1.
Srinivasa Ramanujan
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Srinivasa Iyengar Ramanujan FRS was an Indian mathematician and autodidact who lived during the British Raj. Though he had almost no training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series. Ramanujan initially developed his own research in isolation, it was quickly recognized by Indian mathematicians. When his skills became obvious and known to the mathematical community, centred in Europe at the time. The Cambridge professor realized that Srinivasa Ramanujan had produced new theorems in addition to rediscovering previously known ones, during his short life, Ramanujan independently compiled nearly 3,900 results. Nearly all his claims have now been proven correct and his original and highly unconventional results, such as the Ramanujan prime and the Ramanujan theta function, have inspired a vast amount of further research. The Ramanujan Journal, a scientific journal, was established to publish work in all areas of mathematics influenced by Ramanujan. Deeply religious, Ramanujan credited his substantial mathematical capacities to divinity, An equation for me has no meaning, he once said, the name Ramanujan means younger brother of the god Rama. Iyengar is a caste of Hindu Brahmins of Tamil origin whose members follow the Visishtadvaita philosophy propounded by Ramanuja, Ramanujan was born on 22 December 1887 into a Tamil Brahmin Iyengar family in Erode, Madras Presidency, at the residence of his maternal grandparents. His father, K. Srinivasa Iyengar, worked as a clerk in a sari shop and his mother, Komalatammal, was a housewife and also sang at a local temple. They lived in a traditional home on Sarangapani Sannidhi Street in the town of Kumbakonam. The family home is now a museum, when Ramanujan was a year and a half old, his mother gave birth to a son, Sadagopan, who died less than three months later. In December 1889, Ramanujan contracted smallpox, but unlike the thousands in the Thanjavur district who died of the disease that year and he moved with his mother to her parents house in Kanchipuram, near Madras. His mother gave birth to two children, in 1891 and 1894, but both died in infancy. On 1 October 1892, Ramanujan was enrolled at the local school, after his maternal grandfather lost his job as a court official in Kanchipuram, Ramanujan and his mother moved back to Kumbakonam and he was enrolled in the Kangayan Primary School. When his paternal grandfather died, he was sent back to his maternal grandparents and he did not like school in Madras, and tried to avoid attending. His family enlisted a local constable to make sure the boy attended school, within six months, Ramanujan was back in Kumbakonam. Since Ramanujans father was at work most of the day, his mother took care of the boy as a child and he had a close relationship with her
2.
Plato
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Plato was a philosopher in Classical Greece and the founder of the Academy in Athens, the first institution of higher learning in the Western world. He is widely considered the most pivotal figure in the development of philosophy, unlike nearly all of his philosophical contemporaries, Platos entire work is believed to have survived intact for over 2,400 years. Along with his teacher, Socrates, and his most famous student, Aristotle, Plato laid the foundations of Western philosophy. Alfred North Whitehead once noted, the safest general characterization of the European philosophical tradition is that it consists of a series of footnotes to Plato. In addition to being a figure for Western science, philosophy. Friedrich Nietzsche, amongst other scholars, called Christianity, Platonism for the people, Plato was the innovator of the written dialogue and dialectic forms in philosophy, which originate with him. He was not the first thinker or writer to whom the word “philosopher” should be applied, few other authors in the history of Western philosophy approximate him in depth and range, perhaps only Aristotle, Aquinas and Kant would be generally agreed to be of the same rank. Due to a lack of surviving accounts, little is known about Platos early life, the philosopher came from one of the wealthiest and most politically active families in Athens. Ancient sources describe him as a bright though modest boy who excelled in his studies, the exact time and place of Platos birth are unknown, but it is certain that he belonged to an aristocratic and influential family. Based on ancient sources, most modern scholars believe that he was born in Athens or Aegina between 429 and 423 BCE. According to a tradition, reported by Diogenes Laertius, Ariston traced his descent from the king of Athens, Codrus. Platos mother was Perictione, whose family boasted of a relationship with the famous Athenian lawmaker, besides Plato himself, Ariston and Perictione had three other children, these were two sons, Adeimantus and Glaucon, and a daughter Potone, the mother of Speusippus. The brothers Adeimantus and Glaucon are mentioned in the Republic as sons of Ariston, and presumably brothers of Plato, but in a scenario in the Memorabilia, Xenophon confused the issue by presenting a Glaucon much younger than Plato. Then, at twenty-eight, Hermodorus says, went to Euclides in Megara, as Debra Nails argues, The text itself gives no reason to infer that Plato left immediately for Megara and implies the very opposite. Thus, Nails dates Platos birth to 424/423, another legend related that, when Plato was an infant, bees settled on his lips while he was sleeping, an augury of the sweetness of style in which he would discourse about philosophy. Ariston appears to have died in Platos childhood, although the dating of his death is difficult. Perictione then married Pyrilampes, her mothers brother, who had served many times as an ambassador to the Persian court and was a friend of Pericles, Pyrilampes had a son from a previous marriage, Demus, who was famous for his beauty. Perictione gave birth to Pyrilampes second son, Antiphon, the half-brother of Plato and these and other references suggest a considerable amount of family pride and enable us to reconstruct Platos family tree
3.
