Rational approximations and continued fractions Development of calculus Proof of irrationality and transcendence Around 480, the Chinese mathematician Zu Chongzhi gave the approximation and showed that, which would stand as the most accurate value for over the next 900 years. In the following centuries, most significant mathematical development took place in India and China. The method of Archimedes is the first known algorithm for calculating with arbitrary accuracy. Taking the average of the bounds yields the approximation. By calculating the circumferences of two 96-sided polygons, one slightly smaller than a unit circle and the other slightly larger, he rigorously proved the bounds.
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He knew how to calculate the circumference of a regular polygon, and realized that a polygon inscribed in a circle has a smaller circumference than the circle whereas a circumscribed polygon has a greater circumference. The Books of Kings (600 BC) appears to suggest, which is notably worse than other estimates available at the time, although it is disputed whether the text suggests an exact value.Īrchimedes (287–212 BC), in whose honor is sometimes called Archimedes' constant, was the first to treat the number with mathematical rigor. All of these values differ from the true value by less than one percent. The ancient Indian text Shatapatha Brahmana gives as. A Babylonian stone tablet from 1900 BC uses the proportion 25/8 = 3.125, and an Egyptian document from the same time gives the ratio. Preserved documents do not refer to the number explicitly, but contain approximations for as part of methods for calculating circumference or area. The fact that the ratio of the circumference to the diameter of a circle is the same for all circles, and that it is slightly more than 3, was known to ancient Egyptian, Babylonian, Indian and Greek geometers. 4 Rational approximations and continued fractions.3 Proof of irrationality and transcendence.Since π is irrational, it cannot be written as a finite or periodically repeating decimal. The latter property implies that π is not constructible: in particular, the problem of squaring the circle in a finite number of steps using a compass and straightedge is impossible. Pi is an irrational number, which means that it cannot be expressed exactly as a ratio of integers, and transcendental, meaning that it does not satisfy any algebraic equation with rational coefficients. Pi has further uses in many other areas of mathematics. This ratio is also found in many other geometrical objects, such as spheres and cones. In Euclidean geometry, π represents the ratio between the circumference and the diameter of any circle, or equivalently, the ratio between a circle's area and the square of its radius. Pi, denoted by the lower-case Greek letter π, is a mathematical constant that is approximately equal to 3.14159.
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