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number

(nŭm'bər)

Mathematics.
1. A member of the set of positive integers; one of a series of symbols of unique meaning in a fixed order that can be derived by counting.
2. A member of any of the further sets of mathematical objects, such as negative integers and real numbers.
numbers Arithmetic.
1. A symbol or word used to represent a number.
2. A numeral or a series of numerals used for reference or identification: his telephone number; the apartment number.
1. A position in an ordered sequence that corresponds to one of the positive integers: the house that is number three from the corner; ranked number six in her class.
2. One item in a group or series considered to be in numerical order: an old number of a magazine.
A total; a sum: the number of feet in a mile.
An indefinite quantity of units or individuals: The crowd was small in number. A number of people complained.
numbers
1. A large quantity; a multitude: Numbers of people visited the fair.
2. Numerical superiority: The South had leaders, the North numbers.
Grammar. The indication, as by inflection, of the singularity, duality, or plurality of a linguistic form.
numbers
1. Metrical feet or lines; verses: “These numbers will I tear, and write in prose” (Shakespeare).
2. Obsolete. Poetic meter.
numbers Archaic. Musical periods or measures.
numbers (used with a sing. or pl. verb) Games. A numbers game.
Numbers (used with a sing. verb) (Abbr. Num. or Nb) A book of the Bible.
One of the separate offerings in a program of music or other entertainment: The band's second number was a march.
Slang. A frequently repeated, characteristic speech, argument, or performance: suspects doing their usual number—protesting innocence.
Slang. A person or thing singled out for a particular characteristic: a crafty number.

v., -bered, -ber·ing, -bers.

v.tr.

To assign a number to.
To determine the number or amount of; count.
To total in number or amount; add up to.
To include in a group or category: He was numbered among the lost.
To mention one by one; enumerate.
To limit or restrict in number: Our days are numbered.

v.intr.

To call off numbers; count: numbering to ten.
To constitute a group or number: The applicants numbered in the thousands.

idioms:

by the numbers

In unison as numbers are called out by a leader: performing calisthenics by the numbers.
In a strict, step-by-step or mechanical way.

do a number on Slang.

To defeat, abuse, or humiliate in a calculated and thorough way.

get (or have) (someone's) number

To determine or know someone's real character or motives.

without (or beyond) number

Too many to be counted; countless: mosquitoes without number.

[Middle English nombre, from Old French, from Latin numerus.]

numberer num'ber·er n.

USAGE NOTE As a collective noun number may take either a singular or a plural verb. It takes a singular verb when it is preceded by the definite article the: The number of skilled workers is small. It takes a plural verb when preceded by the indefinite article a: A number of the workers are unskilled.

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Thesaurus: number

noun

arithmetic

computation

See

count

verb

To note (items) one by one so as to get a total: count, enumerate, numerate, reckon, tally, tell. See count.
To come to in number or quantity: aggregate, amount, reach, run into, total. Idioms: add up to. See increase/decrease.

Idioms: number

Idioms beginning with number:
number is up, one's

In addition to the idiom beginning with number, also see a number of; any number of; back number; by the numbers; crunch numbers; days are numbered; do a job (number) on; get (have) someone's number; hot number; in round numbers; look out for (number one); opposite number; safety in numbers.

Antonyms: number

Definition: aggregate, bunch
Antonyms: one

n

Definition: unit of the mathematical system
Antonyms: letter

v

Definition: count, calculate
Antonyms: estimate, guess

Hacker Slang: numbers

[scientific computation] Output of a computation that may not be significant results but at least indicate that the program is running. May be used to placate management, grant sponsors, etc. Making numbers means running a program because output — any output, not necessarily meaningful output — is needed as a demonstration of progress. See pretty pictures, math-out, social science number.

Measures and Units: number

The numbers used for counting, namely 1, 2, 3, and so on, are called the natural numbers; they exclude zero. If a and b are two such numbers and a is bigger than b then there must be a natural number c such that a = b + c. Adding or multiplying two natural numbers always produces another natural number, but subtracting them does not necessarily produce a natural number, and dividing them rarely does. There are unboundedly many natural numbers, but they are self-evidently countable; one can put a unique number to each.

If we omit the assumption that a is bigger than b, then c may have to be equal to or less than zero to have a = b + c. The term integer includes zero and negative whole numbers as well as the natural numbers. Adding, subtracting, or multiplying two integers always produces another integer but dividing them rarely produces an integer. There are unboundedly many integers, but they are obviously countable in the sense that one can put a unique natural number to each, e.g. double the unsigned part, then add one if not a natural number, two otherwise.

