1.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
2.
Infinity
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Infinity is an abstract concept describing something without any bound or larger than any number. In mathematics, infinity is treated as a number but it is not the same sort of number as natural or real numbers. Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th, in the theory he developed, there are infinite sets of different sizes. For example, the set of integers is countably infinite, while the set of real numbers is uncountable. Ancient cultures had various ideas about the nature of infinity, the ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept. The earliest recorded idea of infinity comes from Anaximander, a pre-Socratic Greek philosopher who lived in Miletus and he used the word apeiron which means infinite or limitless. However, the earliest attestable accounts of mathematical infinity come from Zeno of Elea, aristotle called him the inventor of the dialectic. He is best known for his paradoxes, described by Bertrand Russell as immeasurably subtle, however, recent readings of the Archimedes Palimpsest have found that Archimedes had an understanding about actual infinite quantities. The Jain mathematical text Surya Prajnapti classifies all numbers into three sets, enumerable, innumerable, and infinite, on both physical and ontological grounds, a distinction was made between asaṃkhyāta and ananta, between rigidly bounded and loosely bounded infinities. European mathematicians started using numbers in a systematic fashion in the 17th century. John Wallis first used the notation ∞ for such a number, euler used the notation i for an infinite number, and exploited it by applying the binomial formula to the i th power, and infinite products of i factors. In 1699 Isaac Newton wrote about equations with an number of terms in his work De analysi per aequationes numero terminorum infinitas. The infinity symbol ∞ is a symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ infinity and in LaTeX as \infty and it was introduced in 1655 by John Wallis, and, since its introduction, has also been used outside mathematics in modern mysticism and literary symbology. Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers, in real analysis, the symbol ∞, called infinity, is used to denote an unbounded limit. X → ∞ means that x grows without bound, and x → − ∞ means the value of x is decreasing without bound. ∑ i =0 ∞ f = ∞ means that the sum of the series diverges in the specific sense that the partial sums grow without bound. Infinity can be used not only to define a limit but as a value in the real number system
3.
Decimal point
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A decimal mark is a symbol used to separate the integer part from the fractional part of a number written in decimal form. Different countries officially designate different symbols for the decimal mark, the choice of symbol for the decimal mark also affects the choice of symbol for the thousands separator used in digit grouping, so the latter is also treated in this article. In mathematics the decimal mark is a type of radix point, in the Middle Ages, before printing, a bar over the units digit was used to separate the integral part of a number from its fractional part, e. g.9995. His Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear, a similar notation remains in common use as an underbar to superscript digits, especially for monetary values without a decimal mark, e. g.9995. Later, a separatrix between the units and tenths position became the norm among Arab mathematicians, e. g. 99ˌ95, when this character was typeset, it was convenient to use the existing comma or full stop instead. The separatrix was also used in England as an L-shaped or vertical bar before the popularization of the period, gerbert of Aurillac marked triples of columns with an arc when using his Hindu–Arabic numeral-based abacus in the 10th century. Fibonacci followed this convention when writing numbers such as in his influential work Liber Abaci in the 13th century, in France, the full stop was already in use in printing to make Roman numerals more readable, so the comma was chosen. Many other countries, such as Italy, also chose to use the comma to mark the decimal units position and it has been made standard by the ISO for international blueprints. However, English-speaking countries took the comma to separate sequences of three digits, in some countries, a raised dot or dash may be used for grouping or decimal mark, this is particularly common in handwriting. In the United States, the stop or period was used as the standard decimal mark. g. However, as the mid dot was already in use in the mathematics world to indicate multiplication. In the event, the point was decided on by the Ministry of Technology in 1968, the three most spoken international auxiliary languages, Ido, Esperanto, and Interlingua, all use the comma as the decimal mark. Interlingua has used the comma as its decimal mark since the publication of the Interlingua Grammar in 1951, Esperanto also uses the comma as its official decimal mark, while thousands are separated by non-breaking spaces,12345678,9. Idos Kompleta Gramatiko Detaloza di la Linguo Internaciona Ido officially states that commas are used for the mark while full stops are used to separate thousands, millions. So the number 12,345,678.90123 for instance, the 1931 grammar of Volapük by Arie de Jong uses the comma as its decimal mark, and uses the middle dot as the thousands separator. In 1958, disputes between European and American delegates over the representation of the decimal mark nearly stalled the development of the ALGOL computer programming language. ALGOL ended up allowing different decimal marks, but most computer languages, the 22nd General Conference on Weights and Measures declared in 2003 that the symbol for the decimal marker shall be either the point on the line or the comma on the line. It further reaffirmed that numbers may be divided in groups of three in order to facilitate reading, neither dots nor commas are ever inserted in the spaces between groups
4.
Infimum and supremum
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In mathematics, the infimum of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists. Consequently, the term greatest lower bound is also commonly used, the supremum of a subset S of a partially ordered set T is the least element in T that is greater than or equal to all elements of S, if such an element exists. Consequently, the supremum is also referred to as the least upper bound, the infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are special cases that are important in analysis. However, the general definitions remain valid in the abstract setting of order theory where arbitrary partially ordered sets are considered. The concepts of infimum and supremum are similar to minimum and maximum, for instance, the positive real numbers ℝ+* does not have a minimum, because any given element of ℝ+* could simply be divided in half resulting in a smaller number that is still in ℝ+*. There is, however, exactly one infimum of the real numbers,0. A lower bound of a subset S of an ordered set is an element a of P such that a ≤ x for all x in S. A lower bound a of S is called an infimum of S if for all lower bounds y of S in P, y ≤ a. Similarly, a bound of a subset S of a partially ordered set is an element b of P such that b ≥ x for all x in S. An upper bound b of S is called a supremum of S if for all upper bounds z of S in P, z ≥ b, infima and suprema do not necessarily exist. Existence of an infimum of a subset S of P can fail if S has no lower bound at all, however, if an infimum or supremum does exist, it is unique. Consequently, partially ordered sets for which certain infima are known to exist become especially interesting, more information on the various classes of partially ordered sets that arise from such considerations are found in the article on completeness properties. If the supremum of a subset S exists, it is unique, if S contains a greatest element, then that element is the supremum, otherwise, the supremum does not belong to S. Likewise, if the infimum exists, it is unique. If S contains a least element, then that element is the infimum, otherwise, the infimum of a subset S of a partially ordered set P, assuming it exists, does not necessarily belong to S. If it does, it is a minimal or least element of S. Similarly, if the supremum of S belongs to S, for example, consider the set of negative real numbers. This set has no greatest element, since for every element of the set, there is another, larger, for instance, for any negative real number x, there is another negative real number x 2, which is greater. On the other hand, every real number greater than or equal to zero is certainly an upper bound on this set, hence,0 is the least upper bound of the negative reals, so the supremum is 0
5.
