1.
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

2.
Positional notation
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Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the symbol for the different orders of magnitude. This greatly simplified arithmetic, leading to the spread of the notation across the world. With the use of a point, the notation can be extended to include fractions. The Hindu–Arabic numeral system, base-10, is the most commonly used system in the world today for most calculations, today, the base-10 system, which is likely motivated by counting with the ten fingers, is ubiquitous. Other bases have been used in the past however, and some continue to be used today, for example, the Babylonian numeral system, credited as the first positional numeral system, was base-60, but it lacked a real 0 value. Zero was indicated by a space between sexagesimal numerals, by 300 BC, a punctuation symbol was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish, the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges, the Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number, thus numbers like 2 and 120,3 and 180,4 and 240, looked the same because the larger numbers lacked a final sexagesimal placeholder. Counting rods and most abacuses have been used to represent numbers in a numeral system. This approach required no memorization of tables and could produce practical results quickly, for four centuries there was strong disagreement between those who believed in adopting the positional system in writing numbers and those who wanted to stay with the additive-system-plus-abacus. Although electronic calculators have largely replaced the abacus, the continues to be used in Japan. After the French Revolution, the new French government promoted the extension of the decimal system, some of those pro-decimal efforts—such as decimal time and the decimal calendar—were unsuccessful. Other French pro-decimal efforts—currency decimalisation and the metrication of weights and measures—spread widely out of France to almost the whole world. According to Joseph Needham and Lam Lay Yong, decimal fractions were first developed and used by the Chinese in the 1st century BC, the written Chinese decimal fractions were non-positional. However, counting rod fractions were positional, the Jewish mathematician Immanuel Bonfils used decimal fractions around 1350, anticipating Simon Stevin, but did not develop any notation to represent them. A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century. A key argument against the system was its susceptibility to easy fraud by simply putting a number at the beginning or end of a quantity, thereby changing 100 into 5100

3.
Decimal
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This article aims to be an accessible introduction. For the mathematical definition, see Decimal representation, the decimal numeral system has ten as its base, which, in decimal, is written 10, as is the base in every positional numeral system. It is the base most widely used by modern civilizations. Decimal fractions have terminating decimal representations and other fractions have repeating decimal representations, Decimal notation is the writing of numbers in a base-ten numeral system. Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, Roman numerals have symbols for the decimal powers and secondary symbols for half these values. Brahmi numerals have symbols for the nine numbers 1–9, the nine decades 10–90, plus a symbol for 100, Chinese numerals have symbols for 1–9, and additional symbols for powers of ten, which in modern usage reach 1072. Positional decimal systems include a zero and use symbols for the ten values to represent any number, positional notation uses positions for each power of ten, units, tens, hundreds, thousands, etc. The position of each digit within a number denotes the multiplier multiplied with that position has a value ten times that of the position to its right. There were at least two independent sources of positional decimal systems in ancient civilization, the Chinese counting rod system. Ten is the number which is the count of fingers and thumbs on both hands, the English word digit as well as its translation in many languages is also the anatomical term for fingers and toes. In English, decimal means tenth, decimate means reduce by a tenth, however, the symbols used in different areas are not identical, for instance, Western Arabic numerals differ from the forms used by other Arab cultures. A decimal fraction is a fraction the denominator of which is a power of ten. g, Decimal fractions 8/10, 1489/100, 24/100000, and 58900/10000 are expressed in decimal notation as 0.8,14.89,0.00024,5.8900 respectively. In English-speaking, some Latin American and many Asian countries, a period or raised period is used as the separator, in many other countries, particularly in Europe. The integer part, or integral part of a number is the part to the left of the decimal separator. The part from the separator to the right is the fractional part. It is usual for a number that consists only of a fractional part to have a leading zero in its notation. Any rational number with a denominator whose only prime factors are 2 and/or 5 may be expressed as a decimal fraction and has a finite decimal expansion. 1/2 =0.5 1/20 =0.05 1/5 =0.2 1/50 =0.02 1/4 =0.25 1/40 =0.025 1/25 =0.04 1/8 =0.125 1/125 =0.008 1/10 =0

4.
Portmanteau
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In linguistics, a portmanteau is defined as a single morph that represents two or more morphemes. A portmanteau also differs from a compound, which not involve the truncation of parts of the stems of the blended words. For instance, starfish is a compound, not a portmanteau, of star and fish, whereas a hypothetical portmanteau of star and fish might be stish. Humpty Dumpty explains the practice of combining words in various ways by telling Alice, for instance, take the two words fuming and furious. Make up your mind that you will say both words, but leave it unsettled which you will say first … if you have the rarest of gifts, in then-contemporary English, a portmanteau was a suitcase that opened into two equal sections. The etymology of the word is the French porte-manteau, from porter, to carry, in modern French, a porte-manteau is a clothes valet, a coat-tree or similar article of furniture for hanging up jackets, hats, umbrellas and the like. It has also used especially in Europe as a formal description for hat racks from the French words porter. An occasional synonym for portmanteau word is frankenword, an autological word exemplifying the phenomenon it describes, blending Frankenstein, many neologisms are examples of blends, but many blends have become part of the lexicon. In Punch in 1896, the word brunch was introduced as a portmanteau word, in 1964, the newly independent African republic of Tanganyika and Zanzibar chose the portmanteau word Tanzania as its name. Similarly Eurasia is a portmanteau of Europe and Asia, a scientific example is a liger, which is a cross between a male lion and a female tiger. Jeoportmanteau. is a category on the American television quiz show Jeopardy. The categorys name is itself a portmanteau of the words Jeopardy, responses in the category are portmanteaus constructed by fitting two words together. The term gerrymander has itself contributed to portmanteau terms bjelkemander and playmander, oxbridge is a common portmanteau for the UKs two oldest universities, those of Oxford and Cambridge. Many portmanteau words receive some use but do not appear in all dictionaries, for example, a spork is an eating utensil that is a combination of a spoon and a fork, and a skort is an item of clothing that is part skirt, part shorts. On the other hand, turducken, a made by inserting a chicken into a duck. Similarly, the word refudiate was first used by Sarah Palin when she misspoke, though initially a gaffe, the word was recognized as the New Oxford American Dictionarys Word of the Year in 2010. The business lexicon is replete with newly coined portmanteau words like permalance, advertainment, advertorial, infotainment, a company name may also be portmanteau as well as a product name. By contrast, the public, including the media, use portmanteaux to refer to their favorite pairings as a way to. giv people an essence of who they are within the same name and this is particularly seen in cases of fictional and real-life supercouples