Dividing two integers produces a rational number, an entity often called a fraction or, in North America, a ratio. Integers represent a special case, but are rational numbers too. If a is divided by b, we get the fraction represented by a/b, such styling being termed a vulgar fraction. If a is less than b this would be called a proper, otherwise it is an improper fraction, expressible as an integer plus a proper fraction. Decimal notation (common practice only in recent centuries) provides for the fractional part to be expressed as a string of one or more decimal digits, following the integer part and separated from it by a ‘decimal point’. It is an essential feature of any rational number that such a string be either finite in length or settle to being a repeating pattern of at most b digits: thus ½ = 0.5; ⅓ = 0.333 3~ with the single digit 3 repeating endlessly; 1⁄7 = 0.142 857~ with this six-digit string repeating endlessly. The rational numbers are unlimitedly numerous, with the integers forming an infinitesimal subset; however, as discussed under infinity, the totality of rational numbers can be equated with the totality of integers by an unambiguous mapping formula, so the rational numbers are countable too.

Not all numbers can be expressed as the quotient of one integer over another, i.e. not all are rational; the well-known entity π = 3.141 6~ is such an exception, as is

= 1.414 2~ (though not its approximants, like ²²/₇). Any such number is called an irrational number; the vast majority of numbers are such. If we think of all possible numbers between 0 and 1, for instance, and visualize them as strings of digits, it is fairly obvious that only an infinitesimal proportion would have a finite number of or a repeating pattern of digits. The irrational numbers are not countable; together with the rational numbers they form the totality of real numbers (q.v. for special discussion re use of this term in informatics).

The number

solves the equation x² = 2, identically x² - 2 = 0. Any irrational number that is a solution of any such polynomial equation, provided it involves only integers for powers of the variable and for coefficients, and is finitely long, is called an algebraic number. All square roots, cube roots, etc., of natural numbers are obviously algebraic (some being natural numbers themselves). The number π (pi) and the number e are among the exceptions; no such polynomial exists of which either is a solution. Such numbers, infinite in number, are called transcendental.

Nor do all polynomials have solutions within the realm of numbers in the ordinary sense, the simple polynomial x² + 1 = 0 being an example. The problem here is that x² is positive for any real number, positive or negative. To evade this restriction, mathematicians created the special entity labelled i, such that i² = -1, that is i =

. Written also as j, particularly by engineers, this forms the basis of complex numbers, these being of the form a + b i, where a and b are real numbers (termed the real part and the imaginary part of the composite number, a complex number with zero real part being called a pure imaginary number). Any polynomial, even one with complex numbers among its coefficients and exponents, can be solved completely within the realm of complex numbers.

In reference to everyday counting, for example as six apples in a bowl or eleven players on a soccer team, a number is a cardinal number. Expressed in an ordering context, such as the sixth apple put in the bowl or the sixth largest of the apples there, the corresponding term is ordinal number.

The term ‘digit’ should be used for the individual numeric characters forming a number; e.g. 3 is the first digit of the value for π shown above.

See also negative number; prime number; infinity; numeral.

Literary Dictionary: numbers

numbers, a term—now obsolete—formerly applied to poetry in general, by association with the counting of feet or syllables in regular verse metres.

Britannica Concise Encyclopedia: number

Basic element of mathematics used for counting, measuring, solving equations, and comparing quantities. They fall into several categories. The counting numbers are the familiar 1, 2, 3 . . . ; whole numbers are the counting numbers and zero; integers are the whole numbers and the negative counting numbers; and the rational numbers are all possible quotients formed by integers, including fractions. These numbers can be symbolically represented by terminating or repeating decimals. Irrational numbers cannot be represented by fractions of integers or repeating decimals and must be represented by special symbols such as √2, e, and p. Together, the rational and irrational numbers constitute the real numbers, which form an algebraic field (see field theory), as do the complex numbers. While the counting numbers and rational numbers come about as the means of counting, calculating, and measuring, the others arose as means of solving equations. See also transcendental number.

For more information on number, visit Britannica.com.

English Folklore: numbers

There is no systematic numerology in English folklore; the only numbers widely regarded as significant are three, seven, nine, thirteen. The latter only acquired importance over the last two or three centuries, but the good luck ascribed to odd numbers below ten is a regular feature of European folklore. Curiously, five has little role in folklore, except as the pentacle. Twelve is the only even number occasionally regarded as significant; it presumably stands for completeness, by association with the twelve hours on a clock-face, the twelve months of the year, and the old way of reckoning by dozens.

Occasionally, it is said there are 365 windows in a very large house (or trees in a copse, or steps in a long flight), ‘one for each day of the year’. Number 666 is regarded as Satanic and unlucky.

