A number is a mathematical object used to count, measure, and label.
The original examples are the natural numbers 1, 2, 3, 4 and so
forth.[1] A notational symbol that represents a number is called a
numeral.[2] In addition to their use in counting and measuring,
numerals are often used for labels (as with telephone numbers), for
ordering (as with serial numbers), and for codes (as with ISBNs). In
common usage, number may refer to a symbol, a word, or a mathematical
abstraction.
In mathematics, the notion of number has been extended over the
centuries to include 0,[3] negative numbers,[4] rational numbers such
as 1/2 and −2/3, real numbers[5] such as √2 and π, and complex
numbers,[6] which extend the real numbers by adding a square root of
−1.[4] Calculations with numbers are done with arithmetical
operations, the most familiar being addition, subtraction,
multiplication, division, and exponentiation. Their study or usage is
called arithmetic. The same term may also refer to number theory, the
study of the properties of numbers.
Besides their practical uses, numbers have cultural significance
throughout the world.[7][8] For example, in Western society, the
number 13 is regarded as unlucky, and "a million" may signify "a
lot."[7] Though it is now regarded as pseudoscience, numerology, the
belief in a mystical significance of numbers, permeated ancient and
medieval thought.[9]
Contents 1 Numerals 2 Main classification 2.1 Natural numbers 2.2 Integers 2.3 Rational numbers 2.4 Real numbers 2.5 Complex numbers 3 Subclasses of the integers 3.1 Even and odd numbers 3.2 Prime numbers 3.3 Other classes of integers 4 Subclasses of the complex numbers 4.1 Algebraic, irrational and transcendental numbers 4.2 Computable numbers 5 Extensions of the concept 5.1 p-adic numbers 5.2 Hypercomplex numbers 5.3 Transfinite numbers 5.4 Nonstandard numbers 6 History 6.1 First use of numbers
6.2 Zero
6.3 Negative numbers
6.4 Rational numbers
6.5 Irrational numbers
6.6 Transcendental numbers and reals
6.7
7 See also 8 Notes 9 References 10 External links Numerals[edit]
Main article: Numeral system
Numbers should be distinguished from numerals, the symbols used to
represent numbers. The Egyptians invented the first ciphered numeral
system, and the Greeks followed by mapping their counting numbers onto
Ionian and Doric alphabets.[11] Roman numerals, a system that used
combinations of letters from the Roman alphabet, remained dominant in
Main number systems N displaystyle mathbb N Natural 0, 1, 2, 3, 4, 5, ... or 1, 2, 3, 4, 5, ... N 0 displaystyle mathbb N _ 0 or N 1 displaystyle mathbb N _ 1 are sometimes used. Z displaystyle mathbb Z Integer ..., −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, ... Q displaystyle mathbb Q Rational a/b where a and b are integers and b is not 0 R displaystyle mathbb R Real The limit of a convergent sequence of rational numbers C displaystyle mathbb C Complex a + bi where a and b are real numbers and i is a formal square root of −1 There is generally no problem in identifying each number system with a proper subset of the next one (by abuse of notation), because each of these number systems is canonically isomorphic to a proper subset of the next one.[citation needed] The resulting hierarchy allows, for example, to talk, formally correctly, about real numbers that are rational numbers, and is expressed symbolically by writing N ⊂ Z ⊂ Q ⊂ R ⊂ C displaystyle mathbb N subset mathbb Z subset mathbb Q subset mathbb R subset mathbb C . Natural numbers[edit] Main article: Natural number The natural numbers, starting with 1 The most familiar numbers are the natural numbers (sometimes called whole numbers or counting numbers): 1, 2, 3, and so on. Traditionally, the sequence of natural numbers started with 1 (0 was not even considered a number for the Ancient Greeks.) However, in the 19th century, set theorists and other mathematicians started including 0 (cardinality of the empty set, i.e. 0 elements, where 0 is thus the smallest cardinal number) in the set of natural numbers.[14][15] Today, different mathematicians use the term to describe both sets, including 0 or not. The mathematical symbol for the set of all natural numbers is N, also written N displaystyle mathbb N , and sometimes N 0 displaystyle mathbb N _ 0 or N 1 displaystyle mathbb N _ 1 when it is necessary to indicate whether the set should start with 0 or 1, respectively. In the base 10 numeral system, in almost universal use today for mathematical operations, the symbols for natural numbers are written using ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The radix or base is the number of unique numerical digits, including zero, that a numeral system uses to represent numbers (for the decimal system, the radix is 10).