Anumber is amathematical object used tocount,measure, and label. The most basic examples are thenatural numbers: 1, 2, 3, 4, 5, and so forth.[1] Individual numbers can be represented in language with number words or by dedicated symbols callednumerals; for example, "five" is a number word and "5" is the corresponding numeral. As only a limited list of symbols can be memorized, anumeral system is used to represent any number in an organized way. The most common representation is theHindu–Arabic numeral system, which can display anynon-negative integer using a combination of ten symbols, callednumerical digits.[2][a] Numerals can be used for counting (as withcardinal number of a collection orset), labels (as with telephone numbers), for ordering (as withserial numbers), and for codes (as withISBNs). In common usage, anumeral is not clearly distinguished from thenumber that it represents.
Viewing the concept of zero as a number required a fundamental shift in philosophy, identifying nothingness with a value. During the 19th century, mathematicians began to develop the various systems now calledalgebraic structures, which share certain properties of numbers, and may be seen as extending the concept. Some algebraic structures are explicitly referred to as numbers (such as thep-adic numbers andhypercomplex numbers) while others are not, but this is more a matter of convention than a mathematical distinction.[7]
Bones and other artifacts have been discovered with marks cut into them that many believe aretally marks.[9] Some historians suggest that theLebombo bone (dated about 43,000 years ago) and theIshango bone (dated about 22,000 to 30,000 years ago) are the oldest arithmetic artifacts but this interpretation is disputed.[10][11] These tally marks may have been used for counting elapsed time, such as numbers of days, lunar cycles or keeping records ofquantities, such as of animals.[12] Aperceptual system for quantity thought to underlie numeracy, is shared with other species, a phylogenetic distribution suggesting it would have existed before the emergence of language.[13][10]
A tallying system has no concept of place value (as in moderndecimal notation), which limits its representation of large numbers. Nonetheless, tallying systems are considered the first kind of abstract numeral system.[14]
The earliest unambiguous numbers in the archaeological record are theMesopotamian base 60 (sexagesimal) system (c. 3400 BC);[15] place value emerged in the 3rd millennium BCE.[16] The earliest known base 10 system dates to 3100 BC inEgypt.[17] A Babylonian clay tablet dated to1900–1600 BC provides an estimate of the circumference of a circle to its diameter of = 3.125, possibly the oldest approximation of π.[18]
Numbers should be distinguished fromnumerals, the symbols used to represent numbers. The Egyptians invented the first ciphered numeral system, and the Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets.[19] (However, in 300 BC,Archimedes first demonstrated the use of apositional numeral system to display extremely large numbers inThe Sand Reckoner.[20]) Roman numerals, a system that used combinations of letters from the Roman alphabet, remained dominant in Europe until the spread of theHindu–Arabic numeral system around the late 14th century, and the Hindu–Arabic numeral system remains the most common system for representing numbers in the world today.[21] The key to the effectiveness of the system was the symbol forzero, which was developed by ancientIndian mathematicians around 500 AD.[21]
Zero
The number 605 inKhmer numerals, from an inscription from 683 AD. Early use of zero as a decimal figure.[22]
The first known recorded use ofzero as aninteger dates to AD 628, and appeared in theBrāhmasphuṭasiddhānta, the main work of theIndian mathematicianBrahmagupta. He is usually considered the first to formulate the mathematical concept of zero. Brahmagupta treated 0 as a number and discussed operations involving it, includingdivision by zero. He gave rules of using zero with negative and positive numbers, such as "zero plus a positive number is a positive number, and a negative number plus zero is the negative number". By this time (the 7th century), the concept had clearly reached Cambodia in the form ofKhmer numerals,[22] and documentation shows the idea later spreading to China and theIslamic world. The concept began reaching Europe through Islamic sources around the year 1000.[23]
There are other uses of zero before Brahmagupta, though the documentation is not as complete as it is in theBrāhmasphuṭasiddhānta.[24] The earliest uses of zero was as simply a placeholder numeral inplace-value systems, representing another number as was done by the Babylonians.[25] Many ancient texts used 0, including Babylonian and Egyptian texts. Egyptians used the wordnfr to denote zero balance indouble entry accounting. Indian texts used aSanskrit wordShunye orshunya to refer to the concept ofvoid. In mathematics texts this word often refers to the number zero.[26] In a similar vein,Pāṇini (5th century BC) used the null (zero) operator in theAshtadhyayi,[24] an early example of analgebraic grammar for the Sanskrit language (also seePingala).
