A number is a
mathematical object used to
count,
label, and
measure. In
mathematics, the definition of number has been extended over the
years to include such numbers as
0,
negative numbers,
rational numbers,
irrational numbers, and
complex numbers.
Mathematical operations are certain procedures that take one or more
numbers as input and produce a number as output.
Unary operations take a single input number and produce a single
output number. For example, the
successor operation adds 1 to an
integer,
thus the successor of 4 is 5.
Binary operations take two input numbers and produce a single output
number. Examples of binary operations include
addition,
subtraction,
multiplication,
division, and
exponentiation. The study of numerical operations is called
arithmetic.
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 (telephone
numbers), for ordering (serial
numbers), and for codes (e.g.,
ISBNs).
In common usage, the word number can mean the abstract object,
the symbol, or the
word for the number.
Classification of numbers
Different types of numbers are used in many cases. Numbers can be
classified into
sets, called number systems. (For different methods of expressing
numbers with symbols, such as the
Roman numerals, see
numeral systems.)
Important number systems
|
Natural |
0, 1, 2, 3, 4, ... or 1, 2, 3, 4, ... |
|
Integers |
..., −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, ... |
|
Rational |
a⁄b
where a and b are integers and b is not 0 |
|
Real |
The limit of a convergent sequence of rational numbers |
|
Complex |
a + bi or a + ib where a
and b are real numbers and i is the square root
of −1 |
Natural numbers
Main article:
Natural number
The most familiar numbers are the
natural 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.[citation
needed] 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
,
and sometimes
or
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. 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,[1]
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
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
(which includes 0), the result is defined as the set of integer numbers,
also called
integers,
Z also written
.
Here the letter Z comes from
German Zahl, meaning "number". The set of integers forms a
ring with operations addition and multiplication.[2]
Rational numbers
A rational number is a number that can be expressed as a
fraction with an integer numerator and a non-zero Integer number
denominator. Fractions are written as two numbers, the numerator and the
denominator, with a dividing bar between them. In the fraction written
m⁄n
or
-
m represents equal parts, where n equal parts of that
size make up m wholes. Two different fractions may correspond to
the same rational number; for example 1⁄2
and 2⁄4 are
equal, that is:
-
If the
absolute value of m is greater than n, then the
absolute value of the fraction is greater than 1. Fractions can be
greater than, less than, or equal to 1 and can also be positive,
negative, or 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
.
Real numbers
Main article:
Real number
The real numbers include all of the measuring numbers. Real numbers
are usually written using
decimal
numerals, in which a
decimal point is placed to the right of the digit with place value
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. Thus
-
represents 1 hundred, 2 tens, 3 ones, 4 tenths, 5 hundredths, and 6
thousandths. In saying the number, the decimal is read "point", thus:
"one two three point four five six". In the US and UK and a number of
other countries, the decimal point is represented by a
period, whereas in continental Europe and certain other countries
the decimal point is represented by a
comma. Zero is often written as 0.0 when it must be treated as a
real number rather than an integer. In the US and UK a number between −1
and 1 is always written with a leading 0 to emphasize the decimal.
Negative real numbers are written with a preceding
minus sign:
-
Every rational number is also a real number. It is not the case,
however, that every real number is rational. If a real number cannot be
written as a fraction of two integers, it is called
irrational. A decimal that can be written as a fraction either ends
(terminates) or forever
repeats, because it is the answer to a problem in division. Thus the
real number 0.5 can be written as 1⁄2
and the real number 0.333... (forever repeating 3s, otherwise written 0.3)
can be written as 1⁄3.
On the other hand, the real number π (pi),
the ratio of the
circumference of any circle to its
diameter, is
-
Since the decimal neither ends nor forever repeats, it cannot be
written as a fraction, and is an example of an irrational number. Other
irrational numbers include
-
(the
square root of 2, that is, the positive number whose square is 2).
Thus 1.0 and
0.999... are two different decimal numerals representing the natural
number 1. There are infinitely many other ways of representing the
number 1, for example 2⁄2,
3⁄3, 1.00, 1.000,
and so on.
Every real number is either rational or irrational. Every real number
corresponds to a point on the
number line. The real numbers also have an important but highly
technical property called the
least upper bound property. The symbol for the real numbers is R,
also written as
.
When a real number represents a
measurement, there is always a
margin of error. This is often indicated by
rounding or
truncating a decimal, so that digits that suggest a greater accuracy
than the measurement itself are removed. The remaining digits are called
significant digits. For example, measurements with a ruler can
seldom be made without a margin of error of at least 0.001 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 of
5.6088 square meters. Since only the first two digits after the
decimal place are significant, this is usually rounded to 5.61.
In
abstract algebra, it can be shown that any
complete
ordered field is isomorphic to the real numbers. The real numbers
are not, however, an
algebraically closed field.
Complex numbers
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
quartic polynomials. This led to expressions involving the square
roots of negative numbers, and eventually to the definition of a new
number: the 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
-
or
-
where a and b are real numbers. In the expression
a + bi, the real number a
is called the
real part and b is called the
imaginary part. If the real part of a complex number is 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
.
