From Wikipedia, the free encyclopedia
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
0,
negative numbers,
rational numbers,
irrational numbers, and
complex numbers.
Types of numbers
.^ The worksheet uses intersections of two circles in two totally different types of activities. Dr. Andrew Nestler's Analysis of NUMB3RS 12 January 2010 2:25 UTC homepage.smc.edu [Source type: FILTERED WITH BAYES]
^ Babylonians stamped numbers in clay by using a stick and depressing it into the clay at different angles or pressures and the Egyptians painted on pottery and cut numbers into stone.
^ We can use each of the divisibility tests to check if a number is divisible by 22: its units digit is even, and the difference between the sums of every other digit is divisible by 11. Whole Numbers and Their Basic Properties 10 January 2010 11:011 UTC www.mathleague.com [Source type: Reference]
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
Main article:
Natural number
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
.
.^ Measured numbers are an estimated amount, measured to a certain number of significant figures without the benefit of any natural unit or a number that comes from a mathematical operation such as averaging. CHEMTUTOR NUMBERS 10 January 2010 11:011 UTC www.chemtutor.com [Source type: FILTERED WITH BAYES]
.^ The set I of all irrational numbers is not countable. What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
^ The set of all algebraic numbers is countable. What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
^ The set Q of all rational numbers is equivalent to the set N of all integers. What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
Alternatively, in
Peano Arithmetic, the number 3 is represented as sss0, where s is the "successor" function.
.^ Put another way, the function F(m/n) = (m + n  1)(m + n  2)/2 + m enumerates all possible representations of positive rational numbers. What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
Integers
Negative numbers are numbers that are less than zero. They are the opposite of positive numbers.
.^ A surreal number (see a short introduction or in relations to games ) is a pair of sets {X L , X R } where indices indicate the relative position (left and right) of the sets in the pair. What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
^ While Kua numbers just give you the hint to your positive and negative directions (where you should sleep and work). Chinese Numerology  Good or Bad number?  Best Feng Shui Site Free Info 10 January 2010 11:011 UTC messagenote.com [Source type: General]
^ It is the only integer (actually, the only real number) that is neither negative nor positive. Numberopedia: what's special about this number? 10 January 2010 11:011 UTC www.archimedeslab.org [Source type: Reference]
.^ Returns 1 if the sign of NUMBER is negative, 0 if NUMBER is 0, and > 0 if NUMBER is positive.
^ If the number is negative, the sign is dropped. PHP: number_format  Manual 10 January 2010 11:011 UTC www.php.net [Source type: FILTERED WITH BAYES]
 PHP: number_format  Manual 10 January 2010 11:011 UTC www.php.net [Source type: FILTERED WITH BAYES]
^ The words are: minus, plus, times, equal, less than equal, less than, greater than, greater than or equal, percent, number sign.
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
. 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 nonzero natural number denominator. The fraction
m/
n or
.^ In fact, in the episode what Charlie says is, "By using a multiattribute compositional model, I can analyze houses by looking at the individual parts that make up the whole. Dr. Andrew Nestler's Analysis of NUMB3RS 12 January 2010 2:25 UTC homepage.smc.edu [Source type: FILTERED WITH BAYES]
^ Equally as serious, if the Northeastern study proves to be accurate, it indicates a deliberate coverup on the part of the government. Numbers of illegal aliens in the U.S. by Fred Elbel  THE AMERICAN RESISTANCE FOUNDATION  Information on immigration counters  10 January 2010 11:011 UTC www.theamericanresistance.com [Source type: FILTERED WITH BAYES]
^ Using a MultiAttribute Compositional Model, I can analyze houses by looking at the individual parts that make up the whole. Dr. Andrew Nestler's Analysis of NUMB3RS 12 January 2010 2:25 UTC homepage.smc.edu [Source type: FILTERED WITH BAYES]
Two different fractions may correspond to the same rational number; for example 1/2 and 2/4 are equal, that is:
.^ Negative numbers will begin with a  sign and numbers less than zero will begin with "0.".
^ Positive values shift the decimal place to the right and negative values to the left.
^ The words are: minus, plus, times, equal, less than equal, less than, greater than, greater than or equal, percent, number sign.
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 one.
.^ The 'right place' is to the right of the first digit. CHEMTUTOR NUMBERS 10 January 2010 11:011 UTC www.chemtutor.com [Source type: FILTERED WITH BAYES]
^ Positive values shift the decimal place to the right and negative values to the left.