Records show that the Ancient Greeks seemed unsure about the status of 0 as a number: they asked themselves "How can 'nothing' be something?" leading to interestingphilosophical and, by the Medieval period, religious arguments about the nature and existence of 0 and the vacuum. Theparadoxes ofZeno of Elea depend in part on the uncertain interpretation of 0.[27] (The ancient Greeks even questioned whether 1 was a number.[28])
The lateOlmec people of south-central Mexico began to use a placeholder symbol for zero, a shellglyph, in the New World, by 38 BC.[30] It would be theMaya who developed zero as a cardinal number, employing it in theirnumeral system and in theMaya calendar.[31] Maya used abase 20 numerical system by combining a number of dots (base 5) with a number of bars (base 4).[29]George I. Sánchez in 1961 reported a base 4, base 5 "finger" abacus.[32][33]
By 130 AD,Ptolemy, influenced byHipparchus and the Babylonians, was using a symbol for 0 (a small circle with a long overbar) within asexagesimal numeral system otherwise using alphabeticGreek numerals.[34] Because it was used alone, not as just a placeholder, thisHellenistic zero was the firstdocumented use of a true zero in the Old World. In laterByzantine manuscripts of hisSyntaxis Mathematica (Almagest), the Hellenistic zero had morphed into the Greek letterOmicron[35] (otherwise meaning 70 inisopsephy[36]).
A true zero was used in tables alongsideRoman numerals by 525 (first known use byDionysius Exiguus), but as a word,nulla meaningnothing, not as a symbol.[37] When division produced 0 as a remainder,nihil, also meaningnothing, was used. These medieval zeros were used by all future medievalcomputists (calculators of Easter).[citation needed] An isolated use of their initial, N, was used in a table of Roman numerals byBede or a colleague about 725, a true zero symbol.
The abstract concept of negative numbers was recognized as early as 100–50 BC in China.The Nine Chapters on the Mathematical Art contains methods for finding the areas of figures; red rods were used to denote positivecoefficients, black for negative.[38] The first reference in a Western work was in the 3rd century AD in Greece.Diophantus referred to the equation equivalent to4x + 20 = 0 (the solution is negative) inArithmetica, saying that the equation gave an absurd result.[39]
During the 600s, negative numbers were in use in India to represent debts. Diophantus' previous reference was discussed more explicitly by Indian mathematicianBrahmagupta, inBrāhmasphuṭasiddhānta in 628, who used negative numbers to produce the general formquadratic formula that remains in use today. However, in the 12th century in India,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".[40]
European mathematicians, for the most part, resisted the concept of negative numbers until the 17th century,[40] althoughFibonacci allowed negative solutions in financial problems where they could be interpreted as debts (chapter 13 ofLiber Abaci, 1202) and later as losses (inFlos).René Descartes called them false roots as they cropped up in algebraic polynomials yet he found a way to swap true roots and false roots as well.[41] At the same time, the Chinese were indicating negative numbers by drawing a diagonal stroke through the right-most non-zero digit of the corresponding positive number's numeral.[42] An early European experimenter with negative numbers wasNicolas Chuquet during the 15th century. He used them asexponents,[43] 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.
Rational numbers
Archimedes' method of confining the value of pi using the perimeters of circumscribed and inscribed polygons results in rational number estimates.[44]
It is likely that the concept of fractional numbers dates toprehistoric times.[40] TheAncient Egyptians used theirEgyptian fraction notation for rational numbers in mathematical texts such as theRhind Mathematical Papyrus and theKahun Papyrus.[45] The Rhind Papyrus includes an example of deriving the area of a circle from its diameter, which yields an estimate of π as ≈ 3.16049....[18] Classical Greek and Indian mathematicians made studies of the theory of rational numbers, as part of the general study ofnumber theory.[46][40] A particularly influential example of these isEuclid'sElements, dating to roughly 300 BC.[47] Of the Indian texts, the most relevant is theSthananga Sutra, which also covers number theory as part of a general study of mathematics.[40]
The concept ofdecimal fractions is closely linked with decimal place-value notation; the two seem to have developed in tandem. For example, it is common for the Jain mathsutra to include calculations of decimal-fraction approximations topi or thesquare root of 2.[citation needed] Similarly, Babylonian math texts used sexagesimal (base 60) fractions.[48]
Real numbers and irrational numbers
Babylonian clay tablet YBC 7289 showing the first foursexagesimalplace values for an approximation of the square root of 2:[49]1 24 51 10
