TheInfoList

A number is a
mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical proofs ...
used to
count Count (feminine: countess) is a historical title of nobility Nobility is a social class normally ranked immediately below Royal family, royalty and found in some societies that have a formal aristocracy (class), aristocracy. Nobility ...
, , and
label A label (as distinct from signage) is a piece of paper, plastic film, cloth, metal, or other material affixed to a Packaging and labelling, container or Product (business), product, on which is written or printing, printed information or symbo ...
. The original examples are the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s , , , , and so forth. Numbers can be represented in language with
number words In linguistics Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them. The traditional areas of linguistic analysis include ph ...
. More universally, individual numbers can be represented by
symbol A symbol is a mark, sign, or word In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or pragmatics, practical meaning (linguistics), m ...

s, called ''numerals''; for example, "5" is a numeral that represents the . As only a relatively small number of symbols can be memorized, basic numerals are commonly organized in a
numeral system A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using Numerical digit, digits or other symbols in a consistent manner. The same s ...
, which is an organized way to represent any number. The most common numeral system is the
Hindu–Arabic numeral system The Hindu–Arabic numeral system or Indo-Arabic numeral system Audun HolmeGeometry: Our Cultural Heritage 2000 (also called the Arabic numeral system or Hindu numeral system) is a positional notation, positional decimal numeral system, and is t ...
, which allows for the representation of any number using a combination of ten fundamental numeric symbols, called
digit Digit may refer to: Mathematics and science * Numerical digit, as used in mathematics or computer science ** Arabic numerals, the most common modern representation of numerical digits * Digit (anatomy), one of several most distal parts of a limb ...
s. In addition to their use in counting and measuring, numerals are often used for labels (as with
telephone number A telephone number is a sequence of digits assigned to a fixed-line telephone subscriber station connected to a telephone line A telephone line or telephone circuit (or just line or circuit industrywide) is a single-user circuitCircuit ...
s), for ordering (as with
serial number A serial number is a unique identifier A unique identifier (UID) is an identifier that is guaranteed to be unique among all identifiers used for those objects and for a specific purpose. The concept was formalized early in the development o ...

s), and for codes (as with s). In common usage, a ''numeral'' is not clearly distinguished from the ''number'' that it represents. In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the notion of a number has been extended over the centuries to include ,
negative number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s,
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s such as
one half One half is the irreducible fraction An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction in which the numerator and denominator are integer An integer (from the Latin wikt:integer#Lati ...
$\left\left(\tfrac\right\right)$,
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s such as the
square root of 2 The square root of 2 (approximately 1.4142) is a positive real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they ...
$\left\left(\sqrt\right\right)$ and

, and
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s which extend the real numbers with a square root of (and its combinations with real numbers by adding or subtracting its multiples).
Calculation A calculation is a deliberate process that transforms one or more inputs into one or more results. The term is used in a variety of senses, from the very definite arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#A ...

s with numbers are done with
arithmetical operations Arithmetic (from the Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 ...
, the most familiar being
addition Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ...

,
subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...

,
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

,
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting o ...
, and
exponentiation Exponentiation is a mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...
. Their study or usage is called
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
, a term which may also refer to
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

, the study of the properties of numbers. Besides their practical uses, numbers have cultural significance throughout the world.Gilsdorf, Thomas E. ''Introduction to Cultural Mathematics: With Case Studies in the Otomies and Incas'', John Wiley & Sons, Feb 24, 2012.Restivo, S. ''Mathematics in Society and History'', Springer Science & Business Media, Nov 30, 1992. For example, in Western society, the is often regarded as unlucky, and "
a million ''A Million'' () is 2009 South Korean thriller film. Plot Eight people enter a reality TV show to win (approximately ) if they survive 7 days in the Australian Outback. But they don't know the game is murderous trap by an insane TV director who wi ...