^ The original numbers have the same value as the exponential forms, but the exponential forms have the decimal point in the right place. CHEMTUTOR NUMBERS 10 January 2010 11:011 UTC www.chemtutor.com [Source type: FILTERED WITH BAYES]
Thus
represents
.^ The twelve tens would be replaced by two tens and one onehundred.
^ From one perspective it is counterintuitive to assume that the Internet will evolve from tens of thousands of distinct routing domains to one of hundreds of thousands or even millions of distinct routing domains. Exploring Autonomous System Numbers  The Internet Protocol Journal  Volume 9, Number 1  Cisco Systems 10 January 2010 11:011 UTC www.cisco.com [Source type: Reference]
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.
.^ Cantor applied the notion of the 11 correspondence to infinite sets such as the sets of integers, rational, irrational or real numbers. What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
^ Square roots of negative numbers other than 1 can be written under the form: n = i n . Numberopedia: what's special about this number? 10 January 2010 11:011 UTC www.archimedeslab.org [Source type: Reference]
^ In reality every number can be written in many different ways. What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
Negative real numbers are written with a preceding
minus sign:
.^ Cantor applied the notion of the 11 correspondence to infinite sets such as the sets of integers, rational, irrational or real numbers. What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
^ It would be a very legitimate question to ask whether every rational number has an irreducible representation. What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
^ As I was walking down the hall with the ticket, it seemed like with every step I took I realized Id matched another number. Powerball  Home Page 10 January 2010 11:011 UTC www.powerball.com [Source type: News]
.^ (The rules of math permit you to move a fraction from the denominator to the numerator if you invert the fraction. CHEMTUTOR NUMBERS 10 January 2010 11:011 UTC www.chemtutor.com [Source type: FILTERED WITH BAYES]
^ Begin with the KNOWN QUANTITY. Place all the known quantity in the numerator of the beginning fraction if there is no denominator. CHEMTUTOR NUMBERS 10 January 2010 11:011 UTC www.chemtutor.com [Source type: FILTERED WITH BAYES]
^ The more 3s you write the closer the decimal fraction comes to 1/3 but no finite expansion will ever equal 1/3 exactly. What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
.^ However, there are numbers which are neither rational or irrational (for example, infinitesimal numbers are neither rational nor irrational). What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
^ Cantor applied the notion of the 11 correspondence to infinite sets such as the sets of integers, rational, irrational or real numbers. What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
^ It would be a very legitimate question to ask whether every rational number has an irreducible representation. What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
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.
.^ In reality every number can be written in many different ways. What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
^ Thus a real number is either rational or irrational. What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
^ When it's desirable to emphasize the algebraic nature of complex numbers it's customary to write (x, y) = x + iy thus making distinction between the real part x and the imaginary part y more palpable. What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
(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
.^ However, there are numbers which are neither rational or irrational (for example, infinitesimal numbers are neither rational nor irrational). What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
^ Irrational numbers have been defined as decimal (nonperiodic) fractions. What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
^ The number 5, written , as a fraction 1/5 would be written .
Other irrational numbers include
Thus 1.0 and
0.999... are two different decimal numerals representing the natural number
.^ However, there are numbers which are neither rational or irrational (for example, infinitesimal numbers are neither rational nor irrational). What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
^ In all four statements assertion of existence of a single number can be replaced with the assertion of existence of infinitely many numbers. What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
^ In reality every number can be written in many different ways. What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
.^ As I was walking down the hall with the ticket, it seemed like with every step I took I realized Id matched another number. Powerball  Home Page 10 January 2010 11:011 UTC www.powerball.com [Source type: News]
^ Some operations, which can produce an irrational number for rational arguments (e.g., sqrt ), may produce inexact results even for exact arguments.
^ In the case of complex numbers, either the real and imaginary parts are both exact or inexact, or the number has an exact zero real part and an inexact imaginary part; a complex number with an exact zero imaginary part is a real number.
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
.
.^ Since then he has always been in debt and quite depressed, is there any link between the numbers, double numbers and triple number players????? Chinese Numerology  Good or Bad number?  Best Feng Shui Site Free Info 10 January 2010 11:011 UTC messagenote.com [Source type: General]
^ Zvezdochka, are there any cures for the house number, if it represents “not easy”? Chinese Numerology  Good or Bad number?  Best Feng Shui Site Free Info 10 January 2010 11:011 UTC messagenote.com [Source type: General]
^ Note that although most of the real numbers are transcendental (since there is only countably many algebraic numbers), transcendentality is proven a number at a time. What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
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.
