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]
Numerology
Numerology heavily influenced the development of
Greek mathematics, stimulating the investigation of many problems in
number theory which are still of interest today.[9]
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.[10]
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
Infinity
Infinity and infinitesimals
6.8 Complex numbers
6.9 Prime numbers
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
Europe
Europe until the spread of the superior Arabic numeral system around
the late 14th century, and the Arabic numeral system remains the most
common system for representing numbers in the world today.[12] The key
to the effectiveness of the system was the symbol for zero, which was
developed by ancient mathematicians in the Indian subcontinent around
500 AD.[12]
Main classification[edit]
"
Number
Number system" redirects here. For systems for expressing numbers,
see Numeral system.
See also: List of types of numbers
Numbers can be classified into sets, called number systems, such as
the natural numbers and the real numbers.[13] 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.[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
Integer
Integer sequence.
Subclasses of the complex numbers[edit]
Algebraic, irrational and transcendental numbers[edit]
Algebraic numbers
Algebraic numbers are those that are a solution to a polynomial
equation with integer coefficients.
Real numbers
Real numbers that are not rational
numbers are called irrational numbers.
Complex numbers
Complex numbers which are not
algebraic are called transcendental numbers. The algebraic numbers
that are solutions of a monic polynomial equation with integer
coefficients are called algebraic integers.
Computable numbers[edit]
Main article: Computable number
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,
Turing machines
Turing machines or λ-calculus. The computable
numbers are stable for all usual arithmetic operations, including the
computation of the roots of a polynomial, and thus form a real closed
field that contains the real algebraic numbers.
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[edit]
p-adic numbers[edit]
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[edit]
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 William Rowan Hamilton, in which multiplication is not
commutative, the octonions, in which multiplication is not associative
in addition to not being commutative, and the sedenions, in which
multiplication is not alternative, neither associative nor
commutative.
Transfinite numbers[edit]
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[edit]
Hyperreal numbers
Hyperreal numbers are used in non-standard analysis. The hyperreals,
or nonstandard reals (usually denoted as *R), denote an ordered field
that is a proper extension of the ordered field of real numbers R and
satisfies the transfer principle. This principle allows true
first-order statements about R to be reinterpreted as true first-order
statements about *R.
Superreal and surreal numbers extend the real numbers by adding
infinitesimally small numbers and infinitely large numbers, but still
form fields.
A relation number is defined as the class of relations consisting of
all those relations that are similar to one member of the
class.[19][clarification needed]
History[edit]
First use of numbers[edit]
Main article: History of ancient numeral systems
Bones and other artifacts have been discovered with marks cut into
them that many believe are tally marks.[20] 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 Egypt.[21]
Zero [edit]
An early documented use of the zero by
Brahmagupta
Brahmagupta (in the
Brāhmasphuṭasiddhānta) dates to AD 628. He treated 0 as a
number and discussed operations involving it, including division. By
this time (the 7th century) the concept had clearly reached
Cambodia as Khmer numerals, and documentation shows the idea later
spreading to
China
China and the Islamic world.
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
Brahmagupta
Brahmagupta is usually considered the first to
formulate the concept of zero. He gave rules of using zero with
negative and positive numbers. Zero plus a positive number is the
positive number and negative number plus zero is a negative number
etc. The Brahmasphutasiddhanta is the earliest known text to treat
zero as a number in its own right, rather than as simply a placeholder
digit in representing another number as was done by the Babylonians or
as a symbol for a lack of quantity as was done by
Ptolemy
Ptolemy and the
Romans.
The use of 0 as a number should be distinguished from its use as a
placeholder numeral in place-value systems. Many ancient texts
used 0. Babylonian and Egyptian texts used it. Egyptians used the
word nfr to denote zero balance in double entry accounting.
Indian texts used a
Sanskrit
Sanskrit word Shunye or shunya to refer to the
concept of void. In mathematics texts this word often refers to the
number zero.[22] In a similar vein,
Pāṇini
Pāṇini (5th century BC) used
the null (zero) operator in the Ashtadhyayi, an early example of an
algebraic grammar for the
Sanskrit
Sanskrit language (also see Pingala).
There are other uses of zero before Brahmagupta, though the
documentation is not as complete as it is in the
Brahmasphutasiddhanta.