In
abstract algebra, the complex numbers are an example of an
algebraically closed field, meaning that every
polynomial with complex
coefficients can be
factored into linear factors. Like the real number system, the
complex number system is a
field and is
complete, but unlike the real numbers it is not
ordered. That is, there is no meaning in saying that i is
greater than 1, nor is there any meaning in saying that i is less
than 1. In technical terms, the complex numbers lack the
trichotomy property.
Complex numbers correspond to points on the
complex plane, sometimes called the Argand plane.
Each of the number systems mentioned above is a
proper subset of the next number system. Symbolically,
.
Computable numbers
Moving to problems of computation, the
computable numbers are determined in the set of the real numbers.
The computable numbers, also known as the recursive numbers or the
computable reals, are the
real numbers that can be computed to within any desired precision by
a finite, terminating
algorithm. Equivalent definitions can be given using
μ-recursive functions,
Turing machines or
λ-calculus as the formal representation of algorithms. The
computable numbers form a
real closed field and can be used in the place of real numbers for
many, but not all, mathematical purposes.
Other types
Algebraic numbers are those that can be expressed as the solution to
a polynomial equation with integer coefficients. The complement of the
algebraic numbers are the
transcendental 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.
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.
For dealing with infinite collections, the natural numbers have been
generalized to the
ordinal numbers and to the
cardinal numbers. The former gives the ordering of the collection,
while the latter gives its size. For the finite set, the ordinal and
cardinal numbers are equivalent, but they differ in the infinite case.
A
relation number is defined as the class of
relations consisting of all those relations that are similar to one
member of the class.[3]
Sets of numbers that are not subsets of the complex numbers are
sometimes called
hypercomplex numbers. They include the
quaternions H, invented by Sir
William Rowan Hamilton, in which multiplication is not
commutative, and the
octonions, in which multiplication is not
associative. Elements of
function fields of non-zero
characteristic behave in some ways like numbers and are often
regarded as numbers by number theorists.
Specific uses
There are also other sets of numbers with specialized uses. Some are
subsets of the complex numbers. For example,
algebraic numbers are the roots of
polynomials with rational
coefficients. Complex numbers that are not algebraic are called
transcendental numbers.
An
even number is an integer that is "evenly divisible" by 2, i.e.,
divisible by 2 without remainder; an odd number is an integer that is
not evenly divisible by 2. (The old-fashioned term "evenly divisible" is
now almost always shortened to "divisible".)
A formal definition of an odd number is that it is an integer of the
form n = 2k + 1, where k
is an integer. An even number has the form n
= 2k where k is an
integer.
A
perfect number is a
positive integer that is the sum of its proper positive
divisors—the
sum of the positive divisors not including the number itself.
Equivalently, a perfect number is a number that is half the sum of all
of its positive divisors, or
σ(n) = 2n. The first perfect number is
6, because 1, 2, and 3 are its proper positive divisors and
1 + 2 + 3 = 6. The next perfect number is
28 = 1 + 2 + 4 + 7 + 14. The next perfect numbers are
496 and
8128 (sequence
A000396 in
OEIS). These first four perfect numbers were the only ones known to
early
Greek mathematics.
A
figurate number is a number that can be represented as a regular and
discrete
geometric pattern (e.g. dots). If the pattern is
polytopic, the figurate is labeled a polytopic number, and may be a
polygonal number or a polyhedral number. Polytopic numbers for
r = 2, 3, and 4 are:
Numerals
Numbers should be distinguished from
numerals, the symbols used to represent numbers. Boyer showed
that Egyptians created the first ciphered numeral system.[citation
needed] Greeks followed by mapping their counting
numbers onto Ionian and Doric alphabets. The number five can be
represented by both the base 10 numeral "5", by the
Roman numeral "Ⅴ" and ciphered letters.
Notations used to represent numbers are discussed in the article
numeral systems. An important development in the history of numerals
was the development of a positional system, like modern decimals, which
can represent very large numbers. The Roman numerals require extra
symbols for larger numbers.
History
First use of
numbers
Bones and other artifacts have been discovered with marks cut into
them that many believe are
tally marks.[4]
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.[5]
Zero
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 (Modern
Iraq) and Egyptian texts used it. Egyptians used the word nfr to
denote zero balance in
double entry accounting entries. Indian texts used a
Sanskrit word Shunye or
shunya to refer to the
concept of void. In mathematics texts this word often refers to
the number zero.[6]
Records show that the
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 depend in large part on the uncertain interpretation of
0. (The ancient Greeks even questioned whether
1 was a number.)
The late
Olmec people of south-central
Mexico
began to use a true zero (a shell
glyph) in
the New World possibly by the 4th century BC
but certainly by 40 BC, which became an integral part of
Maya numerals and the
Maya calendar. 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 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 or a colleague about 725, a true zero symbol.
An early documented use of the zero by
Brahmagupta (in the
Brāhmasphuṭasiddhānta) dates to 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 and
the
Islamic world.
