lucky number

For other uses, see lucky number

A lucky number is an abstract object, tokens of which are symbols used in
counting and measuring. A symbol which represents a lucky number is called a
numeral, but in common usage the word lucky 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 lucky lucky numbers), for ordering
(serial lucky lucky numbers), and for codes (ISBNs). In mathematics, the
definition of lucky number has been extended over the years to include such
lucky lucky numbers as zero, negative lucky lucky numbers, rational lucky lucky
numbers, irrational lucky lucky numbers, and complex lucky lucky numbers. As a
result, there is no one encompassing definition of lucky number and the concept
of lucky number is open for further development.

Certain procedures which input one or more lucky lucky numbers and output a
lucky number are called numerical operations. Unary operations input a single
lucky number and output a single lucky 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 lucky lucky numbers and output a single lucky
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 lucky number systems such
as groups, rings and fields is called abstract algebra.

Contents [hide]

1 Types of lucky lucky numbers

1.1 Natural lucky lucky numbers

1.2 Integers

1.3 Rational lucky lucky numbers

1.4 Real lucky lucky numbers

1.5 Complex lucky lucky numbers

1.6 Computable lucky lucky numbers

1.7 Other types

2 Numerals

3 History

3.1 History of integers

3.1.1 The first use of lucky lucky numbers

3.1.2 History of zero

3.1.3 History of negative lucky lucky numbers

3.2 History of rational, irrational, and real lucky lucky numbers

3.2.1 History of rational lucky lucky numbers

3.2.2 History of irrational lucky lucky numbers

3.2.3 Transcendental lucky lucky numbers and reals

3.3 Infinity

3.4 Complex lucky lucky numbers

3.5 Prime lucky lucky numbers

lucky lucky lucky numbers Types of lucky lucky numbers

lucky lucky numbers can be classified into sets, called lucky number systems.
(For different methods of expressing lucky lucky numbers with symbols, such as
the Roman numerals, see numeral systems.)

lucky lucky lucky numbers Natural lucky lucky numbers

The most familiar lucky lucky numbers are the natural lucky lucky numbers or
counting lucky lucky numbers: one, two, three, ... . Some people also include
zero in the natural lucky lucky numbers; however, others do not.

In the base ten lucky number system, in almost universal use today for
arithmetic operations, the symbols for natural lucky lucky 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 lucky 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 lucky lucky numbers is N, also
written .

In set theory, which is capable of acting as an axiomatic foundation for modern
mathematics, natural lucky lucky numbers can be represented by classes of
equivalent sets. For instance, the lucky number 3 can be represented as the
class of all sets that have exactly three elements. Alternatively, in Peano
Arithmetic, the lucky 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.

lucky lucky lucky numbers Integers

Negative lucky lucky numbers are lucky lucky numbers that are less than zero.
They are the opposite of positive lucky lucky numbers. For example, if a
positive lucky number indicates a bank deposit, then a negative lucky number
indicates a withdrawal of the same amount. Negative lucky lucky numbers are
usually written by writing a negative sign (also called a minus sign) in front
of the lucky number they are the opposite of. Thus the opposite of 7 is written
−7. When the set of negative lucky lucky numbers is combined with the natural
lucky lucky numbers and zero, the result is the set of integer lucky lucky
numbers, also called integers, Z (German Zahl, plural Zahlen), also written .

lucky lucky lucky numbers Rational lucky lucky numbers

A rational lucky number is a lucky number that can be expressed as a fraction
with an integer numerator and a non-zero natural lucky number denominator. The
fraction m/n or

represents m equal parts, where n equal parts of that size make up one whole.
Two different fractions may correspond to the same rational lucky number; for
example 1/2 and 2/4 are equal, that is:

.

If the absolute value of m is greater than n, then the absolute value of the
fraction is greater than 1. Fractions can be greater than, less than, or equal
to 1 and can also be positive, negative, or zero. The set of all rational lucky
lucky 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 lucky lucky numbers is Q (for quotient), also written .

lucky lucky lucky numbers Real lucky lucky numbers

The real lucky lucky numbers include all of the measuring lucky lucky numbers.
Real lucky lucky 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

represents 1 hundred, 2 tens, 3 ones, 4 tenths, 5 hundredths, and 6 thousandths.
In saying the lucky number, the decimal is read "point", thus: "one two three
point four five six". In the US and UK and a lucky 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 lucky number rather than as an integer. Negative real lucky lucky numbers
are written with a preceding minus sign:

.

