![NumberSetinC](https://upload.wikimedia.org/wikipedia/commons/a/a0/NumberSetinC.svg)
A number is a
mathematical object used to
count
Count (feminine: countess) is a historical title of nobility in certain European countries, varying in relative status, generally of middling rank in the hierarchy of nobility. Pine, L. G. ''Titles: How the King Became His Majesty''. New York: ...
,
measure
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Mea ...
, and
label. 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 ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
s
1,
2,
3,
4, and so forth. Numbers can be represented in language with
number words. More universally, individual numbers can be represented by
symbol
A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different conc ...
s, called ''numerals''; for example, "5" is a numeral that represents the
number five. As only a relatively small number of symbols can be memorized, basic numerals are commonly organized in a
numeral system, which is an organized way to represent any number. The most common numeral system is the
Hindu–Arabic numeral system, which allows for the representation of any number using a combination of ten fundamental numeric symbols, called
digits. 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 landline telephone subscriber station connected to a telephone line or to a wireless electronic telephony device, such as a radio telephone or a mobile telephone, or to other devices f ...
s), for ordering (as with
serial number
A serial number is a unique identifier assigned incrementally or sequentially to an item, to ''uniquely'' identify it.
Serial numbers need not be strictly numerical. They may contain letters and other typographical symbols, or may consist enti ...
s), and for codes (as with
ISBN
The International Standard Book Number (ISBN) is a numeric commercial book identifier that is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.
An ISBN is assigned to each separate edition and ...
s). In common usage, a ''numeral'' is not clearly distinguished from the ''number'' that it represents.
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the notion of a number has been extended over the centuries to include
zero
0 (zero) is a number representing an empty quantity. In place-value notation
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
(0),
negative number
In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed m ...
s,
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s such as
one half
One half ( : halves) is the irreducible fraction resulting from dividing one by two or the fraction resulting from dividing any number by its double. Multiplication by one half is equivalent to division by two, or "halving"; conversely, d ...
,
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s such as the
square root of 2
The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princip ...
and
, and
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
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 mathematical process that transforms one or more inputs into one or more outputs or ''results''. The term is used in a variety of senses, from the very definite arithmetical calculation of using an algorithm, to th ...
s with numbers are done with
arithmetical operations, the most familiar being
addition
Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...
,
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 mathematical operations of arithmetic, with the other ones being additi ...
,
division, and
exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
. Their study or usage is called
arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
, a term which may also refer to
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
, the study of the properties of numbers.
Besides their practical uses, numbers have cultural significance throughout the world.
For example, in Western society, the
number 13 Number 13 can refer to:
* 13 (number)
* Number 13 (comics) a comic strip in ''The Beano''
* ''Number 13'' (1922 film), a film by Alfred Hitchcock and starring Ernest Thesiger which was shot but never completed and is believed to be lost
* "Number ...
is often regarded as
unlucky
Luck is the phenomenon and belief that defines the experience of improbable events, especially improbably positive or negative ones. The naturalistic interpretation is that positive and negative events may happen at any time, both due to rand ...
, 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.
Cas ...
" may signify "a lot" rather than an exact quantity.
Though it is now regarded as
pseudoscience
Pseudoscience consists of statements, beliefs, or practices that claim to be both scientific and factual but are incompatible with the scientific method. Pseudoscience is often characterized by contradictory, exaggerated or falsifiability, unfa ...
, belief in a mystical significance of numbers, known as
numerology
Numerology (also known as arithmancy) is the belief in an occult, divine or mystical relationship between a number and one or more coinciding events. It is also the study of the numerical value, via an alphanumeric system, of the letters in ...
, permeated ancient and medieval thought.
Numerology heavily influenced the development of
Greek mathematics
Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathem ...
, 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 numbers, which consist of various extensions or modifications of the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
system. In modern mathematics, number systems are considered important special examples of more general algebraic structures such as
rings and
fields
Fields may refer to:
Music
*Fields (band), an indie rock band formed in 2006
*Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song by ...
, 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 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 representing an empty quantity. In place-value notation
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
, which was developed by ancient
Indian mathematicians
chronology of Indian mathematicians spans from the Indus Valley civilisation and the Vedas to Modern India.
Indian mathematicians have made a number of contributions to mathematics that have significantly influenced scientists and mathematicians ...
around 500 AD.