Records show that the
Ancient Greeks
Ancient Greeks seemed unsure about the status
of 0 as a number: they asked themselves "how can 'nothing' be
something?" leading to interesting philosophical and, by the Medieval
period, religious arguments about the nature and existence of 0
and the vacuum. The paradoxes of
Zeno of Elea
Zeno of Elea depend in part on the
uncertain interpretation of 0. (The ancient Greeks even
questioned whether 1 was a number.)
The late
Olmec
.jpg/300px-Las_Bocas_Olmec_baby-face_figurine_(Bookgrrrl).jpg)
Olmec people of south-central
Mexico
Mexico began to use a symbol
for zero, a shell glyph, in the New World, possibly by the 4th century
BC but certainly by 40 BC, which became an integral part of Maya
numerals and the Maya calendar. Mayan arithmetic used base 4 and
base 5 written as base 20. Sanchez in 1961 reported a
base 4, base 5 "finger" abacus.
By 130 AD, Ptolemy, influenced by
Hipparchus
Hipparchus and the Babylonians, was
using a symbol for 0 (a small circle with a long overbar) within
a sexagesimal numeral system otherwise using alphabetic Greek
numerals. Because it was used alone, not as just a placeholder, this
Hellenistic zero was the first documented use of a true zero in the
Old World. In later Byzantine manuscripts of his Syntaxis Mathematica
(Almagest), the Hellenistic zero had morphed into the Greek letter
omicron (otherwise meaning 70).
Another true zero was used in tables alongside Roman numerals by 525
(first known use by Dionysius Exiguus), but as a word, nulla meaning
nothing, not as a symbol. When division produced 0 as a
remainder, nihil, also meaning nothing, was used. These medieval zeros
were used by all future medieval computists (calculators of Easter).
An isolated use of their initial, N, was used in a table of Roman
numerals by
Bede
Bede or a colleague about 725, a true zero symbol.
Negative numbers [edit]
Further information: History of negative numbers
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.[23] The first
reference in a Western work was in the 3rd century AD in Greece.
Diophantus
Diophantus referred to the equation equivalent to 4x + 20 = 0 (the
solution is negative) in Arithmetica, saying that the equation gave an
absurd result.
During the 600s, negative numbers were in use in
India
India to represent
debts. Diophantus' previous reference was discussed more explicitly by
Indian mathematician Brahmagupta, in
Brāhmasphuṭasiddhānta 628,
who used negative numbers to produce the general form quadratic
formula that remains in use today. However, in the 12th century
in India, Bhaskara gives negative roots for quadratic equations but
says the negative value "is in this case not to be taken, for it is
inadequate; people do not approve of negative roots."
European mathematicians, for the most part, resisted the concept of
negative numbers until the 17th century, although Fibonacci
allowed negative solutions in financial problems where they could be
interpreted as debts (chapter 13 of Liber Abaci, 1202) and later
as losses (in Flos). At the same time, the Chinese were indicating
negative numbers by drawing a diagonal stroke through the right-most
non-zero digit of the corresponding positive number's numeral.[24] The
first use of negative numbers in a European work was by Nicolas
Chuquet during the 15th century. He used them as exponents, but
referred to them as "absurd numbers".
As recently as the 18th century, it was common practice to ignore any
negative results returned by equations on the assumption that they
were meaningless, just as
René Descartes
René Descartes did with negative solutions
in a Cartesian coordinate system.
Rational numbers [edit]
It is likely that the concept of fractional numbers dates to
prehistoric times. The
Ancient Egyptians
Ancient Egyptians used their Egyptian fraction
notation for rational numbers in mathematical texts such as the Rhind
Mathematical Papyrus and the Kahun Papyrus. Classical Greek and Indian
mathematicians made studies of the theory of rational numbers, as part
of the general study of number theory. The best known of these is
Euclid's Elements, dating to roughly 300 BC. Of the Indian texts,
the most relevant is the Sthananga Sutra, which also covers number
theory as part of a general study of mathematics.
The concept of decimal fractions is closely linked with decimal
place-value notation; the two seem to have developed in tandem. For
example, it is common for the
Jain
Jain math sutra to include calculations
of decimal-fraction approximations to pi or the square root of 2.
Similarly, Babylonian math texts had always used sexagesimal
(base 60) fractions with great frequency.
