A Number is a Mathematical Object Used in Counting and Measuring

8/3/2019 A Number is a Mathematical Object Used in Counting and Measuring

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A number is amathematical objectused incountingandmeasuring. A notational symbol whichrepresents a number is called anumeral, but in common usage the word number is used for boththe abstract object and the symbol, as well as for thewordfor the number. In addition to their usein counting and measuring, numerals are often used for labels (telephone numbers), for ordering(serial numbers), and for codes (e.g.,ISBNs). Inmathematics, the definition of number has been

extended over the years to include such numbers aszero,negative numbers,rational numbers,irrational numbers, andcomplex numbers.

Certain procedures which take one or more numbers as input and produce a number as output arecalled numericaloperations.Unary operationstake a single input number and produce a singleoutput number. For example, thesuccessoroperation adds one to an integer, thus the successorof 4 is 5. More common arebinary operationswhich take two input numbers and produce asingle output number. Examples of binary operations includeaddition,subtraction,multiplication,division, andexponentiation. The study of numerical operations is calledarithmetic.

Classification of numbers

See also:List of types of numbers

Different types of numbers are used in different cases. Numbers can be classified intosets, callednumber systems. (For different methods of expressing numbers with symbols, such as theRomannumerals, seenumeral systems.)

Number systems

Natural (0), 1, 2, 3, 4, 5, 6, 7, ..., n

Integers n, ..., 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, ..., nPositive

integers1, 2, 3, 4, 5, ..., n

Rationalab where a and b are integers and b is not zero

RealThe limit of a convergent sequence of rational

numbers

Complexa + bi where a and b are real numbers and i is the

square root of 1

[edit] Natural numbers

Main article:Natural number

The most familiar numbers are thenatural numbersor counting numbers: one, two, three, and soon. Traditionally, the sequence of natural numbers started with 1 (0 was not even considered anumber for theAncient Greeks.) However, in the 19th century,set theoristsand othermathematicians started the convention of including 0 (cardinalityof theempty set, i.e. 0
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elements, where 0 is thus the smallestcardinal number) in the set of natural numbers. Themathematical symbolfor the set of all natural numbers is N, also written .

In thebase tennumeral system, in almost universal use today by humans for arithmeticoperations, the symbols for natural numbers are written using tendigits: 0, 1, 2, 3, 4, 5, 6, 7, 8,

and 9. In this base ten system, the rightmost digit of a natural number has a place valueof one,and every other digit has a place value ten times that of the place value of the digit to its right.

Inset theory, which is capable of acting as an axiomatic foundation for modern mathematics[1],natural numbers can be represented by classes of equivalent sets. For instance, the number 3 canbe represented as the class of all sets that have exactly three elements. Alternatively, inPeanoArithmetic, the number 3 is represented as sss0, where s is the "successor" function (i.e., 3 is thethird successor of 0). Many different representations are possible; all that is needed to formallyrepresent 3 is to inscribe a certain symbol or pattern of symbols three times.

[edit] Integers

Main article:Integer

Thenegativeof a natural number is defined as a number that produces zero when it is added tothe number. Negative numbers are said to be less than zero and are usually written with anegative sign (also called aminus sign). As an example, the negative of 7 is written 7, and 7 +(7) = 0. When thesetof the negatives of the natural numbers is combined with the naturalnumbers, the result is defined as the set of integer numbers, also calledintegers,Z also written.Here the letter Z comes fromGermanZahl, meaning "number". The set of integers forms aringwith operations addition and multiplication.[2]

[edit] Rational numbers

Main article:Rational number

A rational number is a number that can be expressed as afractionwith an integer numerator anda non-zero natural number denominator. Fractions are written as two numbers, the numerator andthe denominator, with a dividing bar between them. In the fraction written mn or

m represents equal parts, where n equal parts of that size make up one whole. Two differentfractions may correspond to the same rational number; for example 12 and24 are equal, that is:

If theabsolute valueofm is greater than n, then the absolute value of the fraction is greater than1. Fractions can be greater than, less than, or equal to 1 and can also be positive, negative, or
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zero. The set of all rational numbers includes the integers, since every integer can be written as afraction with denominator 1. For example 7 can be written 71. The symbol for the rationalnumbers is Q (forquotient), also written .