60 (number)
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60 is the natural number following 59 and preceding 61. Being three times 20, it is called three score in older literature. It is a number, with divisors 1,2,3,4,5,6,10,12,15,20,30. Because it is the sum of its divisors, it is a unitary perfect number. Being ten times a number, it is a semiperfect number. It is the smallest number divisible by the numbers 1 to 6 and it is the smallest number with exactly 12 divisors. It is the sum of a pair of twin primes and the sum of four consecutive primes and it is adjacent to two primes. It is the smallest number that is the sum of two odd primes in six ways, the smallest non-solvable group has order 60. There are four Archimedean solids with 60 vertices, the icosahedron, the rhombicosidodecahedron, the snub dodecahedron. The skeletons of these polyhedra form 60-node vertex-transitive graphs, there are also two Archimedean solids with 60 edges, the snub cube and the icosidodecahedron. The skeleton of the forms a 60-edge symmetric graph. There are 60 one-sided hexominoes, the polyominoes made from six squares, in geometry, it is the number of seconds in a minute, and the number of minutes in a degree. In normal space, the three angles of an equilateral triangle each measure 60 degrees, adding up to 180 degrees. Because it is divisible by the sum of its digits in base 10, a number system with base 60 is called sexagesimal. It is the smallest positive integer that is written only the smallest. The first fullerene to be discovered was buckminsterfullerene C60, an allotrope of carbon with 60 atoms in each molecule and this ball is known as a buckyball, and looks like a soccer ball. The atomic number of neodymium is 60, and cobalt-60 is an isotope of cobalt. The electrical utility frequency in western Japan, South Korea, Taiwan, the Philippines, Saudi Arabia, the United States, and several other countries in the Americas is 60 Hz
4.
Divisor
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In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some other integer to produce n. In this case one says also that n is a multiple of m, an integer n is divisible by another integer m if m is a divisor of n, this implies dividing n by m leaves no remainder. Under this definition, the statement m ∣0 holds for every m, as before, but with the additional constraint k ≠0. Under this definition, the statement m ∣0 does not hold for m ≠0, in the remainder of this article, which definition is applied is indicated where this is significant. Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4, they are 1,2,4, −1, −2, and −4,1 and −1 divide every integer. Every integer is a divisor of itself, every integer is a divisor of 0. Integers divisible by 2 are called even, and numbers not divisible by 2 are called odd,1, −1, n and −n are known as the trivial divisors of n. A divisor of n that is not a divisor is known as a non-trivial divisor. A non-zero integer with at least one divisor is known as a composite number, while the units −1 and 1. There are divisibility rules which allow one to recognize certain divisors of a number from the numbers digits, the generalization can be said to be the concept of divisibility in any integral domain. 7 is a divisor of 42 because 7 ×6 =42 and it can also be said that 42 is divisible by 7,42 is a multiple of 7,7 divides 42, or 7 is a factor of 42. The non-trivial divisors of 6 are 2, −2,3, the positive divisors of 42 are 1,2,3,6,7,14,21,42. 5 ∣0, because 5 ×0 =0, if a ∣ b and b ∣ a, then a = b or a = − b. If a ∣ b and a ∣ c, then a ∣ holds, however, if a ∣ b and c ∣ b, then ∣ b does not always hold. If a ∣ b c, and gcd =1, then a ∣ c, if p is a prime number and p ∣ a b then p ∣ a or p ∣ b. A positive divisor of n which is different from n is called a proper divisor or a part of n. A number that does not evenly divide n but leaves a remainder is called an aliquant part of n, an integer n >1 whose only proper divisor is 1 is called a prime number
5.