Decimal number
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This article aims to be an accessible introduction. For the mathematical definition, see Decimal representation, the decimal numeral system has ten as its base, which, in decimal, is written 10, as is the base in every positional numeral system. It is the base most widely used by modern civilizations. Decimal fractions have terminating decimal representations and other fractions have repeating decimal representations, Decimal notation is the writing of numbers in a base-ten numeral system. Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, Roman numerals have symbols for the decimal powers and secondary symbols for half these values. Brahmi numerals have symbols for the nine numbers 1–9, the nine decades 10–90, plus a symbol for 100, Chinese numerals have symbols for 1–9, and additional symbols for powers of ten, which in modern usage reach 1072. Positional decimal systems include a zero and use symbols for the ten values to represent any number, positional notation uses positions for each power of ten, units, tens, hundreds, thousands, etc. The position of each digit within a number denotes the multiplier multiplied with that position has a value ten times that of the position to its right. There were at least two independent sources of positional decimal systems in ancient civilization, the Chinese counting rod system. Ten is the number which is the count of fingers and thumbs on both hands, the English word digit as well as its translation in many languages is also the anatomical term for fingers and toes. In English, decimal means tenth, decimate means reduce by a tenth, however, the symbols used in different areas are not identical, for instance, Western Arabic numerals differ from the forms used by other Arab cultures. A decimal fraction is a fraction the denominator of which is a power of ten. g, Decimal fractions 8/10, 1489/100, 24/100000, and 58900/10000 are expressed in decimal notation as 0.8,14.89,0.00024,5.8900 respectively. In English-speaking, some Latin American and many Asian countries, a period or raised period is used as the separator, in many other countries, particularly in Europe. The integer part, or integral part of a number is the part to the left of the decimal separator. The part from the separator to the right is the fractional part. It is usual for a number that consists only of a fractional part to have a leading zero in its notation. Any rational number with a denominator whose only prime factors are 2 and/or 5 may be expressed as a decimal fraction and has a finite decimal expansion. 1/2 =0.5 1/20 =0.05 1/5 =0.2 1/50 =0.02 1/4 =0.25 1/40 =0.025 1/25 =0.04 1/8 =0.125 1/125 =0.008 1/10 =0
6.
Mathematical proof
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In mathematics, a proof is an inferential argument for a mathematical statement. In the argument, other previously established statements, such as theorems, in principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference. Axioms may be treated as conditions that must be met before the statement applies, Proofs are examples of exhaustive deductive reasoning or inductive reasoning and are distinguished from empirical arguments or non-exhaustive inductive reasoning. A proof must demonstrate that a statement is true, rather than enumerate many confirmatory cases. An unproved proposition that is believed to be true is known as a conjecture, Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to examination of current and historical mathematical practice, quasi-empiricism in mathematics. The philosophy of mathematics is concerned with the role of language and logic in proofs, the word proof comes from the Latin probare meaning to test. Related modern words are the English probe, probation, and probability, the Spanish probar, Italian provare, the early use of probity was in the presentation of legal evidence. A person of authority, such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof. It is likely that the idea of demonstrating a conclusion first arose in connection with geometry, the development of mathematical proof is primarily the product of ancient Greek mathematics, and one of the greatest achievements thereof. Thales proved some theorems in geometry, eudoxus and Theaetetus formulated theorems but did not prove them. Aristotle said definitions should describe the concept being defined in terms of other concepts already known and his book, the Elements, was read by anyone who was considered educated in the West until the middle of the 20th century. Further advances took place in medieval Islamic mathematics, while earlier Greek proofs were largely geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more general proofs that no longer depended on geometry. In the 10th century CE, the Iraqi mathematician Al-Hashimi provided general proofs for numbers as he considered multiplication, division and he used this method to provide a proof of the existence of irrational numbers. An inductive proof for arithmetic sequences was introduced in the Al-Fakhri by Al-Karaji, alhazen also developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate. Modern proof theory treats proofs as inductively defined data structures, there is no longer an assumption that axioms are true in any sense, this allows for parallel mathematical theories built on alternate sets of axioms
7.
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
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Radix
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In mathematical numeral systems, the radix or base is the number of unique digits, including zero, used to represent numbers in a positional numeral system. For example, for the system the radix is ten. For example,10 represents the one hundred, while 2 represents the number four. Radix is a Latin word for root, root can be considered a synonym for base in the arithmetical sense. In the system with radix 13, for example, a string of such as 398 denotes the number 3 ×132 +9 ×131 +8 ×130. More generally, in a system with radix b, a string of digits d1 … dn denotes the number d1bn−1 + d2bn−2 + … + dnb0, commonly used numeral systems include, For a larger list, see List of numeral systems. The octal and hexadecimal systems are used in computing because of their ease as shorthand for binary. Every hexadecimal digit corresponds to a sequence of four binary digits, a similar relationship holds between every octal digit and every possible sequence of three binary digits, since eight is the cube of two. However, other systems are possible, e. g. golden ratio base. Base Radix economy Non-standard positional numeral systems Base Convert, a floating-point base calculator MathWorld entry on base
9.
Mathematics education
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In contemporary education, mathematics education is the practice of teaching and learning mathematics, along with the associated scholarly research. This article describes some of the history, influences and recent controversies, elementary mathematics was part of the education system in most ancient civilisations, including Ancient Greece, the Roman empire, Vedic society and ancient Egypt. In most cases, an education was only available to male children with a sufficiently high status. In Platos division of the arts into the trivium and the quadrivium. This structure was continued in the structure of education that was developed in medieval Europe. Teaching of geometry was almost universally based on Euclids Elements, apprentices to trades such as masons, merchants and money-lenders could expect to learn such practical mathematics as was relevant to their profession. The first mathematics textbooks to be written in English and French were published by Robert Recorde, however, there are many different writings on mathematics and mathematics methodology that date back to 1800 BCE. These were mostly located in Mesopotamia where the Sumerians were practicing multiplication and division, there are also artifacts demonstrating their own methodology for solving equations like the quadratic equation. After the Sumerians some of the most famous ancient works on come from Egypt in the form of the Rhind Mathematical Papyrus. The more famous Rhind Papyrus has been dated to approximately 1650 BCE and this papyrus was essentially an early textbook for Egyptian students. In the Renaissance, the status of mathematics declined, because it was strongly associated with trade. Although it continued to be taught in European universities, it was seen as subservient to the study of Natural, Metaphysical and Moral Philosophy, however, it was uncommon for mathematics to be taught outside of the universities. Isaac Newton, for example, received no formal mathematics teaching until he joined Trinity College, Cambridge in 1661, in the 18th and 19th centuries, the industrial revolution led to an enormous increase in urban populations. Basic numeracy skills, such as the ability to tell the time, count money and carry out simple arithmetic, within the new public education systems, mathematics became a central part of the curriculum from an early age. By the twentieth century, mathematics was part of the curriculum in all developed countries. During the twentieth century, mathematics education was established as an independent field of research. S. A, had generated more than 4000 articles after 1920, so in 1941 William L. Schaaf published a classified index, sorting them into their various subjects. While previous approach focused on working with specialized problems in arithmetic, at different times and in different cultures and countries, mathematics education has attempted to achieve a variety of different objectives. The teaching of heuristics and other problem-solving strategies to solve non-routine problems, the method or methods used in any particular context are largely determined by the objectives that the relevant educational system is trying to achieve
10.
