What is the mathematical proof behind the value of Pi?

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SUMMARY

The mathematical constant Pi (π) is defined as the ratio of a circle's circumference to its diameter, consistently yielding approximately 3.14159. The forum discussion highlights various methods to approximate Pi, including Archimedes' polygon method, which uses a 96-sided polygon to estimate its value. Additionally, it emphasizes that Pi is an irrational number, meaning it cannot be expressed as a simple fraction and has an infinite number of non-repeating decimal places. The conversation also touches on integral calculus as a means to derive related geometric formulas, such as the volume and surface area of spheres.

PREREQUISITES
  • Understanding of basic geometry, specifically circles and polygons
  • Familiarity with integral calculus concepts
  • Knowledge of irrational numbers and their properties
  • Basic mathematical series and sequences
NEXT STEPS
  • Study Archimedes' method for approximating Pi using polygons
  • Learn about integral calculus applications in deriving geometric formulas
  • Explore the properties of irrational numbers and their significance
  • Investigate various series that converge to Pi, such as the Leibniz formula
USEFUL FOR

Mathematicians, educators, students in geometry and calculus, and anyone interested in the properties and applications of Pi.

  • #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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