" may signify "a lot" rather than an exact quantity. Though it is now regarded as
pseudoscience Pseudoscience consists of statements, belief A belief is an attitude Attitude may refer to: Philosophy and psychology * Attitude (psychology) In psychology Psychology is the science of mind and behavior. Psychology include ...
, belief in a mystical significance of numbers, known as
numerology Numerology is the pseudoscientific belief in a divine or mysticism, mystical relationship between a number and one or more Coincidence#Interpretation, coinciding events. It is also the study of the numerical value of the letters in words, names, ...
, permeated ancient and medieval thought.Ore, Oystein. ''Number Theory and Its History'', Courier Dover Publications. Numerology heavily influenced the development of
Greek mathematics Greek mathematics refers to mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geomet ...
, stimulating the investigation of many problems in number theory which are still of interest today. During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers, and may be seen as extending the concept. Among the first were the
hypercomplex number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, which consist of various extensions or modifications of the
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

system. In modern mathematics, number systems ( sets) are considered important special examples of more general categories such as
rings Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to: *Ring (jewellery) A ring is a round band, usually of metal A metal (from Ancient Greek, Greek μέταλλον ''métallon'', "mine ...
and
fields File:A NASA Delta IV Heavy rocket launches the Parker Solar Probe (29097299447).jpg, FIELDS heads into space in August 2018 as part of the ''Parker Solar Probe'' FIELDS is a science instrument on the ''Parker Solar Probe'' (PSP), designed to mea ...
, and the application of the term "number" is a matter of convention, without fundamental significance.

# History

## Numerals

Numbers should be distinguished from numerals, 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. Roman numerals, a system that used combinations of letters from the Roman alphabet, remained dominant in Europe until the spread of the superior
Hindu–Arabic numeral system The Hindu–Arabic numeral system or Indo-Arabic numeral system Audun HolmeGeometry: Our Cultural Heritage 2000 (also called the Arabic numeral system or Hindu numeral system) is a positional notation, positional decimal numeral system, and is t ...
around the late 14th century, and the Hindu–Arabic numeral system remains the most common system for representing numbers in the world today. The key to the effectiveness of the system was the symbol for
zero 0 (zero) is a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

, which was developed by ancient
Indian mathematicians The chronology of Indian mathematicians spans from the Indus Valley Civilization oxen for pulling a cart and the presence of the chicken The chicken (''Gallus gallus domesticus''), a subspecies of the red junglefowl, is a type of d ...

## First use of numbers

Bones and other artifacts have been discovered with marks cut into them that many believe are
tally marks frame, Brahmi numerals (lower row) in India in the 1st century CE. Note the similarity of the first three numerals to the Chinese characters for one through three (一 二 三), plus the resemblance of both sets of numerals to horizontal tally ma ...

. 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 The decimal numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is t ...
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 Mesopotamian base 60 system ( BC) and the earliest known base 10 system dates to 3100 BC in
Egypt Egypt ( ar, مِصر, Miṣr), officially the Arab Republic of Egypt, is a transcontinental country This is a list of countries located on more than one continent A continent is one of several large landmasses. Generally identi ...

.

## Zero

The first known documented use of
zero 0 (zero) is a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

dates to AD 628, and appeared in the ''
Brāhmasphuṭasiddhānta The ''Brāhmasphuṭasiddhānta'' ("Correctly Established Doctrine of Brahma Brahma ( sa, ब्रह्मा, Brahmā) is one of the Hindu deities, principal deities of Hinduism, though his importance has declined in recent centuries. He i ...
'', the main work of the Indian mathematician
Brahmagupta Brahmagupta ( – ) was an Indian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures ...

. He treated 0 as a number and discussed operations involving it, including
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting o ...
. By this time (the 7th century) the concept had clearly reached Cambodia as
Khmer numerals Khmer numerals are the numerals A numeral is a figure, symbol, or group of figures or symbols denoting a number. It may refer to: * Numeral system used in mathematics * Numeral (linguistics), a part of speech denoting numbers (e.g. ''one'' and ...

