1.
Non-standard analysis
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The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals, Non-standard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers. Non-standard analysis was originated in the early 1960s by the mathematician Abraham Robinson and he wrote, the idea of infinitely small or infinitesimal quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the stages of the Differential and Integral Calculus. Robinson continued, However, neither he nor his disciples and successors were able to give a rational development leading up to a system of this sort, as a result, the theory of infinitesimals gradually fell into disrepute and was replaced eventually by the classical theory of limits. The key to our method is provided by the analysis of the relation between mathematical languages and mathematical structures which lies at the bottom of contemporary model theory. In 1973, intuitionist Arend Heyting praised non-standard analysis as a model of important mathematical research. A non-zero element of an ordered field F is infinitesimal if and only if its value is smaller than any element of F of the form 1 n, for n. Ordered fields that have infinitesimal elements are also called non-Archimedean, more generally, non-standard analysis is any form of mathematics that relies on non-standard models and the transfer principle. A field which satisfies the principle for real numbers is a hyperreal field. Robinsons original approach was based on these models of the field of real numbers. His classic foundational book on the subject Non-standard Analysis was published in 1966 and is still in print, on page 88, Robinson writes, The existence of non-standard models of arithmetic was discovered by Thoralf Skolem. Skolems method foreshadows the ultrapower construction Several technical issues must be addressed to develop a calculus of infinitesimals, for example, it is not enough to construct an ordered field with infinitesimals. See the article on numbers for a discussion of some of the relevant ideas. In this section we outline one of the simplest approaches to defining a hyperreal field ∗ R, let R be the field of real numbers, and let N be the semiring of natural numbers. Denote by R N the set of sequences of real numbers, a field ∗ R is defined as a suitable quotient of R N, as follows. Take a nonprincipal ultrafilter F ⊂ P, in particular, F contains the Fréchet filter. There are at least three reasons to consider non-standard analysis, historical, pedagogical, and technical, much of the earliest development of the infinitesimal calculus by Newton and Leibniz was formulated using expressions such as infinitesimal number and vanishing quantity

2.
Integer part
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In mathematics and computer science, the floor and ceiling functions map a real number to the greatest preceding or the least succeeding integer, respectively. More precisely, floor = ⌊ x ⌋ is the greatest integer less than or equal to x, carl Friedrich Gauss introduced the square bracket notation for the floor function in his third proof of quadratic reciprocity. This remained the standard in mathematics until Kenneth E. Iverson introduced the names floor and ceiling, both notations are now used in mathematics, this article follows Iverson. e. The value of x rounded to an integer towards 0, the language APL uses ⌊x, other computer languages commonly use notations like entier, INT, or floor. In mathematics, it can also be written with boldface or double brackets, the ceiling function is usually denoted by ceil or ceiling in non-APL computer languages that have a notation for this function. The J Programming Language, a follow on to APL that is designed to use standard symbols, uses >. for ceiling. In mathematics, there is another notation with reversed boldface or double brackets ] ] x x[\. x[, the fractional part is the sawtooth function, denoted by for real x and defined by the formula = x − ⌊ x ⌋. HTML4.0 uses the names, &lfloor, &rfloor, &lceil. Unicode contains codepoints for these symbols at U+2308–U+230B, ⌈x⌉, ⌊x⌋, in the following formulas, x and y are real numbers, k, m, and n are integers, and Z is the set of integers. Floor and ceiling may be defined by the set equations ⌊ x ⌋ = max, ⌈ x ⌉ = min. Since there is exactly one integer in an interval of length one. Then ⌊ x ⌋ = m and ⌈ x ⌉ = n may also be taken as the definition of floor and these formulas can be used to simplify expressions involving floors and ceilings. In the language of order theory, the function is a residuated mapping. These formulas show how adding integers to the arguments affect the functions, negating the argument complements the fractional part, + = {0 if x ∈ Z1 if x ∉ Z. The floor, ceiling, and fractional part functions are idempotent, the result of nested floor or ceiling functions is the innermost function, ⌊ ⌈ x ⌉ ⌋ = ⌈ x ⌉, ⌈ ⌊ x ⌋ ⌉ = ⌊ x ⌋. If m and n are integers and n ≠0,0 ≤ ≤1 −1 | n |. If n is a positive integer ⌊ x + m n ⌋ = ⌊ ⌊ x ⌋ + m n ⌋, ⌈ x + m n ⌉ = ⌈ ⌈ x ⌉ + m n ⌉. For m =2 these imply n = ⌊ n 2 ⌋ + ⌈ n 2 ⌉

3.
Thoralf Skolem
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Thoralf Albert Skolem was a Norwegian mathematician who worked in mathematical logic and set theory. Although Skolems father was a school teacher, most of his extended family were farmers. Skolem attended secondary school in Kristiania, passing the university examinations in 1905. He then entered Det Kongelige Frederiks Universitet to study mathematics, also taking courses in physics, chemistry, zoology, in 1913, Skolem passed the state examinations with distinction, and completed a dissertation titled Investigations on the Algebra of Logic. He also traveled with Birkeland to the Sudan to observe the zodiacal light, in 1916 he was appointed a research fellow at Det Kongelige Frederiks Universitet. In 1918, he became a Docent in Mathematics and was elected to the Norwegian Academy of Science, Skolem did not at first formally enroll as a Ph. D. candidate, believing that the Ph. D. was unnecessary in Norway. He later changed his mind and submitted a thesis in 1926, titled Some theorems about integral solutions to certain algebraic equations and his notional thesis advisor was Axel Thue, even though Thue had died in 1922. In 1927, he married Edith Wilhelmine Hasvold, Skolem continued to teach at Det kongelige Frederiks Universitet until 1930 when he became a Research Associate in Chr. This senior post allowed Skolem to conduct research free of administrative, in 1938, he returned to Oslo to assume the Professorship of Mathematics at the university. There he taught the courses in algebra and number theory. Skolems Ph. D. student Øystein Ore went on to a career in the USA, Skolem served as president of the Norwegian Mathematical Society, and edited the Norsk Matematisk Tidsskrift for many years. He was also the editor of Mathematica Scandinavica. After his 1957 retirement, he made trips to the United States. He remained intellectually active until his sudden and unexpected death, for more on Skolems academic life, see Fenstad. Skolem published around 180 papers on Diophantine equations, group theory, lattice theory and he mostly published in Norwegian journals with limited international circulation, so that his results were occasionally rediscovered by others. An example is the Skolem–Noether theorem, characterizing the automorphisms of simple algebras, Skolem published a proof in 1927, but Emmy Noether independently rediscovered it a few years later. Skolem was among the first to write on lattices, in 1912, he was the first to describe a free distributive lattice generated by n elements. In 1919, he showed that every lattice is distributive and, as a partial converse