The Babylonians, as early as 1800 BCE, demonstrated numerical approximations of irrational quantities such as √2 on clay tablets, with an accuracy analogous to six decimal places, as in the tabletYBC 7289.[49] These values were primarily used for practical calculations in geometry and land measurement.[50] There were practical approximations of irrational numbers in theIndianShulba Sutras composed between 800 and 500 BC.[51]
The first existence proofs of irrational numbers is usually attributed toPythagoras, more specifically to thePythagoreanHippasus, who produced a (most likely geometrical) proof of the irrationality of thesquare root of 2.[52] 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. He could not disprove the existence of irrational numbers, or accept them, so according to legend, he sentenced Hippasus to death by drowning, to impede the spread of this unsettling news.[53]
The 16th century brought final European acceptance of negative integers and fractional numbers. By the 17th century, mathematicians generally used decimal fractions with modern notation. The concept ofreal numbers was introduced in the 17th century byRené Descartes.[54] While studyingcompound interest, in 1683Jacob Bernoulli found that as the compounding intervals grew ever shorter, the rate ofexponential growth converged to abase of 2.71828...; this key mathematical constant would later be namedEuler's number (e).[55] Irrational numbers began to be studied systematically in the 18th century, withLeonhard Euler who proved that the irrational numbers are those numbers whosesimple continued fractions is not finite and that Euler's number (e) is irrational.[56] Theirrationality ofπ was proved in 1761 byJohann Lambert.[57]
In mathematics,infinity is considered an abstractconcept rather than a number; instead of being "greater than any number", infinite is the property of having no end.[65] The earliest known conception of mathematical infinity appears in theYajurveda, an ancient Indian script, which at one point states, "If [the whole] was subtract from [the whole], the leftover will still be [the whole]".[66] Infinity was a popular topic of philosophical study among theJain 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.[67]
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.[71][72] The system ofhyperreal numbers represents a rigorous method of treating the ideas aboutinfinite andinfinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention ofinfinitesimal calculus byNewton andLeibniz.[73]
A modern geometrical version of infinity is given byprojective geometry, which introduces "idealpoints 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 inperspective drawing.[74]
The earliest fleeting reference to square roots of negative numbers occurred in the work of the mathematician and inventorHeron of Alexandria in the1st century AD, when he considered the volume of an impossiblefrustum of apyramid.[75] 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 asNiccolò Fontana Tartaglia andGerolamo 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.[76]
This was doubly unsettling since they did not even consider negative numbers to be on firm ground at the time.René Descartes is sometimes credited with coining the term "imaginary" for these quantities in 1637, intending it as derogatory.[77] (Seeimaginary number for a discussion of the "reality" of complex numbers.) A further source of confusion was that the equation
seemed capriciously inconsistent with the algebraic identity
which is valid for positive real numbersa andb, and was also used in complex number calculations with one ofa,b positive and the other negative. The incorrect use of this identity, and the related identity
in the case when botha andb are negative even bedeviledEuler.[78] This difficulty eventually led him to the convention of using the special symboli in place of to guard against this mistake.
showing a profound connection between the most fundamental numbers in mathematics.[80]
The existence of complex numbers was not completely accepted untilCaspar 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.[81] However, the idea of the graphic representation of complex numbers had appeared as early as 1685, inWallis'sDe algebra tractatus.[82]
In the same year, Gauss provided the first generally accepted proof of thefundamental theorem of algebra,[citation needed] showing that every polynomial over the complex numbers has a full set of solutions in that realm. Gauss studied complex numbers of the forma +bi, wherea andb are integers (now calledGaussian integers) or rational numbers.[83] His student,Gotthold Eisenstein, studied the typea +bω, whereω is a complex root ofx3 − 1 = 0 (now calledEisenstein integers). Other such classes (calledcyclotomic fields) of complex numbers derive from theroots of unityxk − 1 = 0 for higher values ofk. This generalization is largely due toErnst Kummer, who also inventedideal numbers, which were expressed as geometrical entities byFelix Klein in 1893.