, and documentation shows the idea later spreading to China and the
Islamic world The terms Muslim world and Islamic world commonly refer to the Islamic Islam (; ar, اَلْإِسْلَامُ, al-’Islām, "submission o God Oh God may refer to: * An exclamation; similar to "oh no", "oh yes", "oh my", "aw goodne ...
. Brahmagupta's ''Brāhmasphuṭasiddhānta'' is the first book that mentions zero as a number, hence Brahmagupta is usually considered the first to formulate the concept of 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." The ''Brāhmasphuṭasiddhānta'' is the earliest known text to treat zero as a number in its own right, rather than as simply a placeholder digit in representing another number as was done by the Babylonians or as a symbol for a lack of quantity as was done by Ptolemy and the Romans. The use of 0 as a number should be distinguished from its use as a placeholder numeral in
place-value system Positional notation (or place-value notation, or positional numeral system) denotes usually the extension to any radix, base of the Hindu–Arabic numeral system (or decimal, decimal system). More generally, a positional system is a numeral system ...
s. Many ancient texts used 0. Babylonian and Egyptian texts used it. Egyptians used the word ''nfr'' to denote zero balance in
double entry accounting Double-entry bookkeeping, in accounting, is a system of book keeping where every entry to an account requires a corresponding and opposite entry to a different account. The double-entry system has two equal and corresponding sides known as debit ...
. Indian texts used a
Sanskrit Sanskrit (; attributively , ; nominalization, nominally , , ) is a classical language of South Asia that belongs to the Indo-Aryan languages, Indo-Aryan branch of the Indo-European languages. It arose in South Asia after its predecessor langua ...

word or to refer to the concept of ''void''. In mathematics texts this word often refers to the number zero. In a similar vein,
Pāṇini (Devanagari: पाणिनि, ) was a Sanskrit Sanskrit (; attributively , ; nominalization, nominally , , ) is a classical language of South Asia that belongs to the Indo-Aryan languages, Indo-Aryan branch of the Indo-European language ...
(5th century BC) used the null (zero) operator in the '' Ashtadhyayi'', an early example of an algebraic grammar for the Sanskrit language (also see
Pingala Acharya Pingala ('; c. 3rd/2nd century BCE) was an ancient Indian poet and mathematician, and the author of the ' (also called ''Pingala-sutras''), the earliest known treatise on Sanskrit prosody. The ' is a work of eight chapters in the late Sū ...
). There are other uses of zero before Brahmagupta, though the documentation is not as complete as it is in the ''Brāhmasphuṭasiddhānta''. Records show that the
Ancient Greeks Ancient Greece ( el, Ἑλλάς, Hellás) was a civilization belonging to a period of History of Greece, Greek history from the Greek Dark Ages of the 12th–9th centuries BC to the end of Classical Antiquity, antiquity ( AD 600). This era was ...
seemed unsure about the status of 0 as a number: they asked themselves "how can 'nothing' be something?" leading to interesting
philosophical Philosophy (from , ) is the study of general and fundamental questions, such as those about existence Existence is the ability of an entity to interact with physical or mental reality Reality is the sum or aggregate of all that is real o ...

and, by the Medieval period, religious arguments about the nature and existence of 0 and the
vacuum A vacuum is a space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Gree ...

. The
paradoxes A paradox, also known as an antinomy, is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-con ...
of depend in part on the uncertain interpretation of 0. (The ancient Greeks even questioned whether  was a number.) The late
Olmec The Olmecs () were the earliest known major Mesoamerica Mesoamerica is a historical region and cultural area in North America North America is a continent entirely within the Northern Hemisphere and almost all within the Western ...
people of south-central Mexico began to use a symbol for zero, a shell
glyph The term glyph is used in typography Typography is the art and technique of arranging type to make written language A written language is the representation of a spoken or gestural language A language is a structured system o ...
, in the New World, possibly by the but certainly by 40 BC, which became an integral part of
Maya numerals The Mayan numeral system was the system to represent number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. ...

and the
Maya calendar The Maya calendar is a system of calendar A calendar is a system of organizing days. This is done by giving names to periods of time, typically days, weeks, months and years. A calendar date, date is the designation of a single, specifi ...
. Maya arithmetic used base 4 and base 5 written as base 20. George I. Sánchez in 1961 reported a base 4, base 5 "finger" abacus. By 130 AD,
Ptolemy Claudius Ptolemy (; grc-koi, Κλαύδιος Πτολεμαῖος, , ; la, Claudius Ptolemaeus; AD) was a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes ...
, influenced by
Hipparchus Hipparchus of Nicaea (; el, Ἵππαρχος, ''Hipparkhos'';  BC) was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of precession of the ...
and the Babylonians, was using a symbol for 0 (a small circle with a long overbar) within a
sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ...
numeral system otherwise using alphabetic
Greek numerals Greek numerals, also known as Ionic, Ionian, Milesian, or Alexandrian numerals, are a system of writing numbers using the letters of the Greek alphabet The Greek alphabet has been used to write the Greek language since the late ninth or ear ...
. Because it was used alone, not as just a placeholder, this Hellenistic zero was the first ''documented'' use of a true zero in the Old World. In later
Byzantine The Byzantine Empire, also referred to as the Eastern Roman Empire, or Byzantium, was the continuation of the Roman Empire in its eastern provinces during Late Antiquity and the Middle Ages, when its capital city was Constantinople. It survi ...