Classical Literature Companion: numbers

numbers 1. Greek. The Greek names for the numbers 1 to 10 are: hēīs, duo, trēīs, tessarğs (or tettares), pentğ, hex, hepta, octo, ennğa, deka. The system commonly found in papyri and manuscripts for writing numbers is based on the alphabet and is as follows:

1–5	ɑ–ɛ
6	ζ (the minuscule, i.e. small-letter, version of the digamma)
7–9	ζ–θ
10–80 (by tens)	ι–π
90	ρ (koppa, not in the Attic Greek alphabet)
100–800 (by hundreds)	ϱ–ω
900	ϡ (sampi. in the Carian alphabet, equivalent to σσ or ττ)

It is customary to add a stroke above the line when writing numbers up to 999, e.g. ɑ′, β′, etc. Higher numbers are written as follows:

1, 000–9, 000 (by thousands)	, ɑ–, θ
10, 000	M

Multiples of 10, 000 are indicated by writing the multiplier on top. Thus (e.g.):

	β
21, 527	M, ɑφϰζ

(For Archimedes' system of expressing very large numbers see ARCHIMEDES.)

2. Roman. The Latin names for the numbers 1 to 10 are: unus, duo, tres, quattuor, quinque, sex, septem, octo, novem, decem. Roman numerical signs, originally special characters, gradually developed into letters of the alphabet as follows:

I = 1; V = 5; X = 10; L = 50; C = 100; D = 500; M = 1, 000. Horizontal lines were often used to denote thousands; thus V̄ = 5, 000. A notation was constructed by conventions for adding or subtracting according to the following rule: when two figures stand side by side, if the right-hand figure is the larger, the left-hand figure is to be subtracted from it; if the left-hand figure is the larger, the right-hand figure is to be added: e.g. IX = 9; XI = 11; MCM = 1, 900. When, as in this last example, a smaller numeral occurs between two larger, it is subtracted from the numeral on the right.

Philosophy Dictionary: number

The natural numbers are 0, 1, 2, 3… The integers are…-3, -2, -1, 0, 1, 2, 3…The rationals, as their name implies, measure ratios: any number that can be written as a/b, where a and b are integers, and b ≠ 0 (in other words, values of x that give solutions of an equation bx - a = 0 where a and b are integers). The real numbers contain all the rational numbers, but also numbers such as √2 or π that are not rational. The reals can be thought of as the points of a line, with the integers equally spaced along the line. Every real number can be expressed as an infinite decimal. There are more reals than rationals (Cantor's theorem), and a number of rigorous ways of defining the set of reals (see also diagonal argument, Dedekind cut). Transfinite numbers measure the size of infinite sets. The first transfinite number is aleph-null, written ℵ₀, which measures the set of natural numbers. See also continuum hypothesis. The cardinal numbers measure the size of sets: the cardinality of a set is the number of its elements. The ordinal numbers measure the length of a well-ordering (see ordering relation). The difference is not apparent in finite cases, where an ordering is bigger simply if it has more members, but in transfinite cases the notions come apart. Thus the natural numbers can be ordered in the standard way: 1, 2, 3, 4…. The length of this ordering is ω. But they can also be ordered in ways that themselves tail off to infinite successions 1, 3, 5…; 2, 4, 6…. Even although this ordering contains just the same elements, there is no order-preserving one-to-one correspondence between members of the two orderings, and it is of greater length, 2ω.

Columbia Encyclopedia: number,

entity describing the magnitude or position of a mathematical object or extensions of these concepts.

The Natural Numbers

Cardinal numbers describe the size of a collection of objects; two such collections have the same (cardinal) number of objects if their members can be matched in a one-to-one correspondence. Ordinal numbers refer to position relative to an ordering, as first, second, third, etc. The finite cardinal and ordinal numbers are called the natural numbers and are represented by the symbols 1, 2, 3, 4, etc. Both types can be generalized to infinite collections, but in this case an essential distinction occurs that requires a different notation for the two types (see transfinite number).

The Integers and Rational Numbers

To the natural numbers one adjoins their negatives and zero to form the integers. The ratios a/b of the integers, where a and b are integers and b≠0, constitute the rational numbers; the integers are those rational numbers for which b=1. The rational numbers may also be represented by repeating decimals; e.g., 1/2=0.5000..., 2/3=0.6666..., 2/7=0.285714285714... (see decimal system).

