number

num·ber

N0191300 (nŭm′bər)n.1. Mathematics a. 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.b. A member of any of the following sets of mathematical objects: integers, rational numbers, real numbers, and complex numbers. These sets can be derived from the positive integers through various algebraic and analytic constructions.2. numbers Arithmetic.3. a. A symbol or word used to represent a number.b. A numeral or a series of numerals used for reference or identification: his telephone number; the apartment number.4. a. 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.b. One item in a group or series considered to be in numerical order: an old number of a magazine.5. A total; a sum: the number of feet in a mile.6. An indefinite quantity of units or individuals: The crowd was small in number. A number of people complained.7. numbersa. A large quantity; a multitude: Numbers of people visited the fair.b. Numerical superiority: The South had leaders, the North numbers.8. Grammar The indication, as by inflection, of the singularity, duality, or plurality of a linguistic form.9. numbersa. Metrical feet or lines; verses: "These numbers will I tear, and write in prose" (Shakespeare).b. Obsolete Poetic meter.10. numbers Archaic Musical periods or measures.11. numbers(used with a sing. or pl. verb) Games A numbers game.12. Numbers(used with a sing. verb) See Table at Bible.13. One of the separate offerings in a program of music or other entertainment: The band's second number was a march.14. Slang A frequently repeated, characteristic speech, argument, or performance: suspects doing their usual number—protesting innocence.15. Slang A person or thing singled out for a particular characteristic: a crafty number.v. num·bered, num·ber·ing, num·bers v.tr.1. To assign a number to or mark with a number: Did you number the pages of the report?2. To determine the number or amount of; count: Tickets sold for the show were numbered at 500.3. To total in number or amount; add up to: The ships in the harbor number around 100.4. To include in a group or category: He was numbered among the lost.5. To limit or restrict in number: Our days are numbered.v.intr.1. To call off numbers; count: numbering to ten.2. To have as a total; amount to a number: The applicants numbered in the thousands.Idioms: by the numbers1. In unison as numbers are called out by a leader: performing calisthenics by the numbers.2. 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/have (someone's) number To determine or know someone's real character or motives. without/beyond number Too many to be counted; countless: mosquitoes without number.

[Middle English nombre, from Old French, from Latin numerus; see nem- in Indo-European roots.]

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 increasing. It takes a plural verb when preceded by the indefinite article a: A number of the workers have learned new skills.

number

(ˈnʌmbə) n1. (Mathematics) a concept of quantity that is or can be derived from a single unit, the sum of a collection of units, or zero. Every number occupies a unique position in a sequence, enabling it to be used in counting. It can be assigned to one or more sets that can be arranged in a hierarchical classification: every number is a complex number; a complex number is either an imaginary number or a real number, and the latter can be a rational number or an irrational number; a rational number is either an integer or a fraction, while an irrational number can be a transcendental number or an algebraic number. See complex number, imaginary number, real number, rational number, irrational number, integer, fraction, transcendental number, algebraic number See also cardinal number, ordinal number2. (Mathematics) the symbol used to represent a number; numeral3. a numeral or string of numerals used to identify a person or thing, esp in numerical order: a telephone number. 4. the person or thing so identified or designated: she was number seven in the race. 5. the sum or quantity of equal or similar units or things: a large number of people. 6. one of a series, as of a magazine or periodical; issue7. (Pop Music) a self-contained piece of pop or jazz music8. (Classical Music) a self-contained part of an opera or other musical score, esp one for the stage9. a group or band of people, esp an exclusive group: he was not one of our number. 10. slang a person, esp a woman: who's that nice little number?. 11. informal an admired article, esp an item of clothing for a woman: that little number is by Dior. 12. (Recreational Drugs) slang a cannabis cigarette: roll another number. 13. (Grammar) a grammatical category for the variation in form of nouns, pronouns, and any words agreeing with them, depending on how many persons or things are referred to, esp as singular or plural in number and in some languages dual or trial14. any number of several or many15. (Military) by numbers military (of a drill procedure, etc) performed step by step, each move being made on the call of a number16. do a number on someone slang US to manipulate or trick someone17. get someone's number have someone's number informal to discover someone's true character or intentions18. in numbers in large numbers; numerously19. one's number is up informal Brit one is finished; one is ruined or about to die20. without number beyond number of too great a quantity to be counted; innumerablevb (mainly tr) 21. to assign a number to22. to add up to; total23. (also intr) to list (items) one by one; enumerate24. (also intr) to put or be put into a group, category, etc: they were numbered among the worst hit. 25. to limit the number of: his days were numbered. [C13: from Old French nombre, from Latin numerus]

num•ber

(ˈnʌm bər)