4.
Number
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Numbers that answer the question How many. Are 0,1,2,3 and so on, when used to indicate position in a sequence they are ordinal numbers. To the Pythagoreans and Greek mathematician Euclid, the numbers were 2,3,4,5, Euclid did not consider 1 to be a number. Numbers like 3 +17 =227, expressible as fractions in which the numerator and denominator are whole numbers, are rational numbers and these make it possible to measure such quantities as two and a quarter gallons and six and a half miles. What we today would consider a proof that a number is irrational Euclid called a proof that two lengths arising in geometry have no common measure, or are incommensurable, Euclid included proofs of incommensurability of lengths arising in geometry in his Elements. In the Rhind Mathematical Papyrus, a pair of walking forward marked addition. They were the first known civilization to use negative numbers, negative numbers came into widespread use as a result of their utility in accounting. They were used by late medieval Italian bankers, by 1740 BC, the Egyptians had a symbol for zero in accounting texts. In Maya civilization zero was a numeral with a shape as a symbol. The ancient Egyptians represented all fractions in terms of sums of fractions with numerator 1, for example, 2/5 = 1/3 + 1/15. Such representations are known as Egyptian Fractions or Unit Fractions. The earliest written approximations of π are found in Egypt and Babylon, in Babylon, a clay tablet dated 1900–1600 BC has a geometrical statement that, by implication, treats π as 25/8 =3.1250. In Egypt, the Rhind Papyrus, dated around 1650 BC, astronomical calculations in the Shatapatha Brahmana use a fractional approximation of 339/108 ≈3.139. Other Indian sources by about 150 BC treat π as √10 ≈3.1622 The first references to the constant e were published in 1618 in the table of an appendix of a work on logarithms by John Napier. However, this did not contain the constant itself, but simply a list of logarithms calculated from the constant and it is assumed that the table was written by William Oughtred. The discovery of the constant itself is credited to Jacob Bernoulli, the first known use of the constant, represented by the letter b, was in correspondence from Gottfried Leibniz to Christiaan Huygens in 1690 and 1691. Leonhard Euler introduced the letter e as the base for natural logarithms, Euler started to use the letter e for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons, and the first appearance of e in a publication was Eulers Mechanica. While in the subsequent years some researchers used the letter c, e was more common, the first numeral system known is Babylonian numeric system, that has a 60 base, it was introduced in 3100 B. C. and is the first Positional numeral system known

5.
Countable set
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In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A countable set is either a set or a countably infinite set. Some authors use countable set to mean countably infinite alone, to avoid this ambiguity, the term at most countable may be used when finite sets are included and countably infinite, enumerable, or denumerable otherwise. Georg Cantor introduced the term countable set, contrasting sets that are countable with those that are uncountable, today, countable sets form the foundation of a branch of mathematics called discrete mathematics. A set S is countable if there exists a function f from S to the natural numbers N =. If such an f can be found that is also surjective, in other words, a set is countably infinite if it has one-to-one correspondence with the natural number set, N. As noted above, this terminology is not universal, some authors use countable to mean what is here called countably infinite, and do not include finite sets. Alternative formulations of the definition in terms of a function or a surjective function can also be given. In 1874, in his first set theory article, Cantor proved that the set of numbers is uncountable. In 1878, he used one-to-one correspondences to define and compare cardinalities, in 1883, he extended the natural numbers with his infinite ordinals, and used sets of ordinals to produce an infinity of sets having different infinite cardinalities. A set is a collection of elements, and may be described in many ways, one way is simply to list all of its elements, for example, the set consisting of the integers 3,4, and 5 may be denoted. This is only effective for small sets, however, for larger sets, even in this case, however, it is still possible to list all the elements, because the set is finite. Some sets are infinite, these sets have more than n elements for any integer n, for example, the set of natural numbers, denotable by, has infinitely many elements, and we cannot use any normal number to give its size. Nonetheless, it out that infinite sets do have a well-defined notion of size. To understand what this means, we first examine what it does not mean, for example, there are infinitely many odd integers, infinitely many even integers, and infinitely many integers overall. However, it out that the number of even integers. This is because we arrange things such that for every integer, or, more generally, n→2n, see picture. However, not all sets have the same cardinality

6.
Natural number
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In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers, the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1,10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory

7.
Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain

8.
Rational number
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In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number, the decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number and these statements hold true not just for base 10, but also for any other integer base. A real number that is not rational is called irrational, irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating, since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. Rational numbers can be defined as equivalence classes of pairs of integers such that q ≠0, for the equivalence relation defined by ~ if. The rational numbers together with addition and multiplication form field which contains the integers and is contained in any field containing the integers, finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the numbers form a dense subset of the real numbers. The real numbers can be constructed from the numbers by completion, using Cauchy sequences, Dedekind cuts. The term rational in reference to the set Q refers to the fact that a number represents a ratio of two integers. In mathematics, rational is often used as a noun abbreviating rational number, the adjective rational sometimes means that the coefficients are rational numbers. However, a curve is not a curve defined over the rationals. Any integer n can be expressed as the rational number n/1, a b = c d if and only if a d = b c. Where both denominators are positive, a b < c d if and only if a d < b c. If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations, − a − b = a b, two fractions are added as follows, a b + c d = a d + b c b d. A b − c d = a d − b c b d, the rule for multiplication is, a b ⋅ c d = a c b d. Where c ≠0, a b ÷ c d = a d b c, note that division is equivalent to multiplying by the reciprocal of the divisor fraction, a d b c = a b × d c. Additive and multiplicative inverses exist in the numbers, − = − a b = a − b and −1 = b a if a ≠0

9.
Constructible number
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A point in the Euclidean plane is a constructible point if, given a fixed coordinate system, the point can be constructed with unruled straightedge and compass. A complex number is a number if its corresponding point in the Euclidean plane is constructible from the usual x-. It can then be shown that a number r is constructible if and only if, given a line segment of unit length. It can also be shown that a number is constructible if and only if its real. In terms of algebra, a number is constructible if and only if it can be using the four basic arithmetic operations. This has the effect of transforming geometric questions about compass and straightedge constructions into algebra and this transformation leads to the solutions of many famous mathematical problems, which defied centuries of attack. The geometric definition of a point is as follows. First, for any two distinct points P and Q in the plane, let L denote the line through P and Q. Since the order of E, F, G, and H in the definition is irrelevant. Now, let A and A′ be any two distinct fixed points in the plane, a point Z is constructible if either Z = A, Z = A′, there exist points P0. Pn, with Z = Pn, such that for all j ≥1, Pj +1 is constructible from points in the set where P0 = A, for example, the center point of A and A′ is defined as follows. The circles C and C intersect in two points, these points determine a unique line, and the center is defined to be the intersection of this line with L. All rational numbers are constructible, and all numbers are algebraic numbers. Also, if a and b are constructible numbers with b ≠0, then a ± b, a×b, a⁄b, in particular, the set K of all constructible complex numbers forms a field, a subfield of the field of algebraic numbers. A complex number is constructible if and only if the real, furthermore, K is closed under square roots and complex conjugation. These facts can be used to characterize the field of numbers, because, in essence. The characterization is the following, a number is constructible if and only if it lies in a field at the top of a finite tower of quadratic extensions. Trigonometric numbers are irrational cosines or sines of angles that are multiples of π

10.
Algebraic number
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An algebraic number is any complex number that is a root of a non-zero polynomial in one variable with rational coefficients. All integers and rational numbers are algebraic, as are all roots of integers, the same is not true for all real and complex numbers because they also include transcendental numbers such as π and e. Almost all real and complex numbers are transcendental, the rational numbers, expressed as the quotient of two integers a and b, b not equal to zero, satisfy the above definition because x = a/b is the root of bx − a. The quadratic surds are algebraic numbers, if the quadratic polynomial is monic then the roots are quadratic integers. The constructible numbers are numbers that can be constructed from a given unit length using straightedge. These include all quadratic surds, all numbers, and all numbers that can be formed from these using the basic arithmetic operations. Any expression formed from algebraic numbers using any combination of the arithmetic operations. Polynomial roots that cannot be expressed in terms of the arithmetic operations. This happens with many, but not all, polynomials of degree 5 or higher, gaussian integers, those complex numbers a + bi where both a and b are integers are also quadratic integers. Trigonometric functions of rational multiples of π, that is, the trigonometric numbers, for example, each of cos π/7, cos 3π/7, cos 5π/7 satisfies 8x3 − 4x2 − 4x +1 =0. This polynomial is irreducible over the rationals, and so these three cosines are conjugate algebraic numbers. Likewise, tan 3π/16, tan 7π/16, tan 11π/16, tan 15π/16 all satisfy the irreducible polynomial x4 − 4x3 − 6x2 + 4x +1, and so are conjugate algebraic integers. Some irrational numbers are algebraic and some are not, The numbers √2 and 3√3/2 are algebraic since they are roots of polynomials x2 −2 and 8x3 −3, the golden ratio φ is algebraic since it is a root of the polynomial x2 − x −1. The numbers π and e are not algebraic numbers, hence they are transcendental, the set of algebraic numbers is countable. Hence, the set of numbers has Lebesgue measure zero. Given an algebraic number, there is a monic polynomial of least degree that has the number as a root. This polynomial is called its minimal polynomial, if its minimal polynomial has degree n, then the algebraic number is said to be of degree n. An algebraic number of degree 1 is a rational number, a real algebraic number of degree 2 is a quadratic irrational