Archimedean property
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Roughly speaking, it is the property of having no infinitely large or infinitely small elements. It was Otto Stolz who gave the axiom of Archimedes its name because it appears as Axiom V of Archimedes’ On the Sphere and Cylinder. An algebraic structure in any two non-zero elements are comparable, in the sense that neither of them is infinitesimal with respect to the other, is said to be Archimedean. A structure which has a pair of elements, one of which is infinitesimal with respect to the other, is said to be non-Archimedean. For example, an ordered group that is Archimedean is an Archimedean group. This can be made precise in various contexts with different formulations. The concept was named by Otto Stolz after the ancient Greek geometer, the Archimedean property appears in Book V of Euclids Elements as Definition 4, Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another. Because Archimedes credited it to Eudoxus of Cnidus it is known as the Theorem of Eudoxus or the Eudoxus axiom. Archimedes used infinitesimals in heuristic arguments, although he denied that those were finished mathematical proofs, Let x and y be positive elements of a linearly ordered group G. Then x is infinitesimal with respect to y if, for natural number n, the multiple nx is less than y, that is. The group G is Archimedean if there is no x, y such that x is infinitesimal with respect to y. Additionally, if K is a structure with a unit — for example. If x is infinitesimal with respect to 1, then x is an infinitesimal element, likewise, if y is infinite with respect to 1, then y is an infinite element. The algebraic structure K is Archimedean if it has no infinite elements, an ordered field has some additional properties. One may assume that the numbers are contained in the field. If x is infinitesimal, then 1/x is infinite, and vice versa, therefore, to verify that a field is Archimedean it is enough to check only that there are no infinitesimal elements, or to check that there are no infinite elements. If x is infinitesimal and r is a number, then r x is also infinitesimal. As a result, given an element c, the three numbers c/2, c, and 2c are either all infinitesimal or all non-infinitesimal
11.
Series (mathematics)
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In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a sequence has defined first and last terms. To emphasize that there are a number of terms, a series is often called an infinite series. In order to make the notion of an infinite sum mathematically rigorous, given an infinite sequence, the associated series is the expression obtained by adding all those terms together, a 1 + a 2 + a 3 + ⋯. These can be written compactly as ∑ i =1 ∞ a i, by using the summation symbol ∑. The sequence can be composed of any kind of object for which addition is defined. A series is evaluated by examining the finite sums of the first n terms of a sequence, called the nth partial sum of the sequence, and taking the limit as n approaches infinity. If this limit does not exist, the infinite sum cannot be assigned a value, and, in this case, the series is said to be divergent. On the other hand, if the partial sums tend to a limit when the number of terms increases indefinitely, then the series is said to be convergent, and the limit is called the sum of the series. An example is the series from Zenos dichotomy and its mathematical representation, ∑ n =1 ∞12 n =12 +14 +18 + ⋯. The study of series is a part of mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures, in addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For any sequence of numbers, real numbers, complex numbers, functions thereof. By definition the series ∑ n =0 ∞ a n converges to a limit L if and this definition is usually written as L = ∑ n =0 ∞ a n ⇔ L = lim k → ∞ s k. When the index set is the natural numbers I = N, a series indexed on the natural numbers is an ordered formal sum and so we rewrite ∑ n ∈ N as ∑ n =0 ∞ in order to emphasize the ordering induced by the natural numbers. Thus, we obtain the common notation for a series indexed by the natural numbers ∑ n =0 ∞ a n = a 0 + a 1 + a 2 + ⋯. When the semigroup G is also a space, then the series ∑ n =0 ∞ a n converges to an element L ∈ G if. This definition is usually written as L = ∑ n =0 ∞ a n ⇔ L = lim k → ∞ s k, a series ∑an is said to converge or to be convergent when the sequence SN of partial sums has a finite limit
12.
Limit (mathematics)
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In mathematics, a limit is the value that a function or sequence approaches as the input or index approaches some value. Limits are essential to calculus and are used to define continuity, derivatives, the concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory. In formulas, a limit is usually written as lim n → c f = L and is read as the limit of f of n as n approaches c equals L. Here lim indicates limit, and the fact that function f approaches the limit L as n approaches c is represented by the right arrow, suppose f is a real-valued function and c is a real number. Intuitively speaking, the lim x → c f = L means that f can be made to be as close to L as desired by making x sufficiently close to c. The first inequality means that the distance x and c is greater than 0 and that x ≠ c, while the second indicates that x is within distance δ of c. Note that the definition of a limit is true even if f ≠ L. Indeed. Now since x +1 is continuous in x at 1, we can now plug in 1 for x, in addition to limits at finite values, functions can also have limits at infinity. In this case, the limit of f as x approaches infinity is 2, in mathematical notation, lim x → ∞2 x −1 x =2. Consider the following sequence,1.79,1.799,1.7999 and it can be observed that the numbers are approaching 1.8, the limit of the sequence. Formally, suppose a1, a2. is a sequence of real numbers, intuitively, this means that eventually all elements of the sequence get arbitrarily close to the limit, since the absolute value | an − L | is the distance between an and L. Not every sequence has a limit, if it does, it is called convergent, one can show that a convergent sequence has only one limit. The limit of a sequence and the limit of a function are closely related, on one hand, the limit as n goes to infinity of a sequence a is simply the limit at infinity of a function defined on the natural numbers n. On the other hand, a limit L of a function f as x goes to infinity, if it exists, is the same as the limit of any sequence a that approaches L. Note that one such sequence would be L + 1/n, in non-standard analysis, the limit of a sequence can be expressed as the standard part of the value a H of the natural extension of the sequence at an infinite hypernatural index n=H. Thus, lim n → ∞ a n = st , here the standard part function st rounds off each finite hyperreal number to the nearest real number. This formalizes the intuition that for very large values of the index. Conversely, the part of a hyperreal a = represented in the ultrapower construction by a Cauchy sequence, is simply the limit of that sequence
13.
