|
|
| Line 1: |
Line 1: |
| {{pp-pc1}}
| | Solomon is how heIs called but he never really liked that name. West Virginia is where his house is and he does not plan on changing it. He became an information officer After being from his career for years but heIs already used for another. One of his favorite interests does magic but he's struggling to find time for this.<br><br>Take a look at my web-site [https://www.larrainvial.com/Content/descargas/fondos/FI/prospecto_deuda_subsidio.pdf jaun pablo schiappacasse canepa] |
| {{other uses}}
| |
| {{pp-move|small=yes}}
| |
| A '''number''' is a [[mathematical object]] used to [[counting|count]], label, and [[measurement|measure]]. In [[mathematics]], the definition of number has been extended over the years to include such numbers as {{num|0}}, [[negative number]]s, [[rational number]]s, [[irrational number]]s, and [[complex number]]s.
| |
| | |
| [[Mathematical operation]]s are certain procedures that take one or more numbers as input and produce a number as output. [[Unary operation]]s take a single input number and produce a single output number. For example, the [[successor ordinal|successor]] operation adds 1 to an [[integer]], thus the successor of 4 is 5. [[Binary operation]]s take two input numbers and produce a single output number. Examples of binary operations include [[addition]], [[subtraction]], [[multiplication]], [[division (mathematics)|division]], and [[exponentiation]]. The study of numerical operations is called [[arithmetic]].
| |
| | |
| A notational symbol that represents a number is called a [[numeral system|numeral]]. In addition to their use in counting and measuring, numerals are often used for labels ([[telephone number]]s), for ordering ([[serial number]]s), and for codes (e.g., [[ISBN]]s).
| |
| | |
| In common usage, the word ''number'' can mean the abstract object, the symbol, or the [[numeral (linguistics)|word for the number]].
| |
| | |
| ==Classification of numbers==
| |
| {{See also|List of types of numbers}}
| |
| Different types of numbers are used in many cases. Numbers can be classified into [[set (mathematics)|sets]], called '''number systems'''. (For different methods of expressing numbers with symbols, such as the [[Roman numerals]], see [[numeral system]]s.)
| |
| | |
| <center>
| |
| {|class="wikitable" style="text-align: center; width: 400px; height: 200px;"
| |
| |+ Important number systems
| |
| |-
| |
| !<math>\mathbb{N}</math>
| |
| !Natural
| |
| |0, 1, 2, 3, 4, ... '''or''' 1, 2, 3, 4, ...
| |
| |-
| |
| !<math>\mathbb{Z}</math>
| |
| !Integers
| |
| |..., −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, ...
| |
| |-
| |
| !<math>\mathbb{Q}</math>
| |
| !Rational
| |
| |{{sfrac|''a''|''b''}} where ''a'' and ''b'' are integers and ''b'' is not 0
| |
| |-
| |
| !<math>\mathbb{R}</math>
| |
| !Real
| |
| |The limit of a convergent sequence of rational numbers
| |
| |-
| |
| !<math>\mathbb{C}</math>
| |
| !Complex
| |
| |''a'' + ''bi'' or ''a'' + ''ib'' where ''a'' and ''b'' are real numbers and ''i'' is the square root of −1
| |
| |}
| |
| </center>
| |
| | |
| ===Natural numbers===
| |
| {{Main|Natural number}}
| |
| The most familiar numbers are the [[natural number]]s or counting numbers: 1, 2, 3, and so on. Traditionally, the sequence of natural numbers started with 1 (0 was not even considered a number for the [[Ancient Greeks]].) However, in the 19th century, [[set theory|set theorists]] and other mathematicians started including 0 ([[cardinality]] of the [[empty set]], i.e. 0 elements, where 0 is thus the smallest [[cardinal number]]) in the set of natural numbers.{{citation needed|date=March 2011}} Today, different mathematicians use the term to describe both sets, including 0 or not. The [[mathematical symbol]] for the set of all natural numbers is '''N''', also written [[Blackboard bold|<math>\mathbb{N}</math>]], and sometimes <math>\mathbb{N}_0</math> or <math>\mathbb{N}_1</math> when it is necessary to indicate whether the set should start with 0 or 1, respectively.
| |
| | |
| In the [[base 10]] numeral system, in almost universal use today for mathematical operations, the symbols for natural numbers are written using ten [[numerical digit|digits]]: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. In this base 10 system, the rightmost digit of a natural number has a [[place value]] of 1, and every other digit has a place value ten times that of the place value of the digit to its right.
| |
| | |
| In [[set theory]], which is capable of acting as an axiomatic foundation for modern mathematics,<ref>{{Cite book |last=Suppes |first=Patrick |authorlink=Patrick_Suppes |title=Axiomatic Set Theory |publisher=Courier Dover Publications |year=1972 |page=1 |isbn=0-486-61630-4}}</ref> natural numbers can be represented by classes of equivalent sets. For instance, the number 3 can be represented as the class of all sets that have exactly three elements. Alternatively, in [[Peano Arithmetic]], the number 3 is represented as sss0, where s is the "successor" function (i.e., 3 is the third successor of 0). Many different representations are possible; all that is needed to formally represent 3 is to inscribe a certain symbol or pattern of symbols three times.