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 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
where
a and
b are real numbers.
.^ Real numbers that are not algebraic are called transcendental . What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
^ This should probably be called a really imaginary number (see Ref [7] ). What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
^ (In passing, Ian Stewart questions the wisdom of calling irrational numbers real : How can things be real if you can't even write them down fully? What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
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.
.^ Complex numbers are pairs c = (x, y) of two real numbers. What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
^ Real and Complex numbers . What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
^ An impetus for developing the theory of complex numbers came originally from unsolvability of a simple equation x 2 + 1 = 0 in the set of real numbers. What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
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
.
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
μrecursive functions,
Turing machines or
λcalculus 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
The idea behind
padic 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.
.^ Actually we may have gained an insight into the fundamental differences between finite and infinite sets. What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
^ Cantor applied the notion of the 11 correspondence to infinite sets such as the sets of integers, rational, irrational or real numbers. What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
^ The set Q of all rational numbers is equivalent to the set N of all integers. What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
.^ AS numbers are also used in other contexts. Exploring Autonomous System Numbers  The Internet Protocol Journal  Volume 9, Number 1  Cisco Systems 10 January 2010 11:011 UTC www.cisco.com [Source type: Reference]
^ Again this can be achieved through the use of distinct AS numbers of each routing policy set. Exploring Autonomous System Numbers  The Internet Protocol Journal  Volume 9, Number 1  Cisco Systems 10 January 2010 11:011 UTC www.cisco.com [Source type: Reference]
^ The other Pythagoras worksheet lists a calculator among the required materials, and not only is it not necessary, there is no mention of the use of a calculator in the activity. Dr. Andrew Nestler's Analysis of NUMB3RS 12 January 2010 2:25 UTC homepage.smc.edu [Source type: FILTERED WITH BAYES]
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 oldfashioned 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.
.^ Average numbers of people would make a measured number, even though people naturally come only in integers. CHEMTUTOR NUMBERS 10 January 2010 11:011 UTC www.chemtutor.com [Source type: FILTERED WITH BAYES]
^ For example, the sets of odd numbers, even numbers, complete squares or cubes, the set of integers greater than 1996 all can be brought into a 11 correspondence with the set of all integers (of which they are subsets. What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
^ Leonhard Euler (17071783) in a paper published posthumously, showed that every even perfect number has Euclidean form. What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cuttheknot.org [Source type: Reference]
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 (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:
A relation number is defined as the class of
relations consisting of all those relations that are similar to one member of the class.
^{[1]}
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 '
Ⅴ' and ciphered letters.
.^ This is the nth time i used your credit card numbers to test our system. Graham King » Credit card numbers 10 January 2010 11:011 UTC www.darkcoding.net [Source type: General]
^ The worksheet continues, "He discusses how a simple continued fraction can be used to approximate the irrational number needed to ensure that the flute will have an equal temperament." Dr. Andrew Nestler's Analysis of NUMB3RS 12 January 2010 2:25 UTC homepage.smc.edu [Source type: FILTERED WITH BAYES]
^ Your scientific calculator will use the numbers in the shortened form, usually best represented by the E form. CHEMTUTOR NUMBERS 10 January 2010 11:011 UTC www.chemtutor.com [Source type: FILTERED WITH BAYES]
.^ It implies the onset of decline, therefore combination numbers like 39, 69 and 89 , while they sound good, imply over development. Chinese Numerology  Good or Bad number?  Best Feng Shui Site Free Info 10 January 2010 11:011 UTC messagenote.com [Source type: General]
^ Pam , 12815 is a balanced number with the lucky 8 in the middle:) All other numbers also look great because of 8 and 9 numbers.I like this one 4801 it looks positive and interesting. Chinese Numerology  Good or Bad number?  Best Feng Shui Site Free Info 10 January 2010 11:011 UTC messagenote.com [Source type: General]
^ Assumption: the NUMBER value represents a positive integer Returns: a new NUMBER object Throws: java.sql.SQLException  if Java implementation is not available .
The Roman numerals require extra symbols for larger numbers.
