Recall, `pi` is defined as the ratio of the circumference (say c ) of a circle to its diameter (say d). That is , `pi` = c/d . This seems to contradict the fact that `pi` is irrational. How will you resolve this contradiction ?

Answer

To resolve the contradiction between the definition of π (pi) as the ratio of the circumference (c) of a circle to its diameter (d) and the fact that π is an irrational number, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Definition of π**:
- π is defined as the ratio of the circumference of a circle (c) to its diameter (d).
- Mathematically, this is expressed as:
\[ \pi = \frac{c}{d} \]

2. **Recognizing the Nature of c and d**:
- For π to be rational, both c and d must be integers, and d must not be zero.
- However, the circumference (c) of a circle is calculated using the formula:
\[ c = 2\pi r \]
where r is the radius of the circle.

3. **Identifying the Issue**:
- The diameter (d) of a circle is simply twice the radius:
\[ d = 2r \]
- When we substitute these into the equation for π, we get:
\[ \pi = \frac{2\pi r}{2r} = \pi \]
- This shows that while π is defined as a ratio, it does not imply that c and d are integers.

4. **Conclusion About Rationality**:
- The values of c and d can be any real numbers depending on the size of the circle.
- Since π cannot be expressed as a fraction of two integers, it is classified as an irrational number.
- Therefore, the initial appearance of π as a ratio does not contradict its irrationality because c and d are not constrained to be integers.

5. **Final Statement**:
- Thus, the definition of π does not contradict its nature as an irrational number because the circumference and diameter can take on non-integer values.