67-446: 8 ( eight ) is the natural number following 7 and preceding 9 . English eight , from Old English eahta , æhta , Proto-Germanic *ahto is a direct continuation of Proto-Indo-European *oḱtṓ(w) - , and as such cognate with Greek ὀκτώ and Latin octo- , both of which stems are reflected by the English prefix oct(o)- , as in the ordinal adjective octaval or octavary ,
134-675: A and b with b ≠ 0 there are natural numbers q and r such that The number q is called the quotient and r is called the remainder of the division of a by b . The numbers q and r are uniquely determined by a and b . This Euclidean division is key to the several other properties ( divisibility ), algorithms (such as the Euclidean algorithm ), and ideas in number theory. The addition (+) and multiplication (×) operations on natural numbers as defined above have several algebraic properties: Two important generalizations of natural numbers arise from
201-425: A + c = b . This order is compatible with the arithmetical operations in the following sense: if a , b and c are natural numbers and a ≤ b , then a + c ≤ b + c and ac ≤ bc . An important property of the natural numbers is that they are well-ordered : every non-empty set of natural numbers has a least element. The rank among well-ordered sets is expressed by an ordinal number ; for
268-466: A + 1 = S ( a ) and a × 1 = a . Furthermore, ( N ∗ , + ) {\displaystyle (\mathbb {N^{*}} ,+)} has no identity element. In this section, juxtaposed variables such as ab indicate the product a × b , and the standard order of operations is assumed. A total order on the natural numbers is defined by letting a ≤ b if and only if there exists another natural number c where
335-562: A Proto-Turkic stem *sekiz , which has been suggested as originating as a negation of eki "two", as in "without two fingers" (i.e., "two short of ten; two fingers are not being held up"); this same principle is found in Finnic *kakte-ksa , which conveys a meaning of "two before (ten)". The Proto-Indo-European reconstruction *oḱtṓ(w) - itself has been argued as representing an old dual, which would correspond to an original meaning of "twice four". Proponents of this "quaternary hypothesis" adduce
402-588: A tally mark for each object is another primitive method. Later, a set of objects could be tested for equality, excess or shortage—by striking out a mark and removing an object from the set. 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 powers of 10 up to over 1 million. A stone carving from Karnak , dating back from around 1500 BCE and now at
469-401: A × ( b + c ) = ( a × b ) + ( a × c ) . These properties of addition and multiplication make the natural numbers an instance of a commutative semiring . Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative. The lack of additive inverses, which is equivalent to the fact that N {\displaystyle \mathbb {N} }
536-404: A × 0 = 0 and a × S( b ) = ( a × b ) + a . This turns ( N ∗ , × ) {\displaystyle (\mathbb {N} ^{*},\times )} into a free commutative monoid with identity element 1; a generator set for this monoid is the set of prime numbers . Addition and multiplication are compatible, which is expressed in the distribution law :
603-421: A bold N or blackboard bold N {\displaystyle \mathbb {N} } . Many other number sets are built from the natural numbers. For example, the integers are made by adding 0 and negative numbers. The rational numbers add fractions, and the real numbers add infinite decimals. Complex numbers add the square root of −1 . This chain of extensions canonically embeds
670-766: A complete English phrase is in 1763. The 1771 Encyclopaedia Britannica defines natural numbers in the logarithm article. Starting at 0 or 1 has long been a matter of definition. In 1727, Bernard Le Bovier de Fontenelle wrote that his notions of distance and element led to defining the natural numbers as including or excluding 0. In 1889, Giuseppe Peano used N for the positive integers and started at 1, but he later changed to using N 0 and N 1 . Historically, most definitions have excluded 0, but many mathematicians such as George A. Wentworth , Bertrand Russell , Nicolas Bourbaki , Paul Halmos , Stephen Cole Kleene , and John Horton Conway have preferred to include 0. Mathematicians have noted tendencies in which definition
737-460: A natural number as the class of all sets that are in one-to-one correspondence with a particular set. However, this definition turned out to lead to paradoxes, including Russell's paradox . To avoid such paradoxes, the formalism was modified so that a natural number is defined as a particular set, and any set that can be put into one-to-one correspondence with that set is said to have that number of elements. In 1881, Charles Sanders Peirce provided