n. 1. a mathematical unit used to express an amount, quantity, etc., usu. having precise relations with other such units: Six is an even number. 2. a numeral or group of numerals. 3. the sum, total, or aggregate of a collection of units: the number of people with reserved seats. 4. the particular numeral assigned to an object to designate its place in a series: a house number; a license number. 5. one of a series of things distinguished by or marked with numerals. 6. a certain collection or quantity not precisely reckoned, but considerable: a number of times. 7. a collection or company. 8. numbers, a. a considerable quantity; many: Numbers came to the parade. b. numerical strength or superiority. c. metrical feet; verse. d. numbers pool (def. 1). e. Informal. the figures representing the actual cost, expense, profit, etc. f. arithmetic. 9. a tune or arrangement for singing or dancing. 10. a distinct performance within a show, as a song or dance. 11. a single issue of a periodical. 12. a code of numerals, letters, or a combination of these, as that assigned to a particular telephone. 13. Gram. a category of inflection or other variation in the form of a word serving to indicate whether the word has one or more than one referent, as in the distinction between singular and plural and, in some languages, dual or trial. 14. Informal. person; individual: a cute number. 15. Informal. an article of merchandise, esp. of wearing apparel, offered for sale. v.t. 16. to mark with or distinguish by numbers. 17. to amount to or comprise in number; total. 18. to consider or include in a number: I number myself among his friends. 19. to count over one by one; enumerate; tell. 20. to fix the number of. 21. to ascertain the amount or quantity of; count. 22. to apportion or divide. v.i. 23. to make a total; reach an amount. 24. to count. Idioms: 1. by the numbers, a. according to standard procedures; by the book. b. together or in unison to a called-out count. 2. do a number on, Slang. to undermine or humiliate. 3. without number, of unknown or countless number; vast. [1250–1300; (n.) Middle English nombre < Old French < Latin numerus; (v.) Middle English nombren < Old French nombrer < Latin numerāre (derivative of numerus)] num′ber•a•ble, adj. num′ber•er, n. usage: See amount, collective noun.

number

Past participle: numbered
Gerund: numbering

Imperative
number
number

Present
I number
you number
he/she/it numbers
we number
you number
they number

Preterite
I numbered
you numbered
he/she/it numbered
we numbered
you numbered
they numbered

Present Continuous
I am numbering
you are numbering
he/she/it is numbering
we are numbering
you are numbering
they are numbering

Present Perfect
I have numbered
you have numbered
he/she/it has numbered
we have numbered
you have numbered
they have numbered

Past Continuous
I was numbering
you were numbering
he/she/it was numbering
we were numbering
you were numbering
they were numbering

Past Perfect
I had numbered
you had numbered
he/she/it had numbered
we had numbered
you had numbered
they had numbered

Future
I will number
you will number
he/she/it will number
we will number
you will number
they will number

Future Perfect
I will have numbered
you will have numbered
he/she/it will have numbered
we will have numbered
you will have numbered
they will have numbered

Future Continuous
I will be numbering
you will be numbering
he/she/it will be numbering
we will be numbering
you will be numbering
they will be numbering

Present Perfect Continuous
I have been numbering
you have been numbering
he/she/it has been numbering
we have been numbering
you have been numbering
they have been numbering

Future Perfect Continuous
I will have been numbering
you will have been numbering
he/she/it will have been numbering
we will have been numbering
you will have been numbering
they will have been numbering

Past Perfect Continuous
I had been numbering
you had been numbering
he/she/it had been numbering
we had been numbering
you had been numbering
they had been numbering

Conditional
I would number
you would number
he/she/it would number
we would number
you would number
they would number

Past Conditional
I would have numbered
you would have numbered
he/she/it would have numbered
we would have numbered
you would have numbered
they would have numbered