manuscripts of his ''Syntaxis Mathematica'' (''Almagest''), the Hellenistic zero had morphed into the
Greek letter The Greek alphabet has been used to write the Greek language Greek (modern , romanized: ''Elliniká'', Ancient Greek, ancient , ''Hellēnikḗ'') is an independent branch of the Indo-European languages, Indo-European family of languages, nat ...

Omicron Omicron (; uppercase Ο, lowercase ο, ) is the 15th letter of the Greek alphabet. This letter is derived from the Phoenician letter ayin: . In classical Greek, omicron represented the sound in contrast to ''omega'' and ''ου'' . In modern Gr ...

(otherwise meaning 70). Another true zero was used in tables alongside
Roman numerals Roman numerals are a that originated in and remained the usual way of writing numbers throughout Europe well into the . Numbers in this system are represented by combinations of letters from the . Modern style uses seven symbols, each with a ...

by 525 (first known use by
Dionysius Exiguus Dionysius Exiguus (Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Republ ...

), but as a word, meaning ''nothing'', not as a symbol. When division produced 0 as a remainder, , also meaning ''nothing'', was used. These medieval zeros were used by all future medieval computists (calculators of
Easter Easter,Traditional names for the feast in English are "Easter Day", as in the ''Book of Common Prayer''; "Easter Sunday", used by James Ussher''The Whole Works of the Most Rev. James Ussher, Volume 4'' and Samuel Pepys''The Diary of Samuel Pe ...

). An isolated use of their initial, N, was used in a table of Roman numerals by
Bede Bede ( ; ang, Bǣda , ; 672/326 May 735), also known as Saint Bede, The Venerable Bede, and Bede the Venerable ( la, Beda Venerabilis), was an English Benedictine The Benedictines, officially the Order of Saint Benedict ( la, Ordo Sa ...

or a colleague about 725, a true zero symbol.

## Negative numbers

The abstract concept of negative numbers was recognized as early as 100–50 BC in China. ''
The Nine Chapters on the Mathematical Art ''The Nine Chapters on the Mathematical Art'' () is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest surv ...
'' contains methods for finding the areas of figures; red rods were used to denote positive
coefficient In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, black for negative. The first reference in a Western work was in the 3rd century AD in Greece.
Diophantus Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the autho ...
referred to the equation equivalent to (the solution is negative) in ''
Arithmetica ''Arithmetica'' ( grc-gre, Ἀριθμητικά) is an Ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roug ...

'', 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 Brahmagupta ( – ) was an Indian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures ...

, in ''
Brāhmasphuṭasiddhānta The ''Brāhmasphuṭasiddhānta'' ("Correctly Established Doctrine of Brahma Brahma ( sa, ब्रह्मा, Brahmā) is one of the Hindu deities, principal deities of Hinduism, though his importance has declined in recent centuries. He i ...
'' in 628, who used negative numbers to produce the general form
quadratic formula In elementary algebra Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas a ...

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". European mathematicians, for the most part, resisted the concept of negative numbers until the 17th century, although
Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathem ...

allowed negative solutions in financial problems where they could be interpreted as debts (chapter 13 of ''
Liber Abaci ''Liber Abaci'' (also spelled as ''Liber Abbaci''; "The Book of Calculation") is a historic 1202 Latin manuscript on arithmetic by Leonardo of Pisa, posthumously known as Fibonacci. ''Liber Abaci'' was among the first Western books to describe ...
'', 1202) and later as losses (in ).
René Descartes René Descartes ( or ; ; Latinized Latinisation or Latinization can refer to: * Latinisation of names, the practice of rendering a non-Latin name in a Latin style * Latinisation in the Soviet Union, the campaign in the USSR during the 1920s ...