| |
| | |
| ===Integers===
| |
| {{Main|Integer}}
| |
| The [[negative number|negative]] of a positive integer is defined as a number that produces 0 when it is added to the corresponding positive integer. Negative numbers are usually written with a negative sign (a [[minus sign]]). As an example, the negative of 7 is written −7, and {{nowrap|7 + (−7) {{=}} 0}}. When the [[set (mathematics)|set]] of negative numbers is combined with the set of natural numbers (which includes 0), the result is defined as the set of integer numbers, also called [[integer]]s, '''Z''' also written [[Blackboard bold|<math>\mathbb{Z}</math>.]] Here the letter Z comes {{ety|de|Zahl|number}}. The set of integers forms a [[ring (mathematics)|ring]] with operations addition and multiplication.<ref>{{Mathworld|Integer|Integer}}</ref>
| |
| | |
| ===Rational numbers===
| |
| {{Main|Rational number}}
| |
| A rational number is a number that can be expressed as a [[fraction (mathematics)|fraction]] with an integer numerator and a non-zero integer denominator. Fractions are written as two numbers, the numerator and the denominator, with a dividing bar between them. In the fraction written {{sfrac|''m''|''n''}} or
| |
| :<math>m \over n \, </math>
| |
| ''m'' represents equal parts, where ''n'' equal parts of that size make up ''m'' wholes. Two different fractions may correspond to the same rational number; for example {{sfrac|1|2}} and {{sfrac|2|4}} are equal, that is:
| |
| : <math>{1 \over 2} = {2 \over 4}.\,</math>
| |
| | |
| If the [[absolute value]] of ''m'' is greater than ''n'', then the absolute value of the fraction is greater than 1. Fractions can be greater than, less than, or equal to 1 and can also be positive, negative, or 0. The set of all rational numbers includes the integers, since every integer can be written as a fraction with denominator 1. For example −7 can be written {{sfrac|−7|1}}. The symbol for the rational numbers is '''Q''' (for ''[[quotient]]''), also written [[Blackboard bold|<math>\mathbb{Q}</math>.]]
| |
| | |
| ===Real numbers===
| |
| {{Main|Real number}}
| |
| The real numbers include all of the measuring numbers. Real numbers are usually written using [[decimal]] numerals, in which a [[decimal point]] is placed to the right of the digit with place value 1. Each digit to the right of the decimal point has a place value one-tenth of the place value of the digit to its left. Thus
| |
| :<math>123.456\,</math>
| |
| represents 1 hundred, 2 tens, 3 ones, 4 tenths, 5 hundredths, and 6 thousandths. In saying the number, the decimal is read "point", thus: "one two three point four five six". In the US and UK and a number of other countries, the decimal point is represented by a [[full stop|period]], whereas in continental Europe and certain other countries the decimal point is represented by a [[Comma (punctuation)|comma]]. Zero is often written as 0.0 when it must be treated as a real number rather than an integer. In the US and UK a number between −1 and 1 is always written with a leading 0 to emphasize the decimal. Negative real numbers are written with a preceding [[minus sign]]:
| |
| :<math>-123.456.\,</math>
| |
| | |
| Every rational number is also a real number. It is not the case, however, that every real number is rational. If a real number cannot be written as a fraction of two integers, it is called [[irrational number|irrational]]. A decimal that can be written as a fraction either ends (terminates) or forever [[repeating decimal|repeats]], because it is the answer to a problem in division. Thus the real number 0.5 can be written as {{sfrac|1|2}} and the real number 0.333... (forever repeating 3s, otherwise written 0.{{overline|3}}) can be written as {{sfrac|1|3}}. On the other hand, the real number π ([[pi]]), the ratio of the [[circumference]] of any circle to its [[diameter]], is
| |
| :<math>\pi = 3.14159265358979\dots.\,</math>
| |
| | |
| Since the decimal neither ends nor forever repeats, it cannot be written as a fraction, and is an example of an irrational number. Other irrational numbers include
| |
| :<math>\sqrt{2} = 1.41421356237 \dots\,</math>
| |
| (the [[square root of 2]], that is, the positive number whose square is 2).
| |
| | |
| Thus 1.0 and [[0.999...]] are two different decimal numerals representing the natural number 1. There are infinitely many other ways of representing the number 1, for example {{sfrac|2|2}}, {{sfrac|3|3}}, 1.00, 1.000, and so on.
| |
| | |
| Every real number is either rational or irrational. Every real number corresponds to a point on the [[number line]]. The real numbers also have an important but highly technical property called the [[least upper bound]] property. The symbol for the real numbers is '''R''', also written as <math>\mathbb{R}</math>.
| |
| | |
| When a real number represents a [[measurement]], there is always a [[margin of error]]. This is often indicated by [[rounding]] or [[truncating]] a decimal, so that digits that suggest a greater accuracy than the measurement itself are removed. The remaining digits are called [[significant digits]]. For example, measurements with a ruler can seldom be made without a margin of error of at least 0.001 meters. If the sides of a [[rectangle]] are measured as 1.23 meters and 4.56 meters, then multiplication gives an area for the rectangle of {{nowrap|5.6088 square meters}}. Since only the first two digits after the decimal place are significant, this is usually rounded to 5.61.
| |
| | |
| In [[abstract algebra]], it can be shown that any [[completeness (order theory)|complete]] [[ordered field]] is isomorphic to the real numbers. The real numbers are not, however, an [[algebraically closed field]].
