&0.3796&& is a terminating decimal, it can be written as &&\frac{3796}{10000} = \frac{949}{2500}&&. Hence, it is rational.
Q: Is 7.478478… rational or irrational?
A: The decimal &&7.478478…&& repeats the block “478”. It can be converted to &&\frac{7471}{999}&&, so it is rational.
Q: Why is 1.1010010001… irrational?
A: The decimal &&1.1010010001…&& is non-repeating and non-terminating, so it cannot be expressed as &&\frac{p}{q}&&. Hence, it is irrational.
Further Reading
Explore more ncert solutions and official resources at NCERT’s official website.