Solve for the value of pi - Research Paper Example

However, it can be agreed that pi is based on the circle, which has many interesting properties (Gap-System). The circle, regardless of the size, always has the same perfect properties; therefore, the value of pi is constant. The history of pi can not be conclusively decided, since it is untraceable where the circle was decided as the basis. However, biblical references to pi and the discovery of a variation of the constant in ancient texts seem to indicate that the number is old. The vale of pi has been around for a long time; the bible contains two references to pi, though the values given are incorrect, ancient Egyptians and Babylonians had a value for the value of pi, and old-time mathematicians proved the existence of pi.

The value of pi is a fixed value, and is determined to be infinite. The determination of the formula, which gives pi, is important in determining the origin of the value, therefore, this paper will seek to determine the formulas for the determination of pi, from Archimedes to Machin. The first mathematical and theoretical proof of pi was given by Archimedes, a brilliant mathematician in old times. Archimedes showed that pi is a value between two numbers; 223/71 and 22/7. This can be mathematically denoted as: This derivation used by Archimedes is based on the equation of the area of a circle,, which he derived by using a simple system of equations.

In the derivation of pi, Archimedes used a system where regular polygons were inscribed and circumscribed on a circle, from which the diameter and circumference of the circle can be determined by determining the properties of the polygons. The diagram that was used by Archimedes is: 1 In this calculation, consider a circle with a radius OA of 1 unit, over which is circumscribed a regular hexagon (or any regular polygon of 3*2n-1 sides), and in which is inscribed another regular hexagon (or any regular polygon of 3*2n-1 sides).

In this case, we assume that the semi perimeter for the inscribed polygon is bn, and that the semi perimeter for the super scribed polygon is an. The diagram given implies that the semi perimeter for the bigger polygon is ever decreasing, while the sequence for the smaller polygon is increasing, such that they converge at a value pi. Using trigonometric notation, it can be inferred that the semi perimeters of the polygons are given by the formula, , and, where K is the number of sides of the polygon.

It also follows that; , and . Archimedes then used the same trigonometric principles to show that: , and. From these formulas, Archimedes could calculate the values of a and b from n=1, 2… 6. After this calculation, Archimedes concluded that as the semi perimeters of the two polygons changed, the convergence was towards the limit pi, where. The deduction by Archimedes follows a simple principle of trigonometry and mathematical application, where it is known that the inner sides of the hexagon used in the calculation are all equal to the radius of the circle, which means that the perimeter of the hexagon is 6 times the radius of the circle.

Another complicated calculation used by Archimedes is that a line drawn from the middle of a side of the outer polygon is