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. 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. The first use of negative numbers in a European work was by Nicolas Chuquet during the 15th century. He used them as
exponent Exponentiation is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europ ...
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.

## Rational numbers

It is likely that the concept of fractional numbers dates to
prehistoric times Prehistory, also known as pre-literary history, is the period of human history Human history, or world history, is the narrative of Human, humanity's past. It is understood through archaeology, anthropology, genetics, and linguistics, ...
. The
Ancient Egyptians Ancient Egypt was a civilization  A civilization (or civilisation) is a complex society A complex society is a concept that is shared by a range of disciplines including anthropology, archaeology, history and sociology to descri ...
used their
Egyptian fraction An Egyptian fraction is a finite sum of distinct unit fractionA unit fraction is a rational number written as a fraction where the numerator A fraction (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, ...
notation for rational numbers in mathematical texts such as the
Rhind Mathematical Papyrus The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum The British Museum, in the Bloomsbury Bloomsbury is a district in the West End of London The West End of London (commonly referred to as the West End ...

and the
Kahun Papyrus The Kahun Papyri (KP; also Petrie Papyri or Lahun Papyri) are a collection of ancient Egyptian texts discussing administrative, mathematical and medical topics. Its many fragments were discovered by Flinders Petrie Sir William Matthew Flind ...
. Classical Greek and Indian mathematicians made studies of the theory of rational numbers, as part of the general study of
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

. The best known of these is 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 The decimal numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is t ...
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
sutra Sutra ( sa, सूत्र, translit=sūtra, translit-std=IAST, translation=string, threadMonier Williams, ''Sanskrit English Dictionary'', Oxford University Press Oxford University Press (OUP) is the university press 200px, The Pitt Build ...
to include calculations of decimal-fraction approximations to or the
square root of 2 The square root of 2 (approximately 1.4142) is a positive real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they ...
. Similarly, Babylonian math texts used sexagesimal (base 60) fractions with great frequency.

## Irrational numbers

The earliest known use of irrational numbers was in the
Indian Indian or Indians refers to people or things related to India, or to the indigenous people of the Americas, or Aboriginal Australians until the 19th century. People South Asia * Indian people, people of Indian nationality, or people who come ...
Sulba Sutras The ''Shulba Sutras'' or ''Śulbasūtras'' (Sanskrit: शुल्बसूत्र; ': "string, cord, rope") are sutra texts belonging to the Śrauta ritual and containing geometry related to vedi (altar), fire-altar construction. Purpose an ...
composed between 800 and 500 BC. The first existence proofs of irrational numbers is usually attributed to
Pythagoras Pythagoras of Samos, or simply ; in Ionian Greek () was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graec ...

, more specifically to the
Pythagorean Pythagorean, meaning of or pertaining to the ancient Ionian mathematician, philosopher, and music theorist Pythagoras, may refer to: Philosophy * Pythagoreanism, the esoteric and metaphysical beliefs purported to have been held by Pythagoras * Neo ...
Hippasus of Metapontum Hippasus of Metapontum (; grc-gre, Ἵππασος ὁ Μεταποντῖνος, ''Híppasos''; c. 530 – c. 450 BC) was a Greeks, Greek philosopher and early follower of Pythagoras. Little is known about his life or his beliefs, but he is some ...
, who produced a (most likely geometrical) proof of the irrationality of the
square root of 2 The square root of 2 (approximately 1.4142) is a positive real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they ...
. 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, and so, allegedly and frequently reported, he sentenced Hippasus to death by drowning, to impede spreading of this disconcerting news. The 16th century brought final European acceptance of negative integral and 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 the scientific study of irrationals. It had remained almost dormant since
Euclid Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

. In 1872, the publication of the theories of
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematics, mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university withou ...