11.
Gaussian integer
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In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with addition and multiplication of complex numbers, form an integral domain. This integral domain is a case of a commutative ring of quadratic integers. It does not have an ordering that respects arithmetic. Formally, the Gaussian integers are the set Z =, where i 2 = −1, note that when they are considered within the complex plane the Gaussian integers may be seen to constitute the 2-dimensional integer lattice. The norm of a Gaussian integer is the square of its value as a complex number. It is the natural number defined as N = a 2 + b 2 = ¯ =, the norm is multiplicative, since the absolute value of complex numbers is multiplicative, i. e. one has N = N N. The latter can also be verified by a straightforward check, the units of Z are precisely those elements with norm 1, i. e. the set. The Gaussian integers form a principal ideal domain with units, for x ∈ Z, the four numbers ±x, ±ix are called the associates of x. As for every principal ideal domain, Z is also a unique factorization domain and it follows that a Gaussian integer is prime if and only if it is irreducible. The prime elements of Z are also known as Gaussian primes, an associate of a Gaussian prime is also a Gaussian prime. The Gaussian primes are symmetric about the real and imaginary axes, the positive integer Gaussian primes are the prime numbers that are congruent to 3 modulo 4. One should not refer to only these numbers as the Gaussian primes, which refers to all the Gaussian primes, many of which do not lie in Z. In other words, a Gaussian integer is a Gaussian prime if and only if either its norm is a prime number, for example,5 = · and 13 = ·. If p =2, we have 2 = = i2, the ring of Gaussian integers is the integral closure of Z in the field of Gaussian rationals Q consisting of the complex numbers whose real and imaginary part are both rational. It is easy to see graphically that every number is no farther than a distance of 22 from some Gaussian integer. Put another way, every number has a maximal distance of 22 N units to some multiple of z, where z is any Gaussian integer, this turns Z into a Euclidean domain. The ring of Gaussian integers was introduced by Carl Friedrich Gauss in his monograph on quartic reciprocity

12.
Real number
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In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by René Descartes, the real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the numbers, such as π. Real numbers can be thought of as points on a long line called the number line or real line. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, the real line can be thought of as a part of the complex plane, and complex numbers include real numbers. These descriptions of the numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions satisfy the definition and are thus equivalent. The statement that there is no subset of the reals with cardinality greater than ℵ0. Simple fractions were used by the Egyptians around 1000 BC, the Vedic Sulba Sutras in, c.600 BC, around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of number and magnitude into a general idea of real numbers. In the 16th century, Simon Stevin created the basis for modern decimal notation, in the 17th century, Descartes introduced the term real to describe roots of a polynomial, distinguishing them from imaginary ones. In the 18th and 19th centuries, there was work on irrational and transcendental numbers. Johann Heinrich Lambert gave the first flawed proof that π cannot be rational, Adrien-Marie Legendre completed the proof, Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, and Ferdinand von Lindemann, lindemanns proof was much simplified by Weierstrass, still further by David Hilbert, and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871, in 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, the real number system can be defined axiomatically up to an isomorphism, which is described hereafter. Another possibility is to start from some rigorous axiomatization of Euclidean geometry, from the structuralist point of view all these constructions are on equal footing

13.
Complex number
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A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1. In this expression, a is the part and b is the imaginary part of the complex number. If z = a + b i, then ℜ z = a, ℑ z = b, Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point in the complex plane, a complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the numbers are a field extension of the ordinary real numbers. As well as their use within mathematics, complex numbers have applications in many fields, including physics, chemistry, biology, economics, electrical engineering. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers and he called them fictitious during his attempts to find solutions to cubic equations in the 16th century. Complex numbers allow solutions to equations that have no solutions in real numbers. For example, the equation 2 = −9 has no real solution, Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the unit i where i2 = −1. According to the theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. A complex number is a number of the form a + bi, for example, −3.5 + 2i is a complex number. The real number a is called the part of the complex number a + bi. By this convention the imaginary part does not include the unit, hence b. The real part of a number z is denoted by Re or ℜ. For example, Re = −3.5 Im =2, hence, in terms of its real and imaginary parts, a complex number z is equal to Re + Im ⋅ i. This expression is known as the Cartesian form of z. A real number a can be regarded as a number a + 0i whose imaginary part is 0

14.
Quaternion
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In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843, a feature of quaternions is that multiplication of two quaternions is noncommutative. Hamilton defined a quaternion as the quotient of two directed lines in a space or equivalently as the quotient of two vectors. Quaternions are generally represented in the form, a + bi + cj + dk where a, b, c, and d are real numbers, and i, j, and k are the fundamental quaternion units. In practical applications, they can be used other methods, such as Euler angles and rotation matrices, or as an alternative to them. In modern mathematical language, quaternions form a four-dimensional associative normed division algebra over the real numbers, in fact, the quaternions were the first noncommutative division algebra to be discovered. The algebra of quaternions is often denoted by H, or in blackboard bold by H and it can also be given by the Clifford algebra classifications Cℓ0,2 ≅ Cℓ03,0. These rings are also Euclidean Hurwitz algebras, of which quaternions are the largest associative algebra. The unit quaternions can be thought of as a choice of a structure on the 3-sphere S3 that gives the group Spin. Quaternion algebra was introduced by Hamilton in 1843, carl Friedrich Gauss had also discovered quaternions in 1819, but this work was not published until 1900. Hamilton knew that the numbers could be interpreted as points in a plane. Points in space can be represented by their coordinates, which are triples of numbers, however, Hamilton had been stuck on the problem of multiplication and division for a long time. He could not figure out how to calculate the quotient of the coordinates of two points in space. The great breakthrough in quaternions finally came on Monday 16 October 1843 in Dublin, as he walked along the towpath of the Royal Canal with his wife, the concepts behind quaternions were taking shape in his mind. When the answer dawned on him, Hamilton could not resist the urge to carve the formula for the quaternions, i2 = j2 = k2 = ijk = −1, into the stone of Brougham Bridge as he paused on it. On the following day, Hamilton wrote a letter to his friend and fellow mathematician, John T. Graves and this letter was later published in the London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, vol. xxv, pp 489–95. In the letter, Hamilton states, And here there dawned on me the notion that we must admit, in some sense, an electric circuit seemed to close, and a spark flashed forth. Hamilton called a quadruple with these rules of multiplication a quaternion, Hamiltons treatment is more geometric than the modern approach, which emphasizes quaternions algebraic properties

