Everything about Numerical totally explained
A
number is an
abstract object,
tokens of which are
symbols used in
counting and
measuring. A 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. 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. As a result, there's no one encompassing definition of number and the concept of number is open for further development.
Certain procedures which input one or more numbers and output a number are called numerical
operations.
Unary operations input a single number and output a single number. For example, the successor operation adds one to an integer: the successor of 4 is 5. More common are
binary operations which input two numbers and output a single 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 structures of number systems such as
groups,
rings and
fields is called
abstract algebra.
Types of numbers
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, ... . Some people also include zero in the natural numbers; however, others do not.
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
The existence of complex numbers wasn't 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 isn't 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's 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.
Further Information
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