12. Recall, is defined as the ratio of the circumference (say c) of a circle to its diameter This seems to contradict the fact that is irrational. How will (say d). That is, you resolve this contradiction?
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The statement you provided about the definition of \(\pi\) and its irrationality does not actually create a contradiction. Let's clarify:
1. **Definition of \(\pi\)**: \(\pi\) (pi) is defined as the ratio of the circumference \(C\) of a circle to its diameter \(d\):
\[
\pi = \frac{C}{d}
\]
Alternatively, it can also be defined in terms of the area \(A\) of a circle and its radius \(r\):
\[
\pi = \frac{A}{r^2}
\]
2. **Irrationality of \(\pi\)**: \(\pi\) is known to be an irrational number. This means it cannot be expressed as a fraction of two integers (\(\frac{p}{q}\)), where \(p\) and \(q\) are integers and \(q \neq 0\).
Now, let's address the potential confusion:
- **Circumference and Diameter**: The statement mentions that \(\pi\) is defined as the ratio of the circumference \(C\) to the diameter \(d\). This is correct and does not contradict its irrationality.
- **Irrationality and Definition**: The fact that \(\pi\) is irrational means it cannot be exactly expressed as a finite or repeating decimal or as a fraction of integers. However, this does not invalidate its definition as the ratio of circumference to diameter.
To resolve any confusion:
- **Rational vs. Irrational**: \(\pi\) being irrational means its decimal representation goes on forever without repeating, and it cannot be exactly expressed as a fraction. This is a mathematical property that does not change the fact that \(\pi\) is defined as the ratio of circumference to diameter.
- **No Contradiction**: The definition of \(\pi\) as the ratio of circumference to diameter is a geometric and mathematical definition. Its irrationality pertains to the nature of its numerical value, not to the validity of its definition.
Therefore, there is no contradiction between the definition of \(\pi\) as the ratio of circumference to diameter and its status as an irrational number. The definition is a geometric fact, while its irrationality is a number theoretic property. Both aspects coexist without conflict.
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