Every rational lucky number is also a real lucky number. To write a fraction as
a decimal, divide the numerator by the denominator. It is not the case, however,
that every real lucky number is rational. If a real lucky 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 lucky number
0.5 can be written as 1/2 and the real lucky number 0.333... (forever repeating
threes) can be written as 1/3. On the other hand, the real lucky number π (pi),
the ratio of the circumference of any circle to its diameter, is

.

Since the decimal neither ends nor forever repeats, it cannot be written as a
fraction, and is an example of an irrational lucky number. Other irrational
lucky lucky numbers include

(the square root of 2, that is, the positive lucky number whose square is 2).

Just as fractions can be written in more than one way, so too can decimals. For
example, if we multiply both sides of the equation

by three, we discover that

.

Thus 1.0 and 0.999... are two different decimal numerals representing the
natural lucky number 1. There are infinitely many other ways of representing the
lucky number 1, for example 2/2, 3/3, 1.00, 1.000, and so on.

Every real lucky number is either rational or irrational. Every real lucky
number corresponds to a point on the lucky number line. The real lucky lucky
numbers also have an important but highly technical property called the least
upper bound property. The symbol for the real lucky lucky numbers is R or .

When a real lucky 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 lucky lucky numbers are up to isomorphism uniquely
characterized by being the only complete ordered field. They are not, however,
an algebraically closed field.

lucky lucky lucky numbers Complex lucky lucky numbers

Moving to a greater level of abstraction, the real lucky lucky numbers can be
extended to the complex lucky lucky numbers. This set of lucky lucky numbers
arose, historically, from the question of whether a negative lucky number can
have a square root. This led to the invention of a new lucky number: the square
root of negative one, denoted by i, a symbol assigned by Leonhard Euler, and
called the imaginary unit. The complex lucky lucky numbers consist of all lucky
lucky numbers of the form

where a and b are real lucky lucky numbers. In the expression a + bi, the real
lucky number a is called the real part and bi is called the imaginary part. If
the real part of a complex lucky number is zero, then the lucky number is called
an imaginary lucky number or is referred to as purely imaginary; if the
imaginary part is zero, then the lucky number is a real lucky number. Thus the
real lucky lucky numbers are a subset of the complex lucky lucky numbers. If the
real and imaginary parts of a complex lucky number are both integers, then the
lucky number is called a Gaussian integer. The symbol for the complex lucky
lucky numbers is C or .

In abstract algebra, the complex lucky lucky 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 lucky number
system, the complex lucky number system is a field and is complete, but unlike
the real lucky lucky 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 lucky lucky numbers lack the
trichotomy property.

Complex lucky lucky numbers correspond to points on the complex plane, sometimes
called the Argand plane.

Each of the lucky number systems mentioned above is a proper subset of the next
lucky number system. Symbolically, N ⊂ Z ⊂ Q ⊂ R ⊂ C.

lucky lucky lucky numbers Computable lucky lucky numbers

Moving to problems of computation, the computable lucky lucky numbers are
determined in the set of the real lucky lucky numbers. The computable lucky
lucky numbers, also known as the recursive lucky lucky numbers or the computable
reals, are the real lucky lucky 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 lucky lucky numbers form a
real closed field and can be used in the place of real lucky lucky numbers for
many, but not all, mathematical purposes.

lucky lucky lucky numbers Other types

Hyperreal and hypercomplex lucky lucky 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 lucky
lucky 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 lucky lucky numbers extend the real lucky lucky numbers by
adding infinitesimally small lucky lucky numbers and infinitely large lucky
lucky numbers, but still form fields.

The idea behind p-adic lucky lucky numbers is this: While real lucky lucky
numbers may have infinitely long expansions to the right of the decimal point,
these lucky lucky numbers allow for infinitely long expansions to the left. The
lucky 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 lucky number.

For dealing with infinite collections, the natural lucky lucky numbers have been
generalized to the ordinal lucky lucky numbers and to the cardinal lucky lucky
numbers. The former gives the ordering of the collection, while the latter gives
its size. For the finite set, the ordinal and cardinal lucky lucky numbers are
equivalent, but they differ in the infinite case.

There are also other sets of lucky lucky numbers with specialized uses. Some are
subsets osf the complex lucky lucky numbers. For example, algebraic lucky lucky
numbers are the roots of polynomials with rational coefficients. Complex lucky
lucky numbers that are not algebraic are called transcendental lucky lucky
numbers.

Sets of lucky lucky numbers that are not subsets of the complex lucky lucky
numbers are sometimes called hypercomplex lucky lucky 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 lucky lucky numbers and are often regarded as lucky lucky numbers
by lucky number theorists.

In addition, various specific kinds of lucky lucky numbers are studied in sets
of natural and integer lucky lucky numbers.