(by his pupil E. Kossak),
Eduard Heine Heinrich Eduard Heine (16 March 1821 – October 1881) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of German ...
,
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence be ...
, and
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory In algebra, ring theory is the study of ring (mathematics), rings ...
was brought about. In 1869, Charles 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s, separating all
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s into two groups having certain characteristic properties. The subject has received later contributions at the hands of Weierstrass, Leopold Kronecker, Kronecker, and Méray. 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. Continued fractions, 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 first connected the subject with determinants, resulting, with the subsequent contributions of Heine, August Ferdinand Möbius, Möbius, and Günther, in the theory of .

## Transcendental numbers and reals

The existence of transcendental numbers 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 showed that the set of all
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s is uncountable, uncountably infinite but the set of all algebraic numbers is countable, countably infinite, so there is an uncountably infinite number of transcendental numbers.

## Infinity and infinitesimals

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. The symbol $\text$ is often used to represent an infinite quantity. 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 Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence be ...
; in 1895 he published a book about his new set theory, introducing, among other things, transfinite numbers and formulating the continuum hypothesis. 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

The earliest fleeting reference to square roots of negative numbers occurred in the work of the mathematician and inventor Heron of Alexandria in the , 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 Niccolò 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 René Descartes ( or ; ; Latinized Latinisation or Latinization can refer to: * Latinisation of names, the practice of rendering a non-Latin name in a Latin style * Latinisation in the Soviet Union, the campaign in the USSR during the 1920s ...

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 :$\left \left( \sqrt\right \right)^2 =\sqrt\sqrt=-1$ seemed capriciously inconsistent with the algebraic identity :$\sqrt\sqrt=\sqrt,$ 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 :$\frac=\sqrt$ 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 $\sqrt$ to guard against this mistake. The 18th century saw the work of Abraham de Moivre and Leonhard Euler. De Moivre's formula (1730) states: :$\left(\cos \theta + i\sin \theta\right)^ = \cos n \theta + i\sin n \theta$ while Euler's formula of complex analysis (1748) gave us: :$\cos \theta + i\sin \theta = e ^.$ 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 , where ''a'' and ''b'' are integral, or rational (and ''i'' is one of the two roots of ). His student, Gotthold Eisenstein, studied the type , where ''ω'' is a complex root of Other such classes (called cyclotomic fields) of complex numbers derive from the roots of unity for higher values of ''k''. This generalization is largely due to Ernst Kummer, who also invented ideal numbers, which were expressed as geometrical entities by Felix Klein in 1893. 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

Prime numbers 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.

# Main classification

Numbers can be classified into set (mathematics), sets, called number systems, such as the natural numbers and the real numbers. The major categories of numbers are as follows: There is generally no problem in identifying each number system with a proper subset of the next one (by abuse of notation#Equality vs. isomorphism, abuse of notation), because each of these number systems is canonical isomorphism, canonically isomorphic to a proper subset of the next one. The resulting hierarchy allows, for example, to talk, formally correctly, about real numbers that are rational numbers, and is expressed symbolically by writing :$\mathbb \subset \mathbb \subset \mathbb \subset \mathbb \subset \mathbb$.

## Natural numbers

The most familiar numbers are the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s (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 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. 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 $\mathbb$, and sometimes $\mathbb_0$ or $\mathbb_1$ 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. The Radix, radix 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 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, 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

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 (a minus sign). As an example, the negative of 7 is written −7, and . When the set (mathematics), set of negative numbers is combined with the set of natural numbers (including 0), the result is defined as the set of integers, Z also written Blackboard bold, $\mathbb$. Here the letter Z comes . The set of integers forms a ring (mathematics), ring with the operations addition and multiplication. 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 as positive integers, and the natural numbers with zero are referred to as non-negative integers.

## Rational numbers

A rational number is a number that can be expressed as a fraction (mathematics), 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. Fractions are written as two integers, the numerator and the denominator, with a dividing bar between them. The fraction represents ''m'' parts of a whole divided into ''n'' equal parts. Two different fractions may correspond to the same rational number; for example and are equal, that is: : $= .$ In general, :$=$ if and only if $= .$ If the absolute value of ''m'' is greater than ''n'' (supposed to be positive), 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 . The symbol for the rational numbers is Q (for ''quotient''), also written Blackboard bold, $\mathbb$.