[16] In this base 10 system, the rightmost digit of a natural number has a place value of 1, and every other digit has a place value ten times that of the place value of the digit to its right. In set theory, which is capable of acting as an axiomatic foundation for modern mathematics,[17] natural numbers can be represented by classes of equivalent sets. For instance, the number 3 can be represented as the class of all sets that have exactly three elements. Alternatively, in Peano Arithmetic, the number 3 is represented as sss0, where s is the "successor" function (i.e., 3 is the third successor of 0). Many different representations are possible; all that is needed to formally represent 3 is to inscribe a certain symbol or pattern of symbols three times. Integers[edit] Main article: Integer The negative of a positive integer is defined as a number that produces 0 when it is added to the corresponding positive integer. Negative numbers are usually written with a negative sign (a minus sign). As an example, the negative of 7 is written −7, and 7 + (−7) = 0. When the set of negative numbers is combined with the set of natural numbers (including 0), the result is defined as the set of integers, Z also written Z displaystyle mathbb Z . Here the letter Z comes from German Zahl, meaning 'number'. The set of integers forms a ring with the operations addition and multiplication.[18] The natural numbers form a subset of the integers. As there is no common standard for the inclusion or not of zero in the natural numbers, the natural numbers without zero are commonly referred to as positive integers, and the natural numbers with zero are referred to as non-negative integers. Rational numbers[edit] Main article: Rational number A rational number is a number that can be expressed as a fraction with an integer numerator and a positive integer denominator. Negative denominators are allowed, but are commonly avoided, as every rational number is equal to a fraction with positive denominator. Fractions are written as two integers, the numerator and the denominator, with a dividing bar between them. The fraction m/n represents m parts of a whole divided into n equal parts. Two different fractions may correspond to the same rational number; for example 1/2 and 2/4 are equal, that is: 1 2 = 2 4 . displaystyle 1 over 2 = 2 over 4 . In general, a b = c d displaystyle a over b = c over d if and only if a × d = c × b . displaystyle atimes d = ctimes b . If the absolute value of m is greater than n (supposed to be positive), then the absolute value of the fraction is greater than 1. Fractions can be greater than, less than, or equal to 1 and can also be positive, negative, or 0. The set of all rational numbers includes the integers, since every integer can be written as a fraction with denominator 1. For example −7 can be written −7/1. The symbol for the rational numbers is Q (for quotient), also written Q displaystyle mathbb Q . Real numbers[edit] Main article: Real number The symbol for the real numbers is R, also written as R . displaystyle mathbb R . They include all the measuring numbers. Every real number corresponds to a point on the number line. The following paragraph will focus primarily on positive real numbers. The treatment of negative real numbers is according to the general rules of arithmetic and their denotation is simply prefixing the corresponding positive numeral by a minus sign, e.g. -123.456. Most real numbers can only be approximated by decimal numerals, in which a decimal point is placed to the right of the digit with place value 1. Each digit to the right of the decimal point has a place value one-tenth of the place value of the digit to its left. For example, 123.456 represents 123456/1000, or, in words, one hundred, two tens, three ones, four tenths, five hundredths, and six thousandths. A real number can be expressed by a finite number of decimal digits only if it is rational and its fractional part has a denominator whose prime factors are 2 or 5 or both, because these are the prime factors of 10, the base of the decimal system. Thus, for example, one half is 0.5, one fifth is 0.2, one tenth is 0.1, and one fiftieth is 0.02. Representing other real numbers as decimals would require an infinite sequence of digits to the right of the decimal point. If this infinite sequence of digits follows a pattern, it can be written with an ellipsis, or another notation that indicates the repeating pattern. Such a decimal is called a repeating decimal. Thus 1/3 can be written as 0.333..., with an ellipsis to indicate that the pattern continues. Forever repeating 3s are also written as 0.3. It turns out that these repeating decimals (including the repetition of zeroes) denote exactly the rational numbers, i.e., all rational numbers are also real numbers, but it is not the case that every real number is rational. A real number that is not rational is called irrational. A famous irrational real number is the number π, the ratio of the circumference of any circle to its diameter. When pi is written as π = 3.14159265358979 … , displaystyle pi =3.14159265358979dots , as it sometimes is, the ellipsis does not mean that the decimals repeat (they do not), but rather that there is no end to them. It has been proved that π is irrational. Another well known number, proven to be an irrational real number, is 2 = 1.41421356237 … , displaystyle sqrt 2 =1.41421356237dots , the square root of 2, that is, the unique positive real number whose square is 2. Both these numbers have been approximated (by computer) to trillions ( 1 trillion = 1012 = 1,000,000,000,000 ) of digits. Not only these prominent examples, but almost all real numbers are irrational and therefore have no repeating patterns and hence no corresponding decimal numeral. They can only be approximated by decimal numerals, denoting rounded or truncated real numbers. Any rounded or truncated number is necessarily a rational number, of which there are only countably many. All measurements are, by their nature, approximations, and always have a margin of error. Thus 123.456 is considered an approximation of any real number greater or equal to 1234555/10000 and strictly less than 1234565/10000 (rounding to 3 decimals), or of any real number greater or equal to 123456/1000 and strictly less than 123457/1000 (truncation after the 3. decimal). Digits that suggest a greater accuracy than the measurement itself does, should be removed. The remaining digits are then called significant digits. For example, measurements with a ruler can seldom be made without a margin of error of at least 0.001 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 between 5.614591 square meters and 5.603011 square meters. Since not even the second digit after the decimal place is preserved, the following digits are not significant. Therefore the result is usually rounded to 5.61. Just as the same fraction can be written in more than one way, the same real number may have more than one decimal representation. For example, 0.999..., 1.0, 1.00, 1.000, ..., all represent the natural number 1. A given real number has only the following decimal representations: an approximation to some finite number of decimal places, an approximation in which a pattern is established that continues for an unlimited number of decimal places, or an exact value with only finitely many decimal places. In this last case, the last non-zero digit may be replaced by the digit one smaller followed by an unlimited number of 9's, or the last non-zero digit may be followed by an unlimited number of zeros. Thus the exact real number 3.74 can also be written 3.7399999999... and 3.74000000000... . Similarly, a decimal numeral with an unlimited number of 0's can be rewritten by dropping the 0's to the right of the decimal place, and a decimal numeral with an unlimited number of 9's can be rewritten by increasing the rightmost non-9 digit by one, changing all the 9's to the right of that digit to 0's. Finally, an unlimited sequence of 0's to the right of the decimal place can be dropped. For example, 6.849999999999... = 6.85 and 6.850000000000... = 6.85. Finally, if all of the digits in a numeral are 0, the number is 0, and if all of the digits in a numeral are an unending string of 9's, you can drop the nines to the right of the decimal place, and add one to the string of 9s to the left of the decimal place. For example 99.999... = 100. The real numbers also have an important but highly technical property called the least upper bound property. It can be shown that any ordered field, which is also complete, is isomorphic to the real numbers. The real numbers are not, however, an algebraically closed field, because they do not include a solution (often called a square root of minus one) to the algebraic equation x 2 + 1 = 0 displaystyle x^ 2 +1=0 . Complex numbers[edit] Main article: Complex number Moving to a greater level of abstraction, the real numbers can be extended to the complex numbers. This set of numbers arose historically from trying to find closed formulas for the roots of cubic and quadratic polynomials. This led to expressions involving the square roots of negative numbers, and eventually to the definition of a new number: a square root of −1, 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 displaystyle ,a+bi where a and b are real numbers. Because of this, complex numbers correspond to points on the complex plane, a vector space of two real dimensions. In the expression 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 0, then the number is called an imaginary number or is referred to as purely imaginary; if the imaginary part is 0, then the number is a real number. Thus the real numbers are a subset of the complex numbers. If the real and imaginary parts of a complex number are both integers, then the number is called a Gaussian integer. The symbol for the complex numbers is C or C displaystyle mathbb C .
The fundamental theorem of algebra asserts that the complex numbers
form an algebraically closed field, meaning that every polynomial with
complex coefficients has a root in the complex numbers. Like the
reals, the complex numbers form a field, which is complete, but unlike
the real numbers, it is not ordered. That is, there is no consistent
meaning assignable to saying that i is greater than 1, nor is
there any meaning in saying that i is less than 1. In technical
terms, the complex numbers lack of a total order that is compatible
with field operations.