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#1732851244651804-526: A need to improve upon 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—but a consequence of definitions. Later, two classes of such formal definitions emerged, using set theory and Peano's axioms respectively. Later still, they were shown to be equivalent in most practical applications. Set-theoretical definitions of natural numbers were initiated by Frege . He initially defined
871-470: A numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628 CE. However, 0 had been used as a number in the medieval computus (the calculation of the date of Easter), beginning with Dionysius Exiguus in 525 CE, without being denoted by a numeral. Standard Roman numerals do not have a symbol for 0; instead, nulla (or the genitive form nullae ) from nullus ,
938-449: A periodicity of 8. The lie group E 8 is one of 5 exceptional lie groups. The order of the smallest non-abelian group whose subgroups are all normal is 8. Natural number In mathematics , the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers 0, 1, 2, 3, ... , while others start with 1, defining them as
1005-486: A rectangular frame. Another proposes instead that it was based on the notation CIↃ used to represent 1,000. Instead of a Roman numeral, it may alternatively be derived from a variant of ω , the lower-case form of omega , the last letter in the Greek alphabet . Perhaps in some cases because of typographic limitations, other symbols resembling the infinity sign have been used for the same meaning. One paper by Leonhard Euler
1072-509: A set (because of Russell's paradox ). The standard solution is to define a particular set with n elements that will be called the natural number n . The following definition was first published by John von Neumann , although Levy attributes the idea to unpublished work of Zermelo in 1916. As this definition extends to infinite set as a definition of ordinal number , the sets considered below are sometimes called von Neumann ordinals . The definition proceeds as follows: It follows that
1139-574: A subscript (or superscript) "0" is added in the latter case: This section uses the convention N = N 0 = N ∗ ∪ { 0 } {\displaystyle \mathbb {N} =\mathbb {N} _{0}=\mathbb {N} ^{*}\cup \{0\}} . Given the set N {\displaystyle \mathbb {N} } of natural numbers and the successor function S : N → N {\displaystyle S\colon \mathbb {N} \to \mathbb {N} } sending each natural number to
1206-530: A unit is not a number and there are no unique numbers (e.g., any two units from indefinitely many units is a 2). However, in the definition of perfect number which comes shortly afterward, Euclid treats 1 as a number like any other. Independent studies on numbers also occurred at around the same time in India , China, and Mesoamerica . Nicolas Chuquet used the term progression naturelle (natural progression) in 1484. The earliest known use of "natural number" as
1273-509: Is consistent (as it is usually guessed), then Peano arithmetic is consistent. In other words, if a contradiction could be proved in Peano arithmetic, then set theory would be contradictory, and every theorem of set theory would be both true and wrong. The five Peano axioms are the following: These are not the original axioms published by Peano, but are named in his honor. Some forms of the Peano axioms have 1 in place of 0. In ordinary arithmetic,
1340-450: Is a composite number and the first number which is neither prime nor semiprime . By Mihăilescu's Theorem , it is the only nonzero perfect power that is one less than another perfect power. 8 is the first proper Leyland number of the form x + y , where in its case x and y both equal 2. 8 is a Fibonacci number and the only nontrivial Fibonacci number that is a perfect cube . Sphenic numbers always have exactly eight divisors. 8
1407-505: Is a free monoid on one generator. This commutative monoid satisfies the cancellation property , so it can be embedded in a group . The smallest group containing the natural numbers is the integers . If 1 is defined as S (0) , then b + 1 = b + S (0) = S ( b + 0) = S ( b ) . That is, b + 1 is simply the successor of b . Analogously, given that addition has been defined, a multiplication operator × {\displaystyle \times } can be defined via
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#17328512446511474-416: Is a subset of m . In other words, the set inclusion defines the usual total order on the natural numbers. This order is a well-order . Infinity symbol The infinity symbol ( ∞ ) is a mathematical symbol representing the concept of infinity . This symbol is also called a lemniscate, after the lemniscate curves of a similar shape studied in algebraic geometry , or "lazy eight", in