[edit] Real numbers

Main article:Real number

The real numbers include all of the measuring numbers. Real numbers are usually written usingdecimalnumerals, in which adecimal pointis placed to the right of the digit with place valueone. Each digit to the right of the decimal point has a place value one-tenth of the place value ofthe digit to its left. Thus

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 UKand a number of other countries, the decimal point is represented by aperiod, whereas incontinental Europe and certain other countries the decimal point is represented by acomma. Zerois often written as 0.0 when necessary to indicate that it is to be treated as a real number ratherthan as an integer; in the US and UK a number between 1 and 1 is always written with a lead ingzero so that the decimal point is more apparent, in other countries it may not be. Negative realnumbers are written with a precedingminus sign:

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. Adecimal that can be written as a fraction either ends (terminates) or foreverrepeats, because it isthe answer to a problem in division. Thus the real number 0.5 can be written as 12 and the realnumber 0.333... (forever repeating threes, otherwise written 0.3) can be written as 13. On theother hand, the real number (pi), the ratio of thecircumferenceof any circle to itsdiameter, is

Since the decimal neither ends nor forever repeats, it cannot be written as a fraction, and is anexample of an irrational number. Other irrational numbers include

(thesquare root of 2, that is, the positive number whose square is 2).

Thus 1.0 and0.999...are two different decimal numerals representing the natural number 1.There are infinitely many other ways of representing the number 1, for example 22,

33, 1.00,

1.000, and so on.
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Every real number is either rational or irrational. Every real number corresponds to a point on thenumber line. The real numbers also have an important but highly technical property called theleast upper boundproperty. The symbol for the real numbers is R or .

When a real number represents ameasurement, there is always amargin of error. This is often

indicated byroundingortruncatinga decimal, so that digits that suggest a greater accuracy thanthe measurement itself are removed. The remaining digits are calledsignificant digits. Forexample, measurements with a ruler can seldom be made without a margin of error of at least0.01 meters. If the sides of arectangleare measured as 1.23 meters and 4.56 meters, thenmultiplication gives an area for the rectangle of 5.6088 square meters. Since only the first twodigits after the decimal place are significant, this is usually rounded to 5.61.

Inabstract algebra, it can be shown that anycompleteordered fieldis isomorphic to the realnumbers. The real numbers are not, however, analgebraically closed field.

[edit] Complex numbers

Main article:Complex number

Moving to a greater level of abstraction, the real numbers can be extended to the complexnumbers. This set of numbers arose, historically, from trying to find closed formulas for the rootsofcubicandquarticpolynomials. This led to expressions involving the square roots of negativenumbers, and eventually to the definition of a new number: the square root of negative one,denoted byi, a symbol assigned byLeonhard Euler, and called theimaginary unit. The complexnumbers consist of all numbers of the form

where a and b are real numbers. In the expression a + bi, the real number a is called the real partand bi is called the imaginary part. If the real part of a complex number is zero, then the numberis called animaginary numberor is referred to aspurely imaginary; if the imaginary part is zero,then the number is a real number. Thus the real numbers are asubsetof the complex numbers. Ifthe real and imaginary parts of a complex number are both integers, then the number is called aGaussian integer. The symbol for the complex numbers is C or .

Inabstract algebra, the complex numbers are an example of analgebraically closed field,meaning that everypolynomialwith complexcoefficientscan befactoredinto linear factors.Like the real number system, the complex number system is afieldand iscomplete, but unlike

the real numbers it is notordered. 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 numberslack thetrichotomy property.

