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How to Calculate the Number PI

Pi is the fundamental constant that represents the geometric ratio of the circumference of a circle or sphere to its diameter. It is an irrational number, meaning that it can't be expressed as the exact quotient of two integers, and its exact value extends to an infinite number of decimal places without a repeating pattern. Pi has been studied by mathematicians for thousands of years, accumulating numerous formulas in the form of infinite series for its approximation. The most famous of these infinite series is the Gregory Series, or the alternating series that arises when x = 1 is substituted into the Leibniz Series.

Things You'll Need

Pencil
Paper
Calculator

Instructions

- 1
  Write down the Gregory Series. The series appears as: Pi = 4sum[((-1)^(k + 1))/(2k - 1), k = 1 ... infinity]. Expressed in plain English, the relationship says that the constant Pi is equal to four times the summation as k runs from one to infinity of the quantity negative one raised to the power of k plus one divided by the quantity two k minus one.
- 2
  Expand the series out to a satisfying number of terms. This simply means that for the first term simply substitute 1 into the equation for the variable k, write down the term (without computing it), and then continue to make the appropriate substitutions for the successive terms k = 2, k = 3, etc., until you have generated the number of terms that will approximate Pi to the desired accuracy.
- 3
  Add together all of the generated terms. Use the calculator to compute the value of each term as the series terms are added together. Computing the terms as they are summed together is an important step in the calculation as rounding the individual terms will give persistent errors that diminish the accuracy. Check the final sum against a computation of Pi that is accurate to the desired number of significant figures (decimal places.)

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