There are several ways to calculate the decimals of pi, but they generally fall into a few key categories.
Methods for Calculating Pi
One of the oldest approaches for calculating pi to many decimal places uses the power series expansion of the arctangent function. Here's a breakdown of this method:
Using the Arctangent Power Series
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The arctangent function, denoted as atan(x) or arctan(x), can be represented by the following infinite series:
atan(x) = x - x³/3 + x⁵/5 - x⁷/7 + x⁹/9 - ... -
This series converges for values of x between -1 and 1 (inclusive). This means that by adding more terms, you can get closer and closer to the actual value of atan(x).
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A key relationship links the arctangent function and pi:
pi = 4 * atan(1) -
However, using atan(1) is not the most efficient as this series converges very slowly. More advanced formulas combine the arctangent function with different values for 'x' and constants, such as the formula mentioned in our reference:
pi = 16 * atan(1/5) - 4 * atan(1/239) -
This particular formula converges at a rate of about 1.4 decimals per term, meaning that you need to calculate several terms to get a more accurate decimal approximation of pi.
Why is this useful?
- Using a formula like pi = 16*atan(1/5) - 4*atan(1/239), each term you calculate in the arctan series gets you closer to the actual value of pi.
- More terms = more decimals. The more terms you calculate from the atan series, the higher the precision of pi.
- This method, while simple, is very powerful and was widely used historically.
Other Approaches
Although the arctangent series method is highlighted here, other techniques exist:
- Geometric Methods: Such as approximating the circumference of a circle by calculating the perimeter of inscribed polygons.
- Monte Carlo Methods: Using random numbers to simulate the ratio of areas, thus approximating pi.
Conclusion
In summary, calculating pi decimals often involves employing mathematical formulas that relate pi to other functions like the arctangent, combined with the power series expansions of these functions. One way is to use pi = 16*atan(1/5) - 4*atan(1/239) alongside the power series for atan(x).