1541-552: Is based on set theory . It defines the natural numbers as specific sets . More precisely, each natural number n is defined as an explicitly defined set, whose elements allow counting the elements of other sets, in the sense that the sentence "a set S has n elements" means that there exists a one to one correspondence between the two sets n and S . The sets used to define natural numbers satisfy Peano axioms. It follows that every theorem that can be stated and proved in Peano arithmetic can also be proved in set theory. However,
1608-575: 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 weak systems of set theory . One such system is ZFC with the axiom of infinity replaced by its negation. Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include Goodstein's theorem . The set of all natural numbers
1675-411: Is credited with introducing the infinity symbol with its mathematical meaning in 1655, in his De sectionibus conicis . Wallis did not explain his choice of this symbol. It has been conjectured to be a variant form of a Roman numeral , but which Roman numeral is unclear. One theory proposes that the infinity symbol was based on the numeral for 100 million, which resembled the same symbol enclosed within
1742-464: Is from Old Chinese *priāt- , ultimately from Sino-Tibetan b-r-gyat or b-g-ryat which also yielded Tibetan brgyat . It has been argued that, as the cardinal number 7 is the highest number of items that can universally be cognitively processed as a single set, the etymology of the numeral eight might be the first to be considered composite, either as "twice four" or as "two short of ten", or similar. The Turkic words for "eight" are from
1809-410: Is not closed under subtraction (that is, subtracting one natural from another does not always result in another natural), means that N {\displaystyle \mathbb {N} } is not a ring ; instead it is a semiring (also known as a rig ). If the natural numbers are taken as "excluding 0", and "starting at 1", the definitions of + and × are as above, except that they begin with
1876-429: Is standardly denoted N or N . {\displaystyle \mathbb {N} .} Older texts have occasionally employed J as the symbol for this set. Since natural numbers may contain 0 or not, it may be important to know which version is referred to. This is often specified by the context, but may also be done by using a subscript or a superscript in the notation, such as: Alternatively, since
1943-499: Is the cube-octahedron compound . The octonions are a hypercomplex normed division algebra that are an extension of the complex numbers . They are a double cover of special orthogonal group SO(8). The special unitary group SO(3) has an eight-dimensional adjoint representation whose colors are ascribed gauge symmetries that represent the vectors of the eight gluons in the Standard Model . Clifford algebras display
2010-417: Is the base of the octal number system. A polygon with eight sides is an octagon . A regular octagon can fill a plane-vertex with a regular triangle and a regular icositetragon , as well as tessellate two-dimensional space alongside squares in the truncated square tiling . This tiling is one of eight Archimedean tilings that are semi-regular, or made of more than one type of regular polygon , and
2077-445: Is the only stellation with octahedral symmetry . It has eight triangular faces alongside eight vertices that forms a cubic faceting , composed of two self-dual tetrahedra that makes it the simplest of five regular compounds . The cuboctahedron , on the other hand, is a rectified cube or rectified octahedron, and one of only two convex quasiregular polyhedra . It contains eight equilateral triangular faces, whose first stellation
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2144-422: Is used, such as algebra texts including 0, number theory and analysis texts excluding 0, logic and set theory texts including 0, dictionaries excluding 0, school books (through high-school level) excluding 0, and upper-division college-level books including 0. There are exceptions to each of these tendencies and as of 2023 no formal survey has been conducted. Arguments raised include division by zero and
2211-499: The Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622. The Babylonians had a place-value system based essentially on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one—its value being determined from context. A much later advance was the development of