Irrational numbers [edit]
Further information: History of irrational numbers
The earliest known use of irrational numbers was in the Indian Sulba
Sutras composed between 800 and 500 BC.[25] The first existence
proofs of irrational numbers is usually attributed to Pythagoras, more
specifically to the Pythagorean
Hippasus
Hippasus of Metapontum, who produced a
(most likely geometrical) proof of the irrationality of the square
root of 2. The story goes that
Hippasus
Hippasus discovered irrational numbers
when trying to represent the square root of 2 as a fraction. However
Pythagoras
Pythagoras believed in the absoluteness of numbers, and could not
accept the existence of irrational numbers. He could not disprove
their existence through logic, but he could not accept irrational
numbers, and so, allegedly and frequently reported, he sentenced
Hippasus
Hippasus to death by drowning, to impede spreading of this
disconcerting news.[26]
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 Euclid. In 1872, the publication of the theories
of
Karl Weierstrass
Karl Weierstrass (by his pupil E. Kossak),
Eduard Heine (Crelle,
74),
Georg Cantor
Georg Cantor (Annalen, 5), and
Richard Dedekind
Richard Dedekind was brought
about. In 1869,
Charles Méray had taken the same point of departure
as Heine, but the theory is generally referred to the year 1872.
Weierstrass's method was completely set forth by Salvatore Pincherle
(1880), and Dedekind's has received additional prominence through the
author's later work (1888) and endorsement by
Paul Tannery
Paul Tannery (1894).
Weierstrass, Cantor, and Heine base their theories on infinite series,
while Dedekind founds his on the idea of a cut (Schnitt) in the system
of real numbers, separating all rational numbers into two groups
having certain characteristic properties. The subject has received
later contributions at the hands of Weierstrass, Kronecker (Crelle,
101), and Méray.
The search for roots of quintic and higher degree equations was an
important development, the
Abel–Ruffini theorem
Abel–Ruffini theorem (Ruffini 1799, Abel
1824) showed that they could not be solved by radicals (formulas
involving only arithmetical operations and roots). Hence it was
necessary to consider the wider set of algebraic numbers (all
solutions to polynomial equations). Galois (1832) linked polynomial
equations to group theory giving rise to the field of Galois theory.
Continued fractions, closely related to irrational numbers (and due to
Cataldi, 1613), received attention at the hands of Euler, and at the
opening of the 19th century were brought into prominence through
the writings of Joseph Louis Lagrange. Other noteworthy contributions
have been made by Druckenmüller (1837), Kunze (1857), Lemke (1870),
and Günther (1872). Ramus (1855) first connected the subject with
determinants, resulting, with the subsequent contributions of Heine,
Möbius, and Günther, in the theory of Kettenbruchdeterminanten.
Transcendental numbers and reals [edit]
Further information: History of π
The existence of transcendental numbers[27] 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
Infinity and infinitesimals [edit]
Further information: History of infinity
The earliest known conception of mathematical infinity appears in the
Yajur Veda, an ancient Indian script, which at one point states, "If
you remove a part from infinity or add a part to infinity, still what
remains is infinity."
Infinity
Infinity was a popular topic of philosophical
study among the
Jain
Jain mathematicians c. 400 BC. They distinguished
between five types of infinity: infinite in one and two directions,
infinite in area, infinite everywhere, and infinite perpetually.
Aristotle
Aristotle defined the traditional Western notion of mathematical
infinity. He distinguished between actual infinity and potential
infinity—the general consensus being that only the latter had true
value. Galileo Galilei's
Two New Sciences
.png/440px-Galileo_Galilei,_Discorsi_e_Dimostrazioni_Matematiche_Intorno_a_Due_Nuove_Scienze,_1638_(1400x1400).png)
Two New Sciences discussed the idea of
one-to-one correspondences between infinite sets. But the next major
advance in the theory was made by Georg Cantor; in 1895 he published a
book about his new set theory, introducing, among other things,
transfinite numbers and formulating the continuum hypothesis.
In the 1960s,
Abraham Robinson
Abraham Robinson showed how infinitely large and
infinitesimal numbers can be rigorously defined and used to develop
the field of nonstandard analysis. The system of hyperreal numbers
represents a rigorous method of treating the ideas about infinite and
infinitesimal numbers that had been used casually by mathematicians,
scientists, and engineers ever since the invention of infinitesimal
calculus by Newton and Leibniz.
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
Complex numbers [edit]
Further information: History of complex numbers
The earliest fleeting reference to square roots of negative numbers
occurred in the work of the mathematician and inventor Heron of
Alexandria in the 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
Niccolò Fontana Tartaglia
Niccolò Fontana Tartaglia and Gerolamo Cardano. It was soon
realized that these formulas, even if one was only interested in real
solutions, sometimes required the manipulation of square roots of
negative numbers.