Subclasses of the integers[edit]
Even and odd numbers[edit]
Main article: Even and odd numbers
An even number is an integer that is "evenly divisible" by two, that
is divisible by two without remainder; an odd number is an integer
that is not even. (The old-fashioned term "evenly divisible" is now
almost always shortened to "divisible".) Equivalently, another way of
defining an odd number is that it is an integer of the form n = 2k +
1, where k is an integer, and an even number has the form n = 2k where
k is an integer.
Prime numbers[edit]
Main article: Prime number
A prime number is an integer greater than 1 that is not the product of
two smaller positive integers. The first few prime numbers are 2, 3,
5, 7, and 11. The prime numbers have been widely studied for more than
2000 years and have led to many questions, only some of which have
been answered. The study of these questions belong to number theory.
An example of a question that is still unanswered is whether every
even number is the sum of two primes. This is called Goldbach's
conjecture.
A question that has been answered is whether every integer greater
than one is a product of primes in only one way, except for a
rearrangement of the primes. This is called fundamental theorem of
arithmetic. A proof appears in Euclid's Elements.
Other classes of integers[edit]
Many subsets of the natural numbers have been the subject of specific
studies and have been named, often after the first mathematician that
has studied them. Example of such sets of integers are Fibonacci
numbers and perfect numbers. For more examples, see
The number 605 in Khmer numerals, from an inscription from 683 AD. An early use of zero as a decimal figure. Brahmagupta's Brahmasphuṭasiddhanta is the first book that mentions
zero as a number, hence
( − 1 ) 2 = − 1 − 1 = − 1 displaystyle left( sqrt -1 right)^ 2 = sqrt -1 sqrt -1 =-1 seemed capriciously inconsistent with the algebraic identity a b = a b , displaystyle sqrt a sqrt b = sqrt ab , which is valid for positive real numbers a and b, and was also used in complex number calculations with one of a, b positive and the other negative. The incorrect use of this identity, and the related identity 1 a = 1 a displaystyle frac 1 sqrt a = sqrt frac 1 a in the case when both a and b are negative even bedeviled Euler. This difficulty eventually led him to the convention of using the special symbol i in place of − 1 displaystyle sqrt -1 to guard against this mistake.
The 18th century saw the work of
( cos θ + i sin θ ) n = cos n θ + i sin n θ displaystyle (cos theta +isin theta )^ n =cos ntheta +isin ntheta while
cos θ + i sin θ = e i θ . displaystyle cos theta +isin theta =e^ itheta . The existence of complex numbers was not completely accepted until
Wikimedia Commons has media related to Numbers. Concrete number
List of numbers
Notes[edit] ^ "number, n". OED Online. Oxford University Press.
^ "numeral, adj. and n". OED Online. Oxford University Press.
^ Matson, John. "The Origin of Zero". Scientific American. Retrieved
2017-05-16.
^ a b Hodgkin, Luke (2005-06-02). A History of Mathematics: From
Mesopotamia to Modernity. OUP Oxford. pp. 85–88.
ISBN 9780191523830.
^ T. K. Puttaswamy, "The Accomplishments of Ancient Indian
Mathematicians", pp. 410–1. In: Selin, Helaine; D'Ambrosio,
Ubiratan, eds. (2000),
References[edit] Tobias Dantzig, Number, the language of science; a critical survey
written for the cultured non-mathematician, New York, The Macmillan
company, 1930.
Erich Friedman, What's special about this number?
Steven Galovich, Introduction to Mathematical Structures, Harcourt
Brace Javanovich, 23 January 1989, ISBN 0-15-543468-3.
Paul Halmos, Naive Set Theory, Springer, 1974,
ISBN 0-387-90092-6.
Morris Kline, Mathematical Thought from Ancient to Modern Times,
Oxford University Press, 1972.
External links[edit] Wikiquote has quotations related to: Number Look up number in Wiktionary, the free dictionary. Wikiversity has learning resources about Primary mathematics:Numbers Nechaev, V.I. (2001) [1994], "Number", in Hazewinkel, Michiel,
Encyclopedia of Mathematics,
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