2278-407: The cardinal number representing the size of the set of natural numbers , and ω {\displaystyle \omega } ( omega ) denotes the smallest ordinal number which is larger than all natural numbers. The infinity symbol may also be used to represent a point at infinity , especially when there is only one such point under consideration. This usage includes, in particular,
2345-519: The positive integers 1, 2, 3, ... . Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers plus zero. In other cases, the whole numbers refer to all of the integers , including negative integers. The counting numbers are another term for the natural numbers, particularly in primary school education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are six coins on
2412-511: The 10th century were a distinctive western variant of the glyphs used in the Arabic-speaking world, known as ghubār numerals ( ghubār translating to " sand table "). In these digits, the line of the 5 -like glyph used in Indian manuscripts for eight came to be formed in ghubār as a closed loop, which was the 8 -shape that became adopted into European use in the 10th century. Just as in most modern typefaces , in typefaces with text figures
2479-487: The Latin word for "none", was employed to denote a 0 value. 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, sometimes even not as a number at all. Euclid , for example, defined a unit first and then a number as a multitude of units, thus by his definition,
2546-452: The character for the digit 8 usually has an ascender , as, for example, in [REDACTED] . The infinity symbol ∞, described as a "sideways figure eight", is unrelated to the digit 8 in origin; it is first used (in the mathematical meaning "infinity") in the 17th century, and it may be derived from the Roman numeral for "one thousand" CIƆ, or alternatively from the final Greek letter, ω . 8
2613-431: The distributive adjective is octonary . The adjective octuple (Latin octu-plus ) may also be used as a noun, meaning "a set of eight items"; the diminutive octuplet is mostly used to refer to eight siblings delivered in one birth. The Semitic numeral is based on a root *θmn- , whence Akkadian smn- , Arabic ṯmn- , Hebrew šmn- etc. The Chinese numeral , written 八 ( Mandarin : bā ; Cantonese : baat ),
2680-409: The first axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of Dedekind's axioms in his book The principles of arithmetic presented by a new method ( Latin : Arithmetices principia, nova methodo exposita ). This approach is now called Peano arithmetic . It
2747-511: The idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation (within other numbers) dates back as early as 700 BCE by the Babylonians, who omitted such a digit when it would have been the last symbol in the number. The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BCE , but this usage did not spread beyond Mesoamerica . The use of
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2814-516: The infinite in modern mysticism and literature. It is a common element of graphic design , for instance in corporate logos as well as in older designs such as the Métis flag . Both the infinity symbol itself and several variations of the symbol are available in various character encodings . The lemniscate has been a common decorative motif since ancient times; for instance, it is commonly seen on Viking Age combs. The English mathematician John Wallis
2881-558: The infinite include James Joyce , in Ulysses , and David Foster Wallace , in Infinite Jest . The well-known shape and meaning of the infinity symbol have made it a common typographic element of graphic design . For instance, the Métis flag , used by the Canadian Métis people since the early 19th century, is based around this symbol. Different theories have been put forward for
2948-431: The infinite point of a projective line , and the point added to a topological space to form its one-point compactification . In areas other than mathematics, the infinity symbol may take on other related meanings. For instance, it has been used in bookbinding to indicate that a book is printed on acid-free paper and will therefore be long-lasting. On cameras and their lenses , the infinity symbol indicates that
3015-445: The infinity sign is conventionally interpreted as meaning that the variable grows arbitrarily large towards infinity, rather than actually taking an infinite value, although other interpretations are possible. When quantifying actual infinity , infinite entities taken as objects per se, other notations are typically used. For example, ℵ 0 {\displaystyle \aleph _{0}} ( aleph-nought ) denotes