| |
| | |
| ===Complex numbers===
| |
| {{Main|Complex number}}
| |
| Moving to a greater level of abstraction, the real numbers can be extended to the [[complex number]]s. This set of numbers arose, historically, from trying to find closed formulas for the roots of [[cubic equation|cubic]] and [[quartic equation|quartic]] polynomials. This led to expressions involving the square roots of negative numbers, and eventually to the definition of a new number: the square root of −1, denoted by ''[[imaginary unit|i]]'', a symbol assigned by [[Leonhard Euler]], and called the [[imaginary unit]]. The complex numbers consist of all numbers of the form
| |
| :<math>\,a + b i</math> or
| |
| :<math>\,a + i b</math>
| |
| where ''a'' and ''b'' are real numbers. In the expression {{nowrap|''a'' + ''bi''}}, the real number ''a'' is called the [[real part]] and ''b'' is called the [[imaginary part]]. If the real part of a complex number is 0, then the number is called an [[imaginary number]] or is referred to as ''purely imaginary''; if the imaginary part is 0, then the number is a real number. Thus the real numbers are a [[subset]] of the complex numbers. If the real and imaginary parts of a complex number are both integers, then the number is called a [[Gaussian integer]]. The symbol for the complex numbers is '''C''' or <math>\mathbb{C}</math>.
| |
| | |
| In [[abstract algebra]], the complex numbers are an example of an [[algebraically closed field]], meaning that every [[polynomial]] with complex [[coefficient]]s can be [[factorization|factored]] into linear factors. Like the real number system, the complex number system is a [[field (mathematics)|field]] and is [[completeness (order theory)|complete]], but unlike the real numbers it is not [[total order|ordered]]. That is, there is no meaning in saying that ''i'' is greater than 1, nor is there any meaning in saying that ''i'' is less than 1. In technical terms, the complex numbers lack the [[trichotomy property]].
| |
| | |
| Complex numbers correspond to points on the [[complex plane]], sometimes called the Argand plane (for [[Jean-Robert Argand]]).
| |
| | |
| Each of the number systems mentioned above is a [[proper subset]] of the next number system. Symbolically, <math>\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}</math>.
| |
| | |
| ===Computable numbers===
| |
| {{Main|Computable number}}
| |
| Moving to problems of computation, the [[computable number]]s are determined in the set of the real numbers. The computable numbers, also known as the recursive numbers or the computable reals, are the [[real numbers]] that can be computed to within any desired precision by a finite, terminating [[algorithm]]. Equivalent definitions can be given using [[μ-recursive function]]s, [[Turing machines]] or [[λ-calculus]] as the formal representation of algorithms. The computable numbers form a [[real closed field]] and can be used in the place of real numbers for many, but not all, mathematical purposes.
| |
| | |
| ===Other types===
| |
| [[Algebraic numbers]] are those that can be expressed as the solution to a polynomial equation with integer coefficients. The complement of the algebraic numbers are the [[transcendental numbers]].
| |
| | |
| [[Hyperreal number]]s are used in [[non-standard analysis]]. The hyperreals, or nonstandard reals (usually denoted as *'''R'''), denote an [[ordered field]] that is a proper [[Field extension|extension]] of the ordered field of [[real number]]s '''R''' and satisfies the [[transfer principle]]. This principle allows true [[first-order logic|first-order]] statements about '''R''' to be reinterpreted as true first-order statements about *'''R'''.
| |
| | |
| [[Superreal number|Superreal]] and [[surreal number]]s extend the real numbers by adding infinitesimally small numbers and infinitely large numbers, but still form [[field (mathematics)|fields]].
| |
| | |
| The [[p-adic number]]s may have infinitely long expansions to the left of the decimal point, in the same way that real numbers may have infinitely long expansions to the right. The number system that results depends on what [[radix|base]] is used for the digits: any base is possible, but a [[prime number]] base provides the best mathematical properties.
| |
| | |
| For dealing with infinite collections, the natural numbers have been generalized to the [[ordinal number]]s and to the [[cardinal number]]s. The former gives the ordering of the collection, while the latter gives its size. For the finite set, the ordinal and cardinal numbers are equivalent, but they differ in the infinite case.
| |
| | |
| A [[relation number]] is defined as the class of [[relation (mathematics)|relations]] consisting of all those relations that are similar to one member of the class.<ref>{{Cite book
| |
| |last=Russell
| |
| |first=Bertrand
| |
| |authorlink=Bertrand_Russell
| |
| |title=Introduction to Mathematical Philosophy
| |
| |publisher=Routledge
| |
| |year=1919
| |
| |page=56
| |
| |ISBN=0-415-09604-9}}</ref>
| |
| | |
| Sets of numbers that are not subsets of the complex numbers are sometimes called [[hypercomplex number]]s. They include the [[quaternion]]s '''H''', invented by Sir [[William Rowan Hamilton]], in which multiplication is not [[commutative]], and the [[octonion]]s, in which multiplication is not [[associative]]. Elements of [[function field of an algebraic variety|function fields]] of non-zero [[characteristic (algebra)|characteristic]] behave in some ways like numbers and are often regarded as numbers by number theorists.
| |
| | |
| ===Specific uses===
| |
| There are also other sets of numbers with specialized uses. Some are subsets of the complex numbers. For example, [[algebraic numbers]] are the roots of [[polynomials]] with rational [[coefficients]]. Complex numbers that are not algebraic are called [[transcendental numbers]].
| |
| | |
| An [[even number]] is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2. (The old-fashioned term "evenly divisible" is now almost always shortened to "[[divisibility|divisible]]".) A formal definition of an odd number is that it is an integer of the form {{nowrap|''n'' {{=}} 2''k'' + 1,}} where ''k'' is an integer. An even number has the form {{nowrap|''n'' {{=}} 2''k''}} where ''k'' is an [[integer]].