This was doubly unsettling since they did not even consider negative
numbers to be on firm ground at the time. When
René Descartes
René Descartes coined
the term "imaginary" for these quantities in 1637, he intended it as
derogatory. (See imaginary number for a discussion of the "reality" of
complex numbers.) A further source of confusion was that the equation
(
−
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
Abraham de Moivre
Abraham de Moivre and Leonhard Euler.
De Moivre's formula (1730) states:
(
cos
θ
+
i
sin
θ
)
n
=
cos
n
θ
+
i
sin
n
θ
displaystyle (cos theta +isin theta )^ n =cos ntheta +isin ntheta
while
Euler's formula
Euler's formula of complex analysis (1748) gave us:
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
Caspar Wessel described the geometrical interpretation in 1799. Carl
Friedrich Gauss rediscovered and popularized it several years later,
and as a result the theory of complex numbers received a notable
expansion. The idea of the graphic representation of complex numbers
had appeared, however, as early as 1685, in Wallis's De Algebra
tractatus.
Also in 1799, Gauss provided the first generally accepted proof of the
fundamental theorem of algebra, showing that every polynomial over the
complex numbers has a full set of solutions in that realm. The general
acceptance of the theory of complex numbers is due to the labors of
Augustin Louis Cauchy
Augustin Louis Cauchy and Niels Henrik Abel, and especially the
latter, who was the first to boldly use complex numbers with a success
that is well known.
Gauss studied complex numbers of the form a + bi, where a and b are
integral, or rational (and i is one of the two roots of x2 + 1 = 0).
His student, Gotthold Eisenstein, studied the type a + bω, where ω
is a complex root of x3 − 1 = 0. Other such classes (called
cyclotomic fields) of complex numbers derive from the roots of unity
xk − 1 = 0 for higher values of k. This generalization is largely
due to Ernst Kummer, who also invented ideal numbers, which were
expressed as geometrical entities by
Felix Klein
Felix Klein in 1893.
In 1850
Victor Alexandre Puiseux
Victor Alexandre Puiseux took the key step of distinguishing
between poles and branch points, and introduced the concept of
essential singular points. This eventually led to the concept of the
extended complex plane.
Prime numbers [edit]
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
Euclidean algorithm
Euclidean algorithm for finding the
greatest common divisor of two numbers.
In 240 BC,
Eratosthenes
Eratosthenes used the Sieve of
Eratosthenes
Eratosthenes to quickly
isolate prime numbers. But most further development of the theory of
primes in
Europe
Europe dates to the
Renaissance
Renaissance and later eras.
In 1796,
Adrien-Marie Legendre
.jpg/440px-Legendre_and_Fourier_(1820).jpg)
Adrien-Marie Legendre conjectured the prime number theorem,
describing the asymptotic distribution of primes. Other results
concerning the distribution of the primes include Euler's proof that
the sum of the reciprocals of the primes diverges, and the Goldbach
conjecture, which claims that any sufficiently large even number is
the sum of two primes. Yet another conjecture related to the
distribution of prime numbers is the Riemann hypothesis, formulated by
Bernhard Riemann
Bernhard Riemann in 1859. The prime number theorem was finally proved
by
Jacques Hadamard
Jacques Hadamard and
Charles de la Vallée-Poussin
Charles de la Vallée-Poussin in 1896.
Goldbach and Riemann's conjectures remain unproven and unrefuted.
See also[edit]
Wikimedia Commons has media related to Numbers.
Concrete number
List of numbers
List of numbers in various languages
Mathematical constants
Mythical numbers
Numerical cognition
Orders of magnitude
Physical constants
Pi
Prime number
Subitizing and counting
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),
Mathematics
Mathematics Across Cultures: The History of
Non-western Mathematics, Springer, ISBN 1-4020-0260-2 .
^ Descartes, René (1954) [1637], La Géométrie The Geometry of
René Descartes
René Descartes with a facsimile of the first edition, Dover
Publications, ISBN 0-486-60068-8, retrieved 20 April 2011
^ a b Gilsdorf, Thomas E. Introduction to Cultural Mathematics: With
Case Studies in the Otomies and Incas, John Wiley & Sons, Feb 24,
2012.
^ Restivo, S.