Thesaurus

Noun	1.	number - the property possessed by a sum or total or indefinite quantity of units or individuals; "he had a number of chores to do"; "the number of parameters is small"; "the figure was about a thousand"figureamount - the relative magnitude of something with reference to a criterion; "an adequate amount of food for four people"numerosity, numerousness, multiplicity - a large numberpreponderance, prevalence - a superiority in numbers or amount; "a preponderance of evidence against the defendant"countlessness, innumerableness - a number beyond countingbulk, majority - the property resulting from being or relating to the greater in number of two parts; the main part; "the majority of his customers prefer it"; "the bulk of the work is finished"minority - being or relating to the smaller in number of two parts; "when the vote was taken they were in the minority"; "he held a minority position"fewness - the quality of being small in numberroundness - the quality of being round numbers; "he gave us the results in round numbers, but their roundness didn't affect the point he was making"
	2.	number - a concept of quantity involving zero and units; "every number has a unique position in the sequence"arity - the number of arguments that a function can takecoordinate, co-ordinate - a number that identifies a position relative to an axisfolio, page number, pagination, paging - the system of numbering pagesdefinite quantity - a specific measure of amountdecimal - a number in the decimal systemconstant - a number representing a quantity assumed to have a fixed value in a specified mathematical context; "the velocity of light is a constant"oxidation number, oxidation state - the degree of oxidation of an atom or ion or molecule; for simple atoms or ions the oxidation number is equal to the ionic charge; "the oxidation number of hydrogen is +1 and of oxygen is -2"cardinality - (mathematics) the number of elements in a set or group (considered as a property of that grouping)count - the total number counted; "a blood count"factor - any of the numbers (or symbols) that form a product when multiplied togetherFibonacci number - a number in the Fibonacci sequenceprime, prime quantity - a number that has no factor but itself and 1composite number - an integer that is divisible without remainder by at least one positive integer other than itself and onescore - a number that expresses the accomplishment of a team or an individual in a game or contest; "the score was 7 to 0"record - the number of wins versus losses and ties a team has had; "at 9-0 they have the best record in their league"compound number - a quantity expressed in two different units; "one hour and ten minutes"no., ordinal, ordinal number - the number designating place in an ordered sequencecardinal number, cardinal - the number of elements in a mathematical set; denotes a quantity but not the orderradix, base - (numeration system) the positive integer that is equivalent to one in the next higher counting place; "10 is the radix of the decimal system"floating-point number - a number represented in floating-point notationfixed-point number - a number represented in fixed-point notationatomic number - the order of an element in Mendeleyev's table of the elements; equal to the number of protons in the nucleus or electrons in the neutral state of an atom of an elementbaryon number - a number equal to the difference between the number of baryons and the number of antibaryons in any subatomic structure; it is conserved in all types of particle interactionsquota - a prescribed number; "all the salesmen met their quota for the month"linage, lineage - the number of lines in a piece of printed materialnatural number - the number 1 and any other number obtained by adding 1 to it repeatedlyinteger, whole number - any of the natural numbers (positive or negative) or zero; "an integer is a number that is not a fraction"addend - a number that is added to another number (the augend)augend - a number to which another number (the addend) is addedminuend - the number from which the subtrahend is subtractedsubtrahend - the number to be subtracted from the minuendremainder, difference - the number that remains after subtraction; the number that when added to the subtrahend gives the minuendcomplex number, complex quantity, imaginary, imaginary number - (mathematics) a number of the form a+bi where a and b are real numbers and i is the square root of -1second power, square - the product of two equal terms; "nine is the second power of three"; "gravity is inversely proportional to the square of the distance"third power, cube - the product of three equal termsbiquadrate, biquadratic, fourth power, quartic - an algebraic equation of the fourth degreeroot - a number that, when multiplied by itself some number of times, equals a given numberdividend - a number to be divided by another numberdivisor - the number by which a dividend is dividedquotient - the number obtained by divisionremainder - the part of the dividend that is left over when the dividend is not evenly divisible by the divisormultiplier, multiplier factor - the number by which a multiplicand is multipliedmultiplicand - the number that is multiplied by the multiplier
	3.	number - a short theatrical performance that is part of a longer program; "he did his act three times every evening"; "she had a catchy little routine"; "it was one of the best numbers he ever did"routine, act, bit, turnperformance, public presentation - a dramatic or musical entertainment; "they listened to ten different performances"; "the play ran for 100 performances"; "the frequent performances of the symphony testify to its popularity"showstopper, show-stopper, stopper - an act so striking or impressive that the show must be delayed until the audience quiets down
	4.	number - the number is used in calling a particular telephone; "he has an unlisted number"phone number, telephone numbersignal, signaling, sign - any nonverbal action or gesture that encodes a message; "signals from the boat suddenly stopped"
	5.	number - a symbol used to represent a number; "he learned to write the numerals before he went to school"numeralsymbol - an arbitrary sign (written or printed) that has acquired a conventional significanceArabic numeral, Hindu numeral, Hindu-Arabic numeral - one of the symbols 1,2,3,4,5,6,7,8,9,0Roman numeral - a symbol in the old Roman notation; I,V,X,L,C,D,M represent 1,5,10,50,100,500,1000 respectively in Arabic notationantilog, antilogarithm - the number of which a given number is the logarithm
	6.	number - one of a series published periodically; "she found an old issue of the magazine in her dentist's waiting room"issueperiodical - a publication that appears at fixed intervalsserial publication, serial, series - a periodical that appears at scheduled timesedition - an issue of a newspaper; "he read it in yesterday's edition of the Times"
	7.	number - a select company of people; "I hope to become one of their number before I die"company - a social gathering of guests or companions; "the house was filled with company when I arrived"
	8.	number - a numeral or string of numerals that is used for identification; "she refused to give them her Social Security number"identification numberABA transit number, bank identification number, BIN - an identification number consisting of a two-part code assigned to banks and savings associations; the first part shows the location and the second identifies the bank itselflicense number, registration number - the number on the license plate that identifies the car that bears itSocial Security number - the number of a particular individual's Social Security accountpositive identification - evidence proving that you are who you say you are; evidence establishing that you are among the group of people already known to the system; recognition by the system leads to acceptance; "a system for positive identification can prevent the use of a single identity by several people"personal identification number, PIN number, PIN - a number you choose and use to gain access to various accounts
	9.	number - a clothing measurement; "a number 13 shoe"size - the property resulting from being one of a series of graduated measurements (as of clothing); "he wears a size 13 shoe"colloquialism - a colloquial expression; characteristic of spoken or written communication that seeks to imitate informal speech
	10.	number - the grammatical category for the forms of nouns and pronouns and verbs that are used depending on the number of entities involved (singular or dual or plural); "in English the subject and the verb must agree in number"grammatical category, syntactic category - (grammar) a category of words having the same grammatical properties
	11.	number - an item of merchandise offered for sale; "she preferred the black nylon number"; "this sweater is an all-wool number"merchandise, product, ware - commodities offered for sale; "good business depends on having good merchandise"; "that store offers a variety of products"
Verb	1.	number - add up in number or quantity; "The bills amounted to $2,000"; "The bill came to $2,000"total, amount, add up, comework out - be calculated; "The fees work out to less than $1,000"be - have the quality of being; (copula, used with an adjective or a predicate noun); "John is rich"; "This is not a good answer"outnumber - be larger in numberaverage, average out - amount to or come to an average, without loss or gain; "The number of hours I work per work averages out to 40"make - add up to; "four and four make eight"
	2.	number - give numbers to; "You should number the pages of the thesis"paginate, page, foliate - number the pages of a book or manuscriptdenominate, designate - assign a name or title to
	3.	number - enumerate; "We must number the names of the great mathematicians"listitemise, itemize - place on a list of items; "itemize one's tax deductions"identify, name - give the name or identifying characteristics of; refer to by name or some other identifying characteristic property; "Many senators were named in connection with the scandal"; "The almanac identifies the auspicious months"
	4.	number - put into a group; "The academy counts several Nobel Prize winners among its members"countclass, classify, sort out, assort, sort, separate - arrange or order by classes or categories; "How would you classify these pottery shards--are they prehistoric?"
	5.	number - determine the number or amount of; "Can you count the books on your shelf?"; "Count your change"count, numerate, enumeraterecount - count again; "We had to recount all the votes after an accusation of fraud was made"ascertain, determine, find out, find - establish after a calculation, investigation, experiment, survey, or study; "find the product of two numbers"; "The physicist who found the elusive particle won the Nobel Prize"miscount - count wronglycensus - conduct a census; "They censused the deer in the forest"add together, summate, tot, tot up, tote up, total, add up, sum, sum up, tally, add - determine the sum of; "Add all the people in this town to those of the neighboring town"
	6.	number - place a limit on the number ofkeep downcircumscribe, confine, limit - restrict or confine, "I limit you to two visits to the pub a day"