First use of numbers
Bones and other artifacts have been discovered with marks cut into them that many believe are
tally marks. 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 (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
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, مصر , ), officially the Arab Republic of Egypt, is a transcontinental country spanning the northeast corner of Africa and southwest corner of Asia via a land bridge formed by the Sinai Peninsula. It is bordered by the Mediter ...
.
Zero
The first known documented use of
zero
0 (zero) is a number representing an empty quantity. In place-value notation
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
dates to AD 628, and appeared in the ''
Brāhmasphuṭasiddhānta'', the main work of the
Indian mathematician
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta, ...
Brahmagupta
Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical trea ...
. He treated 0 as a number and discussed operations involving it, including
division. By this time (the 7th century) the concept had clearly reached Cambodia as
Khmer numerals
Khmer numerals are the numerals used in the Khmer language. They have been in use since at least the early 7th century, with the earliest known use being on a stele dated to AD 604 found in Prasat Bayang, near Angkor Borei, Cambodia.
Numera ...
, 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 community, which is also known as the Ummah. This consists of all those who adhere to the religious beliefs and laws of Islam or to societies in which Islam is practiced. In ...
.
![Khmer Numerals - 605 from the Sambor inscriptions](https://upload.wikimedia.org/wikipedia/commons/f/fc/Khmer_Numerals_-_605_from_the_Sambor_inscriptions.jpg)
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 systems. Many ancient texts used 0. Babylonian and Egyptian texts used it. Egyptians used the word ''nfr'' to denote zero balance in
double entry accounting
A double is a look-alike or doppelgänger; one person or being that resembles another.
Double, The Double or Dubble may also refer to:
Film and television
* Double (filmmaking), someone who substitutes for the credited actor of a character
* ' ...
. Indian texts used a
Sanskrit
Sanskrit (; attributively , ; nominally , , ) is a classical language belonging to the Indo-Aryan branch of the Indo-European languages. It arose in South Asia after its predecessor languages had diffused there from the northwest in the late ...
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
, era = ;;6th–5th century BCE
, region = Indian philosophy
, main_interests = Grammar, linguistics
, notable_works = ' (Sanskrit#Classical Sanskrit, Classical Sanskrit)
, influenced=
, notable_ideas=Descript ...
(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).
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 northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of cultu ...
seemed unsure about the status of 0 as a number: they asked themselves "how can 'nothing' be something?" leading to interesting
philosophical and, by the Medieval period, religious arguments about the nature and existence of 0 and the
vacuum
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
. The
paradoxes of
Zeno of Elea
Zeno of Elea (; grc, Ζήνων ὁ Ἐλεᾱ́της; ) was a pre-Socratic Greek philosopher of Magna Graecia and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known fo ...
depend in part on the uncertain interpretation of 0. (The ancient Greeks even questioned whether was a number.)
The late
Olmec people of south-central Mexico began to use a symbol for zero, a shell
glyph
A glyph () is any kind of purposeful mark. In typography, a glyph is "the specific shape, design, or representation of a character". It is a particular graphical representation, in a particular typeface, of an element of written language. A g ...
, in the New World, possibly by the but certainly by 40 BC, which became an integral part of
Maya numerals and the
Maya calendar. 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-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...
, influenced by
Hipparchus
Hipparchus (; 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 the precession of the equi ...
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 with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...
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. In modern Greece, they are still used for ordinal numbers and in contexts similar to tho ...
. 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 primarily in its eastern provinces during Late Antiquity and the Middle Ages, when its capital city was Constantinopl ...
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 since the late 9th or early 8th century BCE. It is derived from the earlier Phoenician alphabet, and was the earliest known alphabetic script to have distinct letters for vowels as w ...
Omicron
Omicron (; uppercase Ο, lowercase ο, ell, όμικρον) is the 15th letter of the Greek alphabet. This letter is derived from the Phoenician letter ayin: . In classical Greek, omicron represented the close-mid back rounded vowel in contras ...
(otherwise meaning 70).
Another true zero was used in tables alongside
Roman numerals
Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, eac ...
by 525 (first known use by
Dionysius Exiguus
Dionysius Exiguus (Latin for "Dionysius the Humble", Greek: Διονύσιος; – ) was a 6th-century Eastern Roman monk born in Scythia Minor. He was a member of a community of Scythian monks concentrated in Tomis (present day Constanța, ...