15.
Octonion
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In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold O. There are three lower-dimensional normed division algebras over the reals, the real numbers R themselves, the complex numbers C, the octonions have eight dimensions, twice the number of dimensions of the quaternions, of which they are an extension. They are noncommutative and nonassociative, but satisfy a form of associativity. Octonions are not as known as the quaternions and complex numbers. Despite this, they have interesting properties and are related to a number of exceptional structures in mathematics. Additionally, octonions have applications in such as string theory, special relativity. The octonions were discovered in 1843 by John T. Graves, the octonions were discovered independently by Cayley and are sometimes referred to as Cayley numbers or the Cayley algebra. Hamilton described the history of Graves discovery. Hamilton invented the word associative so that he could say that octonions were not associative, the octonions can be thought of as octets of real numbers. Every octonion is a linear combination of the unit octonions. Addition and subtraction of octonions is done by adding and subtracting corresponding terms and hence their coefficients, multiplication is distributive over addition, so the product of two octonions can be calculated by summing the product of all the terms, again like quaternions. The above definition though is not unique, but is one of 480 possible definitions for octonion multiplication with e0 =1. The others can be obtained by permuting and changing the signs of the basis elements. The 480 different algebras are isomorphic, and there is rarely a need to consider which particular multiplication rule is used. Each of these 480 definitions is invariant up to signs under some 7-cycle of the points, a common choice is to use the definition invariant under the 7-cycle with e1e2 = e4 as it is particularly easy to remember the multiplication. A variation of this sometimes used is to label the elements of the basis by the elements ∞,0,1,2,6, of the projective line over the finite field of order 7. The multiplication is given by e∞ =1 and e1e2 = e4. These are the nonzero codewords of the quadratic residue code of length 7 over the field of 2 elements

16.
Split-complex number
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In abstract algebra, the split-complex numbers are a two-dimensional commutative algebra over the real numbers different from the complex numbers. Every split-complex number has the form x + y j, where x and y are real numbers, the number j is similar to the imaginary unit i, except that j 2 = +1. As an algebra over the reals, the numbers are the same as the direct sum of algebras R ⊕ R under the isomorphism sending x + y j to. The name split comes from this characterization, as a real algebra and it arises, for example, as the real subalgebra generated by an involutory matrix. Geometrically, split-complex numbers are related to the modulus in the way that complex numbers are related to the square of the Euclidean norm. Unlike the complex numbers, the split-complex numbers contain nontrivial idempotents, as well as zero divisors, in interval analysis, a split complex number x + y j represents an interval with midpoint x and radius y. Another application involves using numbers, dual numbers, and ordinary complex numbers. Split-complex numbers have many names, see the synonyms section below. See the article Motor variable for functions of a split-complex number and it is this sign change which distinguishes the split-complex numbers from the ordinary complex ones. The quantity j here is not a number but an independent quantity. The collection of all such z is called the split-complex plane, addition and multiplication of split-complex numbers are defined by + = + j = + j. This multiplication is commutative, associative and distributes over addition, just as for complex numbers, one can define the notion of a split-complex conjugate. If z = x + j y the conjugate of z is defined as z ∗ = x − j y, the conjugate satisfies similar properties to usual complex conjugate. Namely, ∗ = z ∗ + w ∗ ∗ = z ∗ w ∗ ∗ = z and these three properties imply that the split-complex conjugate is an automorphism of order 2. The modulus of a number z = x + j y is given by the isotropic quadratic form ∥ z ∥ = z z ∗ = z ∗ z = x 2 − y 2. It has the composition algebra property, ∥ z w ∥ = ∥ z ∥ ∥ w ∥, however, this quadratic form is not positive-definite but rather has signature, so the modulus is not a norm. The associated bilinear form is given by ⟨ z, w ⟩ = Re = Re = x u − y v, another expression for the modulus is then ∥ z ∥ = ⟨ z, z ⟩. Since it is not positive-definite, this form is not an inner product

17.
Split-quaternion
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In abstract algebra, the split-quaternions or coquaternions are elements of a 4-dimensional associative algebra introduced by James Cockle in 1849 under the latter name. Like the quaternions introduced by Hamilton in 1843, they form a four dimensional vector space equipped with a multiplicative operation. Unlike the quaternion algebra, the split-quaternions contain zero divisors, nilpotent elements, as a mathematical structure, they form an algebra over the real numbers, which is isomorphic to the algebra of 2 × 2 real matrices. For other names for split-quaternions see the Synonyms section below. The products of elements are ij = k = −ji, jk = −i = −kj, ki = j = −ik, i2 = −1, j2 = +1, k2 = +1. It follows from the relations that the set is a group under coquaternion multiplication. A coquaternion q = w + xi + yj + zk, has a conjugate q* = w − xi − yj − zk. Due to the property of its basis vectors, the product of a coquaternion with its conjugate is given by an isotropic quadratic form. Given two coquaterions p and q, one has N = N N, showing that N is a quadratic form admitting composition and this algebra is a composition algebra and N is its norm. Any q ≠0 such that N =0 is a vector, and its presence means that coquaternions form a split composition algebra. When the norm is non-zero, then q has a multiplicative inverse, the set U = is the set of units. The set P of all coquaternions forms a ring with group of units, the coquaternions with N =1 form a non-compact topological group SU, shown below to be isomorphic to SL. Historically coquaternions preceded Cayleys matrix algebra, coquaternions evoked the broader linear algebra, then the complex matrix, represents q in the ring of matrices, i. e. the multiplication of split-quaternions behaves the same way as the matrix multiplication. For example, the determinant of matrix is uu* − vv* = qq*. The appearance of the sign, where there is a plus in H. The use of the split-quaternions of norm one for hyperbolic motions of the Poincaré disk model of geometry is one of the great utilities of the algebra. Besides the complex representation, another linear representation associates coquaternions with 2 ×2 real matrices. Furthermore, note that these three matrices, together with the identity matrix, form a basis for M, one can make the matrix product above correspond to jk = −i in the coquaternion ring