number

noun1. numeral, figure, character, digit, integer None of the doors have numbers on them.2. amount, quantity, collection, total, count, sum, aggregate I have had an enormous number of letters from concerned parents.
amount want, lack, shortage, scarcity, insufficiency, scantiness3. crowd, horde, multitude, throng People turned out to vote in huge numbers.4. group, company, set, band, crowd, gang, coterie We had a stag night for one of our number who had decided to get married.5. issue, copy, edition, imprint, printing an article which appeared in the summer number of the magazineverb1. amount to, come to, total, add up to They told me that their village numbered 100 or so.2. calculate, account, reckon, compute, enumerate One widely cited report numbered the dead at over 10,000.
calculate guess, conjecture, theorize3. include, count He numbered several Americans among his friends.a number of several, a few, various Artillery fire had been heard in a number of border districts.

number

nounArithmetic calculations.Used in plural:arithmetic, computation, figure (used in plural).verb1. To note (items) one by one so as to get a total:count, enumerate, numerate, reckon, tally, tell.2. To come to in number or quantity:aggregate, amount, reach, run into, total.Idiom: add up to.

Translations

数字数目期歌编号

number

(ˈnambə) noun1. (sometimes abbreviated to no – plural nos – when written in front of a figure) a word or figure showing eg how many of something there are, or the position of something in a series etc. Seven was often considered a magic number; Answer nos 1–10 of exercise 2. 数字，号码 2. a (large) quantity or group (of people or things). He has a number of records; There were a large number of people in the room.数目3. one issue of a magazine. the autumn number. 期，号（指期刊） 4. a popular song or piece of music. He sang his most popular number. 歌，乐曲 verb1. to put a number on. He numbered the pages in the top corner.编号2. to include. He numbered her among his closest friends.把…算作3. to come to in total. The group numbered ten.总计ˈnumberless adjective very many. 数不清的ˈnumber-plate noun one of the metal plates carried on the front and back of a motor vehicle showing the registration number of the vehicle. 车辆牌照，汽车号牌 his etc days are numbered he etc won't last much longer. 日子屈指可数without number very many. I've told him times without number (= very often) not to do that. 无数的，多得数不清的

a number of , meaning `several', is plural: A number of boys are absent today .
the number of , meaning `the total quantity of' is singular: The number of girls in the class is small .

number

→ 数字zhCN

May I have your phone number? (US)
Can I have your phone number? (UK) → 我可以知道你的电话号码吗？
Where is carriage number thirty? → 第三十号车厢在哪儿？
Where is cabin number five? → 五号船舱在哪儿？
My driver's license number is ... (US)
My driving licence number is ... (UK) → 我的驾照号码是...
Pump number three, please → 请使用三号油泵
The car is near exit number ... (US)
The car is near junction number ... (UK) → 车靠近第....号路口
License plate number ... (US)
Registration number ... (UK) → 车牌号是....
The key for room number two hundred and two → 这是202号房间的钥匙
What's the number for directory assistance? (US)
What is the number for directory enquiries? (UK) → 电信查询拨打什么号码？
What's the telephone number? → 电话号码是什么？
What's your cell number? (US)
What is the number of your mobile? (UK) → 您的手机号码是什么？
My cellular number is ... (US)
My mobile number is ... (UK) → 我的手机号码是...
You have a wrong number (US)
You have the wrong number (UK) → 您拨错号码了
What is the fax number? → 传真号码是多少？

number

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 numbertransfinite number,
cardinal or ordinal number designating the magnitude (power) or order of an infinite set; the theory of transfinite numbers was introduced by Georg Cantor in 1874.
..... Click the link for more information. ).

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 systemdecimal system
[Lat.,=of tenths], numeration system based on powers of 10. A number is written as a row of digits, with each position in the row corresponding to a certain power of 10.
..... Click the link for more information. ).

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 quaternionsquaternion
, in mathematics, a type of higher complex number first suggested by Sir William R. Hamilton in 1843. A complex number is a number of the form a+bi when a and b are real numbers and i
..... Click the link for more information. .

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 − 1}z^{n − 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 antiquitygeometric problems of antiquity,
three famous problems involving elementary geometric constructions with straight edge and compass, conjectured by the ancient Greeks to be impossible but not proved to be so until modern times.
..... Click the link for more information. ). 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).