), 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 ...
). 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 monk at the monastery of St Peter and its companion monastery of St Paul in the Kingdom o ...
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'' contains methods for finding the areas of figures; red rods were used to denote positive
coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...
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 aut ...
referred to the equation equivalent to (the solution is negative) in ''
Arithmetica
''Arithmetica'' ( grc-gre, Ἀριθμητικά) is an Ancient Greek text on mathematics written by the mathematician Diophantus () in the 3rd century AD. It is a collection of 130 algebraic problems giving numerical solutions of determinate e ...
'', 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 and astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical trea ...
, in ''
Brāhmasphuṭasiddhānta'' in 628, who used negative numbers to produce the general form
quadratic formula
In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, gr ...
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 allowed negative solutions in financial problems where they could be interpreted as debts (chapter 13 of ''
Liber Abaci'', 1202) and later as losses (in ).
René Descartes
René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathem ...
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
Nicolas Chuquet (; born ; died ) was a French mathematician. He invented his own notation for algebraic concepts and exponentiation. He may have been the first mathematician to recognize zero and negative numbers as exponents.
In 1475, Jehan A ...
during the 15th century. He used them as
exponent
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
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 between the use of the first stone tools by hominins 3.3 million years ago and the beginning of recorded history with the invention of writing systems. The use of ...
. The
Ancient Egyptians used their
Egyptian fraction
An Egyptian fraction is a finite sum of distinct unit fractions, such as
\frac+\frac+\frac.
That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from each ...
notation for rational numbers in mathematical texts such as the
Rhind Mathematical Papyrus
The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057 and pBM 10058) is one of the best known examples of ancient Egyptian mathematics. It is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased ...
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 in 1889 and are kept at the U ...
. 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 pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
. 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
Sthananga Sutra (Sanskrit: Sthānāṅgasūtra Prakrit: Ṭhāṇaṃgasutta) (c. 3rd-4th century CE) forms part of the first eleven Angas of the Jaina Canon which have survived despite the bad effects of this Hundavasarpini kala as per the Ś ...
, which also covers number theory as part of a general study of mathematics.
The concept of
decimal 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 math
sutra
''Sutra'' ( sa, सूत्र, translit=sūtra, translit-std=IAST, translation=string, thread)Monier Williams, ''Sanskrit English Dictionary'', Oxford University Press, Entry fo''sutra'' page 1241 in Indian literary traditions refers to an aph ...
to include calculations of decimal-fraction approximations to
pi or the
square root of 2
The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princip ...
. 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 may refer to:
Peoples South Asia
* Indian people, people of Indian nationality, or people who have an Indian ancestor
** Non-resident Indian, a citizen of India who has temporarily emigrated to another country
* South Asia ...
Sulba Sutras composed between 800 and 500 BC. The first existence proofs of irrational numbers is usually attributed to
Pythagoras
Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samos, Samian, or simply ; in Ionian Greek; ) was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymou ...
, more specifically to the
Pythagorean Hippasus of Metapontum, 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 that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princip ...
. 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, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
. In 1872, the publication of the theories of
Karl Weierstrass (by his pupil E. Kossak),
Eduard Heine,
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...
, and
Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and
the axiomatic foundations of arithmetic. His ...
was brought about. In 1869,
Charles Méray
Hugues Charles Robert Méray (12 November 1835, in Chalon-sur-Saône, Saône-et-Loire – 2 February 1911, in Dijon) was a French mathematician. He is noted as the first to publish an arithmetical theory of irrational numbers. His work did not h ...
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
cut (Schnitt) in the system of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s, separating all
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s into two groups having certain characteristic properties. The subject has received later contributions at the hands of Weierstrass,
Kronecker, and Méray.
The search for roots of
quintic
In algebra, a quintic function is a function of the form
:g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\,
where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, a q ...
and higher degree equations was an important development, the
Abel–Ruffini theorem (
Ruffini 1799,
Abel 1824) showed that they could not be solved by
radicals
Radical may refer to:
Politics and ideology Politics
*Radical politics, the political intent of fundamental societal change
*Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe and ...
(formulas involving only arithmetical operations and roots). Hence it was necessary to consider the wider set of
algebraic numbers (all solutions to polynomial equations).