Construction of real numbers
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In mathematics, there are several ways of defining the real number system as an ordered field. The synthetic approach gives a list of axioms for the numbers as a complete ordered field. Under the usual axioms of set theory, one can show that these axioms are categorical, in the sense there is a model for the axioms. Any one of these models must be constructed, and most of these models are built using the basic properties of the rational number system as an ordered field. The synthetic approach axiomatically defines the number system as a complete ordered field. A model for the number system consists of a set R, two distinct elements 0 and 1 of R, two binary operations + and × on R, and a binary relation ≤ on R. In other words, For all x, y, and z in R, x + = + z and x × = × z, for all x and y in R, x + y = y + x and x × y = y × x. For all x, y, and z in R, x × = +, for all x in R, x +0 = x.0 is not equal to 1, and for all x in R, x ×1 = x. For every x in R, there exists an element −x in R, for every x ≠0 in R, there exists an element x−1 in R, such that x × x−1 =1. In other words, For all x in R, x ≤ x, for all x and y in R, if x ≤ y and y ≤ x, then x = y. For all x, y, and z in R, if x ≤ y and y ≤ z, for all x and y in R, x ≤ y or y ≤ x. The field operations + and × on R are compatible with the order ≤, in other words, For all x, y and z in R, if x ≤ y, then x + z ≤ y + z. In other words, If A is a non-empty subset of R, the rational numbers Q satisfy the first three axioms but Q does not satisfy axiom 4. So axiom 4, which requires the order to be Dedekind-complete, is crucial, axiom 4 implies the Archimedean property. Several models for axioms 1-4 are given below, any two models for axioms 1-4 are isomorphic, and so up to isomorphism, there is only one complete ordered Archimedean field. When we say that any two models of the axioms are isomorphic, we mean that for any two models and, there is a bijection f, R → S preserving both the field operations and the order. Explicitly, f is injective and surjective. For all x and y in R, f = f +S f and f = f ×S f, for all x and y in R, x ≤R y if and only if f ≤S f
14.
Number line
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In basic mathematics, a number line is a picture of a graduated straight line that serves as abstraction for real numbers, denoted by R. Every point of a line is assumed to correspond to a real number. Often the integers are shown as specially-marked points evenly spaced on the line, although this image only shows the integers from −9 to 9, the line includes all real numbers, continuing forever in each direction, and also numbers not marked that are between the integers. It is often used as an aid in teaching simple addition and subtraction, a number line is usually represented as being horizontal, but in a Cartesian coordinate plane the vertical axis is also a number line. Another convention uses only one arrowhead which indicates the direction in which numbers grow, if a particular number is farther to the right on the number line than is another number, then the first number is greater than the second. The distance between them is the magnitude of their difference—that is, it measures the first number minus the second one, taking this difference is the process of subtraction. Thus, for example, the length of a segment between 0 and some other number represents the magnitude of the latter number. Two numbers can be added by picking up the length from 0 to one of the numbers and this gives a result that is 3 combined lengths of 5 each, since the process ends at 15, we find that 5 ×3 =15. This puts the right end of the length 2 at the end of the length from 0 to 6. Since three lengths of 2 filled the length 6,2 goes into 6 three times, the section of the number line between two numbers is called an interval. If the section includes both numbers it is said to be an interval, while if it excludes both numbers it is called an open interval. If it includes one of the numbers but not the other one, all the points extending forever in one direction from a particular point are together known as a ray. If the ray includes the point, it is a closed ray. Sometimes it is convenient to scale the numbers on the line with a logarithmic scale. This approach is useful, for example, in illustrating a sequence of events in the history of the universe or of evolution, a line drawn through the origin at right angles to the real number line can be used to represent the imaginary numbers. This line, called line, extends the number line to a complex number plane. Together these lines form what is known as the Cartesian coordinate system, further, the Cartesian coordinate system can itself be extended by visualizing a third number line coming out of the screen, measuring a third variable called z. Positive numbers are closer to the eyes than the screen is, while negative numbers are behind the screen
15.
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
16.
Number system
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A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1,2,3, a notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are used for labels, for ordering. In common usage, number may refer to a symbol, a word, calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation. Their study or usage is called arithmetic, the same term may also refer to number theory, the study of the properties of numbers. Besides their practical uses, numbers have cultural significance throughout the world, for example, in Western society the number 13 is regarded as unlucky, and a million may signify a lot. Though it is now regarded as pseudoscience, numerology, the belief in a significance of numbers, permeated ancient. Numerology heavily influenced the development of Greek mathematics, stimulating the investigation of problems in number theory which are still of interest today. During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers, among the first were the hypercomplex numbers, which consist of various extensions or modifications of the complex number system. Numbers should be distinguished from numerals, the used to represent numbers. Boyer showed that Egyptians created the first ciphered numeral system, Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets. The number five can be represented by digit 5 or by the Roman numeral Ⅴ, notations used to represent numbers are discussed in the article numeral systems. The Roman numerals require extra symbols for larger numbers, different types of numbers have many different uses. Numbers can be classified into sets, called number systems, such as the natural numbers, the same number can be written in many different ways. For different methods of expressing numbers with symbols, such as the Roman numerals, each of these number systems may be considered as a proper subset of the next one. This is expressed, symbolically, by writing N ⊂ Z ⊂ Q ⊂ R ⊂ C, the most familiar numbers are the natural numbers,1,2,3, and so on. Traditionally, the sequence of numbers started with 1 However, in the 19th century, set theorists. Today, different mathematicians use the term to both sets, including 0 or not
17.
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 set of all numbers, often referred to as the rationals, is usually denoted by a boldface Q, it was thus denoted in 1895 by Giuseppe Peano after quoziente. The decimal expansion of a rational number always either terminates after a 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. In abstract algebra, the numbers together with certain operations of addition and multiplication form the archetypical field of characteristic zero. As such, it is characterized as having no proper subfield or, alternatively, 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
18.
Least upper bound
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In mathematics, the infimum of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists. Consequently, the term greatest lower bound is also commonly used, the supremum of a subset S of a partially ordered set T is the least element in T that is greater than or equal to all elements of S, if such an element exists. Consequently, the supremum is also referred to as the least upper bound, the infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are special cases that are important in analysis. However, the general definitions remain valid in the abstract setting of order theory where arbitrary partially ordered sets are considered. The concepts of infimum and supremum are similar to minimum and maximum, for instance, the positive real numbers ℝ+* does not have a minimum, because any given element of ℝ+* could simply be divided in half resulting in a smaller number that is still in ℝ+*. There is, however, exactly one infimum of the real numbers,0. A lower bound of a subset S of an ordered set is an element a of P such that a ≤ x for all x in S. A lower bound a of S is called an infimum of S if for all lower bounds y of S in P, y ≤ a. Similarly, a bound of a subset S of a partially ordered set is an element b of P such that b ≥ x for all x in S. An upper bound b of S is called a supremum of S if for all upper bounds z of S in P, z ≥ b, infima and suprema do not necessarily exist. Existence of an infimum of a subset S of P can fail if S has no lower bound at all, however, if an infimum or supremum does exist, it is unique. Consequently, partially ordered sets for which certain infima are known to exist become especially interesting, more information on the various classes of partially ordered sets that arise from such considerations are found in the article on completeness properties. If the supremum of a subset S exists, it is unique, if S contains a greatest element, then that element is the supremum, otherwise, the supremum does not belong to S. Likewise, if the infimum exists, it is unique. If S contains a least element, then that element is the infimum, otherwise, the infimum of a subset S of a partially ordered set P, assuming it exists, does not necessarily belong to S. If it does, it is a minimal or least element of S. Similarly, if the supremum of S belongs to S, for example, consider the set of negative real numbers. This set has no greatest element, since for every element of the set, there is another, larger, for instance, for any negative real number x, there is another negative real number x 2, which is greater. On the other hand, every real number greater than or equal to zero is certainly an upper bound on this set, hence,0 is the least upper bound of the negative reals, so the supremum is 0
19.
