1.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
2.
Clifford A. Pickover
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Other Fellows have included Carl Sagan and Isaac Asimov. He has been awarded over 300 United States patents, and his The Math Book was winner of the 2011 Neumann Prize. He received his Ph. D. in 1982 from Yale Universitys Department of Molecular Biophysics and Biochemistry, Pickover graduated first in his class from Franklin and Marshall College, after completing the four-year undergraduate program in three years. He joined IBM at the Thomas J. Watson Research Center in 1982, as a member of the speech synthesis group, for much of his career, Pickover has published technical articles in the areas of scientific visualization, computer art, and recreational mathematics. Pickover is still employed at the IBM Thomas J. Watson Research Center and he is currently an associate editor for the scientific journal Computers and Graphics and is an editorial board member for Odyssey and Leonardo. He is also the Brain-Strain columnist for Odyssey magazine, and, for many years, Pickover has received more than 100 IBM invention achievement awards, three research division awards, and four external honor awards. Pickovers primary interest is in finding new ways to expand creativity by melding art, science, mathematics, Pickover is an inventor with over 300 patents, the author of puzzle calendars, and puzzle contributor to magazines geared to children and adults. His Neoreality and Heaven Virus science-fiction series explores the fabric of reality and he also has published articles in the areas of skepticism, psychology, and technical speculation. Additional visualization work includes topics that involve breathing motions of proteins, snow-flake like patterns for speech sounds, cartoon-face representations of data, Pickover has also written extensively on the reported experiences of people on the psychotropic compound DMT. Such apparent entities as Machine Elves are described as well as Insects From A Parallel Universe, on November 4,2006, he began Wikidumper. org, a popular blog featuring articles being considered for deletion by English Wikipedia. Pickover stalks are certain kinds of details that are found in the Mandelbrot set in the study of fractal geometry. In some renditions of this behavior, the closer that the point approaches, the logarithm of the distance is taken to accentuate the details. This work grew from his work with Julia sets and Pickover biomorphs. In Frontiers of Scientific Visualization Pickover explored the art and science of making the unseen workings of nature visible, in Visualizing Biological Information Pickover considered biological data of all kinds, which is proliferating at an incredible rate. According to Pickover, if humans attempt to read data in the form of numbers and letters. If the information is rendered graphically, however, human analysts can assimilate it, the emphasis of this work is on the novel graphical and musical representation of information containing sequences, such as DNA and amino acid sequences, to help us find hidden pattern and meaning. X and y are called the fangs, as an example,1260 is a vampire number because it can be expressed as 21 ×60 =1260. Note that the digits of the factors 21 and 60 can be found, in some scrambled order, similarly,136,948 is a vampire because 136,948 =146 ×938
3.
Recursion
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Recursion occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic, the most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. While this apparently defines a number of instances, it is often done in such a way that no loop or infinite chain of references can occur. The ancestors of ones ancestors are also ones ancestors, the Fibonacci sequence is a classic example of recursion, Fib =0 as base case 1, Fib =1 as base case 2, For all integers n >1, Fib, = Fib + Fib. Many mathematical axioms are based upon recursive rules, for example, the formal definition of the natural numbers by the Peano axioms can be described as,0 is a natural number, and each natural number has a successor, which is also a natural number. By this base case and recursive rule, one can generate the set of all natural numbers, recursively defined mathematical objects include functions, sets, and especially fractals. There are various more tongue-in-cheek definitions of recursion, see recursive humor, Recursion is the process a procedure goes through when one of the steps of the procedure involves invoking the procedure itself. A procedure that goes through recursion is said to be recursive, to understand recursion, one must recognize the distinction between a procedure and the running of a procedure. A procedure is a set of steps based on a set of rules, the running of a procedure involves actually following the rules and performing the steps. An analogy, a procedure is like a recipe, running a procedure is like actually preparing the meal. Recursion is related to, but not the same as, a reference within the specification of a procedure to the execution of some other procedure. For instance, a recipe might refer to cooking vegetables, which is another procedure that in turn requires heating water, for this reason recursive definitions are very rare in everyday situations. An example could be the procedure to find a way through a maze. Proceed forward until reaching either an exit or a branching point, If the point reached is an exit, terminate. Otherwise try each branch in turn, using the procedure recursively, if every trial fails by reaching only dead ends, return on the path led to this branching point. Whether this actually defines a terminating procedure depends on the nature of the maze, in any case, executing the procedure requires carefully recording all currently explored branching points, and which of their branches have already been exhaustively tried. This can be understood in terms of a definition of a syntactic category. A sentence can have a structure in which what follows the verb is another sentence, Dorothy thinks witches are dangerous, so a sentence can be defined recursively as something with a structure that includes a noun phrase, a verb, and optionally another sentence
4.
