1.
1 (number)
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1, is a number, a numeral, and the name of the glyph representing that number. It represents a single entity, the unit of counting or measurement, for example, a line segment of unit length is a line segment of length 1. It is also the first of the series of natural numbers. The word one can be used as a noun, an adjective and it comes from the English word an, which comes from the Proto-Germanic root *ainaz. The Proto-Germanic root *ainaz comes from the Proto-Indo-European root *oi-no-, compare the Proto-Germanic root *ainaz to Old Frisian an, Gothic ains, Danish een, Dutch een, German eins and Old Norse einn. Compare the Proto-Indo-European root *oi-no- to Greek oinos, Latin unus, Old Persian aivam, Old Church Slavonic -inu and ino-, Lithuanian vienas, Old Irish oin, One, sometimes referred to as unity, is the first non-zero natural number. It is thus the integer before two and after zero, and the first positive odd number, any number multiplied by one is that number, as one is the identity for multiplication. As a result,1 is its own factorial, its own square, its own cube, One is also the result of the empty product, as any number multiplied by one is itself. It is also the natural number that is neither composite nor prime with respect to division. The Gupta wrote it as a line, and the Nagari sometimes added a small circle on the left. The Nepali also rotated it to the right but kept the circle small and this eventually became the top serif in the modern numeral, but the occasional short horizontal line at the bottom probably originates from similarity with the Roman numeral I. Where the 1 is written with an upstroke, the number 7 has a horizontal stroke through the vertical line. While the shape of the 1 character has an ascender in most modern typefaces, in typefaces with text figures, many older typewriters do not have a separate symbol for 1 and use the lowercase letter l instead. It is possible to find cases when the uppercase J is used,1 cannot be used as the base of a positional numeral system, as the only digit that would be permitted in such a system would be 0. Since the base 1 exponential function always equals 1, its inverse does not exist, there are two ways to write the real number 1 as a recurring decimal, as 1.000. and as 0.999. There is only one way to represent the real number 1 as a Dedekind cut, in a multiplicative group or monoid, the identity element is sometimes denoted 1, but e is also traditional. However,1 is especially common for the identity of a ring. When such a ring has characteristic n not equal to 0,1 is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number, to name just a few

2.
0.999...
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In mathematics, the repeating decimal 0. 999… denotes a real number that can be shown to be the number one. In other words, the symbols 0. 999… and 1 represent the same number, more generally, every nonzero terminating decimal has an equal twin representation with infinitely many trailing 9s. The terminating decimal representation is preferred, contributing to the misconception that it is the only representation. The same phenomenon occurs in all other bases or in any representation of the real numbers. The equality of 0. 999… and 1 is closely related to the absence of nonzero infinitesimals in the number system. Some alternative number systems, such as the hyperreals, do contain nonzero infinitesimals, the equality 0. 999… =1 has long been accepted by mathematicians and is part of general mathematical education. Nonetheless, some find it sufficiently counterintuitive that they question or reject it. Such skepticism is common enough that the difficulty of convincing them of the validity of this identity has been the subject of studies in mathematics education. Algebraic proofs showing that 0. 999… represents the number 1 use concepts such as fractions, long division, however, these proofs are not rigorous as they do not include a careful analytic definition of 0. 999…. One reason that infinite decimals are an extension of finite decimals is to represent fractions. Using long division, a division of integers like 1⁄9 becomes a recurring decimal,0. 111…. This decimal yields a quick proof for 0. 999… =1, If 0. 999… is to be consistent, it must equal 9⁄9 =1. 0.333 … =390.888 … =890.999 … =99 =1 When a number in decimal notation is multiplied by 10, the digits do not change but each digit moves one place to the left. Thus 10 ×0. 999… equals 9. 999…, which is 9 greater than the original number, in introductory arithmetic, such proofs help explain why 0. 999… =1 but 0. 333… <0.34. In introductory algebra, the proofs help explain why the method of converting between fractions and repeating decimals works. Once a representation scheme is defined, it can be used to justify the rules of decimal arithmetic used in the above proofs. Moreover, one can demonstrate that the decimals 0. 999… and 1. 000… both represent the same real number, it is built into the definition. Since the question of 0. 999… does not affect the development of mathematics

3.
Hapax legomenon
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In corpus linguistics, a hapax legomenon is a word that occurs only once within a context, either in the written record of an entire language, in the works of an author, or in a single text. The term is sometimes used to describe a word that occurs in just one of an authors works. Hapax legomenon is a transliteration of Greek ἅπαξ λεγόμενον, meaning said once, the related terms dis legomenon, tris legomenon, and tetrakis legomenon respectively refer to double, triple, or quadruple occurrences, but are far less commonly used. Hapax legomena are quite common, as predicted by Zipfs law, for large corpora, about 40% to 60% of the words are hapax legomena, and another 10% to 15% are dis legomena. Thus, in the Brown Corpus of American English, about half of the 50,000 words are hapax legomena within that corpus, Hapax legomenon refers to a words appearance in a body of text, not to either its origin or its prevalence in speech. It thus differs from a word, which may never be recorded, may find currency and may be widely recorded, or may appear several times in the work which coins it. Hapax legomena in ancient texts are usually difficult to decipher, since it is easier to infer meaning from multiple contexts than from just one, for example, many of the remaining undeciphered Mayan glyphs are hapax legomena, and Biblical hapax legomena sometimes pose problems in translation. Hapax legomena also pose challenges in natural language processing, some scholars consider Hapax legomena useful in determining the authorship of written works. He argued that the number of hapax legomena in a putative authors corpus indicates his or her vocabulary and is characteristic of the author as an individual, harrisons theory has faded in significance due to a number of problems raised by other scholars. Workman found the numbers of hapax legomena in each Pauline Epistle. At first glance, the last three totals are not out of line with the others, although the Pastoral Epistles have more hapax legomena per page, Workman found the differences to be moderate in comparison to the variation among other Epistles. This was reinforced when Workman looked at several plays by Shakespeare, text topic, if the author writes on different subjects, of course many subject-specific words will occur only in limited contexts. Text audience, if the author is writing to a rather than a student, or their spouse rather than their employer. Time, over the course of years, both the language and a knowledge and use of language will change. There are also subjective questions over whether two forms amount to the word, dog vs. dogs, clue vs. clueless, sign vs. signature. The Jewish Encyclopedia points out that, although there are 1,500 hapaxes in the Hebrew Bible and it would not be especially difficult for a forger to construct a work with any percentage of hapax legomena desired. However, it seems unlikely that much before the 20th century would have conceived such a ploy. In other words, hapax legomena are not a reliable indicator, authorship studies now usually use a wide range of measures to look for patterns rather than rely upon single measurements