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?

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Definition of Pi: Pi is defined as the ratio of the circumference (c) of a circle to its diameter (d), which is expressed as π = c/d. The apparent contradiction arises from the fact that π is known to be an irrational number, yet the ratio of two lengths (c and d) seems like it should be a rational number.Nature of Irrational Numbers: To resolve this contradiction, we must understand the nature of irrational numbers. An irrational number is a number that cannot be expressed as a simple fraction, meaning it cannot be written as a ratio of two integers. The fact that π is irrational means that although it is defined as a ratio of two lengths, these lengths cannot be both whole numbers.Continuous Real Numbers: The circumference and diameter of a circle are both continuous, real numbers. When we measure them, we can only approximate their values. The definition of π as a ratio does not imply that it must be a ratio of two integers; it can be a ratio of two real numbers that do not form a simple fraction.Mathematical Approaches: Furthermore, the irrationality of π is proven through various mathematical approaches, such as infinite series, and is not based solely on its geometric definition. The geometric definition is a way to understand π conceptually, but the true nature of π is revealed through deeper mathematical analysis.Resolution of Contradiction: Therefore, the contradiction is resolved by recognizing that the definition of π as a ratio does not require the ratio to be between two integers. The irrationality of π is a property that emerges from its infinite, non-repeating decimal expansion, which cannot be captured by a simple fraction of any two lengths, regardless of how they are measured.

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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Recall π \pi π is defined as the ratio of the circumference (say ccc) of a circle to its diameter (say ddd). That is π=cd \pi=\frac{c}{d} π=dc​. This seems to contradict the fact that π \pi π 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?