Fibonacci number
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The Fibonacci sequence is named after Italian mathematician Leonardo of Pisa, known as Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics, the sequence described in Liber Abaci began with F1 =1. Fibonacci numbers are related to Lucas numbers L n in that they form a complementary pair of Lucas sequences U n = F n and V n = L n. They are intimately connected with the ratio, for example. Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is a journal dedicated to their study. The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody, in the Sanskrit tradition of prosody, there was interest in enumerating all patterns of long syllables that are 2 units of duration, and short syllables that are 1 unit of duration. Counting the different patterns of L and S of a given duration results in the Fibonacci numbers, susantha Goonatilake writes that the development of the Fibonacci sequence is attributed in part to Pingala, later being associated with Virahanka, Gopāla, and Hemachandra. He dates Pingala before 450 BC, however, the clearest exposition of the sequence arises in the work of Virahanka, whose own work is lost, but is available in a quotation by Gopala, Variations of two earlier meters. For example, for four, variations of meters of two three being mixed, five happens, in this way, the process should be followed in all mātrā-vṛttas. The sequence is also discussed by Gopala and by the Jain scholar Hemachandra, outside India, the Fibonacci sequence first appears in the book Liber Abaci by Fibonacci. The puzzle that Fibonacci posed was, how many pairs will there be in one year, at the end of the first month, they mate, but there is still only 1 pair. At the end of the month the female produces a new pair. At the end of the month, the original female produces a second pair. At the end of the month, the original female has produced yet another new pair. At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs plus the number of pairs alive last month and this is the nth Fibonacci number. The name Fibonacci sequence was first used by the 19th-century number theorist Édouard Lucas, the most common such problem is that of counting the number of compositions of 1s and 2s that sum to a given total n, there are Fn+1 ways to do this. For example, if n =5, then Fn+1 = F6 =8 counts the eight compositions, 1+1+1+1+1 = 1+1+1+2 = 1+1+2+1 = 1+2+1+1 = 2+1+1+1 = 2+2+1 = 2+1+2 = 1+2+2, all of which sum to 5. The Fibonacci numbers can be found in different ways among the set of strings, or equivalently
5.
Usenet
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Usenet is a worldwide distributed discussion system available on computers. It was developed from the general-purpose UUCP dial-up network architecture, tom Truscott and Jim Ellis conceived the idea in 1979, and it was established in 1980. Users read and post messages to one or more categories, known as newsgroups, Usenet resembles a bulletin board system in many respects and is the precursor to Internet forums that are widely used today. Discussions are threaded, as with web forums and BBSs, though posts are stored on the server sequentially, the name comes from the term users network. One notable difference between a BBS or web forum and Usenet is the absence of a server and dedicated administrator. Usenet is distributed among a large, constantly changing conglomeration of servers that store, individual users may read messages from and post messages to a local server operated by a commercial usenet provider, their Internet service provider, university, employer, or their own server. Usenet has significant cultural importance in the world, having given rise to, or popularized, many widely recognized concepts and terms such as FAQ, flame. The name Usenet emphasized its creators hope that the USENIX organization would take a role in its operation. The articles that users post to Usenet are organized into topical categories called newsgroups, for instance, sci. math and sci. physics are within the sci. * hierarchy, for science. Or, talk. origins and talk. atheism are in the talk. * hierarchy, when a user subscribes to a newsgroup, the news client software keeps track of which articles that user has read. In most newsgroups, the majority of the articles are responses to some other article, the set of articles that can be traced to one single non-reply article is called a thread. Most modern newsreaders display the articles arranged into threads and subthreads, when a user posts an article, it is initially only available on that users news server. Each news server talks to one or more servers and exchanges articles with them. In this fashion, the article is copied from server to server, the later peer-to-peer networks operate on a similar principle, but for Usenet it is normally the sender, rather than the receiver, who initiates transfers. Usenet was designed under conditions when networks were much slower and not always available, many sites on the original Usenet network would connect only once or twice a day to batch-transfer messages in and out. This is largely because the POTS network was used for transfers. The format and transmission of Usenet articles is similar to that of Internet e-mail messages, today, Usenet has diminished in importance with respect to Internet forums, blogs and mailing lists. The groups in alt. binaries are still used for data transfer
6.