Negative numbers
The abstract concept of negative numbers was recognized as early as
100 BC – 50 BC. The
Chinese
Nine Chapters on the Mathematical Art (Chinese:
Jiu-zhang Suanshu) contains methods
for finding the areas of figures; red rods were used to denote positive
coefficients, black for negative.[7]
This is the earliest known mention of negative numbers in the East; the
first reference in a Western work was in the 3rd century in
Greece.
Diophantus referred to the equation equivalent to
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 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 either by drawing a diagonal
stroke through the right-most non-zero digit of the corresponding
positive number's numeral.[8]
The first use of negative numbers in a European work was by
Chuquet during the 15th century. He used them as
exponents, but referred to them as "absurd numbers".
As recently as the 18th century, it was common practice to ignore any
negative results returned by equations on the assumption that they were
meaningless, just as
René Descartes did with negative solutions in a
Cartesian coordinate system.
Rational numbers
It is likely that the concept of fractional numbers dates to
prehistoric times. The
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 math sutras 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
The earliest known use of irrational numbers was in the
Indian
Sulba Sutras composed between 800 and 500 BC.[9]
The first existence proofs of irrational numbers is usually attributed
to
Pythagoras, more specifically to the
Pythagorean
Hippasus of Metapontum, who produced a (most likely geometrical)
proof of the irrationality of the
square root of 2. The story goes that Hippasus discovered irrational
numbers when trying to represent the square root of 2 as a fraction.
However Pythagoras believed in the absoluteness of numbers, and could
not accept the existence of irrational numbers. He could not disprove
their existence through logic, but he could not accept irrational
numbers, so he sentenced Hippasus to death by drowning.
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 (by his pupil
Kossak),
Heine (Crelle,
74),
Georg Cantor (Annalen, 5), and
Richard Dedekind was brought about. In 1869,
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 (1894). Weierstrass, Cantor, and Heine base their
theories on infinite series, while Dedekind founds his on the idea of a
cut (Schnitt) in the system of
real numbers, separating all
rational numbers into two groups having certain characteristic
properties. The subject has received later contributions at the hands of
Weierstrass,
Kronecker (Crelle, 101), and Méray.
Continued fractions, closely related to irrational numbers (and due
to Cataldi, 1613), received attention at the hands of
Euler, and at the opening of the 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.
Dirichlet also added to the general theory, as have numerous
contributors to the applications of the subject.
Transcendental numbers and reals
The first results concerning
transcendental numbers were
Lambert's 1761 proof that π cannot be rational, and also that en
is irrational if n is rational (unless n
= 0). (The constant
e was first referred to in
Napier's 1618 work on
logarithms.)
Legendre extended this proof to show that π is not the square root
of a rational number. The search for roots of
quintic and higher degree equations was an important development,
the
Abel–Ruffini theorem (Ruffini
1799,
Abel 1824) showed that they could not be solved by
radicals (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.
The existence of transcendental numbers[10]
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 shows 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
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 was a popular topic of philosophical
study among the
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 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 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. This was the first mathematical model that
represented infinity by numbers and gave rules for operating with these
infinite numbers.
In the 1960s,
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
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
Niccolo 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 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
-
seemed capriciously inconsistent with the algebraic identity
-
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
-
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
to guard against this mistake.
The 18th century saw the work of
Abraham de Moivre and
Leonhard Euler.
De Moivre's formula (1730) states:
-
and to Euler (1748)
Euler's formula of
complex analysis:
-
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 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 in 1893. The general theory of fields was created by
Évariste Galois, who studied the fields generated by the roots of
any polynomial equation F(x) = 0.
In 1850
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
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 for finding the
greatest common divisor of two numbers.
In 240 BC,
Eratosthenes used the
Sieve of Eratosthenes to quickly isolate prime numbers. But most
further development of the theory of primes in Europe dates to the
Renaissance and later eras.
In 1796,
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 in 1859. The
prime number theorem was finally proved by
Jacques Hadamard and
Charles de la Vallée-Poussin in 1896. Goldbach and Riemann's
conjectures remain unproven and unrefuted.
See also
- Numerals by culture
- Other related topics
Notes
-
^
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
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.
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 across cultures:
the history of non-Western mathematics. Kluwer Academic
Publishers. p. 451.
ISBN 0-7923-6481-3.
-
^
Bogomolny, A..
"What's a number?". Interactive Mathematics Miscellany
and Puzzles. Retrieved 11
July 2010.
References
-
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 and
Bertrand Russell,
Principia Mathematica to *56, Cambridge University Press,
1910.
- George I. Sanchez, Arithmetic in Maya, Austin-Texas, 1961.
External links
- Nechaev, V.I.
(2001),
"Number", in Hazewinkel, Michiel,
Encyclopedia of Mathematics,
Springer,
ISBN 978-1-55608-010-4
- Tallant, Jonathan.
"Do Numbers Exist?". Numberphile.
Brady Haran.
-
Important Concepts and Formulas - Numbers
-
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).
-
http://planetmath.org/encyclopedia/MayanMath2.html
-
"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.