The Real Numbers

The real numbers are those representable by an infinite decimal expansion, which may be repeating or nonrepeating; they are in a one-to-one correspondence with the points on a straight line and are sometimes referred to as the continuum. Real numbers that have a nonrepeating decimal expansion are called irrational, i.e., they cannot be represented by any ratio of integers. The Greeks knew of the existence of irrational numbers through geometry; e.g., √2 is the length of the diagonal of a unit square. The proof that √2 is unable to be represented by such a ratio was the first proof of the existence of irrational numbers, and it caused tremendous upheaval in the mathematical thinking of that time.

The Complex Numbers

Numbers of the form z=x+yi, where x and y are real and i=√−1, such as 8+7i (or 8+7√−1), are called complex numbers; x is called the real part of z and yi the imaginary part. The real numbers are thus complex numbers with y=0; e.g., the real number 4 can be expressed as the complex number 4+0i. The complex numbers are in a one-to-one correspondence with the points of a plane, with one axis defining the real parts of the numbers and one axis defining the imaginary parts. Mathematicians have extended this concept even further, as in quaternions.

The Algebraic and Transcendental Numbers

A real or complex number z is called algebraic if it is the root of a polynomial equation zⁿ+a_n−1zⁿ⁻¹+...+a₁z+a₀=0, where the coefficients a₀, a₁,...a_n−1 are all rational; if z cannot be a root of such an equation, it is said to be transcendental. The number √2 is algebraic because it is a root of the equation z²+2=0; similarly, i, a root of z²+1=0, is also algebraic. However, F. Lindemann showed (1882) that π is transcendental, and using this fact he proved the impossibility of “squaring the circle” by straight edge and compass alone (see geometric problems of antiquity). The number e has also been found to be transcendental, although it still remains unknown whether e+π is transcendental.

Bibliography

See G. Ifrah, The Universal History of Numbers (1999).

Grammar Dictionary: number

The grammatical category that classifies a noun, pronoun, or verb as singular or plural. Woman, it, and is are singular; women, they, and are are plural.

Veterinary Dictionary: number

A symbol, as a figure or word, expressive of a certain value or a specified quantity determined by count.

atomic n. — a number expressive of the number of protons in an atomic nucleus, or the positive charge of the nucleus expressed in terms of the electronic charge; symbol A.
Avogadro's n. — see avogadro's number.
mass n. — see mass number.
Reynold's n. — see reynold's number.

Essay: Inventing and writing numbers

Virtually all numeration systems start as simple tallies, using single strokes to represent each additional unit -- / for one, // for two, /////// for seven, and so forth. Evidence of such systems have been found as marks on bone dating from as early as 30,000 bce and becoming common by 15,000 bce. Studies of modern peoples with limited words for numbers -- often just one, two, and many -- show that they also used simple tallies, or at least concrete objects such as sticks, to show specific numbers greater than two. Thus, the tally system for representing numbers can exist even when language has not developed words for numbers. Linguistic evidence suggests that concrete words, such as twin for two people and brace for two dead birds, precede the concept of twoness in language. Yet each could have been indicated with two strokes or two sticks used as tallies.

In the Near East of Neolithic times the number language was a typical "one, two, many" system (as in our monogamous, bigamous, polygamous classification scheme), while tallies were used to represent specific numbers. The token system that is the ultimate ancestor of both cuneiform writing and the Roman alphabet was a later invention. Tokens are small bits of shaped clay that represent specific objects that are being counted. Tokens were gradually replaced by cuneiform writing. The first step was to mark the outside of a clay envelope in which tokens were kept with impressions of the tokens inside. After tokens had developed further, however, certain ones seem to have stood for sets instead of individual objects. By the late fourth millennium bce, some tokens appear to have meant "ten sheep," while others meant "one sheep."

Thus, an envelope containing or marked with two of the ten-sheep tokens and three of the single-sheep tokens would indicate a flock of twenty-three sheep. At this point in time in most of Mesopotamia, however, a different type of token would indicate numbers or measures of a commodity different from sheep. Around this time in Uruk, traders were discovering that the same number could be used to mean ten sheep, ten bags of grain, or ten talents of copper. While this idea already existed for simple tallies, the extension to a more sophisticated numeration system represents the true creation of the abstract concept of number. Also, the tokens inside the envelope were discarded and only the markings were used to indicate numbers.

In Egypt, about 4000 bce, the tallies were also grouped at ten, but ten of those tallies were regrouped at a hundred, and ten of the hundreds at a thousand. This seems more familiar, as it is closer to the system we use. The Greeks adapted the alphabet for numerals, with [[[agr.gif]]] for one, [[[bgr.gif]]] for two, and so forth up to [[[thgr.gif]]] for nine (with an extra symbol thrown in for six). The next nine letters (again with an extra symbol) meant ten through ninety, while the following nine (with an extra symbol) meant a hundred through nine hundred. Many other alphabet-using peoples followed the Greek example.

