1.
Pi
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The number π is a mathematical constant, the ratio of a circles circumference to its diameter, commonly approximated as 3.14159. It has been represented by the Greek letter π since the mid-18th century, being an irrational number, π cannot be expressed exactly as a fraction. Still, fractions such as 22/7 and other numbers are commonly used to approximate π. The digits appear to be randomly distributed, in particular, the digit sequence of π is conjectured to satisfy a specific kind of statistical randomness, but to date no proof of this has been discovered. Also, π is a number, i. e. a number that is not the root of any non-zero polynomial having rational coefficients. This transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass, ancient civilizations required fairly accurate computed values for π for practical reasons. It was calculated to seven digits, using techniques, in Chinese mathematics. The extensive calculations involved have also used to test supercomputers. Because its definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, and spheres. Because of its role as an eigenvalue, π appears in areas of mathematics. It is also found in cosmology, thermodynamics, mechanics, attempts to memorize the value of π with increasing precision have led to records of over 70,000 digits. In English, π is pronounced as pie, in mathematical use, the lowercase letter π is distinguished from its capitalized and enlarged counterpart ∏, which denotes a product of a sequence, analogous to how ∑ denotes summation. The choice of the symbol π is discussed in the section Adoption of the symbol π, π is commonly defined as the ratio of a circles circumference C to its diameter d, π = C d The ratio C/d is constant, regardless of the circles size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d. This definition of π implicitly makes use of geometry, although the notion of a circle can be extended to any curved geometry. Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be defined independently of geometry using limits. An integral such as this was adopted as the definition of π by Karl Weierstrass, definitions of π such as these that rely on a notion of circumference, and hence implicitly on concepts of the integral calculus, are no longer common in the literature. One such definition, due to Richard Baltzer, and popularized by Edmund Landau, is the following, the cosine can be defined independently of geometry as a power series, or as the solution of a differential equation
2.
Area of a circle
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In geometry, the area enclosed by a circle of radius r is πr2. Here the Greek letter π represents a constant, approximately equal to 3.14159, one method of deriving this formula, which originated with Archimedes, involves viewing the circle as the limit of a sequence of regular polygons. The area of a polygon is half its perimeter multiplied by the distance from its center to its sides. Therefore, the area of a disk is the precise phrase for the area enclosed by a circle. Modern mathematics can obtain the area using the methods of calculus or its more sophisticated offspring. However the area of a disk was studied by the Ancient Greeks, eudoxus of Cnidus in the fifth century B. C. had found that the area of a disk is proportional to its radius squared. The circumference is 2πr, and the area of a triangle is half the times the height. A variety of arguments have been advanced historically to establish the equation A = π r 2 of varying degrees of mathematical rigor, the area of a regular polygon is half its perimeter times the apothem. As the number of sides of the regular polygon increases, the polygon tends to a circle, and this suggests that the area of a disk is half the circumference of its bounding circle times the radius. Following Archimedes, compare the area enclosed by a circle to a triangle whose base has the length of the circles circumference. If the area of the circle is not equal to that of the triangle and we eliminate each of these by contradiction, leaving equality as the only possibility. We use regular polygons in the same way, suppose that the area C enclosed by the circle is greater than the area T = 1⁄2cr of the triangle. Let E denote the excess amount, inscribe a square in the circle, so that its four corners lie on the circle. Between the square and the circle are four segments, if the total area of those gaps, G4, is greater than E, split each arc in half. This makes the square into an inscribed octagon, and produces eight segments with a smaller total gap. Continue splitting until the total gap area, Gn, is less than E, now the area of the inscribed polygon, Pn = C − Gn, must be greater than that of the triangle. E = C − T > G n P n = C − G n > C − E P n > T But this forces a contradiction, as follows. Draw a perpendicular from the center to the midpoint of a side of the polygon, its length, also, let each side of the polygon have length s, then the sum of the sides, ns, is less than the circle circumference
3.
Piphilology
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Piphilology comprises the creation and use of mnemonic techniques to remember a span of digits of the mathematical constant π. The word is a play on the word pi itself and of the field of philology. There are many ways to memorize π, including the use of piems, here is an example of a piem, How I need a drink, alcoholic in nature, after the heavy lectures involving quantum mechanics. Notice how the first word has three letters, the word has one, the third has four, the fourth has one, the fifth has five. In longer examples, 10-letter words are used to represent the zero. The short story Cadaeic Cadenza records the first 3834 digits of π in this manner, however, piems prove to be inefficient for large memorizations of π. Other methods include remembering patterns in the numbers and the method of loci, in 1949 a computer was used to calculate π to 2000 places, presenting one of the earliest opportunities for a more difficult challenge. Later computers calculated pi to extraordinary numbers of digits, and people began memorizing more and more of the output, the world record for the number of digits memorized has exploded since the mid-1990s, and it stood at 100,000 as of October 2006. The previous record was set by the person on July 2,2005. An institution from Germany provides the details of the “Pi World Ranking”, see the website at http, the most common mnemonic technique is to memorize a so-called piem in which the number of letters in each word is equal to the corresponding digit of π. This famous example has several variations, including, How I need a drink, alcoholic in nature, short mnemonics such as these, of course, do not take one very far down πs infinite road. Instead, they are intended more as amusing doggerel, If even less accuracy suffices, the following examples can be used, How I wish I could recollect pi easily today. May I have a container of coffee, cream and sugar. This second one gives the value of π as 3.1415926535, indeed, many published poems use truncation instead of one of the several roundings, thereby producing a less-accurate result when the first omitted digit is greater than or equal to five. It is advantageous to use truncation in memorizing if the individual intends to study more places later on, another mnemonic is, The point I said a blind Bulgarian in France would know In this mnemonic the word point represents the decimal point itself. Yet another example is, How I wish I could recollect, of round, the exact relation Arkimedes learned In this example. Longer mnemonics employ the same concept, in the childrens book, Somewhen, a poem is presented as a riddle. Here, the words describe the ratio and as laid out, to side-step the zero at decimal position 32, the word nothing is used
4.