Jean-Pierre Kahane
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Jean-Pierre Kahane is a French mathematician. Kahane attended the École normale supérieure and obtained the agrégation of mathematics in 1949 and he then worked for the CNRS from 1949 to 1954, first as an intern and then as a research assistant. He defended his PhD in 1954, his advisor was Szolem Mandelbrojt and he was assistant professor, then professor of mathematics in Montpellier from 1954 to 1961. Since then, he has been professor until his retirement in 1994 and he was a Plenary Speaker at the ICM in 1962 in Stockholm and an Invited Speaker in 1986 in Berkeley, California. He was elected corresponding member of the French Academy of Sciences in 1982, in 2012 he became a fellow of the American Mathematical Society. Séries de Fourier absolument convergentes, Ergebnisse der Mathematik und ihrer Grenzgebiete,50, Springer Verlag 1970 with Raphaël Salem, Ensembles parfaits et séries trigonométriques, Hermann,1963,1994 Some random series of functions, Lexington, Massachusetts, D. C. Heath 1968, 2nd edition Cambridge University Press 1985 Séries de Fourier aléatoires, Presse de l’ Université de Montreal 1967 editor with A
6.
24 (number)
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24 is the natural number following 23 and preceding 25. The SI prefix for 1024 is yotta, and for 10−24 yocto and these numbers are the largest and smallest number to receive an SI prefix to date. In a 24-hour clock, the hour is in conventional language called twelve or twelve oclock. 24 is the factorial of 4 and a number, being the first number of the form 23q. It follows that 24 is the number of ways to order 4 distinct items and it is the smallest number with exactly eight divisors,1,2,3,4,6,8,12, and 24. It is a composite number, having more divisors than any smaller number. 24 is a number, since adding up all the proper divisors of 24 except 4 and 8 gives 24. Subtracting 1 from any of its divisors yields a number,24 is the largest number with this property. 24 has a sum of 36 and the aliquot sequence. It is therefore the lowest abundant number whose aliquot sum is itself abundant, the aliquot sum of only one number,529 =232, is 24. There are 10 solutions to the equation φ =24, namely 35,39,45,52,56,70,72,78,84 and 90 and this is more than any integer below 24, making 24 a highly totient number. 24 is the sum of the prime twins 11 and 13, the product of any four consecutive numbers is divisible by 24. This is because among any four consecutive numbers there must be two numbers, one of which is a multiple of four, and there must be a multiple of three. The tesseract has 24 two-dimensional faces,24 is the only nontrivial solution to the cannonball problem, that is,12 +22 +32 + … +242 is a perfect square. In 24 dimensions there are 24 even positive definite unimodular lattices, the Leech lattice is closely related to the equally nice length-24 binary Golay code and the Steiner system S and the Mathieu group M24. The modular discriminant Δ is proportional to the 24th power of the Dedekind eta function η, Δ = 12η24, the Barnes-Wall lattice contains 24 lattices. 24 is the number whose divisors — namely 1,2,3,4,6,8,12,24 — are exactly those numbers n for which every invertible element of the commutative ring Z/nZ is a square root of 1. It follows that the multiplicative group × = is isomorphic to the additive group 3 and this fact plays a role in monstrous moonshine
7.
Fundamental theorem of arithmetic
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For example,1200 =24 ×31 ×52 =3 ×2 ×2 ×2 ×2 ×5 ×5 =5 ×2 ×3 ×2 ×5 ×2 ×2 = etc. The requirement that the factors be prime is necessary, factorizations containing composite numbers may not be unique. This theorem is one of the reasons why 1 is not considered a prime number, if 1 were prime. Book VII, propositions 30,31 and 32, and Book IX, proposition 14 of Euclids Elements are essentially the statement, proposition 30 is referred to as Euclids lemma. And it is the key in the proof of the theorem of arithmetic. Proposition 31 is proved directly by infinite descent, proposition 32 is derived from proposition 31, and prove that the decomposition is possible. Book IX, proposition 14 is derived from Book VII, proposition 30, indeed, in this proposition the exponents are all equal to one, so nothing is said for the general case. Article 16 of Gauss Disquisitiones Arithmeticae is a modern statement. < pk are primes and the αi are positive integers and this representation is commonly extended to all positive integers, including one, by the convention that the empty product is equal to 1. This representation is called the representation of n, or the standard form of n. For example 999 = 33×37,1000 = 23×53,1001 = 7×11×13 Note that factors p0 =1 may be inserted without changing the value of n, allowing negative exponents provides a canonical form for positive rational numbers. However, as Integer factorization of large integers is much harder than computing their product, gcd or lcm, these formulas have, in practice, many arithmetical functions are defined using the canonical representation. In particular, the values of additive and multiplicative functions are determined by their values on the powers of prime numbers, the proof uses Euclids lemma, if a prime p divides the product of two natural numbers a and b, then p divides a or p divides b. We need to show that every integer greater than 1 is either prime or a product of primes, for the base case, note that 2 is prime. By induction, assume true for all numbers between 1 and n, if n is prime, there is nothing more to prove. Otherwise, there are integers a and b, where n = ab and 1 < a ≤ b < n, by the induction hypothesis, a = p1p2. pj and b = q1q2. qk are products of primes. But then n = ab = p1p2. pjq1q2. qk is a product of primes, assume that s >1 is the product of prime numbers in two different ways, s = p 1 p 2 ⋯ p m = q 1 q 2 ⋯ q n. We must show m = n and that the qj are a rearrangement of the pi, by Euclids lemma, p1 must divide one of the qj, relabeling the qj if necessary, say that p1 divides q1
8.