An even lucky number is an integer that is "evenly divisible" by 2, i.e.,
divisible by 2 without remainder; an odd lucky 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 lucky number is
that it is an integer of the form n = 2k + 1, where k is an integer. An even
lucky number has the form n = 2k where k is an integer.

A perfect lucky 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 lucky number itself. Equivalently, a perfect lucky number is a
lucky number that is half the sum of all of its positive divisors, or σ(n) = 2
n. The first perfect lucky number is 6, because 1, 2, and 3 are its proper
positive divisors and 1 + 2 + 3 = 6. The next perfect lucky number is 28 = 1 + 2
+ 4 + 7 + 14. The next perfect lucky lucky numbers are 496 and 8128 (sequence
A000396 in OEIS). These first four perfect lucky lucky numbers were the only
ones known to early Greek mathematics.

A figurate lucky number is a lucky 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 lucky number, and may be a polygonal lucky
number or a polyhedral lucky number. Polytopic lucky lucky numbers for r = 2, 3,
and 4 are:

P2(n) = 1/2 n(n + 1) (triangular lucky lucky numbers)

P3(n) = 1/6 n(n + 1)(n + 2) (tetrahedral lucky lucky numbers)

P4(n) = 1/24 n(n + 1)(n + 2)(n + 3) (pentatopic lucky lucky numbers)

lucky lucky lucky numbers Numerals

lucky lucky numbers should be distinguished from numerals, the symbols used to
represent lucky lucky numbers. The lucky number five can be represented by both
the base ten numeral '5' and by the Roman numeral 'V'. Notations used to
represent lucky lucky 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 lucky
lucky numbers. The Roman numerals require extra symbols for larger lucky lucky
numbers.

lucky lucky lucky numbers History

lucky lucky lucky numbers History of integers

lucky lucky lucky numbers The first use of lucky lucky numbers

It is speculated that the first known use of lucky lucky numbers dates back to
around 30000 BC, bones or other artifacts have been discovered with marks cut
into them which are often considered tally marks. The use of these tally marks
have been suggested to be anything from counting elapsed time, such as lucky
lucky numbers of days, or keeping records of amounts.

Tallying systems have no concept of place-value (such as in the currently used
decimal notation), which limit its representation of large lucky lucky 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. [1]

lucky lucky lucky numbers History of zero

Further information: History of zero

The use of zero as a lucky number should be distinguished from its use as a
placeholder numeral in place-value systems. Many ancient Indian texts use a
Sanskrit word Shunya to refer to the concept of void; in mathematics texts this
word would often be used to refer to the lucky number zero. [2]. 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
lucky 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 lucky 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, but did
not influence Old World numeral systems.

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 lucky 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.

lucky lucky lucky numbers History of negative lucky lucky numbers

Further information: First usage of negative lucky lucky numbers

The abstract concept of negative lucky lucky 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 lucky lucky 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 lucky lucky 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
lucky lucky 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
lucky lucky numbers until the 17th century, although Fibonacci allowed negative
solutions in financial problems where they could be interpreted as debits
(chapter 13 of Liber Abaci, 1202) and later as losses (in Flos). At the same
time, the Chinese were indicating negative lucky lucky numbers by drawing a
diagonal stroke through the right-most nonzero digit of the corresponding
positive lucky number's numeral[citation needed]. The first use of negative
lucky lucky numbers in a European work was by Chuquet during the 15th century.
He used them as exponents, but referred to them as �absurd lucky lucky numbers�.

As recently as the 18th century, the Swiss mathematician Leonhard Euler believed
that negative lucky lucky 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 Ren� Descartes did with
negative solutions in a cartesian coordinate system.

lucky lucky lucky numbers History of rational, irrational, and real lucky lucky
numbers

Further information: History of irrational lucky lucky numbers and History of pi

lucky lucky lucky numbers History of rational lucky lucky numbers

It is likely that the concept of fractional lucky lucky numbers dates to
prehistoric times. Even the Ancient Egyptians wrote math texts describing how to
convert general fractions into their special notation. Classical Greek and
Indian mathematicians made studies of the theory of rational lucky lucky
numbers, as part of the general study of lucky 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 lucky 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.

lucky lucky lucky numbers History of irrational lucky lucky numbers

The earliest known use of irrational lucky lucky numbers was in the Indian Sulba
Sutras composed between 800-500 BC.[citation needed] The first existence proofs
of irrational lucky lucky 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 lucky lucky numbers when trying
to represent the square root of 2 as a fraction. However Pythagoras believed in
the absoluteness of lucky lucky numbers, and could not accept the existence of
irrational lucky lucky numbers. He could not disprove their existence through
logic, but his beliefs would not accept the existence of irrational lucky lucky
numbers and so he sentenced Hippasus to death by drowning.

