NCERT Solutions for Class 9 Maths Exercise 1.5 Question 3
Understanding the Question 🧐
This is a very interesting conceptual question. It asks us to think about the definition of Pi (&&\pi&&) and why it doesn’t contradict the fact that &&\pi&& is an irrational number. Let’s look at the problem closely.
Recall, &&\pi&& is defined as the ratio of the circumference (say &&c&&) of a circle to its diameter (say &&d&&). That is, &&\pi = \frac{c}{d}&&. This seems to contradict the fact that &&\pi&& is irrational. How will you resolve this contradiction?
Resolving the Contradiction 📝
The expression &&\pi = \frac{c}{d}&& certainly looks like the definition of a rational number (a number in the form &&\frac{p}{q}&&). However, there is no actual contradiction. The resolution lies in understanding the nature of measurement and the properties of rational and irrational numbers.
Step 1: The Problem with Measurement
When we measure the length of a circumference (&&c&&) or a diameter (&&d&&) using a physical tool like a ruler or a tape measure, we only ever get an approximate rational value. Our tools have limitations and can only measure up to a certain precision (e.g., to the nearest millimeter or centimeter). We never get the exact, true value if that value happens to be irrational.
Step 2: The Condition for a Rational Number
A number is rational only if it can be written as a ratio of two integers. The formula &&\pi = \frac{c}{d}&& doesn’t guarantee that &&\pi&& is rational because we don’t know if &&c&& and &&d&& are integers. In fact, we can prove that they can’t both be rational at the same time.
Step 3: The Logical Conclusion
Let’s think about the possibilities for a real circle:
- Case 1: Assume the diameter (&&d&&) is a rational number. For example, let’s say we have a circle with a diameter of exactly &&10&& cm. The circumference (&&c&&) would be &&c = \pi \times d = \pi \times 10 = 10\pi&&. Since &&\pi&& is irrational, &&10\pi&& is also irrational (because a non-zero rational multiplied by an irrational is irrational). So, in this case, &&\pi = \frac{10\pi}{10} = \frac{\text{irrational}}{\text{rational}}&&, which is irrational.
- Case 2: Assume the circumference (&&c&&) is a rational number. For example, let’s imagine a circle with a circumference of exactly &&22&& cm. The diameter (&&d&&) would be &&d = \frac{c}{\pi} = \frac{22}{\pi}&&. Since &&\pi&& is irrational, &&\frac{22}{\pi}&& is also irrational (because a non-zero rational divided by an irrational is irrational). So, here, &&\pi = \frac{22}{22/\pi} = \frac{\text{rational}}{\text{irrational}}&&, which is consistent with &&\pi&& being irrational.
It is mathematically impossible for both the circumference &&c&& and the diameter &&d&& of a circle to be rational numbers. If they were, their ratio &&\frac{c}{d}&& would be rational, which would mean &&\pi&& is rational. This is a known falsehood.
Conclusion: There is no contradiction. The formula &&\pi = \frac{c}{d}&& is perfectly correct. It simply means that for any given circle, at least one of the measurements (either the circumference &&c&& or the diameter &&d&&, or both) must be an irrational number. This ensures their ratio remains irrational.
- &&\pi&& is a proven irrational number.
- Physical measurements with tools like rulers always yield approximate rational numbers.
- For a number to be rational in the form &&\frac{p}{q}&&, both &&p&& and &&q&& must be integers (&&q \neq 0&&).
- In the formula &&\pi = \frac{c}{d}&&, it is impossible for both &&c&& and &&d&& to be rational numbers. At least one must be irrational.
- The value &&\frac{22}{7}&& is a convenient rational approximation for &&\pi&&, not its exact value.
FAQ ❓
Q: Why does &&\pi = c/d&& look like a rational number?
A: It has the form of a fraction, just like the definition of a rational number (&&\frac{p}{q}&&). However, a number is only guaranteed to be rational if *both* the numerator and denominator are integers. For &&\pi = \frac{c}{d}&&, it’s impossible for both &&c&& and &&d&& to be rational, so the condition isn’t met.
Q: What is the main reason there is no contradiction about &&\pi&& being irrational?
A: The main reason is that for any real circle, it is mathematically impossible for both its circumference &&c&& and its diameter &&d&& to be rational numbers. At least one of them must be irrational, which makes their ratio, &&\pi&&, irrational.
Q: If I measure a circle’s diameter to be 7 cm, is the circumference rational?
A: Your measurement of “7 cm” is a rational approximation. The true circumference would be &&7 \times \pi&& cm. Since &&\pi&& is irrational, the true circumference is also an irrational number.
Further Reading 📖
For a deeper understanding of rational and irrational numbers, you can refer to the official NCERT textbook for Class 9 Maths. More resources are available on the NCERT website at https://ncert.nic.in/.