Galois (1832) linked polynomial equations to
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
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
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
, 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
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
s, resulting, with the subsequent contributions of Heine,
Möbius, and Günther, in the theory of .
Transcendental numbers and reals
The existence of
transcendental numbers was first established by
Liouville (1844, 1851).
Hermite
Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.
Hermi ...
proved in 1873 that ''e'' is transcendental and
Lindemann Lindemann is a German surname.
Persons
Notable people with the surname include:
Arts and entertainment
* Elisabeth Lindemann, German textile designer and weaver
* Jens Lindemann, trumpet player
* Julie Lindemann, American photographer
* Maggie ...
proved in 1882 that π is transcendental. Finally,
Cantor showed that the set of all
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s is
uncountably infinite but the set of all
algebraic number
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
s is
countably infinite
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
, so there is an uncountably infinite number of transcendental numbers.
Infinity and infinitesimals
The earliest known conception of mathematical
infinity
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
appears in the
Yajur Veda
The ''Yajurveda'' ( sa, यजुर्वेद, ', from ' meaning "worship", and ''veda'' meaning "knowledge") is the Veda primarily of prose mantras for worship rituals.Michael Witzel (2003), "Vedas and Upaniṣads", in ''The Blackwell C ...
, 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
Jainism ( ), also known as Jain Dharma, is an Indian religion. Jainism traces its spiritual ideas and history through the succession of twenty-four tirthankaras (supreme preachers of ''Dharma''), with the first in the current time cycle being ...
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
is often used to represent an infinite quantity.
Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phil ...
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
Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...
's ''
Two New Sciences'' discussed the idea of
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 played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...
; in 1895 he published a book about his new
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
, 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
infinite and
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of
infinitesimal calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
by
Newton
Newton most commonly refers to:
* Isaac Newton (1642–1726/1727), English scientist
* Newton (unit), SI unit of force named after Isaac Newton
Newton may also refer to:
Arts and entertainment
* ''Newton'' (film), a 2017 Indian film
* Newton ( ...
and
Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...
.
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 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
A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilat ...
. 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
Gerolamo Cardano (; also Girolamo or Geronimo; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501– 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged through those of mathematician, ...
. 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: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathem ...
coined the term "imaginary" for these quantities in 1637, he intended it as derogatory. (See
imaginary number
An imaginary number is a real number multiplied by the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square of an imaginary number is . Fo ...
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 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
:
in the case when both ''a'' and ''b'' are negative even bedeviled
Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
. This difficulty eventually led him to the convention of using the special symbol ''i'' in place of
to guard against this mistake.
The 18th century saw the work of
Abraham de Moivre
Abraham de Moivre FRS (; 26 May 166727 November 1754) was a French mathematician known for de Moivre's formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory.
He moved ...
and
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
.
De Moivre's formula (1730) states:
:
while
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for an ...
of
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
(1748) gave us:
:
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
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.
Gauss studied
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
essential singular points. This eventually led to the concept of the
extended complex plane
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
.
Prime numbers
Prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s have been studied throughout recorded history. Euclid devoted one book of the ''Elements'' to the theory of primes; in it he proved the infinitude of the primes and the
fundamental theorem of arithmetic, and presented the
Euclidean algorithm
In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an effi ...
for finding the
greatest common divisor of two numbers.
In 240 BC,
Eratosthenes
Eratosthenes of Cyrene (; grc-gre, Ἐρατοσθένης ; – ) was a Greek polymath: a mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the chief librarian at the Library of Alexandria ...
used the
Sieve of Eratosthenes
In mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit.
It does so by iteratively marking as composite (i.e., not prime) the multiples of each prime, starting with the first prime n ...
to quickly isolate prime numbers. But most further development of the theory of primes in Europe dates to the
Renaissance
The Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass ideas ...
and later eras.
In 1796,
Adrien-Marie Legendre conjectured the
prime number theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying ...
, 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
Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers.
The conjecture has been shown to hold ...
, 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
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
, formulated by
Bernhard Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
in 1859. The
prime number theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying ...
was finally proved by
Jacques Hadamard and
Charles de la Vallée-Poussin
Charles is a masculine given name predominantly found in English and French speaking countries. It is from the French form ''Charles'' of the Proto-Germanic name (in runic alphabet) or ''*karilaz'' (in Latin alphabet), whose meaning was " ...
in 1896. Goldbach and Riemann's conjectures remain unproven and unrefuted.
