Newton calculated pi by evaluating an infinite sum. Specifically, in 1666, he computed pi to sixteen decimal places by evaluating the first twenty-two terms of an infinite sum. This approach allowed him to achieve a high degree of accuracy for the value of pi.
Understanding Newton's Method
Newton didn't use the familiar geometrical methods that involve calculating the area of circles. Instead, he used a method based on infinite series, specifically, he derived a formula that produced an infinite sum that converged towards pi.
Steps in Newton's Calculation:
- Derivation of the Infinite Sum: Newton first had to derive the specific infinite sum that could approximate pi. This was a significant mathematical achievement. The specific formula is a complex one, and goes beyond the scope of this simple explanation.
- Term-by-Term Evaluation: The infinite sum he used involved terms that gradually got smaller as the series went on.
- Summation: He added up the first twenty-two terms of this infinite sum.
- Accuracy: After adding these twenty-two terms, the resulting number had sixteen decimal places that matched the true value of pi. This demonstrates the power of his method and that it quickly converged on the correct result.
Practical Implications:
Newton's method provided a far more efficient way to calculate pi than previous geometrical methods, which typically involved complex polygons that approached circles.
| Method | Complexity | Accuracy |
|---|---|---|
| Geometrical Method | More Complex | Lower |
| Newton's Infinite Sum | Less Complex | Higher |
Newton's method not only resulted in calculating the numerical value of pi, but also provided valuable information about the nature of infinite series, which he then used to develop new and advanced mathematics.
