Early calculations of pi, such as those performed by Archimedes, involved geometric methods. He didn't have modern calculators or formulas; instead, he relied on inscribed and circumscribed polygons to approximate the value of pi.
Archimedes' Method
Archimedes' method is a classic example of how pi was first calculated. He essentially trapped the true value of pi between two bounds.
The Process
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Inscribing and Circumscribing Polygons: Archimedes started by drawing a regular hexagon inside a circle (inscribed) and another regular hexagon outside the same circle (circumscribed).
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Doubling the Sides: He then successively doubled the number of sides of these polygons, going from a hexagon (6 sides) to a dodecagon (12 sides), then to a 24-sided polygon, and so on, until he reached a 96-sided polygon.
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Calculating Perimeters: By calculating the perimeters of these polygons, he could determine upper and lower bounds for the circumference of the circle. Since pi is the ratio of a circle's circumference to its diameter, these bounds on the circumference gave him bounds on pi.
The Result
Through this process, Archimedes proved that:
223/71 < π < 22/7
This translates to approximately:
- 1408 < π < 3.1429
Significance
Archimedes' method provides a clear illustration of how early mathematicians approached the problem of calculating pi, relying on geometric principles and careful calculations to approximate its value. This method laid the groundwork for future, more precise calculations.