Complex numbers correspond to points on thecomplex plane, sometimes called the Argandplane.
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_unithttp://en.wikipedia.org/wiki/Quartic_equationhttp://en.wikipedia.org/wiki/Cubic_equationhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/w/index.php?title=Number&action=edit&section=6http://en.wikipedia.org/wiki/Algebraically_closed_fieldhttp://en.wikipedia.org/wiki/Ordered_fieldhttp://en.wikipedia.org/wiki/Completeness_(order_theory)http://en.wikipedia.org/wiki/Abstract_algebrahttp://en.wikipedia.org/wiki/Rectanglehttp://en.wikipedia.org/wiki/Significant_digitshttp://en.wikipedia.org/wiki/Truncatehttp://en.wikipedia.org/wiki/Roundinghttp://en.wikipedia.org/wiki/Margin_of_errorhttp://en.wikipedia.org/wiki/Measurementhttp://en.wikipedia.org/wiki/Least_upper_boundhttp://en.wikipedia.org/wiki/Number_line


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Each of the number systems mentioned above is aproper subsetof the next number system.Symbolically, .

[edit] Computable numbers

Main article:Computable number

Moving to problems of computation, thecomputable numbersare determined in the set of thereal numbers. The computable numbers, also known as the recursive numbers or the computablereals, are thereal numbersthat can be computed to within any desired precision by a finite,terminatingalgorithm. Equivalent definitions can be given using-recursive functions,Turingmachinesor-calculusas the formal representation of algorithms. The computable numbersform areal closed fieldand can be used in the place of real numbers for many, but not all,mathematical purposes.

[edit] Other types

Hyperrealand hypercomplex numbers are used innon-standard analysis. The hyperreals, ornonstandard reals (usually denoted as *R), denote anordered fieldwhich is a properextensionofthe ordered field ofreal numbersR and which satisfies thetransfer principle. This principleallows truefirst orderstatements about R to be reinterpreted as true first order statements about*R.

Superrealandsurreal numbersextend the real numbers by adding infinitesimally small numbersand infinitely large numbers, but still formfields.

Thep-adic numbersmay 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 systemwhich results depends on whatbaseis used for the digits: any base is possible, but a system withthe best mathematical properties is obtained when the base is a prime number.

For dealing with infinite collections, the natural numbers have been generalized to theordinalnumbersand to thecardinal numbers. The former gives the ordering of the collection, while thelatter gives its size. For the finite set, the ordinal and cardinal numbers are equivalent, but theydiffer in the infinite case.

Arelation numberis defined as the class ofrelationsconsisting of all those relations that aresimilar to one member of the class.[3]

Sets of numbers that are not subsets of the complex numbers are sometimes called hypercomplexnumbers. They include thequaternionsH, invented by SirWilliam Rowan Hamilton, in whichmultiplication is notcommutative, and theoctonions, in which multiplication is notassociative.Elements offunction fieldsof non-zerocharacteristicbehave in some ways like numbers and areoften regarded as numbers by number theorists.

