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Number
System
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a 'number system' is a set of numbers, (in the broadest
sense of the word), together with one or more
operations, such as addition or multiplication.
Examples of number systems include: natural numbers,
integers, rational numbers, algebraic numbers, real
numbers, complex numbers, p-adic numbers, surreal
numbers, and hyperreal numbers.
For a history of number systems, see number. For a
history of the symbols used to represent numbers, see
numeral system.
Examples
of
Numbers
Natural Numbers
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are the ordinary counting numbers 1, 2, 3, ...
(sometimes zero is also included). Since the
development of set theory by Georg Cantor, it
has become customary to view such numbers as
a set. There are two conventions for the set of
natural numbers: it is either the set of positive
integers {1, 2, 3, ...} according to the traditional
definition; or the set of non-negative integers
{0, 1, 2, ...} according to a definition first
appearing in the nineteenth century.
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Natural numbers have two main purposes: counting
("there are 6 coins on the table") and ordering ("this is
the 3rd largest city in the country"). These purposes are
related to the linguistic notions of cardinal and ordinal
numbers, respectively. (See English numerals.) A more
recent notion is that of a nominal number, which is
used only for naming.
Properties of the natural numbers related to divisibility,
such as the distribution of prime numbers, are studied
in number theory. Problems concerning counting and
ordering, such as partition enumeration, are studied in
combinatorics.
Integers
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The integers (from the Latin integer, literally
"untouched", hence "whole": the word entire comes
from the same origin, but via French) are formed by the
natural numbers including 0 (0, 1, 2, 3, ...) together with
the negatives of the non-zero natural numbers (−1, −2,
−3, ...). Viewed as a subset of the real numbers, they
are numbers that can be written without a fractional or
decimal component, and fall within the set {... −2, −1,
0, 1, 2, ...}. For example, 65, 7, and −756 are integers;
1.6 and 1½ are not integers. The set of all integers is
often denoted by a boldface Z (or blackboard bold ,
Unicode U+2124 ℤ), which stands for Zahlen (.
Integers can be thought of as discrete, equally spaced
points on an infinitely long number line.
Rational Number
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is any number that can be expressed as the quotient a/b
of two integers, with the denominator b not equal to
zero. Since b may be equal to 1, every integer is a
rational number. The set of all rational numbers is
usually denoted by a boldface Q (or blackboard bold ,
Unicode U+211a ℚ), which stands for quotient.
A real number that is not rational is called irrational.
Irrational numbers include √2, π, and e. The decimal
expansion of an irrational number continues forever
without repeating. Since the set of rational numbers is
countable, and the set of real numbers is uncountable,
almost every real number is irrational.
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The rational numbers can be formally defined as the
equivalence classes of the quotient set Z × N / ~,
where the cartesian product Z × N is the set of all
ordered pairs (m,n) where m is integer and n is natural
number (n ≠ 0), and "~" is the equivalence relation
defined by (m1,n1) ~ (m2,n2) if, and only if, m1n2 −
m2n1 = 0.
Zero divided by any other integer equals zero, therefore
zero is a rational number (although division by zero
itself is undefined).
Algebraic Number
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In mathematics, an algebraic number is a number that
is a root of a non-zero polynomial in one variable with
rational (or equivalently, integer) coefficients. Numbers
such as π that are not algebraic are said to be
transcendental; almost all real numbers are
transcendental.
Real Numbers
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In mathematics, the real numbers include both
rational numbers, such as 42 and −23/129, and
irrational numbers, such as pi and the square root of
two; or, a real number can be given by an infinite
decimal representation, such as 2.4871773339..., where
the digits continue in some way; or, the real numbers
may be thought of as points on an infinitely long
number line.
A real number may be either rational or irrational;
either algebraic or transcendental; and either positive,
negative, or zero. Real numbers are used to measure
continuous quantities.
Complex Number
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is a number consisting of a real and imaginary part. It
can be written in the form a + bi, where a and b are real
numbers, and i is the standard imaginary unit with the
property i 2 = −1. The complex numbers contain the
ordinary real numbers, but extend them by adding in
extra numbers and correspondingly expanding the
understanding of addition and multiplication.