| |
| | |
| A [[perfect number]] is a [[positive integer]] that is the sum of its proper positive [[divisor]]s—the sum of the positive divisors not including the number itself. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors, or {{nowrap|[[divisor function|σ]](''n'') {{=}} 2''n''}}. The first perfect number is {{num|6}}, because 1, 2, and 3 are its proper positive divisors and {{nowrap|1 + 2 + 3 {{=}} 6}}. The next perfect number is {{nowrap|{{num|28}} {{=}} 1 + 2 + 4 + 7 + 14}}. The next perfect numbers are {{num|496}} and {{num|8128}} {{OEIS|id=A000396}}. These first four perfect numbers were the only ones known to early [[Greek mathematics]].
| |
| | |
| A [[figurate number]] is a number that can be represented as a regular and discrete [[geometric]] pattern (e.g. dots). If the pattern is [[polytope|polytopic]], the figurate is labeled a polytopic number, and may be a [[polygonal number]] or a polyhedral number. Polytopic numbers for {{nowrap|''r'' {{=}} 2}}, 3, and 4 are:
| |
| *{{math|1=''P''<sub>2</sub>(n) = {{sfrac|1|2}} ''n''(''n'' + 1)}} ([[triangular number]]s)
| |
| *{{math|1=''P''<sub>3</sub>(n) = {{sfrac|1|6}} ''n''(''n'' + 1)(''n'' + 2)}} ([[tetrahedral number]]s)
| |
| *{{math|1=''P''<sub>4</sub>(n) = {{sfrac|1|24}} ''n''(''n'' + 1)(''n'' + 2)(''n'' + 3)}} ([[pentatopic number]]s)
| |
| | |
| ==Numerals==
| |
| Numbers should be distinguished from ''[[numeral (linguistics)|numerals]]'', the symbols used to represent numbers. Boyer showed that Egyptians created the first ciphered numeral system.{{Citation needed|date=July 2010}} Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets. The number five can be represented by both the base 10 numeral "5", by the [[Roman numeral]] "{{unicode|Ⅴ}}" and ciphered letters. Notations used to represent numbers are discussed in the article [[numeral system]]s. An important development in the history of numerals was the development of a positional system, like modern decimals, which can represent very large numbers. The Roman numerals require extra symbols for larger numbers.
| |
| | |
| ==History==
| |
| ===First use of numbers===
| |
| {{See also|History of writing ancient numbers}}
| |
| Bones and other artifacts have been discovered with marks cut into them that many believe are [[tally marks]].<ref>Marshak, A., ''The Roots of Civilisation; Cognitive Beginnings of Man’s First Art, Symbol and Notation'', (Weidenfeld & Nicolson, London: 1972), 81ff.</ref> These tally marks may have been used for counting elapsed time, such as numbers of days, lunar cycles or keeping records of quantities, such as of animals.
| |
| | |
| A tallying system has no concept of place value (as in modern [[decimal]] notation), which limits its representation of large numbers. Nonetheless tallying systems are considered the first kind of abstract numeral system.
| |
| | |
| The first known system with place value was the [[Ancient Mesopotamian units of measurement|Mesopotamian base 60]] system ([[circa|ca.]] 3400 BC) and the earliest known base 10 system dates to 3100 BC in [[Egypt]].<ref>{{cite web |url=http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptpapyrus.html#berlin |title=Egyptian Mathematical Papyri – Mathematicians of the African Diaspora |publisher=Math.buffalo.edu |date= |accessdate=2012-01-30}}</ref>
| |
| | |
| ===Zero {{anchor|History of zero}}===
| |
| {{further2|[[History of zero]]}}
| |
| The use of 0 as a number should be distinguished from its use as a placeholder numeral in [[place-value system]]s. Many ancient texts used 0. Babylonian (Modern Iraq) and Egyptian texts used it. Egyptians used the word ''nfr'' to denote zero balance in [[double-entry bookkeeping system|double entry accounting]] entries. Indian texts used a [[Sanskrit]] word {{lang|sa|''Shunye''}} or {{lang|sa|''shunya''}} to refer to the concept of ''void''. In mathematics texts this word often refers to the number zero.<ref>{{cite web |url=http://sunsite.utk.edu/math_archives/.http/hypermail/historia/apr99/0197.html |title=Historia Matematica Mailing List Archive: Re: [HM] The Zero Story: a question |publisher=Sunsite.utk.edu |date=1999-04-26 |accessdate=2012-01-30}}</ref><!--The following appears to be an anachronism; no one had defined formal grammars in the modern sense in the 5th century BC. Could be that *retrospectively* we can interpret it that way, but that's different. If restored, please clarify and give a citation.--><!--In a similar vein, [[Pāṇini]] (5th century BC) used the null (zero) operator (ie a [[lambda production]]) in the [[Ashtadhyayi]], his [[formal grammar|algebraic grammar]] for the Sanskrit language. (also see [[Pingala]])-->
| |
| | |
| Records show that the [[Ancient Greece|Ancient Greeks]] seemed unsure about the status of 0 as a number: they asked themselves "how can 'nothing' be something?" leading to interesting [[philosophical]] and, by the Medieval period, religious arguments about the nature and existence of 0 and the [[vacuum]]. The [[Zeno's paradoxes|paradoxes]] of [[Zeno of Elea]] depend in large part on the uncertain interpretation of 0. (The ancient Greeks even questioned whether {{num|1}} was a number.)
