What is the mathematical proof behind the value of Pi?

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Discussion Overview

The discussion revolves around the mathematical understanding and proofs related to the value of Pi (π). Participants explore various methods of approximating Pi, its definition, and its relationship to geometric concepts such as the circumference and area of circles and spheres. The scope includes theoretical and mathematical reasoning, as well as references to historical approaches.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the proof of why Pi is approximately 3.14, referencing Archimedes' method of dividing a circle into 96 slices.
  • Another participant states that Pi is defined as the ratio of a circle's circumference to its diameter, emphasizing that it cannot be expressed as a simple fraction or decimal.
  • Several participants mention various series and formulas that approximate or express Pi, such as the series involving 4/1 - 4/3 + 4/5 - 4/7.
  • There is a discussion about the relationship between the volume and area of spheres and circles, with one participant proposing that the volume of a sphere is derived from the area of a sphere through integration.
  • One participant suggests a practical method for measuring Pi using a string around a cylindrical object, while another humorously critiques this approach.
  • Some participants clarify misconceptions about the "Euler number" and its distinction from Pi.
  • Historical references are made to Archimedes' approximations of Pi and the irrational nature of Pi, which cannot be expressed as a terminating or repeating decimal.

Areas of Agreement / Disagreement

Participants generally agree on the definition of Pi and its significance in geometry, but there are multiple competing views on how to derive or approximate its value. The discussion remains unresolved regarding the best methods to explain or prove the value of Pi.

Contextual Notes

Some participants express uncertainty about the terminology used (e.g., "96-sided triangle" vs. "96-sided polygon") and the implications of various mathematical definitions and proofs. There are also references to the limitations of certain approaches and the need for further clarification on the relationships between different mathematical concepts.

  • #31
[Edit] OTHERS HAVE GIVEN THIS ANSWER, BUT I CAN'T DELETE IT.
There are trig functions that are known to equal π. Their series expansions are used to approximate π to the accuracy desired, but they also prove that the value of π is 3.14159...

For instance, knowing that π/2 = arcsin( 1 ) means that you can use the Taylor series of arcsin to prove the value of π to any accuracy you want. I don't know if this example is practical, but there are many other similar ways. See https://en.wikipedia.org/wiki/Pi#Infinite_series
 

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