[edit] Specific uses
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ociativehttp://en.wikipedia.org/wiki/Function_field_of_an_algebraic_varietyhttp://en.wikipedia.org/wiki/Function_field_of_an_algebraic_varietyhttp://en.wikipedia.org/wiki/Function_field_of_an_algebraic_varietyhttp://en.wikipedia.org/wiki/Characteristic_(algebra)http://en.wikipedia.org/wiki/Characteristic_(algebra)http://en.wikipedia.org/wiki/Characteristic_(algebra)http://en.wikipedia.org/w/index.php?title=Number&action=edit&section=9http://en.wikipedia.org/w/index.php?title=Number&action=edit&section=9http://en.wikipedia.org/w/index.php?title=Number&action=edit&section=9http://en.wikipedia.org/w/index.php?title=Number&action=edit&section=9http://en.wikipedia.org/wiki/Characteristic_(algebra)http://en.wikipedia.org/wiki/Function_field_of_an_algebraic_varietyhttp://en.wikipedia.org/wiki/Associativehttp://en.wikipedia.org/wiki/Octonionhttp://en.wikipedia.org/wiki/Commutativehttp://en.wikipedia.org/wiki/William_Rowan_Hamiltonhttp://en.wikipedia.org/wiki/Quaternionhttp://en.wikipedia.org/wiki/Hypercomplex_numberhttp://en.wikipedia.org/wiki/Hypercomplex_numberhttp://en.wikipedia.org/wiki/Number#cite_note-2http://en.wikipedia.org/wiki/Relation_(mathematics)http://en.wikipedia.org/w/index.php?title=Relation_number&action=edit&redlink=1http://en.wikipedia.org/wiki/Cardinal_numberhttp://en.wikipedia.org/wiki/Ordinal_numberhttp://en.wikipedia.org/wiki/Ordinal_numberhttp://en.wikipedia.org/wiki/Prime_numberhttp://en.wikipedia.org/wiki/Radixhttp://en.wikipedia.org/wiki/P-adic_numberhttp://en.wikipedia.org/wiki/Field_(mathematics)http://en.wikipedia.org/wiki/Surreal_numberhttp://en.wikipedia.org/wiki/Superreal_numberhttp://en.wikipedia.org/wiki/First-order_logichttp://en.wikipedia.org/wiki/Transfer_principlehttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Field_extensionhttp://en.wikipedia.org/wiki/Ordered_fieldhttp://en.wikipedia.org/wiki/Non-standard_analysishttp://en.wikipedia.org/wiki/Hyperreal_numberhttp://en.wikipedia.org/w/index.php?title=Number&action=edit&section=8http://en.wikipedia.org/wiki/Real_closed_fieldhttp://en.wikipedia.org/wiki/Lambda_calculushttp://en.wikipedia.org/wiki/Turing_machineshttp://en.wikipedia.org/wiki/Turing_machineshttp://en.wikipedia.org/wiki/Mu-recursive_functionhttp://en.wikipedia.org/wiki/Algorithmhttp://en.wikipedia.org/wiki/Real_numbershttp://en.wikipedia.org/wiki/Computable_numberhttp://en.wikipedia.org/wiki/Computable_numberhttp://en.wikipedia.org/w/index.php?title=Number&action=edit&section=7http://en.wikipedia.org/wiki/Proper_subset


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There are also other sets of numbers with specialized uses. Some are subsets of the complexnumbers. For example,algebraic numbersare the roots ofpolynomialswith rationalcoefficients.Complex numbers that are not algebraic are calledtranscendental numbers.

Aneven numberis 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 "divisible".) A formal definition of an oddnumber is that it is an integer of the form n = 2k+ 1, where kis an integer. An even number hasthe form n = 2kwhere kis aninteger.

Aperfect numberis defined as apositive integerwhich is the sum of its proper positivedivisors,that is, the sum of the positive divisors not including the number itself. Equivalently, a perfectnumber is a number that is half the sum of all of its positive divisors, or (n) = 2 n. The firstperfect number is6, because 1, 2, and 3 are its proper positive divisors and 1 + 2 + 3 = 6. Thenext perfect number is28= 1 + 2 + 4 + 7 + 14. The next perfect numbers are496and8128(sequenceA000396inOEIS). These first four perfect numbers were the only ones known to

earlyGreek mathematics.Afigurate numberis a number that can be represented as a regular and discretegeometricpattern(e.g. dots). If the pattern ispolytopic, the figurate is labeled a polytopic number, and may be apolygonal numberor a polyhedral number. Polytopic numbers for r = 2, 3, and 4 are:

P2(n) = 12n(n + 1)(triangular numbers) P3(n) = 16n(n + 1)(n + 2)(tetrahedral numbers) P4(n) = 124n(n + 1)(n + 2)(n + 3)(pentatopic numbers)

[edit] NumeralsNumbers should be distinguished fromnumerals, the symbols used to represent numbers. Boyershowed that Egyptians created the first ciphered numeral system.[citation needed] Greeks followed bymapping their counting numbers onto Ionian and Doric alpabets. The number five can be

represented by both the base ten numeral '5', by theRoman numeral'' and ciphered letters.Notations used to represent numbers are discussed in the articlenumeral systems. An importantdevelopment in the history of numerals was the development of a positional system, like moderndecimals, which can represent very large numbers. The Roman numerals require extra symbolsfor larger numbers.