| |
| | |
| The late [[Olmec]] people of south-central [[Mexico]] began to use a true zero (a shell [[glyph]]) in the New World possibly by the {{nowrap|4th century BC}} but certainly by 40 BC, which became an integral part of [[Maya numerals]] and the [[Maya calendar]]. Mayan arithmetic used base 4 and base 5 written as base 20. Sanchez in 1961 reported a base 4, base 5 "finger" abacus.
| |
| | |
| By 130 AD, [[Ptolemy]], influenced by [[Hipparchus]] and the Babylonians, was using a symbol for 0 (a small circle with a long overbar) within a sexagesimal numeral system otherwise using alphabetic [[Greek numerals]]. Because it was used alone, not as just a placeholder, this [[Greek numerals#Hellenistic zero|Hellenistic zero]] was the first ''documented'' use of a true zero in the Old World. In later [[Byzantine Empire|Byzantine]] manuscripts of his ''Syntaxis Mathematica'' (''Almagest''), the Hellenistic zero had morphed into the [[Greek alphabet|Greek letter]] [[omicron]] (otherwise meaning 70).
| |
| | |
| Another true zero was used in tables alongside [[Roman numerals#Zero|Roman numerals]] by 525 (first known use by [[Dionysius Exiguus]]), but as a word, {{lang|la|''nulla''}} meaning ''nothing'', not as a symbol. When division produced 0 as a remainder, {{lang|la|''nihil''}}, also meaning ''nothing'', was used. These medieval zeros were used by all future medieval [[computus|computists]] (calculators of [[Easter]]). An isolated use of their initial, N, was used in a table of Roman numerals by [[Bede]] or a colleague about 725, a true zero symbol.
| |
| | |
| An early documented use of the zero by [[Brahmagupta]] (in the ''[[Brāhmasphuṭasiddhānta]]'') dates to 628. He treated 0 as a number and discussed operations involving it, including [[division by zero|division]]. By this time (the 7th century) the concept had clearly reached Cambodia as [[Khmer numerals]], and documentation shows the idea later spreading to [[China]] and the [[Islamic world]].
| |
| | |
| ===Negative numbers {{anchor|History of negative numbers}}===
| |
| {{further2|[[History of negative numbers]]}}
| |
| The abstract concept of negative numbers was recognized as early as 100 BC – 50 BC. The [[China|Chinese]] ''[[Nine Chapters on the Mathematical Art]]'' ({{lang-zh|Jiu-zhang Suanshu}}) contains methods for finding the areas of figures; red rods were used to denote positive [[coefficient]]s, black for negative.<ref>{{Cite book |last=Staszkow |first=Ronald |coauthors=Robert Bradshaw |title=The Mathematical Palette (3rd ed.) |publisher=Brooks Cole |year=2004 |page=41 |isbn=0-534-40365-4}}</ref> This is the earliest known mention of negative numbers in the East; the first reference in a Western work was in the 3rd century in [[Greece]]. [[Diophantus]] referred to the equation equivalent to {{nowrap|4''x'' + 20 {{=}} 0}} (the solution is negative) in ''[[Arithmetica]]'', saying that the equation gave an absurd result.
| |
| | |
| During the 600s, negative numbers were in use in [[India]] to represent debts. Diophantus' previous reference was discussed more explicitly by Indian mathematician [[Brahmagupta]], in ''[[Brāhmasphuṭasiddhānta]]'' 628, who used negative numbers to produce the general form [[quadratic formula]] that remains in use today. However, in the 12th century in India, [[Bhāskara II|Bhaskara]] gives negative roots for quadratic equations but says the negative value "is in this case not to be taken, for it is inadequate; people do not approve of negative roots."
| |
| | |
| [[Europe]]an mathematicians, for the most part, resisted the concept of negative numbers until the 17th century, although [[Fibonacci]] allowed negative solutions in financial problems where they could be interpreted as debts (chapter 13 of ''[[Liber Abaci]]'', 1202) and later as losses (in {{lang|la|''Flos''}}). At the same time, the Chinese were indicating negative numbers either by drawing a diagonal stroke through the right-most non-zero digit of the corresponding positive number's numeral.<ref>{{Cite book
| |
| |last=Smith
| |
| |first=David Eugene
| |
| |authorlink=David_Eugene_Smith
| |
| |title=History of Modern Mathematics
| |
| |publisher=Dover Publications
| |
| |year=1958
| |
| |page=259
| |
| |isbn=0-486-20429-4}}</ref> The first use of negative numbers in a European work was by [[Chuquet]] during the 15th century. He used them as [[exponent]]s, but referred to them as "absurd numbers".
| |
| | |
| As recently as the 18th century, it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless, just as [[René Descartes]] did with negative solutions in a [[Cartesian coordinate system]].
| |
| | |
| ===Rational numbers {{anchor|History of rational numbers}}===
| |
| It is likely that the concept of fractional numbers dates to [[prehistoric times]]. The [[Ancient Egyptians]] used their [[Egyptian fraction]] notation for rational numbers in mathematical texts such as the [[Rhind Mathematical Papyrus]] and the [[Kahun Papyrus]]. Classical Greek and Indian mathematicians made studies of the theory of rational numbers, as part of the general study of [[number theory]]. The best known of these is [[Euclid's Elements|Euclid's ''Elements'']], dating to roughly 300 BC. Of the Indian texts, the most relevant is the [[Sthananga Sutra]], which also covers number theory as part of a general study of mathematics.
| |
| | |
| The concept of [[decimal fraction]]s is closely linked with decimal place-value notation; the two seem to have developed in tandem. For example, it is common for the Jain math sutras to include calculations of decimal-fraction approximations to [[pi]] or the [[square root of 2]]. Similarly, Babylonian math texts had always used sexagesimal (base 60) fractions with great frequency.