History
The first use of numbers
.^ RFC 4520 x Private Use (can not be registered) e Reserved for Experiments, First Come First Serve All others: Expert Review (Expert  Kurt Zeilenga) . IANA — Protocol Registries 10 January 2010 11:011 UTC www.iana.org [Source type: Reference]
^ Traversal Using Relays around NAT (TURN) Channel Numbers . IANA — Protocol Registries 10 January 2010 11:011 UTC www.iana.org [Source type: Reference]
.^ I used one of these fake credit card numbers to order a pizza delivered to my front door. Graham King » Credit card numbers 10 January 2010 11:011 UTC www.darkcoding.net [Source type: General]
^ These numbers remain unassigned because they are socalled 'flag' numbers, kept for special purposes such as emergency or operator services. Numberopedia: what's special about this number? 10 January 2010 11:011 UTC www.archimedeslab.org [Source type: Reference]
^ These are well useful to feed into the endless stream of PayPal, eBay and other phishing scam emails that you receive these days. Graham King » Credit card numbers 10 January 2010 11:011 UTC www.darkcoding.net [Source type: General]
.^ The balanced ternary base, is a numeral system which uses 3 values or digits: 1, 0, and 1. Numberopedia: what's special about this number? 10 January 2010 11:011 UTC www.archimedeslab.org [Source type: Reference]
^ The Shadok's numbers are a kind of ternary or base3 numeration system: . Numberopedia: what's special about this number? 10 January 2010 11:011 UTC www.archimedeslab.org [Source type: Reference]
^ Total Fertility Rate (TFR): "The average number of children a woman would have assuming that current agespecific birth rates remain constant throughout her childbearing years (usually considered to be ages 15 to 49)". Numbers of illegal aliens in the U.S. by Fred Elbel  THE AMERICAN RESISTANCE FOUNDATION  Information on immigration counters  10 January 2010 11:011 UTC www.theamericanresistance.com [Source type: FILTERED WITH BAYES]
The first known system with placevalue was the
Mesopotamian base 60 system (ca.
.^ The first known use of numbers dates back to around 30,000 BC when tally marks were precisely used by stone age people. Numberopedia: what's special about this number? 10 January 2010 11:011 UTC www.archimedeslab.org [Source type: Reference]
^{[2]}
History of zero
The use of zero as a number should be distinguished from its use as a placeholder numeral in
placevalue 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.
.^ Shunya, the dot, was originally not zero the number, but merely a mark to indicate empty space. Numberopedia: what's special about this number? 10 January 2010 11:011 UTC www.archimedeslab.org [Source type: Reference]
^ Who first thought of using a dot ( bindu , in sanskrit) as the tenth number is not known. Numberopedia: what's special about this number? 10 January 2010 11:011 UTC www.archimedeslab.org [Source type: Reference]
^ Honest numbers are numbers n that can be described using exactly n letters in standard mathematical English. Numberopedia: what's special about this number? 10 January 2010 11:011 UTC www.archimedeslab.org [Source type: Reference]
^{[3]}
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 southcentral
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.
.^ In German, the expression in Null Komma nichts (in zero point nothing) means 'in a trice' . Numberopedia: what's special about this number? 10 January 2010 11:011 UTC www.archimedeslab.org [Source type: Reference]
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” (
Jiuzhang Suanshu) contains methods for finding the areas of figures; red rods were used to denote positive
coefficients, black for negative.
.^ While Kua numbers just give you the hint to your positive and negative directions (where you should sleep and work). Chinese Numerology  Good or Bad number?  Best Feng Shui Site Free Info 10 January 2010 11:011 UTC messagenote.com [Source type: General]
^ The first known use of numbers dates back to around 30,000 BC when tally marks were precisely used by stone age people. Numberopedia: what's special about this number? 10 January 2010 11:011 UTC www.archimedeslab.org [Source type: Reference]
^ Just like the first apartment, the number 9 will not be mentioned in the address. Chinese Numerology  Good or Bad number?  Best Feng Shui Site Free Info 10 January 2010 11:011 UTC messagenote.com [Source type: General]
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.
.^ Your scientific calculator will use the numbers in the shortened form, usually best represented by the E form. CHEMTUTOR NUMBERS 10 January 2010 11:011 UTC www.chemtutor.com [Source type: FILTERED WITH BAYES]
^ Negative numbers are represented by a leading "".
Diophantus’ previous reference was discussed more explicitly by Indian mathematician
Brahmagupta, in
BrahmaSphutaSiddhanta 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 rightmost nonzero digit of the corresponding positive number's numeral
^{[citation needed]}. 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^{[citation needed]}, 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 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.
.^ If the denominator of a rational number is not divisible by 3 , then the repeating part of its decimal expansion is an integer divisible by 9. Numberopedia: what's special about this number? 10 January 2010 11:011 UTC www.archimedeslab.org [Source type: Reference]
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 decimalfraction 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 800500 BC.