Padovan sequence
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The Padovan sequence is the sequence of integers P defined by the initial values P = P = P =1, and the recurrence relation P = P + P. The first few values of P are 1,1,1,2,2,3,4,5,7,9,12,16,21,28,37,49,65,86,114,151,200,265. The Padovan sequence is named after Richard Padovan who attributed its discovery to Dutch architect Hans van der Laan in his 1994 essay Dom, Hans van der Laan, Modern Primitive. The sequence was described by Ian Stewart in his Scientific American column Mathematical Recreations in June 1996 and he also writes about it in one of his books, Math Hysteria, Fun Games With Mathematics. The above definition is the one given by Ian Stewart and by MathWorld, other sources may start the sequence at a different place, in which case some of the identities in this article must be adjusted with appropriate offsets. This is a property of recurrence relations, the Perrin sequence can be obtained from the Padovan sequence by the following formula, P e r r i n = P + P. e. The Padovan sequence also satisfies the identity P2 − P P = P. The Padovan sequence is related to sums of binomial coefficients by the following identity, P = ∑2 m + n = k = ∑ m = ⌈ k /3 ⌉ ⌊ k /2 ⌋. For example, for k =12, the values for the pair with 2m + n =12 which give non-zero binomial coefficients are, and, and, + + =1 +10 +1 =12 = P. The Padovan sequence numbers can be written in terms of powers of the roots of the equation x 3 − x −1 =0 and this equation has 3 roots, one real root p and two complex conjugate roots q and r. Given these three roots, the Padovan sequence can be expressed by a formula involving p, q and r, P = a p n + b q n + c r n where a, b and c are constants. Since the magnitudes of the complex roots q and r are both less than 1, the powers of these roots approach 0 for large n, and P − a p n tends to zero. For all n ≥0, P is the integer closest to p n −1 s, the ratio of successive terms in the Padovan sequence approaches p, which has a value of approximately 1.324718. This constant bears the same relationship to the Padovan sequence and the Perrin sequence as the ratio does to the Fibonacci sequence. P is the number of ways of writing n +2 as a sum in which each term is either 2 or 3. This can be used to prove identities involving products of the Padovan sequence with geometric terms, such as, ∑ n =0 ∞ P α n = α2 α3 − α −1. A Padovan prime is P that is prime, the first few Padovan primes are 2,3,5,7,37,151,3329,23833. Also, if you count the number of As, Bs and Cs in each string, then for the nth string, you have P As, P Bs, the count of BB pairs, AA pairs and CC pairs are also Padovan numbers
7.
Natural number
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In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers, the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1,10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory
8.