| |
| | |
| ===Irrational numbers {{anchor|History of irrational numbers}}===
| |
| {{further2|[[History of irrational numbers]]}}
| |
| The earliest known use of irrational numbers was in the [[Indian mathematics|Indian]] [[Sulba Sutras]] composed between 800 and 500 BC.<ref>{{Cite book |editor-last=Selin |editor-first=Helaine |editor-link=Helaine Selin |title=Mathematics across cultures: the history of non-Western mathematics |publisher=Kluwer Academic Publishers |year=2000 |page=451 |isbn=0-7923-6481-3}}</ref> The first existence proofs of irrational numbers is usually attributed to [[Pythagoras]], more specifically to the [[Pythagoreanism|Pythagorean]] [[Hippasus|Hippasus of Metapontum]], who produced a (most likely geometrical) proof of the irrationality of the [[square root of 2]]. The story goes that Hippasus discovered irrational numbers when trying to represent the square root of 2 as a fraction. However Pythagoras believed in the absoluteness of numbers, and could not accept the existence of irrational numbers. He could not disprove their existence through logic, but he could not accept irrational numbers, so he sentenced Hippasus to death by drowning.
| |
| | |
| The 16th century brought final European acceptance of [[negative number|negative]] integral and [[fraction (mathematics)|fractional]] numbers. By the 17th century, mathematicians generally used decimal fractions with modern notation. It was not, however, until the 19th century that mathematicians separated irrationals into algebraic and transcendental parts, and once more undertook scientific study of irrationals. It had remained almost dormant since [[Euclid]]. In 1872, the publication of the theories of [[Karl Weierstrass]] (by his pupil [[Kossak]]), [[Eduard Heine|Heine]] (''[[Crelle]]'', 74), [[Georg Cantor]] (Annalen, 5), and [[Richard Dedekind]] was brought about. In 1869, [[Méray]] had taken the same point of departure as Heine, but the theory is generally referred to the year 1872. Weierstrass's method was completely set forth by [[Salvatore Pincherle]] (1880), and Dedekind's has received additional prominence through the author's later work (1888) and endorsement by [[Paul Tannery]] (1894). Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a [[Dedekind cut|cut (Schnitt)]] in the system of [[real number]]s, separating all [[rational number]]s into two groups having certain characteristic properties. The subject has received later contributions at the hands of Weierstrass, [[Kronecker]] (Crelle, 101), and Méray.
| |
| | |
| [[Continued fraction]]s, closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of [[Euler]], and at the opening of the 19th century were brought into prominence through the writings of [[Joseph Louis Lagrange]]. Other noteworthy contributions have been made by [[Druckenmüller]] (1837), Kunze (1857), Lemke (1870), and Günther (1872). Ramus (1855) first connected the subject with [[determinant]]s, resulting, with the subsequent contributions of Heine, [[August Ferdinand Möbius|Möbius]], and Günther, in the theory of Kettenbruchdeterminanten. Dirichlet also added to the general theory, as have numerous contributors to the applications of the subject.
| |
| | |
| ===Transcendental numbers and reals {{anchor|History of transcendental numbers and reals}}===
| |
| {{further2|[[History of π]]}}
| |
| The first results concerning [[transcendental number]]s were [[Johann Heinrich Lambert|Lambert's]] 1761 proof that π cannot be rational, and also that ''e''<sup>''n''</sup> is irrational if ''n'' is rational (unless {{nowrap|''n'' {{=}} 0}}). (The constant [[e (mathematical constant)|''e'']] was first referred to in [[John Napier|Napier's]] 1618 work on [[logarithms]].) [[Adrien-Marie Legendre|Legendre]] extended this proof to show that π is not the square root of a rational number. The search for roots of [[Quintic equation|quintic]] and higher degree equations was an important development, the [[Abel–Ruffini theorem]] ([[Paolo Ruffini|Ruffini]] 1799, [[Niels Henrik Abel|Abel]] 1824) showed that they could not be solved by [[nth root|radicals]] (formulas involving only arithmetical operations and roots). Hence it was necessary to consider the wider set of [[algebraic numbers]] (all solutions to polynomial equations). [[Évariste Galois|Galois]] (1832) linked polynomial equations to [[group theory]] giving rise to the field of [[Galois theory]].
| |
| | |
| The existence of transcendental numbers<ref>{{cite web |last=Bogomolny |first=A. |authorlink=Cut-the-Knot |title=What's a number? |work=Interactive Mathematics Miscellany and Puzzles |url=http://www.cut-the-knot.org/do_you_know/numbers.shtml |accessdate=11 July 2010}}</ref> was first established by [[Joseph Liouville|Liouville]] (1844, 1851). [[Charles Hermite|Hermite]] proved in 1873 that ''e'' is transcendental and [[Ferdinand von Lindemann|Lindemann]] proved in 1882 that π is transcendental. Finally [[Cantor's first uncountability proof|Cantor]] shows that the set of all [[real number]]s is [[uncountable|uncountably infinite]] but the set of all [[algebraic number]]s is [[countable|countably infinite]], so there is an uncountably infinite number of transcendental numbers.
| |
| | |
| ===Infinity and infinitesimals {{anchor|History of infinity and infinitesimals}}===
| |
| {{further2|[[History of infinity]]}}
| |
| The earliest known conception of mathematical [[infinity]] appears in the [[Yajur Veda]], an ancient Indian script, which at one point states, "If you remove a part from infinity or add a part to infinity, still what remains is infinity." Infinity was a popular topic of philosophical study among the [[Jain]] mathematicians c. 400 BC. They distinguished between five types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually.