Prime numbers may have been studied throughout recorded history. They are natural numbers that are not a product of two smaller natural numbers. It has been suggested that the Ishango bone includes a list of the prime numbers between 10 and 20.[84] The Rhind papyrus display different forms for prime numbers. But the formal study of prime numbers is first documented by the ancient Greek. Euclid devoted one book of theElements to the theory of primes; in it he proved the infinitude of the primes and thefundamental theorem of arithmetic, and presented theEuclidean algorithm for finding thegreatest common divisor of two numbers.[85]
In 240 BC,Eratosthenes used theSieve of Eratosthenes to quickly isolate prime numbers. But most further development of the theory of primes in Europe dates to theRenaissance and later eras. At around 1000 AD,Ibn al-Haytham discoveredWilson's theorem.Ibn al-Banna' al-Marrakushi found a way to speed up the Sieve of Eratosthenes by only testing up to the square root of the number. Fibonacci communicated Islamic mathematical contributions to Europe, and in 1202 was the first to describe the method oftrial division.[85]
A Shanghai apartment is missing floors 0, 4, 13, and 14
Numbers have held cultural, symbolic and religious significance throughout history and in many cultures.[12][90][91][92] In Ancient Greece,number symbolism heavily influenced the development ofGreek mathematics, stimulating the investigation of many problems in number theory which are still of interest today.[12] According toPlato,Pythagoreans attributed specific characteristics and meaning to particular numbers, and believed that "things themselves are numbers".[93]
Folktales in different cultures exhibit preferences for particular numbers, with three and seven holding special significance in European culture, while four and five are more prominent in Chinese folktales.[94] Numbers are sometimes associated with luck: in Western society, thenumber 13 is consideredunlucky while in Chinese culture thenumber eight is considered auspicious.[95]
Main classification
"Number system" redirects here. For systems which express numbers, seeNumeral system.
Numbers can be classified intosets, callednumber sets ornumber systems, such as thenatural numbers and thereal numbers. The main number systems are as follows:[96]
a +bi wherea andb are real numbers andi is a formal square root of −1
Each of these number systems is asubset of the next one. So, for example, a rational number is also a real number, and every real number is also a complex number. This can be expressed symbolically as:[96]
The most familiar numbers are thenatural numbers (sometimes called whole numbers 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 theorists and other mathematicians started including 0 (cardinality of theempty set, i.e. 0 elements, where 0 is thus the smallestcardinal number) in the set of natural numbers.[97][98] Today, various mathematicians use the term to describe both sets, including 0 or not. Themathematical symbol for the set of all natural numbers isN, also written,[96] and sometimes[99] or[100] when it is necessary to indicate whether the set should start with 0 or 1, respectively.
In thebase 10 numeral system, in almost universal use today for mathematical operations, the symbols for natural numbers are written using tendigits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Theradix or base is the number of unique numerical digits, including zero, that a numeral system uses to represent numbers (for the decimal system, the radix is 10). In this base 10 system, the rightmost digit of a natural number has aplace value of 1, and every other digit has a place value ten times that of the place value of the digit to its right.[101]
Inset theory, which is capable of acting as an axiomatic foundation for modern mathematics,[102] 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, inPeano Arithmetic, the number 3 is represented asS(S(S(0))), whereS is the "successor" function (i.e., 3 is the third successor of 0).[103] 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.
TheInca Empire used knotted strings, orquipus, for numerical records and other uses[104]
The 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 (aminus sign). As an example, the negative of 7 is written −7, and7 + (−7) = 0. When theset of negative numbers is combined with the set of natural numbers (including 0), the result is defined as the set ofintegers,Z also written.[96] Here the letter Z comes from German Zahl'number'. The set of integers forms aring with the operations addition and multiplication.[105]
The natural numbers form a subset of the integers. As there is no common standard for the inclusion or not of zero in the natural numbers, the natural numbers without zero are commonly referred to aspositive integers, and the natural numbers with zero are referred to asnon-negative integers.
A rational number is a number that can be expressed as a fraction with an integer numerator and a positive integer denominator. Negative denominators are allowed, but are commonly avoided, as every rational number is equal to a fraction with positive denominator.[106] Fractions are written as two integers, the numerator and the denominator, with a dividing bar between them. The fractionm/n representsm parts of a whole divided inton equal parts. Two different fractions may correspond to the same rational number; for example1/2 and2/4 are equal, that is:[107]
If theabsolute value ofm is greater thann (supposed to be positive), then the absolute value of the fraction is greater than 1 and it is termed animproper or top-heavy fraction.[108] Fractions can be greater than, less than, or equal to 1[106] 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 −7/1. The symbol for the rational numbers isQ (forquotient), also written.[96]
The symbol for the real numbers isR, also written as[96] They include all the measuring numbers. Every real number corresponds to a point on thenumber line. The treatment of negative real numbers is according to the general rules of arithmetic and their denotation is simply prefixing the corresponding positive numeral by aminus sign, e.g. −123.456.