Number

in linguistics, the grammatical category indicating the number of participants in an action—subjects and objects— through morphological means. The primary distinction in the category of number is between singular and plural. Some languages also have a dual number and, more rarely, a triple number. In the historical development of a language, the dual may weaken and be absorbed by the plural, as happened in the Slavic languages; for example, in the second person personal pronoun Old Church Slavonic distinguished singular (ty, “thou”), dual (va, “you two”), and plural (vy, “you”).

The forms and meanings of the plural include the distributive plural, which indicates that plurality is thought of as consisting of individual objects (listy, “leaves”), and the collective plural, which indicates that plurality is thought of as a single aggregate (list’ia, “leaves”). Collective meaning can also be expressed by a singular form (triap’e, “rags,” voron’e, “crows”). Plural forms can also indicate the concept of class (generic plural) (v etoi mestnosti vodiatsia volki, “there are wolves in this area”). Plural forms may sometimes be used with the meaning of the singular, such as the polite, or honorific, form of the second person personal pronoun (vy, “you,” in addressing one person) and the plural form in the first person (my, “we”) as used in the speech of sovereigns.

Number is an independent category in nouns and personal pronouns. Other parts of speech, including verbs, adjectives, and the other types of pronouns, acquire marking for number through agreement (syntactic number). Number agreement is obligatory in the Indo-European languages (on rabotaet, “he works,” oni rabotaiut, “they work”). However, as morphology becomes simpler, agreement may also disappear. For example, in English there is no number agreement between adjective and noun (“clever child,” “clever children”). There are various ways of expressing plural number: affixation (stol, “table,” stoly, “tables”), suppletion (chelovek, “person,” liudi, “people”), internal inflection (Arabic radžulun, “man,” ridžālun, “men,”), in which the root vowel changes, and reduplication (Indonesian [Malay] orang, “person,” orang-orang, “people”). In Indo-European languages the plural form is required if a noun is modified by a word denoting quantity (desiat’ knig, “ten books,” mnogo knig, “many books”); in other languages the noun may have the form of the singular in such constructions (Hungarian könyv, “book,” tu könyv, “ten books,” sok könyv, “many books”). In many Asian and American languages the plural of nouns used in constructions containing a numeral is expressed by means of special classifiers, which differ according to the lexical group to which the noun belongs. In such instances the nouns do not change their form (Vietnamese hai con meo, “two cats,” where con is the classifier).

REFERENCES

Sapir, E. Iazyk. Moscow-Leningrad, 1934. (Translated from English.)
Jespersen, O. Filosofiia grammatiki. Moscow, 1958. (Translated from English.)
Reformatskii, A. A. “Chislo i grammatika.” In the collection Voprosy grammatiki. Moscow-Leningrad, 1960.
Vinogradov, V. V. Russkii iazyk, 2nd ed. Moscow, 1972.

V. A. VINOGRADOV

Number

the most important mathematical concept. The concept of number arose in simplest form in primitive society and over the ages has undergone changes, gradually growing richer in content with the expansion of the range of human activities and the range of problems requiring quantitative description and investigation. In the first stages of development, the concept of number was determined by the requirements of counting and measurement in man’s daily activities. Subsequently, number became the basic concept of mathematics, and the further development of the concept has been determined by the needs of mathematical science.

The concept of natural number, or positive integer, arose as far back as prehistoric times in connection with the need to count objects. In general, its formation and development proceeded as follows. At the lower stage of primitive society, the concept of abstract number was nonexistent. This did not mean that primitive man was unable to ascertain the number of objects in a given set, such as the number of people involved in a hunt or the number of lakes in which fish could be caught. However, the consciousness of primitive man still could not perceive that which was common to various groups of objects, such as “three people” or “three lakes.” Analyses of the languages of primitive peoples have shown that different phrases were used in counting different objects. The word “three” was conveyed differently in the contexts of “three people” and “three boats.” Of course, such named number series were very short and terminated in a nonindividualized concept (“many”) of a large number of some object, which was also named, that is, expressed by different words for different kinds of objects, such as “crowd,” “herd,” and “pile.”

The concept of abstract number emerged out of the primitive way of counting objects, consisting in comparing the objects of a given specific set with objects of some defined set that acts as a standard. Among most peoples, fingers served as the first such standard (“finger counting”), confirmed by linguistic analyses of the names of the first numbers. At this stage, number becomes abstract, independent of the nature of the objects being counted, and at the same time acts as a fully concrete embodiment associated with the nature of the reference set. The expanding needs of counting forced people to use other counting standards, such as notches on sticks. For recording comparatively large numbers, a new idea came to be used—the designation of some specific number (ten among most peoples) by a new symbol, for example, a notch on a different stick.

With the development of writing, the possibilities of reproducing numbers expanded significantly. At first, numbers were denoted by lines on the writing material, such as papyrus or clay tablet. Later, other symbols were introduced to denote large numbers. The Babylonian cuneiform notations for numbers, like the Roman numerals that have been preserved to this day, clearly show that it was precisely in this manner that number notation developed. A major step forward was the Hindu positional numeration system, which made it possible to write any natural number by means of ten symbols—digits (seeNUMERATION SYSTEM). Thus, as writing developed, the concept of natural number assumed increasingly abstract form. Also, the abstract concept of number, expressed by special words in speech and denoted by special symbols in writing, became increasingly entrenched.