[edit] History

[edit] The first use of numbers

It is speculated that the first known use of numbers dates back to around 35,000 BC. Bones andother artifacts have been discovered with marks cut into them which many consider to be tally
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e_numberhttp://en.wikipedia.org/wiki/Greek_mathematicshttp://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttp://oeis.org/A000396http://en.wikipedia.org/wiki/8128_(number)http://en.wikipedia.org/wiki/496_(number)http://en.wikipedia.org/wiki/28_(number)http://en.wikipedia.org/wiki/6_(number)http://en.wikipedia.org/wiki/Divisor_functionhttp://en.wikipedia.org/wiki/Divisorhttp://en.wikipedia.org/wiki/Positive_integerhttp://en.wikipedia.org/wiki/Perfect_numberhttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Divisibilityhttp://en.wikipedia.org/wiki/Even_numberhttp://en.wikipedia.org/wiki/Transcendental_numbershttp://en.wikipedia.org/wiki/Coefficientshttp://en.wikipedia.org/wiki/Polynomialshttp://en.wikipedia.org/wiki/Algebraic_numbers


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marks. The uses of these tally marks may have been for counting elapsed time, such as numbersof days, or keeping records of quantities, such as of animals.

Tallying systems have no concept of place value (such as in the currently used decimal notation),which limit its representation of large numbers but are nonetheless considered the first kind of

abstract numeral system.The first known system with place value was theMesopotamian base 60system (ca.3400 BC)and the earliest known base 10 system dates to 3100 BC inEgypt.[4]

[edit] Zero

Further information:History of zero

The use of zero as a number should be distinguished from its use as a placeholder numeral inplace-value systems. Many ancient texts used zero. Babylonian and Egyptian texts used it.

Egyptians used the word nfrto denote zero balance in double entry accounting entries. Indiantexts used aSanskritword Shunya to refer to the concept ofvoid; in mathematics texts this wordwould often be used to refer to the number zero.[5]

Records show that theAncient Greeksseemed unsure about the status of zero as a number: theyasked themselves "how can 'nothing' be something?" leading to interestingphilosophicaland, bythe Medieval period, religious arguments about the nature and existence of zero and thevacuum.TheparadoxesofZeno of Eleadepend in large part on the uncertain interpretation of zero. (Theancient Greeks even questioned whether1was a number.)

The lateOlmecpeople of south-centralMexicobegan to use a true zero (a shellglyph) in the

New World possibly by the 4th century BC but certainly by 40 BC, which became an integralpart ofMaya numeralsand theMaya calendar. Mayan arithmetic used base 4 and base 5 writtenas base 20. Sanchez in 1961 reported a base 4, base 5 'finger' abacus.

By 130 AD,Ptolemy, influenced byHipparchusand the Babylonians, was using a symbol forzero (a small circle with a long overbar) within a sexagesimal numeral system otherwise usingalphabeticGreek numerals. Because it was used alone, not as just a placeholder, thisHellenisticzerowas the first documenteduse of a true zero in the Old World. In laterByzantinemanuscriptsof his Syntaxis Mathematica (Almagest), the Hellenistic zero had morphed into theGreek letteromicron(otherwise meaning 70).

Another true zero was used in tables alongsideRoman numeralsby 525 (first known use byDionysius Exiguus), but as a word, nulla meaning nothing, not as a symbol. When divisionproduced zero as a remainder, nihil, also meaning nothing, was used. These medieval zeros wereused by all future medievalcomputists(calculators ofEaster). An isolated use of their initial, N,was used in a table of Roman numerals byBedeor a colleague about 725, a true zero symbol.

An early documented use of the zero byBrahmagupta(in theBrahmasphutasiddhanta) dates to628. He treated zero as a number and discussed operations involving it, includingdivision. By
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this time (the 7th century) the concept had clearly reached Cambodia asKhmer numerals, anddocumentation shows the idea later spreading toChinaand theIslamic world.

[edit] Negative numbers

Further information:History of negative numbers

The abstract concept of negative numbers was recognised as early as 100 BC 50 BC. TheChineseNine Chapters on the Mathematical Art(Chinese: Jiu-zhang Suanshu) containsmethods for finding the areas of figures; red rods were used to denote positive coefficients, blackfor negative.[6]This is the earliest known mention of negative numbers in the East; the firstreference in a Western work was in the 3rd century inGreece.Diophantusreferred to theequation equivalent to 4x + 20 = 0 (the solution would be negative) inArithmetica, saying thatthe equation gave an absurd result.