Roman numerals today are also alphabetical, but they did not originate as such. Early representations show that the X for ten originated in a manner similar to the use of a slanting stroke to connect four tally marks to indicate five -- after each nine tallies, the Romans drew a slant mark across the tenth one, forming something similar to an X. This led to half an X for five, for which we use a V today. The L that we use for fifty has as its ancestor a symbol that was like an upside-down capital T. Number historian Karl Menninger has proposed that early Roman counters circled the tenth X to indicate a hundred or possibly to show a thousand; the left half of the circle remains as the C that is used to indicate a hundred. Similarly, the right half of the circled X became the D for five hundred. The whole symbol somehow became a thousand. Since the Latin words for hundred and thousand start with C and M, the connection with those letters was reinforced.

All of these systems are no longer in use, except for Roman numerals in some traditional uses. A better system that arose in India near the end of this period eventually replaced all other numeration systems. While the Indians used a derivative of the alphabet for words, the Greek idea of alphabetic numerals never reached them. Instead, they used horizontal tallies for one, two, three, and special symbols for four through nine. Originally, this system continued in a way similar to the Greek alphabet system, with special symbols for ten, twenty, and so forth. But around 600 ce (or some say as early as 200 bce) the Indians started using place value, so that instead of writing the equivalent of 100 + 80 + 7 they wrote symbols together, as we write 187, to mean the same thing. Only the first nine digits had to be used along with a symbol for zero, which they probably appropriated from astronomers' marking of empty places in large numbers. A famous inscription dated 870 ce contains the first zero that has survived. Arabs picked up the new system from the Indians and it was soon known as Arabic numerals in Europe. At the very end of the Middle Ages, the pope commanded that all Christians use what we now call Hindu-Arabic numeration.

The Mesopotamians had also invented a place-value system, but one based on sixty instead of ten. Although that did not survive to the present in its original form, it was used by astronomers and others in technical fields. From it we inherited sixty minutes to the hour and to the degree, as well as sixty seconds to the minute and the 360° circle.

Word Tutor: number

IN BRIEF: The aspect of things existing in countable units; digits used in math.

The measure of a man is not the number of his servants but in the number of people whom he serves. — Dr. Paul D. Moody.

Wikipedia: number

Number systems in mathematics
Basic
$\mathbb{N}\sub\mathbb{Z}\sub\mathbb{Q}\sub\mathbb{R}\sub\mathbb{C}$ Natural numbers $\mathbb{N}$ Negative numbers Integers $\mathbb{Z}$ Rational numbers $\mathbb{Q}$ Irrational numbers Real numbers $\mathbb{R}$ Imaginary numbers $\mathbb{I}$ Complex numbers $\mathbb{C}$ Algebraic numbers $\mathbb{A}$ Transcendental numbers $\mathbb{T}$
Complex extensions
Quaternions $\mathbb{H}$ Octonions $\mathbb{O}$ Sedenions $\mathbb{S}$ Cayley-Dickson construction Split-complex numbers $\mathbb{R}^{1,1}$ Bicomplex numbers Biquaternions Coquaternions Tessarines Hypercomplex numbers
Other extensions
Musean hypernumbers Superreal numbers Hyperreal numbers Surreal numbers Dual numbers Transfinite numbers
Other
Nominal numbers Serial numbers Ordinal numbers Cardinal numbers Prime numbers p-adic numbers Constructible numbers Computable numbers Integer sequences Mathematical constants Large numbers π = 3.141592654… e = 2.718281828… i (Imaginary unit) $i 2 = - 1$ ∞ (infinity)

A number is an abstract idea used in counting and measuring. A symbol which represents a number is called a numeral, but in common usage the word number is used for both the idea and the symbol. In addition to their use in counting and measuring, numerals are often used for labels (telephone numbers), for ordering (serial numbers), and for codes (ISBNs). In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers.

Certain procedures which input one or more numbers and output a number are called numerical operations. Unary operations input a single number and output a single number. For example, the successor operation adds one to an integer: the successor of 4 is 5. More common are binary operations which input two numbers and output a single number. Examples of binary operations include addition, subtraction, multiplication, division, and exponentiation. The study of numerical operations is called arithmetic.

The branch of mathematics that studies abstract number systems such as groups, rings and fields is called abstract algebra.

Types of numbers

Numbers can be classified into sets, called number systems. (For different methods of expressing numbers with symbols, such as the Roman numerals, see numeral systems.)

