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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?

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?

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Find instantaneous rates of change

Full solution

Q. 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?

Nature of Irrational Numbers: Pi is known to be an irrational number, which means it cannot be expressed as a simple fraction of two integers. However, the definition of $\pi$ as the ratio of a circle's circumference to its diameter seems to suggest that it could be a rational number since both circumference and diameter are lengths that can be measured. To resolve this apparent contradiction, we need to understand the nature of irrational numbers and the concept of measurement.
Misunderstanding of Ratios: The contradiction arises from the misunderstanding that a ratio of two measurable quantities must always be a rational number. In reality, while both the circumference and the diameter of a circle can be measured, the exact ratio between these two measurements is $\pi$ , which is an irrational number. This means that any measurement we make will only approximate the true value of $\pi$ , and no matter how precise our measurements, we will never be able to express $\pi$ as a ratio of two integers.
Limitations of Measurement Precision: Furthermore, the fact that $\pi$ is irrational means that its decimal representation goes on forever without repeating. When we measure the circumference and diameter of a circle, we are limited by the precision of our measuring tools. Therefore, the values we obtain for the circumference and diameter are only approximations, and the calculated ratio will be an approximation of $\pi$ , not the exact value of $\pi$ itself.
Pi as a Mathematical Abstraction: To fully grasp this concept, one must accept that the physical act of measuring cannot capture the true essence of certain mathematical constants. Pi is a mathematical abstraction that is defined as the ratio of the circumference to the diameter of a circle in a perfectly Euclidean space, where both the circumference and diameter are considered to be infinitely precise. In practical terms, we use approximations of pi (like $3.14$ , $\frac{22}{7}$ , etc.) for calculations involving real-world measurements.

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