## Real numbers

The symbol for the real numbers is R, also written as $\mathbb.$ They include all the measuring numbers. Every real number corresponds to a point on the number line. The following paragraph will focus primarily on positive real numbers. 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 a minus sign, e.g. −123.456. Most real numbers can only be ''approximated'' by
decimal The decimal numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is t ...
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. For example, 123.456 represents , 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 its fractional 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. Representing other real numbers as decimals would require an infinite sequence of digits to the right of the decimal point. If this infinite sequence of digits follows a pattern, it can be written with an ellipsis or another notation that indicates the repeating pattern. Such a decimal is called a repeating decimal. Thus can be written as 0.333..., with an ellipsis to indicate that the pattern continues. Forever repeating 3s are also written as 0.. It turns out that these repeating decimals (including the Trailing zero, repetition of zeroes) denote exactly the rational numbers, i.e., all rational numbers are also real numbers, but it is not the case that every real number is rational. A real number that is not rational is called irrational number, irrational. A famous irrational real number is the number

, the ratio of the circumference of any circle to its diameter. When pi is written as :$\pi = 3.14159265358979\dots,$ 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 that proof that pi is irrational, is irrational. Another well-known number, proven to be an irrational real number, is :$\sqrt = 1.41421356237\dots,$ the
square root of 2 The square root of 2 (approximately 1.4142) is a positive real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they ...
, that is, the unique positive real number whose square is 2. Both these numbers have been approximated (by computer) to trillions of digits. Not only these prominent examples but almost all real numbers are irrational and therefore have no repeating patterns and hence no corresponding decimal numeral. They can only be approximated by decimal numerals, denoting rounding, rounded or truncation, truncated real numbers. Any rounded or truncated number is necessarily a rational number, of which there are only countably many. All measurements are, by their nature, approximations, and always have a margin of error. Thus 123.456 is considered an approximation of any real number greater or equal to and strictly less than (rounding to 3 decimals), or of any real number greater or equal to and strictly less than (truncation after the 3. decimal). Digits that suggest a greater accuracy than the measurement itself does, should be removed. The remaining digits are then called significant digits. For example, measurements with a ruler can seldom be made without a margin of error of at least 0.001 Metre, m. If the sides of a rectangle are measured as 1.23 m and 4.56 m, then multiplication gives an area for the rectangle between and . Since not even the second digit after the decimal place is preserved, the following digits are not ''significant''. Therefore, the result is usually rounded to 5.61. 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, 1.00, 1.000, ..., all represent the natural number 1. A given real number has only the following decimal representations: an approximation to some finite number of decimal places, an approximation in which a pattern is established that continues for an unlimited number of decimal places or an exact value with only finitely many decimal places. In this last case, the last non-zero digit may be replaced by the digit one smaller followed by an unlimited number of 9's, or the last non-zero digit may be followed by an unlimited number of zeros. Thus the exact real number 3.74 can also be written 3.7399999999... and 3.74000000000.... Similarly, a decimal numeral with an unlimited number of 0's can be rewritten by dropping the 0's to the right of the decimal place, and a decimal numeral with an unlimited number of 9's can be rewritten by increasing the rightmost -9 digit by one, changing all the 9's to the right of that digit to 0's. Finally, an unlimited sequence of 0's to the right of the decimal place can be dropped. For example, 6.849999999999... = 6.85 and 6.850000000000... = 6.85. Finally, if all of the digits in a numeral are 0, the number is 0, and if all of the digits in a numeral are an unending string of 9's, you can drop the nines to the right of the decimal place, and add one to the string of 9s to the left of the decimal place. For example, 99.999... = 100. The real numbers also have an important but highly technical property called the least upper bound property. It can be shown that any ordered field, which is also completeness of the real numbers, complete, is isomorphic to the real numbers. The real numbers are not, however, an algebraically closed field, because they do not include a solution (often called a square root of minus one) to the algebraic equation $x^2+1=0$.