| |
| | |
| [[Aristotle]] defined the traditional Western notion of mathematical infinity. He distinguished between [[actual infinity]] and [[potential infinity]]—the general consensus being that only the latter had true value. [[Galileo Galilei]]'s ''[[Two New Sciences]]'' discussed the idea of [[bijection|one-to-one correspondences]] between infinite sets. But the next major advance in the theory was made by [[Georg Cantor]]; in 1895 he published a book about his new [[set theory]], introducing, among other things, [[transfinite number]]s and formulating the [[continuum hypothesis]]. This was the first mathematical model that represented infinity by numbers and gave rules for operating with these infinite numbers.
| |
| | |
| In the 1960s, [[Abraham Robinson]] showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of nonstandard analysis. The system of [[hyperreal numbers]] represents a rigorous method of treating the ideas about [[infinity|infinite]] and [[infinitesimal]] numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of [[infinitesimal calculus]] by [[Isaac Newton|Newton]] and [[Gottfried Leibniz|Leibniz]].
| |
| | |
| A modern geometrical version of infinity is given by [[projective geometry]], which introduces "ideal points at infinity", one for each spatial direction. Each family of parallel lines in a given direction is postulated to converge to the corresponding ideal point. This is closely related to the idea of vanishing points in [[perspective (graphical)|perspective]] drawing.
| |
| | |
| ===Complex numbers {{anchor|History of complex numbers}}===
| |
| {{further2|[[History of complex numbers]]}}
| |
| The earliest fleeting reference to square roots of negative numbers occurred in the work of the mathematician and inventor [[Heron of Alexandria]] in the {{nowrap|1st century AD}}, when he considered the volume of an impossible [[frustum]] of a [[pyramid]]. They became more prominent when in the 16th century closed formulas for the roots of third and fourth degree polynomials were discovered by Italian mathematicians such as [[Niccolo Fontana Tartaglia]] and [[Gerolamo Cardano]]. It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers.
| |
| | |
| This was doubly unsettling since they did not even consider negative numbers to be on firm ground at the time. When [[René Descartes]] coined the term "imaginary" for these quantities in 1637, he intended it as derogatory. (See [[imaginary number]] for a discussion of the "reality" of complex numbers.) A further source of confusion was that the equation
| |
| :<math>\left ( \sqrt{-1}\right )^2 =\sqrt{-1}\sqrt{-1}=-1</math>
| |
| seemed capriciously inconsistent with the algebraic identity
| |
| :<math>\sqrt{a}\sqrt{b}=\sqrt{ab},</math>
| |
| which is valid for positive real numbers ''a'' and ''b'', and was also used in complex number calculations with one of ''a'', ''b'' positive and the other negative. The incorrect use of this identity, and the related identity
| |
| :<math>\frac{1}{\sqrt{a}}=\sqrt{\frac{1}{a}}</math>
| |
| in the case when both ''a'' and ''b'' are negative even bedeviled [[Euler]]. This difficulty eventually led him to the convention of using the special symbol ''i'' in place of <math>\sqrt{-1}</math> to guard against this mistake.
| |
| | |
| The 18th century saw the work of [[Abraham de Moivre]] and [[Leonhard Euler]]. [[De Moivre's formula]] (1730) states:
| |
| :<math>(\cos \theta + i\sin \theta)^{n} = \cos n \theta + i\sin n \theta \,</math>
| |
| and to Euler (1748) [[Euler's formula]] of [[complex analysis]]:
| |
| :<math>\cos \theta + i\sin \theta = e ^{i\theta }. \,</math>
| |
| | |
| The existence of complex numbers was not completely accepted until [[Caspar Wessel]] described the geometrical interpretation in 1799. [[Carl Friedrich Gauss]] rediscovered and popularized it several years later, and as a result the theory of complex numbers received a notable expansion. The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in [[John Wallis|Wallis]]'s ''De Algebra tractatus''.
| |
| | |
| Also in 1799, Gauss provided the first generally accepted proof of the [[fundamental theorem of algebra]], showing that every polynomial over the complex numbers has a full set of solutions in that realm. The general acceptance of the theory of complex numbers is due to the labors of [[Augustin Louis Cauchy]] and [[Niels Henrik Abel]], and especially the latter, who was the first to boldly use complex numbers with a success that is well-known.
| |
| | |
| [[Carl Friedrich Gauss|Gauss]] studied [[Gaussian integer|complex numbers of the form]] {{nowrap|''a'' + ''bi''}}, where ''a'' and ''b'' are integral, or rational (and ''i'' is one of the two roots of {{nowrap|''x''<sup>2</sup> + 1 {{=}} 0}}). His student, [[Gotthold Eisenstein]], studied the type {{nowrap|''a'' + ''bω''}}, where ''ω'' is a complex root of {{nowrap|''x''<sup>3</sup> − 1 {{=}} 0.}} Other such classes (called [[cyclotomic fields]]) of complex numbers derive from the [[roots of unity]] {{nowrap|''x''<sup>''k''</sup> − 1 {{=}} 0}} for higher values of ''k''. This generalization is largely due to [[Ernst Kummer]], who also invented [[ideal number]]s, which were expressed as geometrical entities by [[Felix Klein]] in 1893. The general theory of fields was created by [[Évariste Galois]], who studied the fields generated by the roots of any polynomial equation {{nowrap|''F''(''x'') {{=}} 0}}.