3082-725: The left line and the upper half of the right line removed. However, the digit for eight used in India in the early centuries of the Common Era developed considerable graphic variation, and in some cases took the shape of a single wedge, which was adopted into the Perso-Arabic tradition as ٨ (and also gave rise to the later Devanagari form ८ ); the alternative curved glyph also existed as a variant in Perso-Arabic tradition, where it came to look similar to our digit 5. The digits as used in Al-Andalus by
3149-469: The lens's focal length is set to an infinite distance , and is "probably one of the oldest symbols to be used on cameras". In modern mysticism, the infinity symbol has become identified with a variation of the ouroboros , an ancient image of a snake eating its own tail that has also come to symbolize the infinite, and the ouroboros is sometimes drawn in figure-eight form to reflect this identification—rather than in its more traditional circular form. In
3216-406: The meaning of the symbol on this flag, including the hope for an infinite future for Métis culture and its mix of European and First Nations traditions, but also evoking the geometric shapes of Métic dances, , Celtic knots , or Plains First Nations Sign Language . A rainbow -coloured infinity symbol is also used by the autism rights movement , as a way to symbolize the infinite variation of
3283-446: The natural numbers are defined iteratively as follows: It can be checked that the natural numbers satisfy the Peano axioms . With this definition, given a natural number n , the sentence "a set S has n elements" can be formally defined as "there exists a bijection from n to S ." This formalizes the operation of counting the elements of S . Also, n ≤ m if and only if n
3350-458: The natural numbers in the other number systems. Natural numbers are studied in different areas of math. Number theory looks at things like how numbers divide evenly ( divisibility ), or how prime numbers are spread out. Combinatorics studies counting and arranging numbered objects, such as partitions and enumerations . The most primitive method of representing a natural number is to use one's fingers, as in finger counting . Putting down
3417-403: The natural numbers naturally form a subset of the integers (often denoted Z {\displaystyle \mathbb {Z} } ), they may be referred to as the positive, or the non-negative integers, respectively. To be unambiguous about whether 0 is included or not, sometimes a superscript " ∗ {\displaystyle *} " or "+" is added in the former case, and
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#17328512446513484-435: The natural numbers, this is denoted as ω (omega). In this section, juxtaposed variables such as ab indicate the product a × b , and the standard order of operations is assumed. While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure of division with remainder or Euclidean division is available as a substitute: for any two natural numbers
3551-439: The next one, one can define addition of natural numbers recursively by setting a + 0 = a and a + S ( b ) = S ( a + b ) for all a , b . Thus, a + 1 = a + S(0) = S( a +0) = S( a ) , a + 2 = a + S(1) = S( a +1) = S(S( a )) , and so on. The algebraic structure ( N , + ) {\displaystyle (\mathbb {N} ,+)} is a commutative monoid with identity element 0. It
3618-458: The numeral 9 , which might be built on the stem new- , meaning "new" (indicating the beginning of a "new set of numerals" after having counted to eight). The modern digit 8, like all modern Arabic numerals other than zero, originates with the Brahmi numerals . The Brahmi digit for eight by the 1st century was written in one stroke as a curve └┐ looking like an uppercase H with the bottom half of
3685-494: The only tiling that can admit a regular octagon. The Ammann–Beenker tiling is a nonperiodic tesselation of prototiles that feature prominent octagonal silver eightfold symmetry, that is the two-dimensional orthographic projection of the four-dimensional 8-8 duoprism . An octahedron is a regular polyhedron with eight equilateral triangles as faces . is the dual polyhedron to the cube and one of eight convex deltahedra . The stella octangula , or eight-pointed star ,
3752-595: The ordinary natural numbers via the ultrapower construction . Other generalizations are discussed in Number § Extensions of the concept . Georges Reeb used to claim provocatively that "The naïve integers don't fill up N {\displaystyle \mathbb {N} } ". There are two standard methods for formally defining natural numbers. The first one, named for Giuseppe Peano , consists of an autonomous axiomatic theory called Peano arithmetic , based on few axioms called Peano axioms . The second definition
