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. A notational symbol that represents a number is called a numeral . 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 , negative numbers , rational numbers such as 1/2 and −2/3, real numbers such as √2 and π , and complex numbers , which extend the real numbers by adding a square root of −1 . 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. For example, in Western society, the number 13
is regarded as unlucky, and "a million " may signify "a lot." Though
it is now regarded as pseudoscience , numerology , the belief in a
mystical significance of numbers, permeated ancient and medieval
thought.
During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers and may be seen as extending the concept. Among the first were the hypercomplex numbers , which consist of various extensions or modifications of the complex number system. Today, number systems are considered important special examples of much more general categories such as rings and fields , and the application of the term "number" is a matter of convention, without fundamental significance. CONTENTS * 1 Numerals * 2 Main classification * 2.1
* 3 Subclasses of the integers * 3.1
* 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 Main article:
Numbers should be distinguished from NUMERALS, the symbols used to
represent numbers. The Egyptians invented the first ciphered numeral
system, and the Greeks followed by mapping their counting numbers onto
Ionian and Doric alphabets. Roman numerals, a system that used
combinations of letters from the Roman alphabet, remained dominant in
MAIN CLASSIFICATION "
Numbers can be classified into sets , called NUMBER SYSTEMS, such as the natural numbers and the real numbers . The major categories of numbers are as follows: 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. 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 Main article:
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
In the base 10 numeral system, in almost universal use today for mathematical operations, the symbols for natural numbers are written using ten digits : 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The radix or base is the number of unique numerical digits, including zero, that a numeral system uses to represent numbers (for the decimal system, the radix is 10). In this base 10 system, the rightmost digit of a natural number has a place value of 1, and every other digit has a place value ten times that of the place value of the digit to its right. In set theory , which is capable of acting as an axiomatic foundation
for modern mathematics, natural numbers can be represented by classes
of equivalent sets. For instance, the number 3 can be represented as
the class of all sets that have exactly three elements. Alternatively,
in Peano
INTEGERS Main article:
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. 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 Main article:
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 Main article:
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 = 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,
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 Main article:
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
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
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 EVEN AND ODD NUMBERS Main article:
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 Main article:
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 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
SUBCLASSES OF THE COMPLEX NUMBERS ALGEBRAIC, IRRATIONAL AND TRANSCENDENTAL NUMBERS
COMPUTABLE NUMBERS Main article:
A COMPUTABLE NUMBER, also known as recursive number, is a real number
such that there exists an algorithm which, given a positive number n
as input, produces the first n digits of the computable number's
decimal representation. Equivalent definitions can be given using
μ-recursive functions ,
The computable numbers may be viewed as the real numbers that may be exactly represented in a computer: a computable number is exactly represented by its first digits and a program for computing further digits. However, the computable numbers are rarely used in practice. One reason is that there is no algorithm for testing the equality of two computable numbers. More precisely, there cannot exist any algorithm which takes any computable number as an input, and decides in every case if this number is equal to zero or not. The set of computable numbers has the same cardinality as the natural numbers. Therefore, almost all real numbers are non-computable. However, it is very difficult to produce explicitly a real number that is not computable. EXTENSIONS OF THE CONCEPT P-ADIC NUMBERS Main article: p-adic number The p-adic numbers may have infinitely long expansions to the left of the decimal point, in the same way that real numbers may have infinitely long expansions to the right. The number system that results depends on what base is used for the digits: any base is possible, but a prime number base provides the best mathematical properties. The set of the p-adic numbers contains the rational numbers, but is not contained in the complex numbers. The elements of an algebraic function field over a finite field and algebraic numbers have many similar properties (see Function field analogy ). Therefore, they are often regarded as numbers by number theorists. The p-adic numbers play an important role in this analogy. HYPERCOMPLEX NUMBERS Main article: hypercomplex number Some number systems that are not included in the complex numbers may
be constructed from the real numbers in a way that generalize the
construction of the complex numbers. They are sometimes called
hypercomplex numbers . They include the quaternions H, introduced by
Sir
TRANSFINITE NUMBERS Main article: transfinite number For dealing with infinite sets , the natural numbers have been generalized to the ordinal numbers and to the cardinal numbers . The former gives the ordering of the set, while the latter gives its size. For finite sets, both ordinal and cardinal numbers are identified with the natural numbers. In the infinite case, many ordinal numbers correspond to the same cardinal number. NONSTANDARD NUMBERS