An important step in the development of the concept of natural number was the recognition of the infiniteness of the sequence of natural numbers, that is, the possibility of its unlimited continuation. A clear understanding of the infiniteness of the sequence of natural numbers is reflected in Greek mathematical works (third century B.C.),specifically in Euclid’s and Archimedes’ works. The unlimited continuability of the sequence of prime numbers is established as early as Euclid’s Elements, and the principles for the construction of names and symbols for any large number, particularly numbers larger than the “number of grains of sand in the world,” are given in Archimedes’ book Sandreckoner.

Operations on numbers came into use with the development of the concept of natural number in connection with the counting of objects. The operations of addition and subtraction originally arose as operations on sets themselves, in the form of the joining of two sets into one and the separation of part of a set. Multiplication, apparently, arose as a result of counting in equal parts (by twos or threes, for example), and division arose as the division of a set into equal parts (seeMULTIPLICATION and DIVISION). The abstract nature of these operations became evident after centuries of experience, as did the independence of the quantitative result of an operation from the nature of the objects forming the set, for example, that two objects and three objects will add up to five objects, regardless of the nature of the objects. Only then did mathematicians begin to develop the rules of operations, to study the operations, and to devise methods of solving problems; in other words, only then did the development of the science of numbers—arithmetic—begin (seeARITHMETIC). Arithmetic developed above all as a system of knowledge with an overt practical orientation. However, in the very process of its development, it became evident that there was a need to study the properties of numbers as such and to elucidate the increasingly complex regularities in their interrelations brought about by the very existence of the operations. The refinement of the concept of natural number began, and various classes, such as even and odd numbers and prime and composite numbers, became distinguished. The study of the deep-seated regularities of the natural numbers continues and constitutes the branch of mathematics known as number theory (seeNUMBERS, THEORY OF).

Natural numbers, in addition to their primary function of characterizing the number of objects, have another function, that of characterizing the order of objects in a sequence. The concept of ordinal number (first, second, and so on), which arises in connection with this function, is closely linked with the concept of cardinal number (one, two, and so on). In particular, the most frequently used method of counting objects since time immemorial has been the placement of the objects being counted in a sequence and then counting them using ordinal numbers (for example, if the last object being counted is seventh, then the total number of objects is seven).

The question of substantiating the concept of natural number is rather recent. The concept is so familiar and simple that the need for its definition in terms of some simpler concepts never arose. It was only in the mid–19th century, under the influence of the development of the axiomatic method in mathematics, on the one hand (seeAXIOMATIC METHOD), and the critical reassessment of the foundations of mathematical analysis, on the other, that the time became ripe for substantiating the concept of cardinal natural number. A clear definition of the concept of natural number based on the concept of set (an aggregate of objects) was provided in the 1870’s by G. Cantor. First, Cantor defines the concept of the equivalence of sets. Specifically, two sets are said to be equivalent if the objects of the sets can be put into one-to-one correspondence. Then the number of objects within a given set is defined as that which the given set and any other set of objects equivalent to it have in common. The definition reflects the essence of the natural number as that which results from counting the objects composing the given set. Indeed, at all historical levels, counting has consisted in comparing one by one the objects being counted with the objects constituting a “reference” set (in the early stages, the fingers of the hands and notches on sticks; today, words and symbols representing numbers). Cantor’s definition was the starting point for the extension of the concept of cardinal number from finite to infinite sets.

Another substantiation of the concept of natural number is based on an analysis of the relation of succession, which, it turns out, can be axiomatized. A system of axioms constructed on the basis of this principle was formulated by G. Peano.

It should be noted that the extension of the concept of ordinal number to infinite sets (transfinite ordinal numbers and, more generally, ordinal types) diverges sharply from the generalized concept of cardinal number, since quantitatively identical (equivalent) sets can be ordered by different methods. (SeeTRANSFINITE ORDINAL NUMBER and SET THEORY.)

Historically, the first extension of the concept of number was the introduction of fractions. The first use of fractions was connected with the need to carry out measurements. The measurement of some quantity consists in comparing it with another qualitatively similar quantity, which is taken as the unit of measurement. This comparison is performed by means of the operation—specific to the method of measurement—of “applying” the unit of measurement to the quantity being measured and counting the number of such applications. Length is measured in this way by applying a segment that is taken as the unit of measurement, and the amount of a liquid is measured by means of a measuring vessel. However, the unit of measurement does not always fit the quantity being measured a whole number of times, a fact that cannot always be ignored, even in the most primitive practical activity. Herein lies the source of the simplest and most “convenient” fractions, such as one-half, one-third, and one-fourth. It was only with the development of arithmetic as the science of numbers that the idea emerged of considering fractions with any natural denominator, as well as the concept of fractional number as a quotient in the division of two natural numbers, of which the dividend is not divisible by the divisor (see FRACTION).

The further extension of the concept of number was now no longer connected with the direct needs of counting and measurement but was a direct consequence of the development of mathematics.

The introduction of negative numbers was brought about of necessity by the development of algebra as the science providing general methods for the solution of arithmetic problems, regardless of content or given numerical data. The need for negative numbers in algebra arose in the solution of problems that reduce to linear equations with one unknown. A possible negative answer in problems of this kind may be interpreted by using as examples very simple directed quantities (oppositely directed segments, motion in the direction opposite to the direction chosen, property-debt). However, in problems that involve the repeated application of the operations of addition and subtraction, a great many cases must be considered in the course of a solution if negative numbers are not used, which may prove to be so burdensome a task that the advantage of an algebraic solution over an arithmetic solution is lost. Thus, the extensive use of algebraic methods in solving problems is extremely difficult unless negative numbers are used. In India negative numbers were used systematically as early as the sixth to 11th centuries in problem solving and were interpreted basically as they are today.