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. For example, 123.456 represents123456/1000, or, in words, one hundred, two tens, three ones, four tenths, five hundredths, and six thousandths. A real number can be expressed by a finite number of decimal digits only if it is rational and itsfractional part has a denominator whose prime factors are 2 or 5 or both, because these are the prime factors of 10, the base of the decimal system. Thus, for example, one half is 0.5, one fifth is 0.2, one-tenth is 0.1, and one fiftieth is 0.02.
Repeating decimal
If the fractional part of a real number has an infinite sequence of digits that follows a cyclical pattern, it can be written with an ellipsis or another notation that indicates the repeating pattern. Such a decimal is called arepeating decimal. Thus3/11 can be written as 0.272727..., with an ellipsis to indicate that the pattern continues. Forever repeating 27s are also written as 0.27.[109] These recurring decimals, including therepetition of zeroes, denote exactly the rational numbers, i.e., all rational numbers are real numbers, but it is not the case that every real number is rational.[110]
For a fractional part with a repeating decimal of consecutive nines, they may be replaced by incrementing the last digit before the nines. Thus, 3.7399999999... or 3.739 is equivalent to 3.74. A fractional part with an unlimited number of 0s can be rewritten by dropping the 0s to the right of the rightmost nonzero digit.[111] Just as the same fraction can be written in more than one way, the same real number may have more than one decimal representation. For example,0.999..., 1.0,[111] 1.00, 1.000, ..., all represent the natural number 1.
Irrational numbers
For real numbers that are not rational numbers, representing them as decimals would require an infinite sequence of varying digits to the right of the decimal point. These real numbers are calledirrational. A famous irrational real number is theπ,[57] the ratio of thecircumference of any circle to itsdiameter. When pi is written as
as it sometimes is, the ellipsis does not mean that the decimals repeat (they do not), but rather that there is no end to them. It has been proved thatπ is irrational. Another well-known number, proven to be an irrational real number, is
thesquare root of 2, that is, the unique positive real number whose square is 2.[112] Both these numbers have been approximated (by computer) to trillions( 1 trillion = 1012 = 1,000,000,000,000 ) of digits.[113][114]
Euclid'sgolden ratio, defined here by is to as is to, is an irrational number 𝜙=1.61803… that tends to appear in many aspects of both art and science.[115]
Almost all real numbers are irrational and therefore have no repeating patterns and hence no corresponding decimal numeral. They can only beapproximated bydecimal numerals, denotingrounded ortruncated real numbers, in which adecimal point is placed to the right of the digit with place value 1. Any rounded or truncated number is necessarily a rational number, of which there are onlycountably many.
All measurements are, by their nature, approximations, and always have amargin of error. Thus 123.456 is considered an approximation of any real number in theinterval:
when rounding to three decimals, or of any real number in the interval:
when truncating after the third decimal. Digits that suggest a greater accuracy than the measurement itself does, should be removed. The remaining digits are then calledsignificant digits.
For example, measurements with a ruler can seldom be made without a margin of error of at least 0.001m. If the sides of a rectangle are measured as 1.23 m and 4.56 m, then multiplication gives an area for the rectangle between5.614591 m2 and5.603011 m2. Since not even the second digit after the decimal place is preserved, the subsequent digits are notsignificant. Therefore, the result is usually rounded to5.61 m2.[116]
Set theory
The real numbers have an important but highly technical property called theleast upper bound property.
