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.