Natural numbers

The most familiar numbers are the natural numbers or counting numbers: one, two, three, ... . Some people also include zero in the natural numbers; however, others do not.

In the base ten number system, in almost universal use today, the symbols for natural numbers are written using ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. In this base ten system, the rightmost digit of a natural number has a place value of one, and every other digit has a place value ten times that of the place value of the digit to its right. The symbol for the set of all natural numbers is N, also written UNIQ4a6bbbd91f2115d3-math-7e7ff903648dd1b800000001.

Integers

Negative numbers are numbers that are less than zero. They are the opposite of positive numbers. For example, if a positive number indicates a bank deposit, then a negative number indicates a withdrawal of the same amount. Negative numbers are usually written by writing a negative sign in front of the number they are the opposite of. Thus the opposite of 7 is written −7. When the set of negative whole numbers are combined with the positive whole numbers and zero, one obtains the integers Z (German Zahl, plural Zahlen), also written UNIQ4a6bbbd91f2115d3-math-7e7ff903648dd1b800000002.

Rational numbers

A rational number is a number that can be expressed as a fraction with an integer numerator and a non-zero natural number denominator. The fraction m/n or

$m \over n \,$

represents m equal parts, where n equal parts of that size make up one whole. Two different fractions may correspond to the same rational number; for example 1/2 and 2/4 are equal, that is:

${1 \over 2} = {2 \over 4}\,$ .

If the absolute value of m is greater than n, then the absolute value of the fraction is greater than 1. Fractions can be greater than, less than, or equal to 1 and can also be positive, negative, or zero. The set of all fractions includes the integers, since every integer can be written as a fraction with denominator 1. For example −7 can be written −7/1. The symbol for the rational numbers is Q (for quotient), also written UNIQ4a6bbbd91f2115d3-math-7e7ff903648dd1b800000005.

Real numbers

The real numbers include all of the measuring numbers. Real numbers are usually written using decimal numerals, in which a decimal point is placed to the right of the digit with place value one. Following the decimal point, each digit has a place value one-tenth the place value of the digit to its left. Thus

$123.456\,$

represents 1 hundred, 2 tens, 3 ones, 4 tenths, 5 hundredths, and 6 thousandths. In saying the number, the decimal is read "point", thus: "one two three point four five six". In, for example, the US and UK, the decimal is represented by a period, in continental Europe by a comma. Zero is often written as 0.0 and negative real numbers are written with a preceding minus sign:

$-123.456\,$ .

Every rational number is also a real number. To write a fraction as a decimal, divide the numerator by the denominator. It is not the case, however, that every real number is rational. If a real number cannot be written as a fraction of two integers, it is called irrational. A decimal that can be written as a fraction either ends (terminates) or forever repeats, because it is the answer to a problem in division. Thus the real number 0.5 can be written as 1/2 and the real number 0.333... (forever repeating threes) can be written as 1/3. On the other hand, the real number π (pi), the ratio of the circumference of any circle to its diameter, is

$\pi = 3.14159265358979...\,$ .

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

$\sqrt{2} = 1.41421356237 ...\,$

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

Just as fractions can be written in more than one way, so too can decimals. For example, if we multiply both sides of the equation

$1/3 = 0.333...\,$

by three, we discover that

$1 = 0.999...\,$ .

Thus 1.0 and 0.999... are two different decimal numerals representing the natural number 1. There are infinitely many other ways of representing the number 1, for example 2/2, 3/3, 1.00, 1.000, and so on.

Every real number is either rational or irrational. Every real number corresponds to a point on the number line. The real numbers also have an important but highly technical property called the least upper bound property. The symbol for the real numbers is R or $\mathbb{R}$ .

When a real number represents a measurement, there is always a margin of error. This is often indicated by rounding or truncating a decimal, so that digits that suggest a greater accuracy than the measurement itself are removed. The remaining digits are called significant digits. For example, measurements with a ruler can seldom be made without a margin of error of at least 0.01 meters. If the sides of a rectangle are measured as 1.23 meters and 4.56 meters, then multiplication gives an area for the rectangle of 5.6088 square meters. Since only the first two digits after the decimal place are significant, this is usually rounded to 5.61.

In abstract algebra, the real numbers are uniquely characterized by being the only completely ordered field. They are not, however, an algebraically closed field.

Complex numbers

Moving to a greater level of abstraction, the real numbers can be extended to the complex numbers. This set of numbers arose, historically, from the question of whether a negative number can have a square root. This led to the invention of a new number: the square root of negative one, denoted by 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. In the expression a + bi, the real number a is called the real part and b is called the imaginary part. If the real part of a complex number is zero, then the number is called an imaginary number or is referred to as purely imaginary; if the imaginary part is zero, 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 $\mathbb{C}$ .