Mathematics
Mathematics in Society and History, Springer Science
& Business Media, Nov 30, 1992.
^ a b Ore, Oystein.
Number
Number Theory and Its History, Courier Dover
Publications.
^ Gouvea, Fernando Q. The Princeton Companion to Mathematics, Chapter
II.1, "The Origins of Modern Mathematics", p. 82. Princeton University
Press, September 28, 2008. ISBN 978-0691118802.
^ Chrisomalis, Stephen (2003-09-01). "The Egyptian origin of the Greek
alphabetic numerals". Antiquity. 77 (297): 485–496.
doi:10.1017/S0003598X00092541. ISSN 0003-598X.
^ a b Bulliet, Richard; Crossley, Pamela; Headrick,, Daniel; Hirsch,
Steven; Johnson, Lyman (2010). The Earth and Its Peoples: A Global
History, Volume 1. Cengage Learning. p. 192.
ISBN 1439084742. Indian mathematicians invented the concept of
zero and developed the "Arabic" numerals and system of place-value
notation used in most parts of the world
today [better source needed]
^ "Eine Menge, ist die Zusammenfassung bestimmter, wohlunterschiedener
Objekte unserer Anschauung oder unseres Denkens – welche Elemente
der Menge genannt werden – zu einem Ganzen." [1]
^ Weisstein, Eric W. "Natural Number". MathWorld.
^ "natural number", Merriam-Webster.com, Merriam-Webster, retrieved 4
October 2014
^ "Base System". www.chalkstreet.com. 29 June 2016. Retrieved 31
August 2016.
^ Suppes, Patrick (1972). Axiomatic Set Theory. Courier Dover
Publications. p. 1. ISBN 0-486-61630-4.
^ Weisstein, Eric W. "Integer". MathWorld.
^ Russell, Bertrand (1919). Introduction to Mathematical Philosophy.
Routledge. p. 56. ISBN 0-415-09604-9.
^ Marshak, A., The Roots of Civilisation; Cognitive Beginnings of
Man’s First Art,
Symbol
Symbol and Notation, (Weidenfeld & Nicolson,
London: 1972), 81ff.
^ "Egyptian Mathematical Papyri – Mathematicians of the African
Diaspora". Math.buffalo.edu. Retrieved 2012-01-30.
^ "Historia Matematica Mailing List Archive: Re: [HM] The Zero Story:
a question". Sunsite.utk.edu. 1999-04-26. Archived from the original
on 2012-01-12. Retrieved 2012-01-30.
^ Staszkow, Ronald; Robert Bradshaw (2004). The Mathematical Palette
(3rd ed.). Brooks Cole. p. 41. ISBN 0-534-40365-4.
^ Smith, David Eugene (1958). History of Modern Mathematics. Dover
Publications. p. 259. ISBN 0-486-20429-4.
^ Selin, Helaine, ed. (2000).
Mathematics
Mathematics across cultures: the history
of non-Western mathematics. Kluwer Academic Publishers. p. 451.
ISBN 0-7923-6481-3.
^ Bernard Frischer (1984). "Horace and the Monuments: A New
Interpretation of the Archytas Ode". In D. R. Shackleton Bailey.
Harvard Studies in Classical Philology. Harvard University Press.
p. 83. ISBN 0-674-37935-7.
^ Bogomolny, A. "What's a number?". Interactive
Mathematics
Mathematics Miscellany
and Puzzles. Retrieved 11 July 2010.
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.
Alfred North Whitehead
Alfred North Whitehead and Bertrand Russell,
Principia Mathematica
Principia Mathematica to
*56, Cambridge University Press, 1910.
George I. Sanchez,
Arithmetic
Arithmetic in Maya, Austin-Texas, 1961.
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,
Springer Science+Business Media
Springer Science+Business Media B.V. /
Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Tallant, Jonathan. "Do Numbers Exist?". Numberphile. Brady
Haran.
Mesopotamian and Germanic numbers
BBC Radio 4, In Our Time: Negative Numbers
'4000 Years of Numbers', lecture by Robin Wilson, 07/11/07, Gresham
College (available for download as MP3 or MP4, and as a text file).
"What's the World's Favorite Number?". 2011-06-22. Retrieved
2011-09-17. ; "Cuddling With 9, Smooching With 8, Winking At 7".
2011-08-11. Retrieved 2011-09-17.
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displaystyle mathbb N
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)
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