18.
Split-octonion
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In mathematics, the split-octonions are an 8-dimensional nonassociative algebra over the real numbers. Unlike the standard octonions, they contain elements which are non-invertible. Also the signatures of their quadratic forms differ, the split-octonions have a split-signature whereas the octonions have a positive-definite signature, up to isomorphism, the octonions and the split-octonions are the only two octonion algebras over the real numbers. There are corresponding split octonion algebras over any field F, the octonions and the split-octonions can be obtained from the Cayley–Dickson construction by defining a multiplication on pairs of quaternions. We introduce a new imaginary unit ℓ and write a pair of quaternions in the form a + ℓb, the product is defined by the rule, = + ℓ where λ = ℓ2. If λ is chosen to be −1, we get the octonions, if, instead, it is taken to be +1 we get the split-octonions. One can also obtain the split-octonions via a Cayley–Dickson doubling of the split-quaternions, here either choice of λ gives the split-octonions. A basis for the split-octonions is given by the set, the red arrows indicate possible direction reversals imposed by negating the lower right quadrant of the parent creating a split octonion with this multiplication table. The quadratic form on x is given by N = x ¯ x = − This norm is the standard norm on R4,4. Due to the signature the norm N is isotropic, meaning there are nonzero x for which N =0. An element x has an inverse x−1 if and only if N ≠0, in this case the inverse is given by x −1 = x ¯ N. The split-octonions, like the octonions, are noncommutative and nonassociative, also like the octonions, they form a composition algebra since the quadratic form N is multiplicative. The split-octonions satisfy the Moufang identities and so form an alternative algebra, therefore, by Artins theorem, the subalgebra generated by any two elements is associative. The set of all elements form a Moufang loop. Since the split-octonions are nonassociative they cannot be represented by ordinary matrices, zorn found a way to represent them as matrices containing both scalars and vectors using a modified version of matrix multiplication. Specifically, define a vector-matrix to be a 2×2 matrix of the form where a and b are real numbers, define multiplication of these matrices by the rule = where · and × are the ordinary dot product and cross product of 3-vectors. With addition and scalar multiplication defined as usual the set of all such matrices forms a nonassociative unital 8-dimensional algebra over the reals, define the determinant of a vector-matrix by the rule det = a b − v ⋅ w. This determinant is a form on the Zorns algebra which satisfies the composition rule

19.
Dual quaternion
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In mathematics and mechanics, the set of dual quaternions is a Clifford algebra that can be used to represent spatial rigid body displacements. A dual quaternion is a pair of quaternions Â =. Because rigid body displacements are defined by six parameters, dual quaternion parameters include two algebraic constraints, in ring theory, dual quaternions are a ring constructed in the same way as the quaternions, except using dual numbers instead of real numbers as coefficients. A dual quaternion can be represented in the form p + ε q where p and q are ordinary quaternions and ε is the dual unit, unlike quaternions they do not form a division ring. Similar to the way that rotations in 3D space can be represented by quaternions of unit length and this fact is used in theoretical kinematics, and in applications to 3D computer graphics, robotics and computer vision. In 1898 Alexander McAulay used Ω with Ω2 =0 to generate the dual quaternion algebra, however, his terminology of octonions did not stick as todays octonions are another algebra. In Russia, Aleksandr Kotelnikov developed dual vectors and dual quaternions for use in the study of mechanics, in 1891 Eduard Study realized that this associative algebra was ideal for describing the group of motions of three-dimensional space. He further developed the idea in Geometrie der Dynamen in 1901, B. L. van der Waerden called the structure Study biquaternions, one of three eight-dimensional algebras referred to as biquaternions. In order to describe operations with dual quaternions, it is helpful to first consider quaternions, a quaternion is a linear combinations of the basis elements 1, i, j, and k. Hamiltons product rule for i, j, and k is often written as i 2 = j 2 = k 2 = i j k = −1. Compute i = −j k = −i, to obtain j k = i, now because j = j i = −k, we see that this product yields i j = −j i, which links quaternions to the properties of determinants. The vector dot and cross operations can now be used to define the product of A = a0 + A and C = c0 + C as G = A C = = +. A dual quaternion is usually described as a quaternion with dual numbers as coefficients, a dual number is an ordered pair â =. Two dual numbers add componentwise and multiply by the rule â ĉ = =, Dual numbers are often written in the form â = a + εb, where ε is the dual unit that commutes with i, j, k and has the property ε2 =0. The result is that a dual quaternion is the pair of quaternions Â =. Two dual quaternions add componentwise and multiply by the rule, A ^ C ^ = =. It is convenient to write a dual quaternion as the sum of a scalar and a dual vector, Â = â0 + A. This notation allows us to write the product of two dual quaternions as G ^ = A ^ C ^ = = +

20.
Cardinal number
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In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a set is a natural number, the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite sets, cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, in the case of finite sets, this agrees with the intuitive notion of size. In the case of sets, the behavior is more complex. It is also possible for a subset of an infinite set to have the same cardinality as the original set. There is a sequence of cardinal numbers,0,1,2,3, …, n, …, ℵ0, ℵ1, ℵ2, …, ℵ α, …. This sequence starts with the natural numbers including zero, which are followed by the aleph numbers, the aleph numbers are indexed by ordinal numbers. Under the assumption of the axiom of choice, this transfinite sequence includes every cardinal number, If one rejects that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs. Cardinality is studied for its own sake as part of set theory and it is also a tool used in branches of mathematics including model theory, combinatorics, abstract algebra, and mathematical analysis. In category theory, the numbers form a skeleton of the category of sets. The notion of cardinality, as now understood, was formulated by Georg Cantor, cardinality can be used to compare an aspect of finite sets, e. g. the sets and are not equal, but have the same cardinality, namely three. Cantor applied his concept of bijection to infinite sets, e. g. the set of natural numbers N =, thus, all sets having a bijection with N he called denumerable sets and they all have the same cardinal number. This cardinal number is called ℵ0, aleph-null and he called the cardinal numbers of these infinite sets transfinite cardinal numbers. Cantor proved that any unbounded subset of N has the same cardinality as N and he later proved that the set of all real algebraic numbers is also denumerable. His proof used an argument with nested intervals, but in an 1891 paper he proved the result using his ingenious. The new cardinal number of the set of numbers is called the cardinality of the continuum. His continuum hypothesis is the proposition that c is the same as ℵ1 and this hypothesis has been found to be independent of the standard axioms of mathematical set theory, it can neither be proved nor disproved from the standard assumptions

