![]() ![]() An important consequence of the transcendence of π is the fact that it is not constructible. This means that there is no polynomial with rational coefficients of which π is a root. This was proven in 1761 by Johann Heinrich Lambert.įurthermore, π is also transcendental, as was proven by Ferdinand von Lindemann in 1882. The constant π is an irrational number that is, it cannot be written as the ratio of two integers. Similarly, the more complex approximations of π given below involve repeated calculations of some sort, yielding closer and closer approximations with increasing numbers of calculations. Results for some values of r are shown in the table below: r When finished, divide the sum, representing the area of a circle of radius r, by r 2 to find the approximation of π. Starting at 0, add 1 for each point whose distance to the origin (0,0) is less than r. Consider all points ( x, y) in which both x and y are integers between -r and r. In other words, begin by choosing a value for r. Mathematically, this formula can be written: The total number of points satisfying that condition thus approximates the area of the circle, which then can be used to calculate an approximation of π. Squares whose centre resides inside the circle can then be counted by testing whether, for each point ( x, y), Mathematical "graph paper" is formed by imagining a 1x1 square centered around each point ( x, y), where x and y are integers between -r and r. The Pythagorean theorem gives the distance from any point ( x, y) to the centre: ![]() If a circle with radius r is drawn with its centre at the point (0,0), any point whose distance from the origin is less than r will fall inside the circle. This process works mathematically as well as experimentally. Since the area of the circle is known to be ![]() One common classroom activity for experimentally measuring the value of π involves drawing a large circle on graph paper, then measuring its approximate area by counting the number of cells inside the circle. It requires more than 600 terms just to narrow its value to 3.14 (two places), and billions of terms to achieve accuracy to ten places. This series is easy to understand, but is impractical in use as it converges to π very slowly. Nevertheless, it is possible to compute π using techniques involving only algebra and geometry. Most formulas given for calculating the digits of π have desirable mathematical properties, but may be difficult to understand without a background in trigonometry and calculus. ![]() See history of numerical approximations of π. Digits of π are available on many web pages, and there is software for calculating π to billions of digits on any personal computer. Despite much analytical work, and supercomputer calculations that have determined over 1 trillion digits of π, no simple pattern in the digits has ever been found. This infinite sequence of digits has fascinated mathematicians and laymen alike, and much effort over the last few centuries has been put into computing more digits and investigating the number's properties. Nevertheless, the exact value of π has an infinite decimal expansion: its decimal expansion never ends and does not repeat, since π is an irrational number (and indeed, a transcendental number). With the 50 digits given here, the circumference of any circle that would fit in the observable universe (ignoring the curvature of space) could be computed with an error less than the size of a proton. The formulæ below illustrate other (equivalent) definitions. The constant π may be defined in other ways that avoid the concepts of arc length and area, for example as twice the smallest positive x for which cos( x) = 0. In Euclidean plane geometry, π is defined either as the ratio of a circle's circumference to its diameter, or as the ratio of a circle's area to the area of a square whose side is the radius. Area of the circle = π × area of the shaded square ![]()
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