Full solution

\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?

Definition of Pi: Pi is defined as the ratio of the circumference of a circle to its diameter. This definition might seem to imply that pi should be a rational number since it is a ratio. However, the fact that pi is irrational means that it cannot be expressed as a simple fraction of two integers. To resolve this apparent contradiction, we need to understand the nature of irrational numbers and the concept of a limit.
Rational vs Irrational Numbers: A rational number is a number that can be expressed as the ratio of two integers. An irrational number, on the other hand, cannot be expressed as a simple fraction; its decimal representation goes on forever without repeating. The fact that $\pi$ is a ratio does not necessarily mean it must be a rational number. The ratio of the circumference to the diameter of a circle is the same for all circles, but this ratio, $\pi$ , is an irrational number.
Calculation of Pi: The value of pi is determined by calculating the circumference of a circle with a known diameter using increasingly precise methods. As the methods become more precise, the calculated value of pi becomes more accurate. However, no matter how precise the method, the value of $pi$ never resolves to a simple fraction of integers. This is because $pi$ is a transcendental number, a type of irrational number that is not the root of any non-zero polynomial equation with rational coefficients.
Resolution of Contradiction: The contradiction is resolved by understanding that the definition of $\pi$ as a ratio does not require the ratio to be between integers. The circumference and diameter of a circle are real numbers, and their ratio, $\pi$ , is a real number as well. The irrationality of $\pi$ means that while we can approximate it with rational numbers, we can never express it exactly as a ratio of two integers.