NCERT Solutions for Class 9 Maths Exercise 1.3 Question 9
Understanding the Question 🧐
This question is a final check of our understanding of the entire exercise. We need to look at five different numbers and decide if they are rational or irrational. Let’s quickly review the rules:
- A Rational Number can be written as a fraction &&\frac{p}{q}&&. Its decimal form will either terminate (end) or be non-terminating but repeating.
- An Irrational Number cannot be written as a fraction. Its decimal form is always non-terminating and non-repeating.
Classify the following numbers as rational or irrational.
(i) &&\sqrt{23}&& 📝
Analysis: We need to check if 23 is a perfect square. A perfect square is a number that is the product of an integer with itself (like &&4^2=16&&, &&5^2=25&&). Since 23 is not a perfect square, its square root cannot be a whole number.
Conclusion: The square root of a non-perfect square is always irrational.
Classification: Irrational
(ii) &&\sqrt{225}&& 📝
Analysis: Let’s find the value of &&\sqrt{225}&&. We know that &&15 \times 15 = 225&&. So, 225 is a perfect square.
&&\sqrt{225} = 15&&
Conclusion: The number 15 can be written as a fraction, &&\frac{15}{1}&&, which fits the &&\frac{p}{q}&& form.
Classification: Rational
(iii) &&0.3796&& 📝
Analysis: This decimal number comes to an end after the digit 6. This means it is a terminating decimal.
Conclusion: All terminating decimals are rational because they can be written as a fraction. For example, &&0.3796 = \frac{3796}{10000}&&.
Classification: Rational
(iv) &&7.478478…&& 📝
Analysis: The ‘…’ tells us the decimal is non-terminating. However, we can see a clear pattern: the block of digits ‘478’ is repeating itself. This can be written as &&7.\overline{478}&&.
Conclusion: A decimal that is non-terminating but has a repeating block of digits is a rational number.
Classification: Rational
(v) &&1.101001000100001…&& 📝
Analysis: The decimal is non-terminating. Let’s look for a repeating pattern. The number of zeros between the ones keeps increasing (one zero, then two, then three, etc.). Because the pattern is always changing, there is no fixed block of digits that repeats.
Conclusion: This is a non-terminating and non-repeating decimal.
Classification: Irrational
Conclusion
This exercise provides a great summary of the concepts covered in this chapter. By applying the definitions, we can confidently classify any given number as either rational or irrational.
- Is it the square root of a non-perfect square? → Irrational
- Is it a terminating decimal? → Rational
- Is it a non-terminating, repeating decimal? → Rational
- Is it a non-terminating, non-repeating decimal? → Irrational
FAQ
Q: What is the main difference between a rational and an irrational number?
A: A rational number can be written as a &&\frac{p}{q}&& fraction and its decimal representation either terminates (ends) or repeats in a cycle. An irrational number cannot be written as a simple fraction and its decimal representation goes on forever without any repeating pattern.
Q: How can you quickly tell if a square root is rational or irrational?
A: If the number inside the square root is a perfect square (like 4, 9, 25, 225), its square root is a whole number and therefore rational. If the number is not a perfect square (like 2, 3, 23), its square root is irrational.
Q: Is a terminating decimal like &&0.3796&& rational or irrational?
A: It is rational. Any terminating decimal can be written as a fraction by putting it over a power of 10 (e.g., &&0.3796 = \frac{3796}{10000}&&).
Q: Is a non-terminating repeating decimal like &&7.478478…&& rational or irrational?
A: It is rational. All non-terminating but repeating decimals can be converted into the &&\frac{p}{q}&& fraction form, which is the definition of a rational number.
Q: Why is &&1.101001…&& irrational while &&7.478478…&& is rational?
A: The number &&7.478478…&& is rational because the block of digits ‘478’ repeats in a predictable, fixed cycle. The number &&1.101001…&& is irrational because its pattern constantly changes (the number of zeros increases), so it never settles into a repeating cycle.
Further Reading
Understanding the classification of numbers is fundamental to mathematics. For more practice, you can refer to the official NCERT textbooks.