Main classification
Numbers can be classified into
sets, called number sets or number systems, such as the
natural numbers
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal n ...
and the
real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
. The main number systems are as follows:
Each of these number system is a
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
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
:
.
A more complete list of number sets appears in the following diagram.
Natural numbers
![Nat num](https://upload.wikimedia.org/wikipedia/commons/e/e1/Nat_num.svg)
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 ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
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
Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of cultu ...
.) However, in the 19th century,
set theorists and other mathematicians started including 0 (
cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
of the
empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
, i.e. 0 elements, where 0 is thus the smallest
cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...
) in the set of natural numbers. Today, different mathematicians use the term to describe both sets, including 0 or not. The
mathematical symbol
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. ...
for the set of all natural numbers is N, also written
, and sometimes
or
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
digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The
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
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
, 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
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly u ...
, 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
The plus and minus signs, and , are mathematical symbols used to represent the notions of positive and negative, respectively. In addition, represents the operation of addition, which results in a sum, while represents subtraction, resulti ...
). As an example, the negative of 7 is written −7, and . When the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of negative numbers is combined with the set of natural numbers (including 0), the result is defined as the set of
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s, Z also written
. Here the letter Z comes . The set of integers forms a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
with the operations addition and multiplication.
The natural numbers form a
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
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
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
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
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
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
.
Real numbers
The symbol for the real numbers is R, also written as
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
The plus and minus signs, and , are mathematical symbols used to represent the notions of positive and negative, respectively. In addition, represents the operation of addition, which results in a sum, while represents subtraction, resulti ...
, e.g. −123.456.
Most real numbers can only be ''approximated'' by
decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
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
The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. If the latter is defined as the largest integer not greater than , called floor of or \lfloor x\rfloor, its fractional part can ...
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
A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. It can be shown that a number is rational if an ...
. 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
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. A famous irrational real number is the
, the ratio of the
circumference
In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out to ...
of any circle to its
diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for ...
. 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 that
is irrational. Another well-known number, proven to be an irrational real number, is
:
the
square root of 2
The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princip ...
, 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
In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the math ...
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
rounded or
truncated real numbers. Any rounded or truncated number is necessarily a rational number, of which there are only
countably many
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
. 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
Significant figures (also known as the significant digits, ''precision'' or ''resolution'') of a number in positional notation are digits in the number that are reliable and necessary to indicate the quantity of something.
If a number expre ...
. For example, measurements with a ruler can seldom be made without a margin of error of at least 0.001
m. If the sides of a
rectangle
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containi ...
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 rightmost nonzero digit, and a decimal numeral with an unlimited number of 9's can be rewritten by increasing by one the rightmost digit less than 9, and 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 a 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
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
, is isomorphic to the real numbers. The real numbers are not, however, an
algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
, because they do not include a solution (often called a
square root of minus one
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
) to the algebraic equation
.
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 extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s. This set of numbers arose historically from trying to find closed formulas for the roots of
cubic
Cubic may refer to:
Science and mathematics
* Cube (algebra), "cubic" measurement
* Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex
** Cubic crystal system, a crystal system w ...
and
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
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
E ...
of −1, denoted by ''
i'', a symbol assigned by
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
, and called the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
. The complex numbers consist of all numbers of the form
:
where ''a'' and ''b'' are real numbers. Because of this, complex numbers correspond to points on the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
, a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
of two real
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
s. 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
An imaginary number is a real number multiplied by the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square of an imaginary number is . Fo ...
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
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
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
.
The
fundamental theorem of algebra asserts that the complex numbers form an
algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
, meaning that every
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
with complex coefficients has a
root
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...
in the complex numbers. Like the reals, the complex numbers form a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
, which is
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
, but unlike the real numbers, it is not
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
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive) ...
that is
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
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 "
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
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
. 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 pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
.
Goldbach's conjecture
Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers.
The conjecture has been shown to hold ...
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
The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postulat ...
.
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 number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.
T ...
s. For more examples, see
Integer sequence.
Subclasses of the complex numbers
Algebraic, irrational and transcendental numbers
Algebraic number
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
s 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 number
In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and .