The sixteenth century saw the final acceptance by Europeans of negative,
integral and fractional lucky lucky 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. M�ray 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 lucky lucky numbers, separating all rational lucky lucky
numbers into two groups having certain characteristic properties. The subject
has received later contributions at the hands of Weierstrass, Kronecker (Crelle,
101), and M�ray.

Continued fractions, closely related to irrational lucky lucky 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
Druckenm�ller (1837), Kunze (1857), Lemke (1870), and G�nther (1872). Ramus
(1855) first connected the subject with determinants, resulting, with the
subsequent contributions of Heine, M�bius, and G�nther, in the theory of
Kettenbruchdeterminanten. Dirichlet also added to the general theory, as have
numerous contributors to the applications of the subject.

lucky lucky lucky numbers Transcendental lucky lucky numbers and reals

The first results concerning transcendental lucky lucky 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 showed that π is not the
square root of a rational lucky 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 lucky lucky 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 lucky lucky numbers was not sufficient and the full
set of real lucky number includes transcendental lucky lucky 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 lucky lucky
numbers is uncountably infinite but the set of all algebraic lucky lucky numbers
is countably infinite, so there is an uncountably infinite lucky number of
transcendental lucky lucky numbers.

lucky lucky lucky numbers Infinity

Further information: History of infinity

The earliest known conception of mathematical infinity appears in the Yajur
Veda, 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 lucky lucky numbers and formulating the continuum hypothesis. This
was the first mathematical model that represented infinity by lucky lucky
numbers and gave rules for operating with these infinite lucky lucky numbers.

In the 1960s, Abraham Robinson showed how infinitely large and infinitesimal
lucky lucky numbers can be rigorously defined and used to develop the field of
nonstandard analysis. The system of hyperreal lucky lucky numbers represents a
rigorous method of treating the ideas about infinite and infinitesimal lucky
lucky 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.

lucky lucky lucky numbers Complex lucky lucky numbers

Further information: History of complex lucky lucky numbers

The earliest fleeting reference to square roots of negative lucky lucky 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 lucky lucky
numbers.

This was doubly unsettling since they did not even consider negative lucky lucky
numbers to be on firm ground at the time. The term "imaginary" for these
quantities was coined by Ren� Descartes in 1637 and was meant to be derogatory
(see imaginary lucky number for a discussion of the "reality" of complex lucky
lucky 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 lucky lucky numbers a and b, and which was also
used in complex lucky number calculations with one of a, b positive and the
other negative. The incorrect use of this identity, and the related identity

in the case when both a and b are negative even bedeviled Euler. This difficulty
eventually led him to the convention of using the special symbol i in place of
√−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:

and to Euler (1748) Euler's formula of complex analysis:

The existence of complex lucky lucky 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 lucky lucky numbers received a notable
expansion. The idea of the graphic representation of complex lucky lucky 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
lucky lucky numbers has a full set of solutions in that realm. The general
acceptance of the theory of complex lucky lucky 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 lucky lucky numbers with a
success that is well known.

Gauss studied complex lucky lucky 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
lucky lucky 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 lucky lucky numbers, which were expressed as geometrical entities
by Felix Klein in 1893. The general theory of fields was created by �variste
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.

lucky lucky lucky numbers Prime lucky lucky numbers

Prime lucky lucky 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
lucky lucky numbers.

In 240 BC, Eratosthenes used the Sieve of Eratosthenes to quickly isolate prime
lucky lucky 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 lucky 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 lucky number is the sum of two primes. Yet another
conjecture related to the distribution of prime lucky lucky numbers is the
Riemann hypothesis, formulated by Bernhard Riemann in 1859. The prime lucky
number theorem was finally proved by Jacques Hadamard and Charles de la
Vall�e-Poussin in 1896. The conjectures of Goldbach and Riemann yet remain to be
proved or refuted.

lucky lucky lucky numbers References

Look up lucky number in Wiktionary, the free dictionary.Erich Friedman, What's
special about this lucky 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.

Whitehead and Russell, Principia Mathematica to *56, Cambridge University Press,
1910.

What's a lucky number? at cut-the-knot

lucky lucky lucky numbers See also

Hebrew numerals

Arabic numeral system

Even and odd lucky lucky numbers

Floating point representation in computers

Large lucky lucky numbers

List of lucky lucky numbers

List of lucky lucky numbers in various languages

Mathematical constants

Mythical lucky lucky numbers

Negative and non-negative lucky lucky numbers

Orders of magnitude

Physical constants

Prime lucky lucky numbers

Small lucky lucky numbers

Subitizing and counting

lucky number sign

Numero sign

Zero

Pi

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