Superreal and surreal numbers extend the real numbers by adding infinitesimally small numbers and infinitely large numbers, but still form fields . A relation number is defined as the class of relations consisting of all those relations that are similar to one member of the class. HISTORY FIRST USE OF NUMBERS Main article:
Bones and other artifacts have been discovered with marks cut into them that many believe are tally marks . These tally marks may have been used for counting elapsed time, such as numbers of days, lunar cycles or keeping records of quantities, such as of animals. A tallying system has no concept of place value (as in modern decimal notation), which limits its representation of large numbers. Nonetheless tallying systems are considered the first kind of abstract numeral system. The first known system with place value was the Mesopotamian base 60
system (ca. 3400 BC) and the earliest known base 10 system dates to
3100 BC in
ZERO An early documented use of the zero by
Brahmagupta's Brahmasphuṭasiddhanta is the first book that mentions
zero as a number, hence
The use of 0 as a number should be distinguished from its use as a
placeholder numeral in place-value systems . Many ancient texts used
0. Babylonian and Egyptian texts used it. Egyptians used the word nfr
to denote zero balance in double entry accounting . Indian texts used
a
There are other uses of zero before Brahmagupta, though the documentation is not as complete as it is in the Brahmasphutasiddhanta. Records show that the
The late
By 130 AD,
Another true zero was used in tables alongside Roman numerals by 525
(first known use by
NEGATIVE NUMBERS Further information:
The abstract concept of negative numbers was recognized as early as
100 BC – 50 BC in China. The Nine Chapters on the Mathematical Art
contains methods for finding the areas of figures; red rods were used
to denote positive coefficients , black for negative. The first
reference in a Western work was in the 3rd century AD in
During the 600s, negative numbers were in use in
European mathematicians, for the most part, resisted the concept of
negative numbers until the 17th century, although
As recently as the 18th century, it was common practice to ignore any
negative results returned by equations on the assumption that they
were meaningless, just as
RATIONAL NUMBERS It is likely that the concept of fractional numbers dates to
prehistoric times . The
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
IRRATIONAL NUMBERS Further information:
The earliest known use of irrational numbers was in the Indian Sulba
Sutras composed between 800 and 500 BC. The first existence proofs of
irrational numbers is usually attributed to
The 16th century brought final European acceptance of negative
integral and fractional numbers. By the 17th century, mathematicians
generally used decimal fractions with modern notation. It was not,
however, until the 19th century that mathematicians separated
irrationals into algebraic and transcendental parts, and once more
undertook scientific study of irrationals. It had remained almost
dormant since
The search for roots of quintic and higher degree equations was an
important development, the
Continued fractions , closely related to irrational numbers (and due
to Cataldi, 1613), received attention at the hands of
TRANSCENDENTAL NUMBERS AND REALS Further information:
The existence of transcendental numbers was first established by Liouville (1844, 1851). Hermite proved in 1873 that e is transcendental and Lindemann proved in 1882 that π is transcendental. Finally, Cantor showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite , so there is an uncountably infinite number of transcendental numbers. INFINITY AND INFINITESIMALS Further information:
The earliest known conception of mathematical infinity appears in the
In the 1960s,
A modern geometrical version of infinity is given by projective geometry , which introduces "ideal points at infinity", one for each spatial direction. Each family of parallel lines in a given direction is postulated to converge to the corresponding ideal point. This is closely related to the idea of vanishing points in perspective drawing. COMPLEX NUMBERS Further information:
The earliest fleeting reference to square roots of negative numbers
occurred in the work of the mathematician and inventor Heron of
Alexandria in the 1st century AD, when he considered the volume of an
impossible frustum of a pyramid . They became more prominent when in
the 16th century closed formulas for the roots of third and fourth
degree polynomials were discovered by Italian mathematicians such as
This was doubly unsettling since they did not even consider negative
numbers to be on firm ground at the time. When
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
The 18th century saw the work of
while Euler\'s formula of complex analysis (1748) gave us: cos + i sin = e i . {displaystyle cos theta +isin theta =e^{itheta }.} The existence of complex numbers was not completely accepted until
Also in 1799, Gauss provided the first generally accepted proof of
the fundamental theorem of algebra , showing that every polynomial
over the complex numbers has a full set of solutions in that realm.
The general acceptance of the theory of complex numbers is due to the
labors of
Gauss studied complex numbers of the form a + bi, where a and b are
integral, or rational (and i is one of the two roots of x2 + 1 = 0).
His student,
In 1850
PRIME NUMBERS Prime numbers have been studied throughout recorded history. Euclid
devoted one book of the Elements to the theory of primes; in it he
proved the infinitude of the primes and the fundamental theorem of
arithmetic , and presented the
In 240 BC,
In 1796,
SEE ALSO Wikimedia Commons has media related to NUMBERS . *
NOTES * ^ "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.
REFERENCES *
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