Real number system
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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
20.
Positive integer
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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
21.
Nonnegative number
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In mathematics, the concept of sign originates from the property of every non-zero real number of being positive or negative. Zero itself is signless, although in some contexts it makes sense to consider a signed zero, along with its application to real numbers, change of sign is used throughout mathematics and physics to denote the additive inverse, even for quantities which are not real numbers. Also, the sign can indicate aspects of mathematical objects that resemble positivity and negativity. A real number is said to be if its value is greater than zero. The attribute of being positive or negative is called the sign of the number, zero itself is not considered to have a sign. Also, signs are not defined for complex numbers, although the argument generalizes it in some sense, in common numeral notation, the sign of a number is often denoted by placing a plus sign or a minus sign before the number. For example, +3 denotes positive three, and −3 denotes negative three, when no plus or minus sign is given, the default interpretation is that a number is positive. Because of this notation, as well as the definition of numbers through subtraction. In this context, it makes sense to write − = +3, any non-zero number can be changed to a positive one using the absolute value function. For example, the value of −3 and the absolute value of 3 are both equal to 3. In symbols, this would be written |−3| =3 and |3| =3, the number zero is neither positive nor negative, and therefore has no sign. In arithmetic, +0 and −0 both denote the same number 0, which is the inverse of itself. Note that this definition is culturally determined, in France and Belgium,0 is said to be both positive and negative. The positive resp. negative numbers without zero are said to be strictly positive resp, in some contexts, such as signed number representations in computing, it makes sense to consider signed versions of zero, with positive zero and negative zero being different numbers. One also sees +0 and −0 in calculus and mathematical analysis when evaluating one-sided limits and this notation refers to the behaviour of a function as the input variable approaches 0 from positive or negative values respectively, these behaviours are not necessarily the same. Because zero is positive nor negative, the following phrases are sometimes used to refer to the sign of an unknown number. A number is negative if it is less than zero, a number is non-negative if it is greater than or equal to zero. A number is non-positive if it is less than or equal to zero, thus a non-negative number is either positive or zero, while a non-positive number is either negative or zero
22.
Number systems
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A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1,2,3, a notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are used for labels, for ordering. In common usage, number may refer to a symbol, a word, calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation. Their study or usage is called arithmetic, the same term may also refer to number theory, the study of the properties of numbers. Besides their practical uses, numbers have cultural significance throughout the world, for example, in Western society the number 13 is regarded as unlucky, and a million may signify a lot. Though it is now regarded as pseudoscience, numerology, the belief in a significance of numbers, permeated ancient. Numerology heavily influenced the development of Greek mathematics, stimulating the investigation of problems in number theory which are still of interest today. During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers, among the first were the hypercomplex numbers, which consist of various extensions or modifications of the complex number system. Numbers should be distinguished from numerals, the used to represent numbers. Boyer showed that Egyptians created the first ciphered numeral system, Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets. The number five can be represented by digit 5 or by the Roman numeral Ⅴ, notations used to represent numbers are discussed in the article numeral systems. The Roman numerals require extra symbols for larger numbers, different types of numbers have many different uses. Numbers can be classified into sets, called number systems, such as the natural numbers, the same number can be written in many different ways. For different methods of expressing numbers with symbols, such as the Roman numerals, each of these number systems may be considered as a proper subset of the next one. This is expressed, symbolically, by writing N ⊂ Z ⊂ Q ⊂ R ⊂ C, the most familiar numbers are the natural numbers,1,2,3, and so on. Traditionally, the sequence of numbers started with 1 However, in the 19th century, set theorists. Today, different mathematicians use the term to both sets, including 0 or not
23.
Real analysis
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Real analysis is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable. The theorems of real analysis rely intimately upon the structure of the number line. The real number system consists of a set, together with two operations and an order, and is, formally speaking, an ordered quadruple consisting of these objects, there are several ways of formalizing the definition of the real number system. The synthetic approach gives a list of axioms for the numbers as a complete ordered field. Under the usual axioms of set theory, one can show that these axioms are categorical, in the sense there is a model for the axioms. Any one of these models must be constructed, and most of these models are built using the basic properties of the rational number system as an ordered field. These constructions are described in detail in the main article. In addition to these notions, the real numbers, equipped with the absolute value function as a metric. Many important theorems in real analysis remain valid when they are restated as statements involving metric spaces and these theorems are frequently topological in nature, and placing them in the more abstract setting of metric spaces may lead to proofs that are shorter, more natural, or more elegant. The real numbers have several important lattice-theoretic properties that are absent in the complex numbers, most importantly, the real numbers form an ordered field, in which addition and multiplication preserve positivity. Moreover, the ordering of the numbers is total. These order-theoretic properties lead to a number of important results in analysis, such as the monotone convergence theorem, the intermediate value theorem. However, while the results in analysis are stated for real numbers. In particular, many ideas in analysis and operator theory generalize properties of the real numbers – such generalizations include the theories of Riesz spaces. Also, mathematicians consider real and imaginary parts of complex sequences, a sequence is a function whose domain is a countable, totally ordered set, usually taken to be the natural numbers or whole numbers. Occasionally, it is convenient to consider bidirectional sequences indexed by the set of all integers. Of interest in analysis, a real-valued sequence, here indexed by the natural numbers, is a map a, N → R, n ↦ a n. Each a = a n is referred to as a term of the sequence, a sequence that tends to a limit is said to be convergent, otherwise it is divergent
24.