In European science the use of negative numbers did not become firmly established until the time of R. Descartes, who provided geometrical interpretation of negative numbers as directed line segments. Descartes’s creation of analytic geometry, which made it possible to consider the roots of an equation as the coordinates of the points of intersection of some curve with the axis of abscissas, at long last eliminated the fundamental difference between the positive and negative roots of an equation, since their interpretation proved to be essentially the same.

Integers and fractions, both positive and negative, as well as the digit zero, were grouped under the general term “rational numbers.” The set of rational numbers is said to be closed with respect to the four arithmetic operations. This means that the sum, difference, product, and quotient (except the quotient in division by zero, which is meaningless) of any two rational numbers is also a rational number. The set of rational numbers is ordered with respect to the concepts of greater than and less than. Furthermore, it has the property of density: there are infinitely many rational numbers between any two different rational numbers. This makes it possible to carry out various measurements, for example, of the length of a line segment using a selected unit of measurement, to any degree of accuracy by means of rational numbers. Thus, the set of rational numbers turns out to be sufficient to satisfy many practical needs. The formal substantiation of the concepts of fraction and negative number was accomplished in the 19th century and, in contrast to the substantiation of the concept of natural number, posed no fundamental difficulties.

The set of rational numbers proved insufficient for the study of continuously changing variables, which necessitated a further extension of the concept of number, consisting in the transition from the set of rational numbers to the set of real numbers. This transition involved the addition to the rational numbers the irrational numbers. A discovery of vast fundamental importance was made by the ancient Greeks: not all precisely defined line segments (the term “precisely defined” itself is an idealization inherent in geometry) are commensurable; that is, the length of a line segment cannot always be expressed by rational numbers if another line segment is taken as the unit. A classic example of incommensurable line segments is the side of a square and its diagonal. That incommensurable line segments exist was not an impediment to the development of geometry. The Greeks worked out a theory of the ratios of segments, presented in Euclid’s Elements, that takes into account the possibility of incommensurability. The Greeks knew how to compare the magnitude of such ratios and to perform arithmetic operations on them (in purely geometrical form); that is, they treated them as numbers. However, they did not fully perceive the idea that the ratio of the lengths of incommensurable line segments may be considered as numbers. This may be attributed to the idealist separation of theoretical mathematics from practical problems prevalent in the school to which Euclid belonged. In Archimedes’ works we find greater interest in practical problems, particularly approximate calculations of the ratios of incommensurable line segments, but even Archimedes did not develop the concept of irrational number as a number that expresses the ratio of the lengths of incommensurable line segments.

In the 17th century, the era of the birth of modern science, particularly modern mathematics, a number of methods of studying continuous processes and methods of approximate calculations were developed. A clear definition of the concept of real number is given by I. Newton, one of the founders of mathematical analysis, in Arithmetica universalis: “By number we mean not so much a set of units as the abstract ratio of some quantity to another quantity of the same kind that we have taken as unity.” This formulation gives a unified definition of a real number, whether rational or irrational. Later, in the 1870’s, the concept of real number was refined on the basis of a detailed analysis of the concept of continuity in the works of R. Dedekind, G. Cantor, and K. Weierstrass.

According to Dedekind, the property of continuity of a straight line consists in the fact that if all the points that make up a straight line are divided into two classes such that every point of the first class lies to the left of every point of the second class (that is, if the straight line is “broken” into two parts), then either the first class contains a rightmost point or the second class contains a leftmost point In either case, the “extreme” point is the point at which the “break” of the straight line occurred.

The set of all rational numbers does not possess the property of continuity. If the set of all rational numbers is divided into two classes such that every number of the first class is smaller than every number of the second, then upon such a subdivision (a Dedekind cut) it may turn out that there will be no largest number in the first class and no smallest number in the second. This will be the case, for example, if all negative rational numbers, zero, and all positive (rational) numbers whose square is less than 2 are placed in the first class and all positive (rational) numbers whose square is greater than 2 are placed in the second class. Such a cut is said to be irrational. Then the following definition of an irrational number is given: with every irrational cut in the set of rational numbers we associate an irrational number assumed to be larger than any number of the first class and smaller than any number of the second class. The set of all real numbers, both rational and irrational, already has the property of continuity.

Cantor’s substantiation of the concept of real number differs from Dedekind’s, although it too is based on an analysis of the concept of continuity. Both Dedekind’s. and Cantor’s definitions use the abstraction of the actual infinite. For example, in Dedekind’s theory an irrational number is defined by means of a cut in the set of all rational numbers, which is conceived as being given in its entirety.

Recent years have seen the development of the concept of computable numbers, that is, numbers approximations to which can be given by means of some algorithm. This concept is defined on the basis of a refined concept of algorithm and without resorting to the abstraction of the actual infinite.

