A
number is a
mathematical object used in
counting and
measuring.
A notational symbol which represents a number is called a
numeral, but in common usage the word number
is used for both the abstract object and the symbol, as well as for
the
word for the number. In
addition to their use in counting and measuring, numerals are often
used for labels (
telephone
numbers), for ordering (
serial
numbers), and for codes (
ISBNs). In
mathematics, the definition of number
has been extended over the years to include such numbers as
zero,
negative numbers,
rational numbers,
irrational numbers, and
complex numbers.
Certain procedures which take one or more numbers as input and
produce a number as output are called numerical
operations.
Unary operations take a single input number
and produce a single output number. For example, the
successor operation adds one to an
integer, thus the successor of 4 is 5. More common are
binary operations which 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.
The branch of
mathematics that studies
structure in number systems, by means of topics such as
groups,
rings and
fields, is called
abstract algebra.
Types of numbers
Different types of numbers are used in different 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.)
Natural numbers
The most familiar numbers are the
natural numbers or counting numbers:
one, two, three, and so on.
In the
base ten number system, in almost
universal use today for arithmetic 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 ten system, the rightmost digit of a natural number has a
place value of one, and every other digit has a place value ten
times that of the place value of the digit to its right. The symbol
for the set of all natural numbers is
N, also
written
\mathbb{N}.
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 Arithmetic, the number 3 is
represented as sss0, where s is the "successor" function. Many
different representations are possible; all that is needed to
formally represent 3 is to inscribe a certain symbol or pattern of
symbols 3 times.
Integers
Negative numbers
are numbers that are less than zero. They are the opposite of
positive numbers. For example, if a positive number indicates a
bank deposit, then a negative number indicates a withdrawal of the
same amount. Negative numbers are usually written by a negative
sign (also called a minus sign) in front of the number they are the
opposite of. Thus the opposite of 7 is written −7. When the
set of negative numbers is
combined with the natural numbers and zero, the result is the set
of integer numbers, also called
integers,
Z also written
\mathbb{Z}. Here the letter Z comes
from the German word
Zahl, (plural
Zahlen).
Rational numbers
A
rational number
is a number that can be expressed as a
fraction with an integer
numerator and a non-zero natural number
denominator. The fraction
m/
n
or
- m \over n \,
m represents equal parts, where
n equal parts of
that size make up one whole. Two different fractions may correspond
to the same rational number; for example 1/2 and 2/4 are equal,
that is:
- {1 \over 2} = {2 \over 4}.\,
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 zero. 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
\mathbb{Q}.
Real numbers
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
one. 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
- 123.456\,
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 necessary to indicate that it is to be treated as a real
number rather than as an integer. Negative real numbers are written
with a preceding
minus sign:
- -123.456.\,
Every rational number is also a real number. To write a fraction as
a decimal, divide the numerator by the denominator. 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 threes) 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
- \pi = 3.14159265358979....\,
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
- \sqrt{2} = 1.41421356237 ...\,
(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 or \mathbb{R}.
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 0.01 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, the real
numbers are up to isomorphism uniquely characterized by being the
only
complete ordered field. They are not, however, an
algebraically closed
field.
Complex numbers
Moving to a greater level of abstraction, the real numbers can be
extended to the
complex
numbers. This set of numbers arose, historically, from
the question of whether a negative number can have a
square root. This led to the invention of a new
number: the square root of negative one, 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
where
a and
b are real numbers. In the expression
a +
bi, the real number
a is called the
real part and
bi is called the
imaginary
part. If the real part of a complex number is zero, then the
number is called an
imaginary
number or is referred to as
purely imaginary; if the
imaginary part is zero, 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 \mathbb{C}.
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 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, \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q}
\subset \mathbb{R} \subset \mathbb{C} .
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
Hyperreal and hypercomplex numbers
are used in
non-standard
analysis. The hyperreals, or
nonstandard reals
(usually denoted as *
R), denote an
ordered field which is a proper
extension of the ordered field of
real numbers R and which
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 idea behind
p-adic numbers is
this:While real numbers may have infinitely long expansions to the
right of the decimal point, these numbers allow for infinitely long
expansions to the left. The number system which results depends on
what
base is used for the digits: any base is
possible, but a system with the best mathematical properties is
obtained when the base is a
prime
number.
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.
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.
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.
In addition, various specific kinds of numbers are studied in sets
of
natural and
integer 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 =
2
k + 1, where
k is an integer. An even number has
the form
n = 2
k where
k is an
integer.
A
perfect number is defined as a
positive integer which is
the sum of its proper positive
divisors,
that is, 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) = 2
n. 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 . 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:
A
relation number is defined as the class of
relations consisting of all
those relations that are
similar to one
member of the class.
Numerals
Numbers should be distinguished from
numerals, the symbols used to represent
numbers. Boyer showed that Egyptians created the first ciphered
numeral system. Greeks followed by mapping their counting numbers
onto Ionian and Doric alpabets. The number five can be represented
by both the base ten numeral '5', by the
Roman numeral ' ' 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
History of integers
The first use of numbers
It is speculated that the first known use of numbers dates back to
around 35,000 BC. Bones and other artifacts have been discovered
with marks cut into them which many consider to be
tally marks. The uses of these tally marks may
have been for counting elapsed time, such as numbers of days, or
keeping records of quantities, such as of animals.