| |
| | |
| In 1850 [[Victor Alexandre Puiseux]] took the key step of distinguishing between poles and branch points, and introduced the concept of [[mathematical singularity|essential singular points]]. This eventually led to the concept of the [[extended complex plane]].
| |
| | |
| ===Prime numbers {{anchor|History of prime numbers}}===
| |
| [[Prime number]]s have been studied throughout recorded history. Euclid devoted one book of the ''Elements'' to the theory of primes; in it he proved the infinitude of the primes and the [[fundamental theorem of arithmetic]], and presented the [[Euclidean algorithm]] for finding the [[greatest common divisor]] of two numbers.
| |
| | |
| In 240 BC, [[Eratosthenes]] used the [[Sieve of Eratosthenes]] to quickly isolate prime numbers. But most further development of the theory of primes in Europe dates to the [[Renaissance]] and later eras.
| |
| | |
| In 1796, [[Adrien-Marie Legendre]] conjectured the [[prime number theorem]], describing the asymptotic distribution of primes. Other results concerning the distribution of the primes include Euler's proof that the sum of the reciprocals of the primes diverges, and the [[Goldbach conjecture]], which claims that any sufficiently large even number is the sum of two primes. Yet another conjecture related to the distribution of prime numbers is the [[Riemann hypothesis]], formulated by [[Bernhard Riemann]] in 1859. The [[prime number theorem]] was finally proved by [[Jacques Hadamard]] and [[Charles de la Vallée-Poussin]] in 1896. Goldbach and Riemann's conjectures remain unproven and unrefuted.
| |
| | |
| ==See also==
| |
| {{Commons|Numbers}}
| |
| {{columns-list|3|
| |
| ;Numerals by culture
| |
| *[[Arabic numerals]]
| |
| *[[Babylonian numerals]]
| |
| *[[Egyptian numerals]]
| |
| *[[Greek numerals]]
| |
| *[[Hebrew numerals]]
| |
| *[[Indian numerals]]
| |
| *[[Roman numerals]]
| |
| | |
| ;Other related topics
| |
| *[[Concrete number]]
| |
| *[[Floating point|Floating point representation in computers]]
| |
| *[[The Foundations of Arithmetic]]
| |
| *[[Integer (computer science)]]
| |
| *[[List of numbers]]
| |
| *[[List of numbers in various languages]]
| |
| *[[Literal (computer science)]]
| |
| *[[Mathematical constant]]s
| |
| *[[Mathematical constants and functions]]
| |
| *[[Mythical number]]s
| |
| *[[Number sign]]
| |
| *[[Numerical cognition]]
| |
| *[[Numero sign]]
| |
| *[[Orders of magnitude]]
| |
| *[[Physical constant]]s
| |
| *[[Subitizing and counting]]}}
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| *[[Tobias Dantzig]], ''Number, the language of science; a critical survey written for the cultured non-mathematician'', New York, The Macmillan company, 1930.
| |
| *Erich Friedman, ''[http://www.stetson.edu/~efriedma/numbers.html What's special about this number?]''
| |
| *Steven Galovich, ''Introduction to Mathematical Structures'', Harcourt Brace Javanovich, 23 January 1989, ISBN 0-15-543468-3.
| |
| *[[Paul Halmos]], ''Naive Set Theory'', Springer, 1974, ISBN 0-387-90092-6.
| |
| *[[Morris Kline]], ''Mathematical Thought from Ancient to Modern Times'', Oxford University Press, 1972.
| |
| *[[Alfred North Whitehead]] and [[Bertrand Russell]], ''[[Principia Mathematica]]'' to *56, Cambridge University Press, 1910.
| |
| *George I. Sanchez, Arithmetic in Maya, Austin-Texas, 1961.
| |
| | |
| ==External links==
| |
| {{Wiktionary|number}}
| |
| {{wikiversity|Primary mathematics:Numbers}}
| |
| *{{SpringerEOM |title=Number |id=Number |oldid=11869 |first=V.I. |last=Nechaev}}
| |
| *{{cite web|last=Tallant|first=Jonathan|title=Do Numbers Exist?|url=http://www.numberphile.com/videos/exist.html|work=Numberphile|publisher=[[Brady Haran]]}}
| |
| *[http://freepages.history.rootsweb.com/~catshaman/13comp/0numer.htm Mesopotamian and Germanic numbers]
| |
| *[http://www.bbc.co.uk/radio4/history/inourtime/inourtime_20060309.shtml BBC Radio 4, In Our Time: Negative Numbers]
| |
| *[http://www.gresham.ac.uk/event.asp?PageId=45&EventId=622 '4000 Years of Numbers'], lecture by Robin Wilson, 07/11/07, [[Gresham College]] (available for download as MP3 or MP4, and as a text file).
| |
| | |
| *{{cite web |url=http://www.npr.org/blogs/krulwich/2011/07/22/138493147/what-s-your-favorite-number-world-wide-survey-v1 |title=What's the World's Favorite Number? |accessdate=2011-09-17 |date=2011-06-22}}; {{cite web |url=http://www.npr.org/templates/transcript/transcript.php?storyId=139797360 |title=Cuddling With 9, Smooching With 8, Winking At 7 |date=2011-08-11 |accessdate=2011-09-17}}
| |
| | |
| {{Number systems}}
| |
| | |
| [[Category:Group theory]]
| |
| [[Category:Numbers| ]]
| |
| [[Category:Mathematical objects]]
| |
| | |
| {{Link GA|ca}}
| |
| {{Link GA|pl}}
| |