## Complex numbers

Moving to a greater level of abstraction, the real numbers can be extended to the
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s. This set of numbers arose historically from trying to find closed formulas for the roots of cubic function, cubic and quadratic function, quadratic polynomials. This led to expressions involving the square roots of negative numbers, and eventually to the definition of a new number: a 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 :$\,a + b i$ where ''a'' and ''b'' are real numbers. Because of this, complex numbers correspond to points on the complex plane, a vector space of two real dimensions. In the expression , 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 $\mathbb$. The fundamental theorem of algebra asserts that the complex numbers form an algebraically closed field, meaning that every polynomial with complex coefficients has a zero of a function, root in the complex numbers. Like the reals, the complex numbers form a field (mathematics), field, which is complete space, complete, but unlike the real numbers, it is not total order, ordered. That is, there is no consistent meaning assignable to 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 a total order that is ordered field, compatible with field operations.

# Subclasses of the integers

## Even and odd numbers

An even number is an integer that is "evenly divisible" by two, that is Euclidean division, divisible by two without remainder; an odd number is an integer that is not even. (The old-fashioned term "evenly divisible" is now almost always shortened to "divisibility, divisible".) Any odd number ''n'' may be constructed by the formula for a suitable integer ''k''. Starting with the first non-negative odd numbers are . Any even number ''m'' has the form where ''k'' is again an integer. Similarly, the first non-negative even numbers are .

## Prime numbers

A prime number, often shortened to just prime, 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. The primes have been widely studied for more than 2000 years and have led to many questions, only some of which have been answered. The study of these questions belongs to
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

. Goldbach's conjecture is an example of a still unanswered question: "Is every even number the sum of two primes?" 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 the fundamental theorem of arithmetic. A proof appears in Euclid's Elements.

## Other classes of integers

Many subsets of the natural numbers have been the subject of specific studies and have been named, often after the first mathematician that has studied them. Example of such sets of integers are Fibonacci numbers and perfect numbers. For more examples, see Integer sequence.

# Subclasses of the complex numbers

## Algebraic, irrational and transcendental numbers

Algebraic numbers are those that are a solution to a polynomial equation with integer coefficients. Real numbers that are not rational numbers are called irrational numbers. Complex numbers which are not algebraic are called transcendental numbers. The algebraic numbers that are solutions of a monic polynomial equation with integer coefficients are called algebraic integers.

## Constructible numbers

Motivated by the classical problems of Straightedge and compass construction, constructions with straightedge and compass, the constructible 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.

## Computable numbers

A computable number, also known as ''recursive number'', is a
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
such that there exists an algorithm which, given a positive number ''n'' as input, produces the first ''n'' digits of the computable number's decimal representation. Equivalent definitions can be given using μ-recursive functions, Turing machines or λ-calculus. The computable numbers are stable for all usual arithmetic operations, including the computation of the roots of a polynomial, and thus form a real closed field that contains the real algebraic numbers. 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.

# Extensions of the concept

The ''p''-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 what radix, base is used for the digits: any base is possible, but a prime number base provides the best mathematical properties. The set of the ''p''-adic numbers contains the rational numbers, but is not contained in the complex numbers. The elements of an algebraic function field over a finite field and algebraic numbers have many similar properties (see Function field analogy). Therefore, they are often regarded as numbers by number theorists. The ''p''-adic numbers play an important role in this analogy.

## Hypercomplex numbers

Some number systems that are not included in the complex numbers may be constructed from the real numbers in a way that generalize the construction of the complex numbers. They are sometimes called
hypercomplex number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s. They include the quaternions H, introduced by Sir William Rowan Hamilton, in which multiplication is not commutative, the octonions, in which multiplication is not associative in addition to not being commutative, and the sedenions, in which multiplication is not Alternative algebra, alternative, neither associative nor commutative.

## Transfinite numbers

For dealing with infinite set (mathematics), sets, the natural numbers have been generalized to the ordinal numbers and to the cardinal 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.

## Nonstandard numbers

Hyperreal numbers 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
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 numbers extend the real numbers by adding infinitesimally small numbers and infinitely large numbers, but still form
fields File:A NASA Delta IV Heavy rocket launches the Parker Solar Probe (29097299447).jpg, FIELDS heads into space in August 2018 as part of the ''Parker Solar Probe'' FIELDS is a science instrument on the ''Parker Solar Probe'' (PSP), designed to mea ...
.

* Concrete number * List of numbers * List of numbers in various languages * List of types of numbers * * Complex numbers * Numerical cognition * Orders of magnitude * * * * * * Subitizing and counting

# 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,