Moving to a greater level of abstraction, the real numbers can be extended to thecomplex numbers. The complete solution set of a polynomial ofdegree two or higher can include the square roots of negative numbers. (An example is.[118]) To conveniently represent this, thesquare root of −1 is denoted byi, a symbol assigned byLeonhard Euler called theimaginary unit.[120] Hence, complex numbers consist of all values of the form:
If the real part of a complex number is 0, then the number is called animaginary number or is referred to aspurely imaginary;[120] if the imaginary part is 0, then the number is a real number. Thus the real numbers are asubset of the complex numbers. If the real and imaginary parts of a complex number are both integers, then the number is called aGaussian integer.[121] The symbol for the complex numbers isC or.[96]
Aneven number is an integer that is "evenly divisible" by two, that isdivisible by two without remainder; anodd number is an integer that is not even.[126] (The old-fashioned term "evenly divisible" is now almost always shortened to "divisible".) This property of an integer is called theparity.[127] Any odd numbern may be constructed by the formulan = 2k + 1, for a suitable integerk. Starting withk = 0, the first non-negative odd numbers are {1, 3, 5, 7, ...}. Any even numberm has the formm = 2k wherek is again aninteger. Similarly, the first non-negative even numbers are {0, 2, 4, 6, ...}. The product of an even number with an integer is another even number; only the product of an odd number with an odd number is another odd number.[126]
Largest known prime numbers by year since 1951[128]
Aprime number, often shortened to justprime, is an integer greater than 1 that is not the product of two smaller positive integers. The first few prime numbers are 2, 3, 5, 7, and 11. There is no such simple formula as for odd and even numbers to generate the prime numbers. A special class are theMersenne primes, which are prime numbers of the form2n − 1, wheren is a positive integer. These hold many records for the largest prime numbers discovered.[129]
The study of primes have led to many questions, only some of which have been answered. The study of these questions belongs tonumber theory.[12]Goldbach's conjecture is an example of a still unanswered question: "Is every even number the sum of two primes?"[88] One answered question, as to whether every integer greater than one is a product of primes in only one way, except for a rearrangement of the primes, was confirmed; this proven claim is called thefundamental theorem of arithmetic. A proof appears inEuclid's Elements.[85]
A period is a complex number that can be expressed as anintegral of analgebraic function over an algebraicdomain. The periods are a class of numbers which includes, alongside the algebraic numbers, many well knownmathematical constants such as thenumberπ. The set of periods form a countablering and bridge the gap between algebraic and transcendental numbers.[137][138]
The periods can be extended by permitting the integrand to be the product of an algebraic function and theexponential of an algebraic function. This gives another countable ring: the exponential periods. Thenumbere as well asEuler's constant are exponential periods.[137][139]
Constructible numbers
Motivated by the classical problems ofconstructions with straightedge and compass, theconstructible numbers are those complex numbers whose real and imaginary parts can be constructed using straightedge and compass, starting from a given segment of unit length, in a finite number of steps.[140] A related subject isorigami numbers, which are points constructed through paper folding.[141]
Acomputable number, also known asrecursive number, is areal number such that there exists analgorithm which, given a positive numbern as input, produces the firstn digits of the computable number's decimal representation.[142] Equivalent definitions can be given usingμ-recursive functions,Turing machines orλ-calculus.[143] The computable numbers are stable for all usual arithmetic operations, including the computation of the roots of apolynomial, and thus form areal closed field that contains the realalgebraic numbers.[144]
The computable numbers may be viewed as the real numbers that may be exactly represented in a computer: a computable number is exactly represented by its first digits and a program for computing further digits. However, the computable numbers are rarely used in practice. One reason is that there is no algorithm for testing the equality of two computable numbers. More precisely, there cannot exist any algorithm which takes any computable number as an input, and decides in every case if this number is equal to zero or not.
The set of computable numbers has the same cardinality as the natural numbers. Therefore,almost all real numbers are non-computable. However, it is very difficult to produce explicitly a real number that is not computable.
Thep-adic numbers 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 whatbase is used for the digits: any base is possible, but aprime number base provides the best mathematical properties. The set of thep-adic numbers contains the rational numbers,[145][146] but is not contained in the complex numbers.
The elements of analgebraic function field over afinite field and algebraic numbers have many similar properties (seeFunction field analogy). Therefore, they are often regarded as numbers by number theorists. Thep-adic numbers play an important role in this analogy.
Higher dimensional number systems may be constructed from the real numbers in a way that generalize the construction of the complex numbers. They are sometimes calledhypercomplex numbers, and are not included in the set of complex numbers. They include thequaternions, introduced by SirWilliam Rowan Hamilton, in which multiplication is notcommutative;[147] theoctonions, in which multiplication is notassociative in addition to not being commutative;[148] and thesedenions, in which multiplication is notalternative, neither associative nor commutative.[149] The hypercomplex numbers include one real unit together with imaginary units, for whichn is a non-negative integer. For example, quaternions can generally represented using the form:
where the coefficientsa,b,c,d are real numbers, andi,j,k are 3 different imaginary units.[148]
Each hypercomplex number system is asubset of the next hypercomplex number system of double dimensions obtained via theCayley–Dickson construction.[150] For example, the 4-dimensional quaternions are a subset of the 8-dimensional octonions, which are in turn a subset of the 16-dimensional sedenions, in turn a subset of the 32-dimensionaltrigintaduonions, andad infinitum with dimensions, withn being any non-negative integer. Including the complex and real numbers and their subsets, this can be expressed symbolically as:[150]
Alternatively, starting from the real numbers, which have zero complex units, this can be expressed as
Quaternions have proven particularly useful for computation of rotations in three dimensions. For example, they are used in control systems for rockets and aircraft, as well as for robotics, computer visualization, navigation, and animation.[152] Octonions appear to have a deeper theoretical connection with physics, particularly instring theory,M-theory andsupergravity.[153]
For dealing with infinitesets, the natural numbers have been generalized to theordinal numbers and to thecardinal numbers. The former gives the ordering of the set, while the latter gives its size. For finite sets, both ordinal and cardinal numbers are identified with the natural numbers. In the infinite case, many ordinal numbers correspond to the same cardinal number.[154]
^Inlinguistics, anumeral can refer to a symbol like 5, but also to a word or a phrase that names a number, like "five hundred"; numerals include also other words representing numbers, like "dozen".
