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Do We Need the Digits of $\pi$?

Are we at best, estimating things when we use formulas involving pi to describe the area, etc. of things?

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marked as duplicate by t.b., Asaf Karagila, J. M., Zev Chonoles Dec 30 '11 at 8:04

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

"how can we ... rely on it for real world calculations?" - In the real world, there is no such thing as a "real" circle. The wheels, cans, and other such things are approximations to the real thing; manufacturing will always give things with a few imperfections. – J. M. Dec 30 '11 at 8:02

Not being able to write the exact value of $\pi$ does not mean we do not know it. It has many representations.

  • Some are well known, e.g. "the ratio between the circumference and the diameter of a circle",

  • Others are less known, e.g. $\sqrt{\sum_{n=1}^\infty \dfrac{6}{n^2}}$

The use of $\pi$ in calculations made in engineering, or physics, or other "approximates" is also fine because you know enough digits of $\pi$ to ensure that the error margin is small enough.

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No: When we say the area of a circle with radius $r$ is $\pi r^2$, and the circumference is $2 \pi r$, those are exact.

Yes: When we say the area and circumference of a circle with radius $2$ are about $12.56637$, those are approximations.

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