^{[citation needed]} 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.
.^ In the late 18th century, James Stirling, a Scottish mathematician, developed an approximation for factorials using the transcendental numbers 'Pi' and 'e': n ! Numberopedia: what's special about this number? 10 January 2010 11:011 UTC www.archimedeslab.org [Source type: Reference]
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 (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 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
e^{n} 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.
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
onetoone 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.
.^ These numbers remain unassigned because they are socalled 'flag' numbers, kept for special purposes such as emergency or operator services. Numberopedia: what's special about this number? 10 January 2010 11:011 UTC www.archimedeslab.org [Source type: Reference]
^ In short, NaN is not really a number but a symbol that represents a numerical quantity whose magnitude cannot be determined by the operating system. Numberopedia: what's special about this number? 10 January 2010 11:011 UTC www.archimedeslab.org [Source type: Reference]
^ Measured numbers are an estimated amount, measured to a certain number of significant figures without the benefit of any natural unit or a number that comes from a mathematical operation such as averaging. CHEMTUTOR NUMBERS 10 January 2010 11:011 UTC www.chemtutor.com [Source type: FILTERED WITH BAYES]
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).
.^ Only the interesting or funny ones make it. Graham King » Credit card numbers 10 January 2010 11:011 UTC www.darkcoding.net [Source type: General]
^ Not one of the numbers work for me, and there is no card verification number, so, all these numbers are useless. Graham King » Credit card numbers 10 January 2010 11:011 UTC www.darkcoding.net [Source type: General]
^ The only real restriction is that the expiry date needs to be sometime in the future – even if only a few days. Graham King » Credit card numbers 10 January 2010 11:011 UTC www.darkcoding.net [Source type: General]
.^ Charlie declares, "Even a lawyer can't argue with math," which is an odd thing to say since no one did any math at that moment. Dr. Andrew Nestler's Analysis of NUMB3RS 12 January 2010 2:25 UTC homepage.smc.edu [Source type: FILTERED WITH BAYES]
^ So not only did they exaggerate the number of giants in the land, they exaggerated the size of the giants as well. Today's Bible: NUMBERS Daily Commentary and Study Guide 10 January 2010 11:011 UTC www.biblereferenceguide.com [Source type: Original source]
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
seemed to be capriciously inconsistent with the algebraic identity
which is valid for positive real numbers
.^ Square roots of negative numbers other than 1 can be written under the form: n = i n . Numberopedia: what's special about this number? 10 January 2010 11:011 UTC www.archimedeslab.org [Source type: Reference]
^ I used one of these fake credit card numbers to order a pizza delivered to my front door. Graham King » Credit card numbers 10 January 2010 11:011 UTC www.darkcoding.net [Source type: General]
^ While Kua numbers just give you the hint to your positive and negative directions (where you should sleep and work). Chinese Numerology  Good or Bad number?  Best Feng Shui Site Free Info 10 January 2010 11:011 UTC messagenote.com [Source type: General]
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 √−1 to guard against this mistake.
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.
.^ DNS KEY Record DiffieHellman WellKnown Prime/Generator Pairs . IANA — Protocol Registries 10 January 2010 11:011 UTC www.iana.org [Source type: Reference]
^ Assigned WellKnown Socket Numbers . IANA — Protocol Registries 10 January 2010 11:011 UTC www.iana.org [Source type: Reference]
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
x^{2} + 1 = 0). His student,
Ferdinand Eisenstein, studied the type
a +
bω, where
ω is a complex root of
x^{3} − 1 = 0. Other such classes (called
cyclotomic fields) of complex numbers are derived from the
roots of unity x^{k} − 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.
Prime numbers
Word alternatives
Some numbers traditionally have alternative words to express them, including the following:
See also
References
 ^ "Introduction to Mathematical Philosophy" by Bertrand Russell, page 56
 ^ http://www.math.buffalo.edu/mad/AncientAfrica/mad_ancient_egyptpapyrus.html#berlin
 ^ http://sunsite.utk.edu/math_archives/.http/hypermail/historia/apr99/0197.html
 Tobias Dantzig, Number, the language of science; a critical survey written for the cultured nonmathematician, 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 0155434683.
 Paul Halmos, Naive Set Theory, Springer, 1974, ISBN 0387900926.
 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,AustinTexas, 1961.
 What's a Number? at cuttheknot
External links
Number systems 

Basic 


Real numbers and
their extensions 


Other number systems 