^This follows from thesubstitution property of equality, by multiplying both fractions with the product of their denominators:. Likewise, the converse is true by dividing with the product.
References
^"number, n."OED Online. Oxford University Press.Archived from the original on 4 October 2018. Retrieved16 May 2017.
^"numeral, adj. and n."OED Online. Oxford University Press.Archived from the original on 30 July 2022. Retrieved16 May 2017.
^Matson, John."The Origin of Zero".Scientific American.Archived from the original on 26 August 2017. Retrieved16 May 2017.
^Gouvêa, Fernando Q. (28 September 2008). "II.1, The Origins of Modern Mathematics".The Princeton Companion to Mathematics. Princeton University Press. p. 82.ISBN978-0-691-11880-2.Today, it is no longer that easy to decide what counts as a 'number.' The objects from the original sequence of 'integer, rational, real, and complex' are certainly numbers, but so are thep-adics. The quaternions are rarely referred to as 'numbers,' on the other hand, though they can be used to coordinatize certain mathematical notions.
^"The Ishango Bone". Institute of Natural Sciences. Retrieved23 October 2025.
^Chrisomalis, Stephen (2018). "The writing of numbers: recounting and recomposing numerical notations".Terrain.70.doi:10.4000/terrain.17506.
^Schmandt-Besserat, Denise (1992).Before Writing: From Counting to Cuneiform (2 vols). University of Texas Press.
^Robson, Eleanor (2008).Mathematics in Ancient Iraq: A Social History. Princeton University Press.
^Williams, Scott W."Egyptian Mathematical Papyri".Mathematicians of the African Diaspora. Mathematics Department, State University of New York at Buffalo.Archived from the original on 7 April 2015. Retrieved30 January 2012.
^abBulliet, Richard; Crossley, Pamela; Headrick, Daniel; Hirsch, Steven; Johnson, Lyman (2010).The Earth and Its Peoples: A Global History. Vol. 1. Cengage Learning. p. 192.ISBN978-1-4390-8474-8.Archived from the original on 28 January 2017. Retrieved16 May 2017.Indian mathematicians invented the concept of zero and developed the 'Arabic' numerals and system of place-value notation used in most parts of the world today
^Freudenhammer, Thomas (2021). "Gerbert of Aurillac and the Transmission of Arabic Numerals to Europe".Sudhoffs Archiv.105 (1):3–19.doi:10.25162/sar-2021-0001.JSTOR48636817.
^abPranoto, Iwan; Nair, Ranjit (2020)."Zero". In Nair, Rukmini Bhaya; deSouza, Peter Ronald (eds.).Keywords for India: A Conceptual Lexicon for the 21st Century. Bloomsbury Publishing. pp. 73–74.ISBN978-1-3500-3925-4.
^Nath, R. (April 2012)."The Mighty Zero"(PDF).Science Reporter:19–22. Retrieved20 October 2025.
^Roero, C. S. (2003)."Egyptian mathematics". In Grattan-Guinness, I. (ed.).Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences. A Johns Hopkins paperback. Vol. 1. JHU Press. pp. 30–36.ISBN978-0-8018-7396-6.
^Von Fritz, Kurt (April 1945). "The Discovery of Incommensurability by Hippasus of Metapontum".Annals of Mathematics.46 (2):242–264.doi:10.2307/1969021.JSTOR1969021.
^Winter, Graham (November 2007). "A Number between 2 and 3".Mathematics in School.36 (5). The Mathematical Association:30–32.JSTOR30216078.
^Cretney, Rosanna (May 2014). "The origins of Euler's early work on continued fractions".Historia Mathematica.41 (2). Elsevier:139–156.doi:10.1016/j.hm.2013.12.004.
