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 of 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 René 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. 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 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 Méray.

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 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.

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 René 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 Évariste Galois, who studied the fields generated by the roots of any polynomial equation F(x) = 0.

In 1850 Victor Alexandre Puiseux took the key step of distinguishing between poles and branch points, and introduced the concept of essential singular points; this would eventually lead to the concept of the extended complex plane.

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 Vallée-Poussin in 1896. The conjectures of Goldbach and Riemann yet remain to be proved or refuted.

lucky lucky lucky numbers References
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

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