Positional notation
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Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the symbol for the different orders of magnitude. This greatly simplified arithmetic, leading to the spread of the notation across the world. With the use of a point, the notation can be extended to include fractions. The Hindu–Arabic numeral system, base-10, is the most commonly used system in the world today for most calculations, today, the base-10 system, which is likely motivated by counting with the ten fingers, is ubiquitous. Other bases have been used in the past however, and some continue to be used today, for example, the Babylonian numeral system, credited as the first positional numeral system, was base-60, but it lacked a real 0 value. Zero was indicated by a space between sexagesimal numerals, by 300 BC, a punctuation symbol was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish, the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges, the Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number, thus numbers like 2 and 120,3 and 180,4 and 240, looked the same because the larger numbers lacked a final sexagesimal placeholder. Counting rods and most abacuses have been used to represent numbers in a numeral system. This approach required no memorization of tables and could produce practical results quickly, for four centuries there was strong disagreement between those who believed in adopting the positional system in writing numbers and those who wanted to stay with the additive-system-plus-abacus. Although electronic calculators have largely replaced the abacus, the continues to be used in Japan. After the French Revolution, the new French government promoted the extension of the decimal system, some of those pro-decimal efforts—such as decimal time and the decimal calendar—were unsuccessful. Other French pro-decimal efforts—currency decimalisation and the metrication of weights and measures—spread widely out of France to almost the whole world. According to Joseph Needham and Lam Lay Yong, decimal fractions were first developed and used by the Chinese in the 1st century BC, the written Chinese decimal fractions were non-positional. However, counting rod fractions were positional, the Jewish mathematician Immanuel Bonfils used decimal fractions around 1350, anticipating Simon Stevin, but did not develop any notation to represent them. A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century. A key argument against the system was its susceptibility to easy fraud by simply putting a number at the beginning or end of a quantity, thereby changing 100 into 5100
25.
Infinite series
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In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a sequence has defined first and last terms. To emphasize that there are a number of terms, a series is often called an infinite series. In order to make the notion of an infinite sum mathematically rigorous, given an infinite sequence, the associated series is the expression obtained by adding all those terms together, a 1 + a 2 + a 3 + ⋯. These can be written compactly as ∑ i =1 ∞ a i, by using the summation symbol ∑. The sequence can be composed of any kind of object for which addition is defined. A series is evaluated by examining the finite sums of the first n terms of a sequence, called the nth partial sum of the sequence, and taking the limit as n approaches infinity. If this limit does not exist, the infinite sum cannot be assigned a value, and, in this case, the series is said to be divergent. On the other hand, if the partial sums tend to a limit when the number of terms increases indefinitely, then the series is said to be convergent, and the limit is called the sum of the series. An example is the series from Zenos dichotomy and its mathematical representation, ∑ n =1 ∞12 n =12 +14 +18 + ⋯. The study of series is a part of mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures, in addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For any sequence of numbers, real numbers, complex numbers, functions thereof. By definition the series ∑ n =0 ∞ a n converges to a limit L if and this definition is usually written as L = ∑ n =0 ∞ a n ⇔ L = lim k → ∞ s k. When the index set is the natural numbers I = N, a series indexed on the natural numbers is an ordered formal sum and so we rewrite ∑ n ∈ N as ∑ n =0 ∞ in order to emphasize the ordering induced by the natural numbers. Thus, we obtain the common notation for a series indexed by the natural numbers ∑ n =0 ∞ a n = a 0 + a 1 + a 2 + ⋯. When the semigroup G is also a space, then the series ∑ n =0 ∞ a n converges to an element L ∈ G if. This definition is usually written as L = ∑ n =0 ∞ a n ⇔ L = lim k → ∞ s k, a series ∑an is said to converge or to be convergent when the sequence SN of partial sums has a finite limit
26.
Convergent series
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In mathematics, a series is the sum of the terms of an infinite sequence of numbers. Given an infinite sequence, the nth partial sum S n is the sum of the first n terms of the sequence, that is, S n = ∑ k =1 n a k. A series is convergent if the sequence of its partial sums tends to a limit, that means that the partial sums become closer and closer to a given number when the number of their terms increases. More precisely, a series converges, if exists a number ℓ such that for any arbitrarily small positive number ε. If the series is convergent, the number ℓ is called the sum of the series, any series that is not convergent is said to be divergent. The reciprocals of the positive integers produce a divergent series,11 +12 +13 +14 +15 +16 + ⋯ → ∞. The reciprocals of triangular numbers produce a convergent series,11 +13 +16 +110 +115 +121 + ⋯ =2. The reciprocals of factorials produce a convergent series,11 +11 +12 +16 +124 +1120 + ⋯ = e. The reciprocals of square numbers produce a convergent series,11 +14 +19 +116 +125 +136 + ⋯ = π26. The reciprocals of powers of 2 produce a convergent series,11 +12 +14 +18 +116 +132 + ⋯ =2. Alternating the signs of reciprocals of powers of 2 also produces a convergent series,11 −12 +14 −18 +116 −132 + ⋯ =23. The reciprocals of Fibonacci numbers produce a convergent series,11 +11 +12 +13 +15 +18 + ⋯ = ψ, there are a number of methods of determining whether a series converges or diverges. The terms of the sequence are compared to those of another sequence, if, for all n,0 ≤ a n ≤ b n, and ∑ n =1 ∞ b n converges, then so does ∑ n =1 ∞ a n. However, if, for all n,0 ≤ b n ≤ a n, assume that for all n, a n >0. Suppose that there exists r such that lim n → ∞ | a n +1 a n | = r, if r <1, then the series converges. If r >1, then the series diverges, if r =1, the ratio test is inconclusive, and the series may converge or diverge. Root test or nth root test, suppose that the terms of the sequence in question are non-negative. Define r as follows, r = lim sup n → ∞ | a n | n, if r <1, then the series converges
27.
Geometric series
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In mathematics, a geometric series is a series with a constant ratio between successive terms. For example, the series 12 +14 +18 +116 + ⋯ is geometric, Geometric series are among the simplest examples of infinite series with finite sums, although not all of them have this property. Historically, geometric series played an important role in the development of calculus. Geometric series are used throughout mathematics, and they have important applications in physics, engineering, biology, economics, computer science, queueing theory, the terms of a geometric series form a geometric progression, meaning that the ratio of successive terms in the series is constant. This relationship allows for the representation of a series using only two terms, r and a. The term r is the ratio, and a is the first term of the series. In the case above, where r is one half, the series has the sum one, if r is greater than one or less than minus one the terms of the series become larger and larger in magnitude. The sum of the terms also gets larger and larger, if r is equal to one, all of the terms of the series are the same. If r is one the terms take two values alternately. The sum of the oscillates between two values. This is a different type of divergence and again the series has no sum, see for example Grandis series,1 −1 +1 −1 + ···. The sum can be computed using the self-similarity of the series, consider the sum of the following geometric series, s =1 +23 +49 +827 + ⋯. This series has common ratio 2/3, if we multiply through by this common ratio, then the initial 1 becomes a 2/3, the 2/3 becomes a 4/9, and so on,23 s =23 +49 +827 +1681 + ⋯. This new series is the same as the original, except that the first term is missing, subtracting the new series s from the original series s cancels every term in the original but the first, s −23 s =1, so s =3. A similar technique can be used to evaluate any self-similar expression, as n goes to infinity, the absolute value of r must be less than one for the series to converge. When a =1, this can be simplified to 1 + r + r 2 + r 3 + ⋯ =11 − r, the formula also holds for complex r, with the corresponding restriction, the modulus of r is strictly less than one. Since = 1−rn+1 and rn+1 →0 for | r | <1, convergence of geometric series can also be demonstrated by rewriting the series as an equivalent telescoping series. Consider the function, g = r K1 − r
28.