^abLaczkovich, M. (May 1997). "On Lambert's Proof of the Irrationality of π".The American Mathematical Monthly.104 (5). Taylor & Francis, Ltd.:439–443.doi:10.2307/2974737.JSTOR2974737.
^Heine, Eduard (14 December 2009). "Die Elemente der Functionenlehre".[Crelle's] Journal für die reine und angewandte Mathematik.1872 (74):172–188.doi:10.1515/crll.1872.74.172.
^Cantor, Georg (December 1883). "Ueber unendliche, lineare Punktmannichfaltigkeiten, pt. 5".Mathematische Annalen.21 (4):545–591.doi:10.1007/BF01446819.
^Dedekind, Richard (1872).Stetigkeit und irrationale Zahlen. Braunschweig: Friedrich Vieweg & Sohn. Subsequently published in:Fricke, Robert; Noether, Emmy; Ore, Öystein, eds. (1932).Gesammelte mathematische Werke. Vol. 3. Braunschweig: Friedrich Vieweg & Sohn. pp. 315–334.
^Gilman, Daniel Coit; et al., eds. (1906)."Number".The New International Encyclopaedia. Vol. 14. Dodd, Mead. p. 676.
^abJohnson, Phillip E. (1972). "The Genesis and Development of Set Theory".The Two-Year College Mathematics Journal.3 (1). Taylor & Francis:55–62.doi:10.2307/3026799.JSTOR3026799.
^Schubring, Gert (2001). Lützen, Jesper (ed.).Argand and the Early Work on Graphical Representation: New Sources and Interpretations. Around Caspar Wessel and the Geometric Representation of Complex Numbers. Proceedings of the Wessel Symposium at The Royal Danish Academy of Sciences and Letters, Copenhagen, August 11-15 1998. Invited Papers. Mathematisk-fysiske meddelelser. Kgl. Danske Videnskabernes Selskab. pp. 140–142.ISBN978-87-7876-236-8.
^Locke, L. Leland (April–June 1912). "The Ancient Quipu, a Peruvian Knot Record".American Anthropologist, New Series.14 (2). Wiley:325–332.doi:10.1525/aa.1912.14.2.02a00070.JSTOR659935.
^Takahashi, Daisuke (July 2018). "Computation of the 100 quadrillionth hexadecimal digit of π on a cluster of Intel Xeon Phi processors".Parallel Computing.75. Elsevier:1–10.doi:10.1016/j.parco.2018.02.002.hdl:2241/00153370.
^Mansfield, Richard (October 1990). "The Irrationals are not Recursively Enumerable".Proceedings of the American Mathematical Society.110 (2):495–497.doi:10.2307/2048094.JSTOR2048094.
^Stein, Robert G. "Exploring the Gaussian Integers".The Two-Year College Mathematics Journal.7 (4):4–10.doi:10.1080/00494925.1976.11974454 (inactive 26 October 2025).{{cite journal}}: CS1 maint: DOI inactive as of October 2025 (link)
^Ziegler, Günter M. (April 2004)."The Great Prime Number Record Races"(PDF).Monthly Notices of the American Mathematical Society.51 (4). Retrieved26 October 2025.
^Vandiver, H. S. (May 1942). "An Arithmetical Theory of the Bernoulli Numbers".Transactions of the American Mathematical Society.51 (3). American Mathematical Society:502–531.doi:10.2307/1990076.JSTOR1990076.
^Pollack, Paul; Shevelev, Vladimir (December 2012). "On perfect and near-perfect numbers".Journal of Number Theory.132 (12). Elsevier:3037–3046.doi:10.1016/j.jnt.2012.06.008.
^Berthelette, Sophie; et al. (29 June 2024). "On computable numbers, with an application to the Druckproblem".Theoretical Computer Science.1002 114573. Elsevier.doi:10.1016/j.tcs.2024.114573.
^Immerman, Neil (18 October 2021)."Computability and Complexity". In Zalta, Edward N. (ed.).Stanford Encyclopedia of Philosophy (Winter 2021 ed.). Retrieved27 October 2025.
^Alling, Norman L. (1985). "Conway's field of surreal numbers".Transactions of the American Mathematical Society.287:365–386.doi:10.1090/S0002-9947-1985-0766225-7.
Further reading
Cory, Leo (2015).A Brief History of Numbers. Oxford University Press.ISBN978-0-19-870259-7.
Dantzig, Tobias (1930).Number, the language of science; a critical survey written for the cultured non-mathematician. New York: The Macmillan Company.
.jpg)