Brady Haran
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Haran is also the co-host of the Hello Internet podcast along with fellow Youtuber CGP Grey. Brady Haran studied journalism for a year before being hired by The Adelaide Advertiser, in 2002, he moved from Australia to Nottingham, United Kingdom. In Nottingham, he worked for the BBC, began to work with film, in 2007, Haran worked as a filmmaker-in-residence for Nottingham Science City, as part of an agreement between the BBC and The University of Nottingham. Haran then left the BBC to work full-time making YouTube videos, following Test Tube, Haran decided to create new YouTube channels. In his first 5 years as an independent filmmaker he made over 1500 videos, in 2012, he was the producer, editor, and interviewer behind 12 YouTube channels such as The Periodic Table of Videos, Sixty Symbols and Numberphile. Working with Poliakoff, Harans videos explaining chemistry and science for non-technical persons received positive recognition, together, they have made over 500 short videos that cover the elements and other chemistry-related topics. Their YouTube channel has had more than 120 million views and their Gold Bullion Vault, shot in the vaults of The Bank of England, was released 7 December 2012, and received more than two million hits in the next two months. Also, Haran and Poliakoff authored an article in the Nature Chemistry journal, Haran frequently collaborates with professionals and experts, who often appear in his videos to discuss subjects relevant to their work. Most notably his series Periodic Videos features chemist Sir Martyn Poliakoff, séquin, scientists Brian Butterworth, Ed Copeland, Laurence Eaves, and Clifford Stoll, and scientific writers and popularizers Alex Bellos, Steve Mould, Matt Parker, Tom Scott, and Simon Singh. In January 2014, Haran launched the podcast Hello Internet along with co-host CGP Grey, the podcast peaked as the No.1 iTunes podcast in United Kingdom, United States, Germany, Canada, and Australia. It was selected as one of Apples best new podcasts of 2014, Grey reported a podcast listenership of approximately a quarter million downloads per episode as of September 2015. The podcast features discussions pertaining to their lives as professional content creators for YouTube, as well as their interests, typical topics include new gadgets, technology etiquette, workplace efficiency, wristwatches, plane accidents, vexillology, and the differences between Harans and Greys personal mannerisms. As a result of their conversations, Haran has been credited for coining the term freebooting to refer to the unauthorized rehosting of online media, the podcast has an official flag called Nail & Gear which was chosen by the listeners. Test tube, behind the scenes in the world of science, teaching chem eng – Martyn Poliakoff and Brady Haran on Nottingham Unis periodic table for the YouTube generation. How to measure the impact of chemistry on the small screen
9.
Exponentiation
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Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent n. The exponent is usually shown as a superscript to the right of the base, Some common exponents have their own names, the exponent 2 is called the square of b or b squared, the exponent 3 is called the cube of b or b cubed. The exponent −1 of b, or 1 / b, is called the reciprocal of b, when n is a positive integer and b is not zero, b−n is naturally defined as 1/bn, preserving the property bn × bm = bn + m. The definition of exponentiation can be extended to any real or complex exponent. Exponentiation by integer exponents can also be defined for a variety of algebraic structures. The term power was used by the Greek mathematician Euclid for the square of a line, archimedes discovered and proved the law of exponents, 10a 10b = 10a+b, necessary to manipulate powers of 10. In the late 16th century, Jost Bürgi used Roman numerals for exponents, early in the 17th century, the first form of our modern exponential notation was introduced by Rene Descartes in his text titled La Géométrie, there, the notation is introduced in Book I. Nicolas Chuquet used a form of notation in the 15th century. The word exponent was coined in 1544 by Michael Stifel, samuel Jeake introduced the term indices in 1696. In the 16th century Robert Recorde used the square, cube, zenzizenzic, sursolid, zenzicube, second sursolid. Biquadrate has been used to refer to the power as well. Some mathematicians used exponents only for greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials, for example, as ax + bxx + cx3 + d, another historical synonym, involution, is now rare and should not be confused with its more common meaning. In 1748 Leonhard Euler wrote consider exponentials or powers in which the exponent itself is a variable and it is clear that quantities of this kind are not algebraic functions, since in those the exponents must be constant. With this introduction of transcendental functions, Euler laid the foundation for the introduction of natural logarithm as the inverse function for y = ex. The expression b2 = b ⋅ b is called the square of b because the area of a square with side-length b is b2, the expression b3 = b ⋅ b ⋅ b is called the cube of b because the volume of a cube with side-length b is b3. The exponent indicates how many copies of the base are multiplied together, for example,35 =3 ⋅3 ⋅3 ⋅3 ⋅3 =243. The base 3 appears 5 times in the multiplication, because the exponent is 5
10.
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