In abstract algebra, the complex numbers are an example of an algebraically closed field, meaning that every polynomial with complex coefficients can be factored into linear factors. Like the real number system, the complex number system is a field and is complete, but unlike the real numbers it is not ordered. That is, there is no meaning in saying that i is greater than 1, nor is there any meaning in saying that that i is less than 1. In technical terms, the complex numbers lack the trichotomy property.

Complex numbers correspond to points on the complex plane, sometimes called the Argand plane.

Each of the number systems mentioned above is a proper subset of the next number system. Symbolically, N ⊂ Z ⊂ Q ⊂ R ⊂ C.

Other types

Superreal, hyperreal and surreal numbers extend the real numbers by adding infinitesimally small numbers and infinitely large numbers, but still form fields.

The idea behind p-adic numbers is this: While real numbers may have infinitely long expansions to the right of the decimal point, these numbers allow for infinitely long expansions to the left. The number system which results depends on what base is used for the digits: any base is possible, but a system with the best mathematical properties is obtained when the base is a prime number.

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

There are also other sets of numbers with specialized uses. Some are subsets of the complex numbers. For example, algebraic numbers are the roots of polynomials with rational coefficients. Complex numbers that are not algebraic are called transcendental numbers.

Sets of numbers that are not subsets of the complex numbers include the quaternions H, invented by Sir William Rowan Hamilton, in which multiplication is not commutative, and the octonions, in which multiplication is not associative. Elements of function fields of finite characteristic behave in some ways like numbers and are often regarded as numbers by number theorists.

Numerals

Numbers should be distinguished from numerals, the symbols used to represent numbers. The number five can be represented by both the base ten numeral '5' and by the Roman numeral 'V'. Notations used to represent numbers are discussed in the article numeral systems. An important development in the history of numerals was the development of a positional system, like modern decimals, which can represent very large numbers. The Roman numerals require extra symbols for larger numbers.

History

History of integers

The first numbers

Further information: History of numeral systems

It is speculated that the first known use of numbers dates back to around 30000 BC, bones or other artifacts have been discovered with marks cut into them which are often considered tally marks. The use of these tally marks have been suggested to be anything from counting elapsed time, such as numbers of days, or keeping records of amounts.

Tallying systems have no concept of place-value (such as in the currently used decimal notation), which limit its representation of large numbers and as such is often considered that this is the first kind of abstract system that would be used, and could be considered a Numeral System.

The first known system with place-value was the Mesopotamian base 60 system (ca. 3400 BC) and the earliest known base 10 system dates to 3100 BC in Egypt. [1]

History of zero

Further information: History of zero

The use of zero as a number should be distinguished from its use as a placeholder numeral in place-value systems. Many ancient Indian texts use a Sanskrit word Shunya to refer to the concept of void; in mathematics texts this word would often be used to refer to the number zero. [2]. In a similar vein, Pāṇini (5th century BC) used the null (zero) operator (ie a lambda production) in the Ashtadhyayi, his algebraic grammar for the Sanskrit language. (also see Pingala)

Records show that the Ancient Greeks seemed unsure about the status of zero 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 zero and the vacuum. The paradoxes of Zeno of Elea depend in large part on the uncertain interpretation of zero. (The ancient Greeks even questioned if 1 was a number.)

The late Olmec people of south-central Mexico began to use a true zero (a shell glyph) in the New World possibly by the 4th century BC but certainly by 40 BC, which became an integral part of Maya numerals and the Maya calendar, but did not influence Old World numeral systems.

By 130, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for zero (a small circle with a long overbar) within a sexagesimal numeral system otherwise using alphabetic Greek numerals. Because it was used alone, not as just a placeholder, this Hellenistic zero was the first documented use of a true zero in the Old World. In later Byzantine manuscripts of his Syntaxis Mathematica (Almagest), the Hellenistic zero had morphed into the Greek letter omicron (otherwise meaning 70).

Another true zero was used in tables alongside Roman numerals by 525 (first known use by Dionysius Exiguus), but as a word, nulla meaning nothing, not as a symbol. When division produced zero as a remainder, nihil, also meaning nothing, was used. These medieval zeros were used by all future medieval computists (calculators of Easter). An isolated use of their initial, N, was used in a table of Roman numerals by Bede or a colleague about 725, a true zero symbol.