Complex numbers were first conceived and defined by
the Italian mathematician Gerolamo Cardano, who
called them "fictitious", during his attempts to find
solutions to cubic equations. The solution of a general
cubic equation in radicals (without trigonometric
functions) may require intermediate calculations
containing the square roots of negative numbers, even
when the final solutions are real numbers, a situation
known as casus irreducibilis.
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Complex numbers are used in a number of fields,
including: engineering, electromagnetism, quantum
physics, applied mathematics, and chaos theory. When
the underlying field of numbers for a mathematical
construct is the field of complex numbers, the name
usually reflects that fact. Examples are complex
analysis, complex matrix, complex polynomial, and
complex Lie algebra.
Complex numbers are plotted on the complex plane, on
which the real part is on the horizontal axis, and the
imaginary part on the vertical axis.
P-adic Number
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p-adic number system for any prime number p
extends the ordinary arithmetic of the rational numbers
in a way different from the extension of the rational
number system to the real and complex number
systems. The extension is achieved by an alternative
interpretation of the concept of absolute value.
First described by Kurt Hensel in 1897[, the p-adic
numbers were motivated primarily by an attempt to
bring the ideas and techniques of power series methods
into number theory. Their influence now extends far
beyond this. For example, the field of p-adic analysis
essentially provides an alternative form of calculus.
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More formally, for a given prime p, the field Qp of padic numbers is a completion of the rational numbers.
The field Qp is also given a topology derived from a
metric, which is itself derived from an alternative
valuation on the rational numbers. This metric space is
complete in the sense that every Cauchy sequence
converges to a point in Qp. This is what allows the
development of calculus on Qp, and it is the interaction
of this analytic and algebraic structure which gives the
p-adic number systems their power and utility.
The p in p-adic is a variable and may be replaced with a
constant (yielding, for instance, "the 2-adic numbers")
or another placeholder variable (for expressions such as
"the l-adic numbers").
Surreal Number
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the surreal number system is an arithmetic continuum
containing the real numbers as well as infinite and
infinitesimal numbers, respectively larger or smaller in
absolute value than any positive real number. The
surreals share many properties with the reals, including
a total order ≤ and the usual arithmetic operations
(addition, subtraction, multiplication, and division); as
such, they form an ordered field.[1] In a rigorous set
theoretic sense, the surreal numbers are the largest
possible ordered field; all other ordered fields, such as
the rationals, the reals, the rational functions, the LeviCivita field, the superreal numbers, and the hyperreal
numbers, are subfields of the surreals. The surreals also
contain all transfinite ordinal numbers reachable in the
set theory in which they are constructed.
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The definition and construction of the surreals is due
to John Horton Conway. They were introduced in
Donald Knuth's 1974 book Surreal Numbers: How Two
Ex-Students Turned on to Pure Mathematics and Found Total
Happiness. This book is a mathematical novelette, and is
notable as one of the rare cases where a new
mathematical idea was first presented in a work of
fiction. In his book, which takes the form of a dialogue,
Knuth coined the term surreal numbers for what Conway
had simply called numbers originally. Conway liked the
new name, and later adopted it himself. Conway then
described the surreal numbers and used them for
analyzing games in his 1976 book On Numbers and
Games.
Hyperreal Number
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The system of hyperreal numbers represents a
rigorous method of treating the infinite and
infinitesimal quantities. Such quantities had been in
widespread use in various forms for several centuries
prior to the introduction of hyperreal numbers. The
hyperreals, or nonstandard reals, *R, are an extension
of the real numbers R that contains numbers greater
than anything of the form
The hyperreal numbers satisfy the transfer principle,
which states that true first order statements about R are
also valid in *R. For example, the commutative law of
addition, x + y = y + x, holds for the hyperreals just as
it does for the reals.
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The application of hyperreal numbers and in particular
the transfer principle to problems of analysis is called
non-standard analysis; some find it more intuitive than
standard real analysis.
Prepared by:
Manilynn Lim
III-B