Though only a few classes ...
s. The algebraic numbers that are solutions of a
monic polynomial
In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form:
:x^n+c_x^+\cd ...
equation with integer coefficients are called
algebraic integers.
Constructible numbers
Motivated by the classical problems of
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, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
such that there exists an
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
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 machine
A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algori ...
s or
λ-calculus. The computable numbers are stable for all usual arithmetic operations, including the computation of the roots of a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
, and thus form a
real closed field that contains the real
algebraic number
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
s.
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
In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the math ...
real numbers are non-computable. However, it is very difficult to produce explicitly a real number that is not computable.
Extensions of the concept
''p''-adic numbers
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
base is used for the digits: any base is possible, but a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
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
In mathematics, an algebraic function field (often abbreviated as function field) of ''n'' variables over a field ''k'' is a finitely generated field extension ''K''/''k'' which has transcendence degree ''n'' over ''k''. Equivalently, an algebraic ...
over a
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
and algebraic numbers have many similar properties (see
Function field analogy
This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of pro ...
). 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 numbers. They include the
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s H, introduced by Sir
William Rowan Hamilton
Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...
, in which multiplication is not
commutative, the
octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have e ...
s, in which multiplication is not
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
in addition to not being commutative, and the
sedenions, in which multiplication is not
alternative
Alternative or alternate may refer to:
Arts, entertainment and media
* Alternative (''Kamen Rider''), a character in the Japanese TV series ''Kamen Rider Ryuki''
* ''The Alternative'' (film), a 1978 Australian television film
* ''The Alternative ...
, neither associative nor commutative.
Transfinite numbers
For dealing with infinite
sets, the natural numbers have been generalized to the
ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least n ...
s and to the
cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...
s. 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
extension
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
* Extensible cardinal
* Extension (model theory)
* Extension (predicate logic), the set of tuples of values that satisfy the predicate
* E ...
of the ordered field of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s R and satisfies the
transfer principle. This principle allows true
first-order
In mathematics and other formal sciences, first-order or first order most often means either:
* "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of high ...
statements about R to be reinterpreted as true first-order statements about *R.
Superreal and
surreal numbers extend the real numbers by adding infinitesimally small numbers and infinitely large numbers, but still form
fields
Fields may refer to:
Music
*Fields (band), an indie rock band formed in 2006
*Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song by ...
.
See also
*
Concrete number
A concrete number or ''numerus numeratus'' is a number associated with the things being counted, in contrast to an abstract number or ''numerus numerans'' which is a number as a single entity. For example, "five apples" and "half of a pie" are con ...
*
List of numbers
*
List of types of numbers
*
*
Complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s
*
Numerical cognition
Numerical cognition is a subdiscipline of cognitive science that studies the cognitive, developmental and neural bases of numbers and mathematics. As with many cognitive science endeavors, this is a highly interdisciplinary topic, and includes r ...
*
Orders of magnitude
*
*
*
*
*
*
Subitizing and counting
Subitizing is the rapid, accurate, and confident judgments of numbers performed for small numbers of items. The term was coined in 1949 by E. L. Kaufman et al., and is derived from the Latin adjective '' subitus'' (meaning "sudden") and captures ...
Notes
References
*
Tobias Dantzig Tobias Dantzig (; February 19, 1884 – August 9, 1956) was an American mathematician, the father of George Dantzig, and the author of '' Number: The Language of Science (A critical survey written for the cultured non-mathematician)'' (1930) and ''A ...
, ''Number, the language of science; a critical survey written for the cultured non-mathematician'', New York, The Macmillan Company, 1930.
* Erich Friedman,
What's special about this number?''
* Steven Galovich, ''Introduction to Mathematical Structures'', Harcourt Brace Javanovich, 1989, .
*
Paul Halmos, ''Naive Set Theory'', Springer, 1974, .
*
Morris Kline, ''Mathematical Thought from Ancient to Modern Times'', Oxford University Press, 1990.
*
Alfred North Whitehead
Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He is best known as the defining figure of the philosophical school known as process philosophy, which today has found applicat ...
and
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
, ''
Principia Mathematica'' to *56, Cambridge University Press, 1910.
* Leo Cory, ''A Brief History of Numbers'', Oxford University Press, 2015, .
External links
*
*
*
*
*;
Online Encyclopedia of Integer Sequences
{{Authority control
Group theory
Mathematical objects