During the 600s, negative numbers were in use inIndiato represent debts. Diophantus previous

reference was discussed more explicitly by Indian mathematicianBrahmagupta, inBrahma-Sphuta-Siddhanta628, who used negative numbers to produce the general formquadraticformulathat remains in use today. However, in the 12th century in India, Bhaskaragivesnegative 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."

Europeanmathematicians, for the most part, resisted the concept of negative numbers until the17th century, althoughFibonacciallowed negative solutions in financial problems where theycould be interpreted as debts (chapter 13 ofLiber Abaci, 1202) and later as losses (in Flos). Atthe same time, the Chinese were indicating negative numbers either by drawing a diagonal strokethrough the right-most nonzero digit of the corresponding positive number's numeral.[7]The first

use of negative numbers in a European work was byChuquetduring the 15th century. He usedthem asexponents, but referred to them as absurd numbers.

As recently as the 18th century, it was common practice to ignore any negative results returnedby equations on the assumption that they were meaningless, just asRen Descartesdid withnegative solutions in aCartesian coordinate system.

[edit] Rational numbers

It is likely that the concept of fractional numbers dates toprehistoric times. TheAncientEgyptiansused theirEgyptian fractionnotation for rational numbers in mathematical texts such

as theRhind Mathematical Papyrusand theKahun Papyrus. Classical Greek and Indianmathematicians made studies of the theory of rational numbers, as part of the general study ofnumber theory. The best known of these isEuclid's Elements, dating to roughly 300 BC. Of theIndian texts, the most relevant is theSthananga Sutra, which also covers number theory as part ofa general study of mathematics.

The concept ofdecimal fractionsis closely linked with decimal place-value notation; the twoseem to have developed in tandem. For example, it is common for the Jain math sutras to include
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calculations of decimal-fraction approximations topior thesquare root of two. Similarly,Babylonian math texts had always used sexagesimal (base 60) fractions with great frequency.

Irrational numbers

Further information:History of irrational numbers

The earliest known use of irrational numbers was in theIndianSulba Sutrascomposed between800500 BC.[8]The first existence proofs of irrational numbers is usually attributed toPythagoras, more specifically to thePythagoreanHippasus of Metapontum, who produced a(most likely geometrical) proof of the irrationality of thesquare root of 2. The story goes thatHippasus discovered irrational numbers when trying to represent the square root of 2 as afraction. However Pythagoras believed in the absoluteness of numbers, and could not accept theexistence of irrational numbers. He could not disprove their existence through logic, but hisbeliefs would not accept the existence of irrational numbers and so he sentenced Hippasus todeath by drowning.

The sixteenth century saw the final acceptance by Europeans ofnegativeintegral andfractionalnumbers. The seventeenth century saw decimal fractions with the modern notation quitegenerally used by mathematicians. But it was not until the nineteenth century that the irrationalswere separated into algebraic and transcendental parts, and a scientific study of theory ofirrationals was taken once more. It had remained almost dormant sinceEuclid. The year 1872saw the publication of the theories ofKarl Weierstrass(by his pupilKossak),Heine(Crelle, 74),Georg Cantor(Annalen, 5), andRichard Dedekind.Mrayhad taken in 1869 the same point ofdeparture as Heine, but the theory is generally referred to the year 1872. Weierstrass's methodhas been completely set forth bySalvatore Pincherle(1880), and Dedekind's has receivedadditional prominence through the author's later work (1888) and the recent endorsement byPaul

Tannery(1894). Weierstrass, Cantor, and Heine base their theories on infinite series, whileDedekind founds his on the idea of acut (Schnitt)in the system ofreal numbers, separating allrational numbersinto two groups having certain characteristic properties. The subject hasreceived later contributions at the hands of Weierstrass,Kronecker(Crelle, 101), and Mray.