21.
Irrational number
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In mathematics, the irrational numbers are all the real numbers, which are not rational numbers, the latter being the numbers constructed from ratios of integers. Irrational numbers may also be dealt with via non-terminating continued fractions, for example, the decimal representation of the number π starts with 3.14159265358979, but no finite number of digits can represent π exactly, nor does it repeat. Mathematicians do not generally take terminating or repeating to be the definition of the concept of rational number, as a consequence of Cantors proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational. The first proof of the existence of numbers is usually attributed to a Pythagorean. The then-current Pythagorean method would have claimed that there must be sufficiently small. However, Hippasus, in the 5th century BC, was able to deduce that there was in no common unit of measure. His reasoning is as follows, Start with a right triangle with side lengths of integers a, b. The ratio of the hypotenuse to a leg is represented by c, b, assume a, b, and c are in the smallest possible terms. By the Pythagorean theorem, c2 = a2+b2 = b2+b2 = 2b2, since c2 = 2b2, c2 is divisible by 2, and therefore even. Since c2 is even, c must be even, since c is even, dividing c by 2 yields an integer. Squaring both sides of c = 2y yields c2 =2, or c2 = 4y2, substituting 4y2 for c2 in the first equation gives us 4y2= 2b2. Dividing by 2 yields 2y2 = b2, since y is an integer, and 2y2 = b2, b2 is divisible by 2, and therefore even. Since b2 is even, b must be even and we have just show that both b and c must be even. Hence they have a factor of 2. However this contradicts the assumption that they have no common factors and this contradiction proves that c and b cannot both be integers, and thus the existence of a number that cannot be expressed as a ratio of two integers. Greek mathematicians termed this ratio of incommensurable magnitudes alogos, or inexpressible. ”Another legend states that Hippasus was merely exiled for this revelation, the discovery of incommensurable ratios was indicative of another problem facing the Greeks, the relation of the discrete to the continuous. Brought into light by Zeno of Elea, who questioned the conception that quantities are discrete and composed of a number of units of a given size. ”However Zeno found that in fact “ in general are not discrete collections of units. That in fact, these divisions of quantity must necessarily be infinite, for example, consider a line segment, this segment can be split in half, that half split in half, the half of the half in half, and so on

22.
Surreal number
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The surreals share many properties with the reals, including the usual arithmetic operations, as such, they form an ordered field. The surreals also contain all transfinite ordinal numbers, the arithmetic on them is given by the natural operations, research on the go endgame by John Horton Conway led to another definition and construction of the surreal numbers. Conways construction was introduced in Donald Knuths 1974 book Surreal Numbers, How Two Ex-Students Turned on to Pure Mathematics, in his book, which takes the form of a dialogue, Knuth coined the term surreal numbers for what Conway had called simply numbers. Conway later adopted Knuths term, and used surreals for analyzing games in his 1976 book On Numbers and Games. In the Conway construction, the numbers are constructed in stages. Different subsets may end up defining the same number, and may define the number even if L ≠ L′. So strictly speaking, the numbers are equivalence classes of representations of form that designate the same number. In the first stage of construction, there are no previously existing numbers so the representation must use the empty set. This representation, where L and R are both empty, is called 0, subsequent stages yield forms like, =1 =2 =3 and = −1 = −2 = −3 The integers are thus contained within the surreal numbers. Similarly, representations arise like, = 1/2 = 1/4 = 3/4 so that the rationals are contained within the surreal numbers. Thus the real numbers are also embedded within the surreals, but there are also representations like = ω = ε where ω is a transfinite number greater than all integers and ε is an infinitesimal greater than 0 but less than any positive real number. The construction consists of three interdependent parts, the rule, the comparison rule and the equivalence rule. A form is a pair of sets of numbers, called its left set. A form with left set L and right set R is written, when L and R are given as lists of elements, the braces around them are omitted. Either or both of the left and right set of a form may be the empty set, the form with both left and right set empty is also written. The numeric forms are placed in classes, each such equivalence class is a surreal number. The elements of the left and right set of a form are drawn from the universe of the surreal numbers, equivalence Rule Two numeric forms x and y are forms of the same number if and only if both x ≤ y and y ≤ x. An ordering relationship must be antisymmetric, i. e. it must have the property that x = y only when x and y are the same object and this is not the case for surreal number forms, but is true by construction for surreal numbers