Leonhard Euler
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He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, Euler was one of the most eminent mathematicians of the 18th century, and is held to be one of the greatest in history. He is also considered to be the most prolific mathematician of all time. His collected works fill 60 to 80 quarto volumes, more than anybody in the field and he spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Eulers influence on mathematics, Read Euler, read Euler, Leonhard Euler was born on 15 April 1707, in Basel, Switzerland to Paul III Euler, a pastor of the Reformed Church, and Marguerite née Brucker, a pastors daughter. He had two sisters, Anna Maria and Maria Magdalena, and a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, Paul Euler was a friend of the Bernoulli family, Johann Bernoulli was then regarded as Europes foremost mathematician, and would eventually be the most important influence on young Leonhard. Eulers formal education started in Basel, where he was sent to live with his maternal grandmother. In 1720, aged thirteen, he enrolled at the University of Basel, during that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupils incredible talent for mathematics. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono, at that time, he was unsuccessfully attempting to obtain a position at the University of Basel. In 1727, he first entered the Paris Academy Prize Problem competition, Pierre Bouguer, who became known as the father of naval architecture, won and Euler took second place. Euler later won this annual prize twelve times, around this time Johann Bernoullis two sons, Daniel and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg on 17 May 1727 and he was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration. Euler mastered Russian and settled life in Saint Petersburg. He also took on a job as a medic in the Russian Navy. The Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia, as a result, it was made especially attractive to foreign scholars like Euler
29.
Quaternary numeral system
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Quaternary is the base-4 numeral system. It uses the digits 0,1,2 and 3 to represent any real number. Four is the largest number within the range and one of two numbers that is both a square and a highly composite number, making quaternary a convenient choice for a base at this scale. Despite being twice as large, its economy is equal to that of binary. However, it no better in the localization of prime numbers. See decimal and binary for a discussion of these properties, as with the octal and hexadecimal numeral systems, quaternary has a special relation to the binary numeral system. Each radix 4,8 and 16 is a power of 2, so the conversion to and from binary is implemented by matching each digit with 2,3 or 4 binary digits, for example, in base 4,302104 =11001001002. Although octal and hexadecimal are widely used in computing and computer programming in the discussion and analysis of binary arithmetic and logic, by analogy with byte and nybble, a quaternary digit is sometimes called a crumb. There is a surviving list of Ventureño language number words up to 32 written down by a Spanish priest ca, the Kharosthi numerals have a partial base 4 counting system from 1 to decimal 10. Quaternary numbers are used in the representation of 2D Hilbert curves, here a real number between 0 and 1 is converted into the quaternary system. Every single digit now indicates in which of the respective 4 sub-quadrants the number will be projected, parallels can be drawn between quaternary numerals and the way genetic code is represented by DNA. The four DNA nucleotides in order, abbreviated A, C, G and T, can be taken to represent the quaternary digits in numerical order 0,1,2. With this encoding, the complementary digit pairs 0↔3, and 1↔2 match the complementation of the pairs, A↔T and C↔G. For example, the nucleotide sequence GATTACA can be represented by the quaternary number 2033010, quaternary line codes have been used for transmission, from the invention of the telegraph to the 2B1Q code used in modern ISDN circuits
30.
Sequence
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In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members, the number of elements is called the length of the sequence. Unlike a set, order matters, and exactly the elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the numbers or the set of the first n natural numbers. The position of an element in a sequence is its rank or index and it depends on the context or of a specific convention, if the first element has index 0 or 1. For example, is a sequence of letters with the letter M first, also, the sequence, which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in these examples, or infinite, the empty sequence is included in most notions of sequence, but may be excluded depending on the context. A sequence can be thought of as a list of elements with a particular order, Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations, Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers. There are a number of ways to denote a sequence, some of which are useful for specific types of sequences. One way to specify a sequence is to list the elements, for example, the first four odd numbers form the sequence. This notation can be used for sequences as well. For instance, the sequence of positive odd integers can be written. Listing is most useful for sequences with a pattern that can be easily discerned from the first few elements. Other ways to denote a sequence are discussed after the examples, the prime numbers are the natural numbers bigger than 1, that have no divisors but 1 and themselves. Taking these in their natural order gives the sequence, the prime numbers are widely used in mathematics and specifically in number theory. The Fibonacci numbers are the integer sequence whose elements are the sum of the two elements. The first two elements are either 0 and 1 or 1 and 1 so that the sequence is, for a large list of examples of integer sequences, see On-Line Encyclopedia of Integer Sequences
31.
Limit of a sequence
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In mathematics, the limit of a sequence is the value that the terms of a sequence tend to. If such a limit exists, the sequence is called convergent, a sequence which does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of analysis ultimately rests, limits can be defined in any metric or topological space, but are usually first encountered in the real numbers. The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involve limiting processes, leucippus, Democritus, Antiphon, Eudoxus and Archimedes developed the method of exhaustion, which uses an infinite sequence of approximations to determine an area or a volume. Archimedes succeeded in summing what is now called a geometric series, Newton dealt with series in his works on Analysis with infinite series, Method of fluxions and infinite series and Tractatus de Quadratura Curvarum. In the latter work, Newton considers the binomial expansion of n which he then linearizes by taking limits, at the end of the century, Lagrange in his Théorie des fonctions analytiques opined that the lack of rigour precluded further development in calculus. Gauss in his etude of hypergeometric series for the first time rigorously investigated under which conditions a series converged to a limit, the modern definition of a limit was given by Bernhard Bolzano and by Karl Weierstrass in the 1870s. In the real numbers, a number L is the limit of the if the numbers in the sequence become closer and closer to L. If x n = c for some constant c, then x n → c, if x n =1 n, then x n →0. If x n =1 / n when n is even, given any real number, one may easily construct a sequence that converges to that number by taking decimal approximations. For example, the sequence 0.3,0.33,0.333,0.3333, note that the decimal representation 0.3333. is the limit of the previous sequence, defined by 0.3333. ≜ lim n → ∞ ∑ i =1 n 310 i, finding the limit of a sequence is not always obvious. Two examples are lim n → ∞ n and the Arithmetic–geometric mean, the squeeze theorem is often useful in such cases. In other words, for measure of closeness ϵ, the sequences terms are eventually that close to the limit. The sequence is said to converge to or tend to the limit x, symbolically, this is, ∀ ϵ >0 ∃ N ∈ R ∀ n ∈ N. If a sequence converges to some limit, then it is convergent, limits of sequences behave well with respect to the usual arithmetic operations. For any continuous function f, if x n → x then f → f, in fact, any real-valued function f is continuous if and only if it preserves the limits of sequences. Some other important properties of limits of sequences include the following
32.