The final stage in the development of the concept of number was the introduction of complex numbers (seeCOMPLEX NUMBERS). The concept of complex number emerged in the course of algebra’s development. Apparently, it first arose among 16th-century Italian mathematicians (G. Cardano, R. Bombelli) in connection with the discovery of the algebraic solution of third-and fourth-degree equations. It is known that even the solution of a quadratic equation can sometimes lead to the operation of extracting the square root of a negative number, which cannot be performed in the domain of real numbers. This occurs only in cases when the equation does not have real roots. A practical problem that reduces to the solution of such a quadratic equation turns out to have no solution. The following fact was observed in connection with the discovery of the algebraic solution of third-degree equations. If all three roots of the equation are real numbers, then it proves necessary, in the course of calculations, to extract the square root of a negative number. The “imaginariness” that arises in this case disappears only if all subsequent operations are performed. This fact was the first stimulus to the study of complex numbers. However, mathematicians were slow to accept the use of complex numbers and operations performed on them. The remnants of disbelief in the legitimacy of complex numbers are reflected in the term “imaginary,” preserved to this day. This disbelief was dissipated only after the establishment, in the late 18th century, of the geometrical interpretation of complex numbers as points in a plane and the establishment of the indisputable benefit derived from the introduction of complex numbers in the theory of algebraic equations, especially after the celebrated work of K. Gauss. Even before Gauss, in the works of L. Euler, complex numbers played a significant role not only in algebra but also in mathematical analysis. They acquired particular importance in the 19th century, in connection with the development of the theory of functions of a complex variable.

The set of all complex numbers, like the set of real numbers and the set of rational numbers, is closed with respect to the operations of addition, subtraction, multiplication, and division. Moreover, the set of all complex numbers has the property of algebraic closure, which means that every algebraic equation with complex coefficients has roots that are also in the domain of all complex numbers. The set of all real numbers, especially the rational numbers, does not have the property of algebraic closure. For example, the equation x² + 1 = 0, with real coefficients, does not have real roots. As established by Weierstrass, the set of all complex numbers cannot be enlarged by the inclusion of new numbers in such a way that all laws of operation that are valid in the set of complex numbers are preserved in the enlarged set.

In addition to the main line of development of the concept of number (natural numbers→rational numbers→real numbers→complex numbers), the specific requirements of certain branches of mathematics have engendered various generalizations of the concept of number in essentially different directions. For example, in branches of mathematics associated with set theory, the aforementioned concepts of cardinal and ordinal transfinite numbers are of major importance, P-adic numbers, systems of which are obtained from systems of rational numbers by the inclusion of new entities different from irrational numbers, have assumed major importance in modern number theory. Various systems of entities possessing properties that are more or less close to those of the set of integers or rational numbers—groups, rings, fields, and algebras (seeGROUP; RING, ALGEBRAIC; and FIELD)—are being studied in algebra. (See alsoHYPERCOMPLEX NUMBERS.)

REFERENCES

Istoriia matematiki, vols. 1–3. Moscow, 1970–72.
Van der Waerden, B. L. Probuzhdaiushchaiasia nauka. Moscow, 1959. (Translated from Dutch.)
Entsiklopediia elementarnoi matematiki, book 1: Arifmetika. Moscow-Leningrad, 1951.
Nechaev, V. I. Chislovye sistemy. Moscow, 1972.

D. K. FADDEEV

number

[′nəm·bər] (mathematics) Any real or complex number. The number of elements in a set is the cardinality of the set.

number

1. a concept of quantity that is or can be derived from a single unit, the sum of a collection of units, or zero. Every number occupies a unique position in a sequence, enabling it to be used in counting. It can be assigned to one or more sets that can be arranged in a hierarchical classification: every number is a complex number; a complex number is either an imaginary number or a real number, and the latter can be a rational number or an irrational number; a rational number is either an integer or a fraction, while an irrational number can be a transcendental number or an algebraic number 2. the symbol used to represent a number; numeral 3. a. a self-contained part of an opera or other musical score, esp one for the stage

Numbers

(dreams)Interpreting numbers that we see in dreams may be difficult. Their meaning may be very personal, such as a reflection of financial concern or any other area of daily life represented by numbers. One way to interpret numbers is to try to see how they are specifically related to you. For example, if you see number 25 in your dream and it does not make any sense to you at first, think about the meaningful numbers in your life that add up to 25. Maybe your house number is 12 and your parent’s house number is 13. Together they make 25, and this dream could have been addressing issues about your relationship with your parents. On the other hand, numbers in dreams may represent global concepts and point to collective dilemmas. Even numbers may represent the feminine and odd numbers the masculine. Some interesting interpretations of common numbers include the following: Number two —psychic development and doubling; something new coming up with the potential for building. Number three —the trinity; it is an active or a process number (something is going on in the psyche). Number four —completion and femininity. Number five —life force; refers to the five fingers and five appendages of the body. Number seven —sacred number in Christianity and Judaism; the highest stage of illumination and spirituality. Number twelve —represents time and may mark the most important cycles in life.

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The Natural Numbers

The Integers and Rational Numbers

The Real Numbers

The Complex Numbers

The Algebraic and Transcendental Numbers

Bibliography

Number

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Number

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Patient discussion about number

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noun numeral

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noun amount

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noun crowd

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noun group

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noun issue

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verb amount to

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verb calculate

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verb include

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phrase a number of

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noun arithmetic calculations

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verb to note (items) one by one so as to get a total

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verb to come to in number or quantity

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noun the property possessed by a sum or total or indefinite quantity of units or individuals

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noun a concept of quantity involving zero and units

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noun a short theatrical performance that is part of a longer program

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noun the number is used in calling a particular telephone

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noun a symbol used to represent a number

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noun one of a series published periodically