23.
Transcendental number
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In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a non-zero polynomial equation with integer coefficients. The best-known transcendental numbers are π and e, though only a few classes of transcendental numbers are known, transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the numbers are countable while the sets of real. All real transcendental numbers are irrational, since all numbers are algebraic. Another irrational number that is not transcendental is the ratio, φ or ϕ. The name transcendental comes from the root trans meaning across and length of numbers, euler was probably the first person to define transcendental numbers in the modern sense. Johann Heinrich Lambert conjectured that e and π were both transcendental numbers in his 1768 paper proving the number π is irrational, and proposed a tentative sketch of a proof of πs transcendence. In other words, the nth digit of this number is 1 only if n is one of the numbers 1. Liouville showed that number is what we now call a Liouville number. Liouville showed that all Liouville numbers are transcendental, the first number to be proven transcendental without having been specifically constructed for the purpose was e, by Charles Hermite in 1873. In 1874, Georg Cantor proved that the numbers are countable. He also gave a new method for constructing transcendental numbers, in 1878, Cantor published a construction that proves there are as many transcendental numbers as there are real numbers. Cantors work established the ubiquity of transcendental numbers, in 1882, Ferdinand von Lindemann published a proof that the number π is transcendental. He first showed that ea is transcendental when a is algebraic, then, since eiπ = −1 is algebraic, iπ and therefore π must be transcendental. This approach was generalized by Karl Weierstrass to the Lindemann–Weierstrass theorem, the transcendence of π allowed the proof of the impossibility of several ancient geometric constructions involving compass and straightedge, including the most famous one, squaring the circle. The affirmative answer was provided in 1934 by the Gelfond–Schneider theorem and this work was extended by Alan Baker in the 1960s in his work on lower bounds for linear forms in any number of logarithms. The set of numbers is uncountably infinite. Since the polynomials with rational coefficients are countable, and since each such polynomial has a number of zeroes

24.
Ordinal number
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In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another. Any finite collection of objects can be put in order just by the process of counting, labeling the objects with distinct whole numbers, Ordinal numbers are thus the labels needed to arrange collections of objects in order. An ordinal number is used to describe the type of a well ordered set. Whereas ordinals are useful for ordering the objects in a collection, they are distinct from cardinal numbers, although the distinction between ordinals and cardinals is not always apparent in finite sets, different infinite ordinals can describe the same cardinal. Like other kinds of numbers, ordinals can be added, multiplied, a natural number can be used for two purposes, to describe the size of a set, or to describe the position of an element in a sequence. When restricted to finite sets these two concepts coincide, there is one way to put a finite set into a linear sequence. This is because any set has only one size, there are many nonisomorphic well-orderings of any infinite set. Whereas the notion of number is associated with a set with no particular structure on it. A well-ordered set is an ordered set in which there is no infinite decreasing sequence, equivalently. Ordinals may be used to label the elements of any given well-ordered set and this length is called the order type of the set. Any ordinal is defined by the set of ordinals that precede it, in fact, the most common definition of ordinals identifies each ordinal as the set of ordinals that precede it. For example, the ordinal 42 is the type of the ordinals less than it, i. e. the ordinals from 0 to 41. Conversely, any set of ordinals that is downward-closed—meaning that for any ordinal α in S and any ordinal β < α, β is also in S—is an ordinal. There are infinite ordinals as well, the smallest infinite ordinal is ω, which is the type of the natural numbers. After all of these come ω·2, ω·2+1, ω·2+2, and so on, then ω·3, now the set of ordinals formed in this way must itself have an ordinal associated with it, and that is ω2. Further on, there will be ω3, then ω4, and so on, and ωω, then ωωω, then later ωωωω and this can be continued indefinitely far. The smallest uncountable ordinal is the set of all countable ordinals, in a well-ordered set, every non-empty subset contains a distinct smallest element. Given the axiom of dependent choice, this is equivalent to just saying that the set is ordered and there is no infinite decreasing sequence

25.
P-adic number
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The extension is achieved by an alternative interpretation of the concept of closeness or absolute value. In particular, p-adic numbers have the property that they are said to be close when their difference is divisible by a high power of p, the higher the power. P-adic numbers were first described by Kurt Hensel in 1897, though, with hindsight, the p-adic numbers were motivated primarily by an attempt to bring the ideas and techniques of power series methods into number theory. Their influence now extends far beyond this, for example, the field of p-adic analysis essentially provides an alternative form of calculus. More formally, for a prime p, the field Qp of p-adic numbers is a completion of the rational numbers. The field Qp is also given a topology derived from a metric, which is derived from the p-adic order. This metric space is complete in the sense that every Cauchy sequence converges to a point in Qp and this is what allows the development of calculus on Qp, and it is the interaction of this analytic and algebraic structure that gives the p-adic number systems their power and utility. The p in p-adic is a variable and may be replaced with a prime or another placeholder variable, the adic of p-adic comes from the ending found in words such as dyadic or triadic. This section is an introduction to p-adic numbers, using examples from the ring of 10-adic numbers. Although for p-adic numbers p should be a prime, base 10 was chosen to highlight the analogy with decimals, the decadic numbers are generally not used in mathematics, since 10 is not prime, the decadics are not a field. More formal constructions and properties are given below, in the standard decimal representation, almost all real numbers do not have a terminating decimal representation. For example, 1/3 is represented as a non-terminating decimal as follows 13 =0.333333 …, informally, non-terminating decimals are easily understood, because it is clear that a real number can be approximated to any required degree of precision by a terminating decimal. If two decimal expansions differ only after the 10th decimal place, they are close to one another. 10-adic numbers use a similar non-terminating expansion, but with a different concept of closeness, whereas two decimal expansions are close to one another if their difference is a large negative power of 10, two 10-adic expansions are close if their difference is a large positive power of 10. Thus 4739 and 5739, which differ by 103, are close in the 10-adic world, more precisely, a positive rational number r can be expressed as r =, p/q·10e, where p and q are positive integers and q is relatively prime to p and to 10. For each r ≠0 there exists the maximal e such that this representation is possible, let the 10-adic «absolute value» of r be | r |10, =110 e. Certainly, we have to define |0|10, =0, now, taking p/q =1 and e =0,1,2. We have |100|10 =100, |101|10 = 10−1, |102|10 = 10−2, with the consequence that we have lim + ∞ ← e 10 e =0