Tallying systems have no concept of place-value (such as in the
currently used decimal notation), which limit its representation of
large numbers and as such is often considered that this is the
first kind of abstract system that would be used, and could be
considered a 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
.
[3463]
History of zero
The use of zero as a number should be distinguished from its use as
a placeholder numeral in
place-value
systems. Many ancient texts used zero. Babylonians 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
Shunya to refer to the
concept of
void; in mathematics texts this word would
often be used to refer to the number zero.
[3464]. In a similar vein,
Pāṇini (5th century BC) used the null
(zero) operator (ie a
lambda
production) in the
Ashtadhyayi, his
algebraic grammar for the
Sanskrit language. (also see
Pingala)
Records show that the
Ancient Greeks
seemed unsure about the status of zero 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 zero
and the
vacuum. The
paradoxes of
Zeno
of Elea depend in large part on the uncertain interpretation of
zero. (The ancient Greeks even questioned if
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,
Ptolemy, influenced by
Hipparchus and the Babylonians, was using a
symbol for zero (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 zero 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
Brahmasphutasiddhanta) dates to 628.
He treated zero as a number and discussed operations involving it,
including
division.
By this time (7th
century) the concept had clearly reached Cambodia, and documentation shows the idea
later spreading to China
and the
Islamic world.
History of negative numbers
The abstract concept of negative numbers was recognised as early as
100 BC - 50 BC.
The Chinese
”Nine Chapters
on the Mathematical Art” (Jiu-zhang Suanshu)
contains methods for finding the areas of figures; red rods were
used to denote positive coefficients,
black for negative. 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 would be 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
Brahma-Sphuta-Siddhanta 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 nonzero digit of the corresponding positive number's
numeral . 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, the Swiss
mathematician Leonhard Euler believed
that negative numbers were greater than infinity , and 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.
History of rational, irrational, and real numbers
History of rational numbers
It is likely that the concept of fractional numbers dates to
prehistoric times. Even the
Ancient Egyptians wrote math texts
describing how to convert general
fractions into their
special notation. The
RMP 2/n table and the
Kahun Papyrus wrote out unit fraction series
by using least common multiples. 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 two. Similarly, Babylonian
math texts had always used sexagesimal fractions with great
frequency.
History of irrational numbers
The earliest known use of irrational numbers was in the
Indian Sulba
Sutras composed between 800-500 BC. 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 his beliefs would not accept the existence of irrational
numbers and so he sentenced Hippasus to death by drowning.
The sixteenth century saw the final acceptance by Europeans of
negative, integral
and
fractional numbers. The
seventeenth century saw decimal fractions with the modern notation
quite generally used by mathematicians. But it was not until the
nineteenth century that the irrationals were separated into
algebraic and transcendental parts, and a scientific study of
theory of irrationals was taken once more. It had remained almost
dormant since
Euclid. The year 1872 saw the
publication of the theories of
Karl
Weierstrass (by his pupil
Kossak),
Heine (
Crelle, 74),
Georg
Cantor (Annalen, 5), and
Richard
Dedekind.
Méray had taken in 1869 the
same point of departure as
Heine, but
the theory is generally referred to the year 1872. Weierstrass's
method has been completely set forth by
Salvatore Pincherle (1880), and
Dedekind's has received additional prominence through the author's
later work (1888) and the recent 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 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 nineteenth 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 (formula
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.
Even the set of algebraic numbers was not sufficient and the full
set of real number includes
transcendental numbers. The existence
of which 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
The earliest known conception of mathematical
infinity appears in the
Yajur
Veda - an ancient script in India, 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 circa 400 BC. They distinguished between five types
of infinity: infinite in one and two directions, infinite in area,
infinite everywhere, and infinite perpetually.
In the West, the traditional notion of mathematical infinity was
defined by
Aristotle, who distinguished
between
actual infinity and
potential infinity; the general consensus
being that only the latter had true value.
Galileo'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
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 (see
Niccolo Fontana Tartaglia,
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. The term
"imaginary" for these quantities was coined by
René Descartes in 1637 and was meant to
be derogatory (see
imaginary number
for a discussion of the "reality" of complex numbers). A further
source of confusion was that the equation
- \left ( \sqrt{-1}\right )^2 =\sqrt{-1}\sqrt{-1}=-1
seemed to be capriciously inconsistent with the algebraic identity
- \sqrt{a}\sqrt{b}=\sqrt{ab},
which is valid for positive real numbers
a and
b,
and which 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
- \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 to guard against this mistake.
The 18th century saw the labors of
Abraham de Moivre and
Leonhard Euler. To De Moivre is due (1730)
the well-known formula which bears his name,
de Moivre's formula:
- (\cos \theta + i\sin \theta)^{n} = \cos n \theta + i\sin n
\theta \,
and to Euler (1748)
Euler's formula
of
complex analysis:
- \cos \theta + i\sin \theta = e ^{i\theta }. \,
The existence of complex numbers was not completely accepted until
the geometrical interpretation had been described by
Caspar Wessel in 1799; it was rediscovered
several years later and popularized by
Carl Friedrich Gauss, 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 not a little 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,
Ferdinand 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 are
derived 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
would eventually lead 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. The conjectures of Goldbach and Riemann
yet remain to be proved or refuted.
Word alternatives
Some numbers traditionally have alternative words to express them,
including the following:
See also
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.
- What's a Number? at cut-the-knot
External links
(available for download as MP3 or MP4, and as a
text file).
Wikis
Quiz
Map
Misc
Search