Method of exhaustion
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The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is constructed, the difference in area between the n-th polygon and the containing shape will become arbitrarily small as n becomes large. As this difference becomes small, the possible values for the area of the shape are systematically exhausted by the lower bound areas successively established by the sequence members. The method of exhaustion typically required a form of proof by contradiction and this amounts to finding an area of a region by first comparing it to the area of a second region. The idea originated in the late 5th century BC with Antiphon, the theory was made rigorous a few decades later by Eudoxus of Cnidus, who used it to calculate areas and volumes. It was later reinvented in China by Liu Hui in the 3rd century AD in order to find the area of a circle, the first use of the term was in 1647 by Grégoire de Saint-Vincent in Opus geometricum quadraturae circuli et sectionum. The method of exhaustion is seen as a precursor to the methods of calculus, the development of analytical geometry and rigorous integral calculus in the 17th-19th centuries subsumed the method of exhaustion so that it is no longer explicitly used to solve problems. Euclid used the method of exhaustion to prove the following six propositions in the book 12 of his Elements, proposition 2 The area of a circle is proportional to the square of its radius. Proposition 5 The volumes of two tetrahedra of the same height are proportional to the areas of their triangular bases, proposition 10 The volume of a cone is a third of the volume of the corresponding cylinder which has the same base and height. Proposition 11 The volume of a cone of the height is proportional to the area of the base. Proposition 12 The volume of a cone that is the similar to another is proportional to the cube of the ratio of the diameters of the bases, proposition 18 The volume of a sphere is proportional to the cube of its diameter. Archimedes used the method of exhaustion as a way to compute the area inside a circle by filling the circle with a polygon of a greater area and greater number of sides. He also provided the bounds 3 + 10/71 < π <3 + 10/70, the Method of Mechanical Theorems The Quadrature of the Parabola Trapezoidal rule
5.
Zu Chongzhi
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Zu Chongzhi, courtesy name Wenyuan, was a prominent Chinese mathematician and astronomer during the Liu Song and Southern Qi Dynasties. Chongzhis ancestry was from modern Baoding, Hebei, to flee from the ravage of war, Zus grandfather Zu Chang moved to the Yangtze, as part of the massive population movement during the Eastern Jin. Zu Chang at one point held the position of Chief Minister for the Palace Buildings within the Liu Song and was in charge of government construction projects, Zus father, Zu Shuozhi also served the court and was greatly respected for his erudition. His family had historically been involved in research, and from childhood Zu was exposed to both astronomy and mathematics. When he was only a youth his talent earned him much repute, when Emperor Xiaowu of Liu Song heard of him, he was sent to an Academy, the Hualin Xuesheng, and later at the Imperial Nanjing University to perform research. In 461 in Nanxu, he was engaged in work at the office of the local governor, Zu Chongzhi, along with his son Zu Gengzhi wrote a mathematical text entitled Zhui Shu. It is said that the treatise contains formulas for the volume of the sphere, cubic equations and this book didnt survive to the present day, it has been lost since the Song Dynasty. His mathematical achievements included, the Daming calendar introduced by him in 465, distinguishing the sidereal year and the tropical year, and he measured 45 years and 11 months per degree between those two, and today we know the difference is 70.7 years per degree. Calculating one year as 365.24281481 days, which is close to 365.24219878 days as we know today. Calculating the Jupiter year as about 11.858 Earth years, deriving two approximations of pi, which held as the most accurate approximation for π for over nine hundred years. His best approximation was between 3.1415926 and 3.1415927, with 355/113 and 22/7 being the other notable approximations and he obtained the result by approximating a circle with a 24,576 sided polygon. Hence Mikami strongly urged that the fraction 355/113 be named after Zu Chongzhi as Zus fraction, in Chinese literature, this fraction is known as Zus ratio. Zus ratio is a best rational approximation to π, and is the closest rational approximation to π from all fractions with denominator less than 16600, finding the volume of a sphere as πD3/6 where D is diameter. Zu was an astronomer who calculated the values of time with unprecedented precision. His methods of interpolating and the use of integration is far ahead of his time, even the astronomer Yi Xings isnt comparable to his value. The Sung dynasty calendar was backwards to the Northern barbarians because they were implementing their daily lives with the Da Ming Li. It is said that his methods of calculation were so advanced, the majority of Zus great mathematical works are recorded in his lost text the Zhui Shu. Most schools argue about his complexity since traditionally the Chinese had developed mathematics as algebraic, logically, scholars assume that the Zhui Shu yields methods of cubic equations