Continued fractions, closely related to irrational numbers (and due to Cataldi, 1613), receivedattention at the hands ofEuler, and at the opening of the nineteenth century were brought intoprominence through the writings ofJoseph Louis Lagrange. Other noteworthy contributions havebeen made byDruckenmller(1837), Kunze (1857), Lemke (1870), and Gnther (1872). Ramus(1855) first connected the subject withdeterminants, resulting, with the subsequent contributionsof Heine,Mbius, and Gnther, 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

Further information:History of pi

The first results concerning transcendental numbers wereLambert's1761 proof that cannot berational, and also that en is irrational ifn is rational (unless n = 0). (The constantewas first
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referred to inNapier's1618 work onlogarithms.)Legendreextended this proof to show that isnot the square root of a rational number. The search for roots ofquinticand higher degreeequations was an important development, theAbelRuffini theorem(Ruffini1799,Abel1824)showed that they could not be solved byradicals(formula involving only arithmetical operationsand roots). Hence it was necessary to consider the wider set ofalgebraic numbers(all solutions

to polynomial equations).Galois(1832) linked polynomial equations togroup theorygiving riseto the field ofGalois theory.

Even the set of algebraic numbers was not sufficient and the full set of real number includestranscendental numbers,[9]the existence of which was first established byLiouville(1844, 1851).Hermiteproved in 1873 that e is transcendental andLindemannproved in 1882 that istranscendental. FinallyCantorshows that the set of allreal numbersisuncountably infinitebutthe set of allalgebraic numbersiscountably infinite, so there is an uncountably infinite numberof transcendental numbers.

[edit] Infinity and infinitesimals

Further information:History of infinity

The earliest known conception of mathematicalinfinityappears in theYajur Veda, an ancientIndian script, which at one point states "if you remove a part from infinity or add a part toinfinity, still what remains is infinity". Infinity was a popular topic of philosophical study amongtheJainmathematicians circa 400 BC. They distinguished between five types of infinity: infinitein one and two directions, infinite in area, infinite everywhere, and infinite perpetually.

In the West, the traditional notion of mathematical infinity was defined byAristotle, whodistinguished betweenactual infinityandpotential infinity; the general consensus being that only

the latter had true value.Galileo'sTwo New Sciencesdiscussed the idea ofone-to-onecorrespondencesbetween infinite sets. But the next major advance in the theory was made byGeorg Cantor; in 1895 he published a book about his newset theory, introducing, among otherthings,transfinite numbersand formulating thecontinuum hypothesis. This was the firstmathematical model that represented infinity by numbers and gave rules for operating with theseinfinite numbers.

In the 1960s,Abraham Robinsonshowed how infinitely large and infinitesimal numbers can berigorously defined and used to develop the field of nonstandard analysis. The system ofhyperreal numbersrepresents a rigorous method of treating the ideas aboutinfiniteandinfinitesimalnumbers that had been used casually by mathematicians, scientists, and engineers

ever since the invention ofinfinitesimal calculusbyNewtonandLeibniz.A modern geometrical version of infinity is given byprojective geometry, which introduces"ideal points at infinity," one for each spatial direction. Each family of parallel lines in a givendirection is postulated to converge to the corresponding ideal point. This is closely related to theidea of vanishing points inperspectivedrawing.

[edit] Complex numbers
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Further information:History of complex numbers

The earliest fleeting reference to square roots of negative numbers occurred in the work of themathematician and inventorHeron of Alexandriain the 1st century AD, when he considered thevolume of an impossiblefrustumof apyramid. They became more prominent when in the 16th

century closed formulas for the roots of third and fourth degree polynomials were discovered byItalian mathematicians such asNiccolo Fontana TartagliaandGerolamo Cardano. It was soonrealized that these formulas, even if one was only interested in real solutions, sometimes requiredthe manipulation of square roots of negative numbers.

This was doubly unsettling since they did not even consider negative numbers to be on firmground at the time. The term "imaginary" for these quantities was coined byRen Descartesin1637 and was meant to be derogatory (seeimaginary numberfor a discussion of the "reality" ofcomplex numbers). A further source of confusion wa

Documents

A Number is a Mathematical Object Used in Counting and Measuring