5.
666 (number)
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666 is the natural number following 665 and preceding 667. Six hundred and sixty-six is called the number of the Beast in chapter 13 of the Book of Revelation, of the New Testament,666 is the sum of the first 36 natural numbers, and thus it is a triangular number. Notice that 36 =15 +21,15 and 21 are also triangular numbers, in base 10,666 is a repdigit and a Smith number. A prime reciprocal magic square based on 1/149 in base 10 has a total of 666. The prime factorization of 666 is 2 •32 •37, some manuscripts of the original Greek use the symbols χξϛ chi xi stigma, while other manuscripts spell out the number in words. In modern popular culture,666 has become one of the most widely recognized symbols for the Antichrist or, alternatively, the number 666 is purportedly used to invoke Satan. Earnest references to the number occur both among apocalypticist Christian groups and in explicitly anti-Christian subcultures, references in contemporary Western art or literature are, more likely than not, intentional references to the Beast symbolism. Such popular references are therefore too numerous to list and it is common to see the symbolic role of the integer 666 transferred to the digit sequence 6-6-6. Some people take the Satanic associations of 666 so seriously that they actively avoid things related to 666 or the digits 6-6-6, in some early biblical manuscripts, including Papyrus 115, the number is cited as 616. In the Bible,666 is the number of talents of gold Solomon collected each year, in the Bible,666 is the number of Adonikams descendants who return to Jerusalem and Judah from the Babylonian exile. In the Bible, there may be a latent reference to 666 in the name of the great sixth-century BC king of Babylon, commonly spelled Nebuchadnezzar, transliterating from the Book of Daniel, the name is Nebuchadrezzar or Nebuchadrezzur in the Book of Jeremiah. The number of name can be calculated, since Hebrew letters double as numbers. Nebuchadrezzar is 663, and Nebuchadrezzur,669, midway between the two variants is 666. If the mysteries of Jeremiah are to be related to those of Revelation, Nebuchadrezzar, using gematria, Neron Caesar transliterated from Greek into Hebrew produces the number 666. The Latin spelling of Nero Caesar transliterated into Hebrew produces the number 616, thus, in the Bible,666 may have been a coded reference to Nero the Roman Emperor from 55 to 68 AD. Is the magic sum, or sum of the constants of a six by six magic square. Is the sum of all the numbers on a roulette wheel, was a winning lottery number in the 1980 Pennsylvania Lottery scandal, in which equipment was tampered to favor a 4 or 6 as each of the three individual random digits. Was the original name of the Macintosh SevenDust computer virus that was discovered in 1998, the number is a frequent visual element of Aryan Brotherhood tattoos

6.
Six nines in pi
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A sequence of six 9s occurs in the decimal representation of π, starting at the 762nd decimal place. The earliest known mention of this occurs in Douglas Hofstadters 1985 book Metamagical Themas, where Hofstadter states I myself once learned 380 digits of π. This sequence of six nines is sometimes called the Feynman point, after physicist Richard Feynman, who has also been claimed to have stated this same idea in a lecture. It is not clear when, or even if, Feynman made such a statement, however, it is not mentioned in published biographies or in his autobiographies, π is conjectured to be, but not known to be, a normal number. For a randomly chosen normal number, the probability of a sequence of six digits occurring this early in the decimal representation is usually only about 0. 08%. However, if the sequence can overlap itself then the probability is less, the probability of six 9s in a row this early is about 10% less, or 0. 0686%. But the probability of a repetition of any digit six times starting in the first 762 digits is ten times greater, the early string of six 9s is also the first occurrence of four and five consecutive identical digits. The next appearance of four consecutive identical digits is of the digit 7 at position 1,589. Other patterns are possible, a repetition of a three times in the first three digits, or four times starting in the first ten digits, or five times in the first 100 digits. Each of these has about a 1% chance, so looking at repeats up to length 12, there is about a 10% chance of finding something as surprising as the six nines. From this point of view, the fact that we really do find a repeat of several digits after 762 digits is not really very surprising, the next sequence of six consecutive identical digits is again composed of 9s, starting at position 193,034. The next distinct sequence of six consecutive identical digits starts with the digit 8 at position 222,299, while strings of nine 9s next occur at position 590,331,982 and 640,787,382. The positions of the first occurrences of 9, alone and in strings of 2,3,4,5,6,7,8, and 9 consecutive 9s, are 5,44,762,762,762,762,1,722,776,36,356,642, and 564,665,206, respectively. The first 1001 digits of π, showing consecutive runs of three or more digits, including the consecutive six 9s underlined and coloured red, are as follows,0.999,9 Mathematical coincidence Repdigit Ramanujans constant Feynman Point Mathworld Article — From the Mathworld project