System of measurement
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A system of measurement is a collection of units of measurement and rules relating them to each other. Systems of measurement have historically been important, regulated and defined for the purposes of science and commerce, systems of measurement in modern use include the metric system, the imperial system, and United States customary units. The French Revolution gave rise to the system, and this has spread around the world. In most systems, length, mass, and time are base quantities, later science developments showed that either electric charge or electric current could be added to extend the set of base quantities by which many other metrological units could be easily defined. Other quantities, such as power and speed, are derived from the set, for example. Such arrangements were satisfactory in their own contexts, the preference for a more universal and consistent system only gradually spread with the growth of science. Changing a measurement system has substantial financial and cultural costs which must be offset against the advantages to be obtained using a more rational system. However pressure built up, including scientists and engineers for conversion to a more rational. The unifying characteristic is that there was some definition based on some standard, eventually cubits and strides gave way to customary units to met the needs of merchants and scientists. In the metric system and other recent systems, a basic unit is used for each base quantity. Often secondary units are derived from the units by multiplying by powers of ten. Thus the basic unit of length is the metre, a distance of 1.234 m is 1,234 millimetres. Metrication is complete or nearly complete in almost all countries, US customary units are heavily used in the United States and to some degree in Liberia. Traditional Burmese units of measurement are used in Burma, U. S. units are used in limited contexts in Canada due to the large volume of trade, there is also considerable use of Imperial weights and measures, despite de jure Canadian conversion to metric. In the United States, metric units are used almost universally in science, widely in the military, and partially in industry, but customary units predominate in household use. At retail stores, the liter is a used unit for volume, especially on bottles of beverages. Some other standard non-SI units are still in use, such as nautical miles and knots in aviation. Metric systems of units have evolved since the adoption of the first well-defined system in France in 1795, during this evolution the use of these systems has spread throughout the world, first to non-English-speaking countries, and then to English speaking countries
9.
On-Line Encyclopedia of Integer Sequences
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The On-Line Encyclopedia of Integer Sequences, also cited simply as Sloanes, is an online database of integer sequences. It was created and maintained by Neil Sloane while a researcher at AT&T Labs, Sloane continues to be involved in the OEIS in his role as President of the OEIS Foundation. OEIS records information on integer sequences of interest to professional mathematicians and amateurs, and is widely cited. As of 30 December 2016 it contains nearly 280,000 sequences, the database is searchable by keyword and by subsequence. Neil Sloane started collecting integer sequences as a student in 1965 to support his work in combinatorics. The database was at first stored on punched cards and he published selections from the database in book form twice, A Handbook of Integer Sequences, containing 2,372 sequences in lexicographic order and assigned numbers from 1 to 2372. The Encyclopedia of Integer Sequences with Simon Plouffe, containing 5,488 sequences and these books were well received and, especially after the second publication, mathematicians supplied Sloane with a steady flow of new sequences. The collection became unmanageable in book form, and when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service, as a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998. The database continues to grow at a rate of some 10,000 entries a year, Sloane has personally managed his sequences for almost 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the database. In 2004, Sloane celebrated the addition of the 100, 000th sequence to the database, A100000, in 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki at OEIS. org was created to simplify the collaboration of the OEIS editors and contributors, besides integer sequences, the OEIS also catalogs sequences of fractions, the digits of transcendental numbers, complex numbers and so on by transforming them into integer sequences. Sequences of rationals are represented by two sequences, the sequence of numerators and the sequence of denominators, important irrational numbers such as π =3.1415926535897. are catalogued under representative integer sequences such as decimal expansions, binary expansions, or continued fraction expansions. The OEIS was limited to plain ASCII text until 2011, yet it still uses a form of conventional mathematical notation. Greek letters are represented by their full names, e. g. mu for μ. Every sequence is identified by the letter A followed by six digits, sometimes referred to without the leading zeros, individual terms of sequences are separated by commas. Digit groups are not separated by commas, periods, or spaces, a represents the nth term of the sequence. Zero is often used to represent non-existent sequence elements, for example, A104157 enumerates the smallest prime of n² consecutive primes to form an n×n magic square of least magic constant, or 0 if no such magic square exists. The value of a is 2, a is 1480028129, but there is no such 2×2 magic square, so a is 0