An early documented use of the zero by Brahmagupta (in the Brahmasphutasiddhanta) dates to 628. He treated zero as a number and discussed operations involving it, including division. By this time (7th century) the concept had clearly reached Cambodia, and documentation shows the idea later spreading to China and the Islamic world.

History of negative numbers

Further information: First usage of negative numbers

The abstract concept of negative numbers was recognised as early as 100 BC - 50 BC. The Chinese ”Nine Chapters on the Mathematical Art” (Jiu-zhang Suanshu) contains methods for finding the areas of figures; red rods were used to denote positive coefficients, black for negative. This is the earliest known mention of negative numbers in the East; the first reference in a western work was in the 3rd century in Greece. Diophantus referred to the equation equivalent to $4 x + 20 = 0$ (the solution would be negative) in Arithmetica, saying that the equation gave an absurd result.

During the 600s, negative numbers were in use in India to represent debts. Diophantus’ previous reference was discussed more explicitly by Indian mathematician Brahmagupta, in Brahma-Sphuta-Siddhanta 628, who used negative numbers to produce the general form quadratic formula that remains in use today. However, in the 12th century in India, Bhaskara gives negative roots for quadratic equations but says the negative value "is in this case not to be taken, for it is inadequate; people do not approve of negative roots."

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 debits (chapter 13 of Liber Abaci, 1202) and later as losses (in Flos). At the same time, the Chinese were indicating negative numbers by drawing a diagonal stroke through the right-most nonzero digit of the corresponding positive number's numeral^{[citation needed]}. The first use of negative numbers in a European work was by Chuquet during the 15th century. He used them as exponents, but referred to them as “absurd numbers”.

As recently as the 18th century, the Swiss mathematician Leonhard Euler believed that negative numbers were greater than infinity^{[citation needed]}, and it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless, just as René Descartes did with negative solutions in a cartesian coordinate system.

History of rational, irrational, and real numbers

Further information: History of irrational numbers and History of pi

History of rational numbers

It is likely that the concept of fractional numbers dates to prehistoric times. Even the Ancient Egyptians wrote math texts describing how to convert general fractions into their special notation. Classical Greek and Indian mathematicians made studies of the theory of rational numbers, as part of the general study of number theory. The best known of these is Euclid's Elements, 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 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 sutras to include calculations of decimal-fraction approximations to pi or the square root of two. Similarly, Babylonian math texts had always used sexagesimal fractions with great frequency.

History of irrational numbers

The earliest known use of irrational numbers was in the Indian Sulba Sutras composed between 800-500 BC. ^{[citation needed]} The first existence proofs of irrational numbers is usually attributed to Pythagoras, 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 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 his beliefs would not accept the existence of irrational numbers and so he sentenced Hippasus to death by drowning.

The sixteenth century saw the final acceptance by Europeans of negative, integral and fractional numbers. The seventeenth century saw decimal fractions with the modern notation quite generally used by mathematicians. But it was not until the nineteenth century that the irrationals were separated into algebraic and transcendental parts, and a scientific study of theory of irrationals was taken once more. It had remained almost dormant since Euclid. The year 1872 saw the publication of the theories of Karl Weierstrass (by his pupil Kossak), Heine (Crelle, 74), Georg Cantor (Annalen, 5), and Richard Dedekind. Méray had taken in 1869 the same point of departure as Heine, but the theory is generally referred to the year 1872. Weierstrass's method has been completely set forth by Salvatore Pincherle (1880), and Dedekind's has received additional prominence through the author's later work (1888) and the recent 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 numbers, separating all rational numbers into two groups having certain characteristic properties. The subject has received later contributions at the hands of Weierstrass, Kronecker (Crelle, 101), and Méray.

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 nineteenth century were brought into prominence through the writings of Joseph Louis Lagrange. Other noteworthy contributions have been made by Druckenmüller (1837), Kunze (1857), Lemke (1870), and Günther (1872). Ramus (1855) first connected the subject with determinants, resulting, with the subsequent contributions of Heine, Möbius, and Günther, in the theory of Kettenbruchdeterminanten. Dirichlet also added to the general theory, as have numerous contributors to the applications of the subject.

Transcendental numbers and reals

The first results concerning transcendental numbers were Lambert's 1761 proof that π cannot be rational, and also that eⁿ is irrational if n is rational (unless n = 0). (The constant e was first referred to in Napier's 1618 work on logarithms.) Legendre extended this proof to showed that π is not the square root of a rational number. The search for roots of quintic 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 (formula 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 giving rise to the field of Galois theory.

Even the set of algebraic numbers was not sufficient and the full set of real number includes transcendental numbers. The existence of which was first established by Liouville (1844, 1851).