3819-664: The people in the movement and of human cognition. The Bakelite company took up this symbol in its corporate logo to refer to the wide range of varied applications of the synthetic material they produced. Versions of this symbol have been used in other trademarks, corporate logos, and emblems including those of Fujitsu , Cell Press , and the 2022 FIFA World Cup . The symbol is encoded in Unicode at U+221E ∞ INFINITY and in LaTeX as \infty : ∞ {\displaystyle \infty } . An encircled version
3886-471: The same natural number, the number of elements of the set. This number can also be used to describe the position of an element in a larger finite, or an infinite, sequence . A countable non-standard model of arithmetic satisfying the Peano Arithmetic (that is, the first-order Peano axioms) was developed by Skolem in 1933. The hypernatural numbers are an uncountable model that can be constructed from
3953-399: The size of the empty set . Computer languages often start from zero when enumerating items like loop counters and string- or array-elements . Including 0 began to rise in popularity in the 1960s. The ISO 31-11 standard included 0 in the natural numbers in its first edition in 1978 and this has continued through its present edition as ISO 80000-2 . In 19th century Europe, there
4020-433: The successor of x {\displaystyle x} is x + 1 {\displaystyle x+1} . Intuitively, the natural number n is the common property of all sets that have n elements. So, it seems natural to define n as an equivalence class under the relation "can be made in one to one correspondence ". This does not work in all set theories , as such an equivalence class would not be
4087-422: The table", in which case they are called cardinal numbers . They are also used to put things in order, like "this is the third largest city in the country", which are called ordinal numbers . Natural numbers are also used as labels, like jersey numbers on a sports team, where they serve as nominal numbers and do not have mathematical properties. The natural numbers form a set , commonly symbolized as
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#17328512446514154-438: The terminology of livestock branding . This symbol was first used mathematically by John Wallis in the 17th century, although it has a longer history of other uses. In mathematics, it often refers to infinite processes ( potential infinity ) rather than infinite values ( actual infinity ). It has other related technical meanings, such as the use of long-lasting paper in bookbinding , and has been used for its symbolic value of
4221-402: The two definitions are not equivalent, as there are theorems that can be stated in terms of Peano arithmetic and proved in set theory, which are not provable inside Peano arithmetic. A probable example is Fermat's Last Theorem . The definition of the integers as sets satisfying Peano axioms provide a model of Peano arithmetic inside set theory. An important consequence is that, if set theory
4288-423: The two uses of counting and ordering: cardinal numbers and ordinal numbers . The least ordinal of cardinality ℵ 0 (that is, the initial ordinal of ℵ 0 ) is ω but many well-ordered sets with cardinal number ℵ 0 have an ordinal number greater than ω . For finite well-ordered sets, there is a one-to-one correspondence between ordinal and cardinal numbers; therefore they can both be expressed by
4355-519: The works of Vladimir Nabokov , including The Gift and Pale Fire , the figure-eight shape is used symbolically to refer to the Möbius strip and the infinite, as is the case in these books' descriptions of the shapes of bicycle tire tracks and of the outlines of half-remembered people. Nabokov's poem after which he entitled Pale Fire explicitly refers to "the miracle of the lemniscate". Other authors whose works use this shape with its symbolic meaning of
4422-430: Was mathematical and philosophical discussion about the exact nature of the natural numbers. Henri Poincaré stated that axioms can only be demonstrated in their finite application, and concluded that it is "the power of the mind" which allows conceiving of the indefinite repetition of the same act. Leopold Kronecker summarized his belief as "God made the integers, all else is the work of man". The constructivists saw
4489-740: Was typeset with an open letterform more closely resembling a reflected and sideways S than a lemniscate (something like S ), and even "O–O" has been used as a stand-in for the infinity symbol itself. In mathematics, the infinity symbol is typically used to represent a potential infinity . For instance, in mathematical expressions with summations and limits such as ∑ n = 0 ∞ 1 2 n = lim x → ∞ 2 x − 1 2 x − 1 = 2 , {\displaystyle \sum _{n=0}^{\infty }{\frac {1}{2^{n}}}=\lim _{x\to \infty }{\frac {2^{x}-1}{2^{x-1}}}=2,}
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