Nested intervals
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In other words, the intervals diminish, with the left-hand end moving only towards the right, and the right-hand end only to the left. The main question to be posed is the nature of the intersection of all the In. Without any further information, all that can be said is that the intersection J of all the In, i. e. the set of all common to the intervals, is either the empty set. The possibility of an empty intersection can be illustrated by the intersection when In is the open interval, here the intersection is empty, because no number x is both greater than 0 and less than every fraction 2−n. The situation is different for closed intervals, the nested intervals theorem states that if each In is a closed and bounded interval, say In = with an ≤ bn then under the assumption of nesting, the intersection of the In is not empty. It may be a set, or another closed interval. More explicitly, the requirement of nesting means that an ≤ an +1, moreover, if the length of the intervals converges to 0, then the intersection of the In is a singleton. One can consider the complement of each interval, written as ∪, by De Morgans laws, the complement of the intersection is a union of two disjoint open sets. By the connectedness of the line there must be something between them. This shows that the intersection of nested, closed, and bounded intervals is nonempty, in two dimensions there is a similar result, nested closed disks in the plane must have a common intersection. This result was shown by Hermann Weyl to classify the behaviour of certain differential equations. Bisection Cantors Intersection Theorem Fridy, J. A.3.3 The Nested Intervals Theorem, Introductory Analysis, The Theory of Calculus, Academic Press, p.29, ISBN9780122676550. Shilov, Georgi E.1.8 The Principle of Nested Intervals, Elementary Real and Complex Analysis, Dover Books on Mathematics, Courier Dover Publications, pp. 21–22, ISBN9780486135007. Theorem 2.1.5, Basic Real Analysis, Springer, p.45, ISBN9780817642112
33.
Nested intervals theorem
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In other words, the intervals diminish, with the left-hand end moving only towards the right, and the right-hand end only to the left. The main question to be posed is the nature of the intersection of all the In. Without any further information, all that can be said is that the intersection J of all the In, i. e. the set of all common to the intervals, is either the empty set. The possibility of an empty intersection can be illustrated by the intersection when In is the open interval, here the intersection is empty, because no number x is both greater than 0 and less than every fraction 2−n. The situation is different for closed intervals, the nested intervals theorem states that if each In is a closed and bounded interval, say In = with an ≤ bn then under the assumption of nesting, the intersection of the In is not empty. It may be a set, or another closed interval. More explicitly, the requirement of nesting means that an ≤ an +1, moreover, if the length of the intervals converges to 0, then the intersection of the In is a singleton. One can consider the complement of each interval, written as ∪, by De Morgans laws, the complement of the intersection is a union of two disjoint open sets. By the connectedness of the line there must be something between them. This shows that the intersection of nested, closed, and bounded intervals is nonempty, in two dimensions there is a similar result, nested closed disks in the plane must have a common intersection. This result was shown by Hermann Weyl to classify the behaviour of certain differential equations. Bisection Cantors Intersection Theorem Fridy, J. A.3.3 The Nested Intervals Theorem, Introductory Analysis, The Theory of Calculus, Academic Press, p.29, ISBN9780122676550. Shilov, Georgi E.1.8 The Principle of Nested Intervals, Elementary Real and Complex Analysis, Dover Books on Mathematics, Courier Dover Publications, pp. 21–22, ISBN9780486135007. Theorem 2.1.5, Basic Real Analysis, Springer, p.45, ISBN9780817642112
34.
Intersection (set theory)
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In mathematics, the intersection A ∩ B of two sets A and B is the set that contains all elements of A that also belong to B, but no other elements. For explanation of the used in this article, refer to the table of mathematical symbols. The intersection of A and B is written A ∩ B, formally, A ∩ B = that is x ∈ A ∩ B if and only if x ∈ A and x ∈ B. For example, The intersection of the sets and is, the number 9 is not in the intersection of the set of prime numbers and the set of odd numbers. More generally, one can take the intersection of sets at once. The intersection of A, B, C, and D, Intersection is an associative operation, thus, A ∩ = ∩ C. Additionally, intersection is commutative, thus A ∩ B = B ∩ A, inside a universe U one may define the complement Ac of A to be the set of all elements of U not in A. We say that A intersects B if A intersects B at some element, a intersects B if their intersection is inhabited. We say that A and B are disjoint if A does not intersect B, in plain language, they have no elements in common. A and B are disjoint if their intersection is empty, denoted A ∩ B = ∅, for example, the sets and are disjoint, the set of even numbers intersects the set of multiples of 3 at 0,6,12,18 and other numbers. The most general notion is the intersection of a nonempty collection of sets. If M is a nonempty set whose elements are themselves sets, then x is an element of the intersection of M if, the notation for this last concept can vary considerably. Set theorists will sometimes write ⋂M, while others will instead write ⋂A∈M A, the latter notation can be generalized to ⋂i∈I Ai, which refers to the intersection of the collection. Here I is a nonempty set, and Ai is a set for every i in I. In the case that the index set I is the set of numbers, notation analogous to that of an infinite series may be seen. When formatting is difficult, this can also be written A1 ∩ A2 ∩ A3 ∩, even though strictly speaking, A1 ∩ (A2 ∩ (A3 ∩. Finally, let us note that whenever the symbol ∩ is placed before other symbols instead of them, it should be of a larger size. Note that in the section we excluded the case where M was the empty set
35.
Suprema
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In mathematics, the infimum of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists. Consequently, the term greatest lower bound is also commonly used, the supremum of a subset S of a partially ordered set T is the least element in T that is greater than or equal to all elements of S, if such an element exists. Consequently, the supremum is also referred to as the least upper bound, the infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are special cases that are important in analysis. However, the general definitions remain valid in the abstract setting of order theory where arbitrary partially ordered sets are considered. The concepts of infimum and supremum are similar to minimum and maximum, for instance, the positive real numbers ℝ+* does not have a minimum, because any given element of ℝ+* could simply be divided in half resulting in a smaller number that is still in ℝ+*. There is, however, exactly one infimum of the real numbers,0. A lower bound of a subset S of an ordered set is an element a of P such that a ≤ x for all x in S. A lower bound a of S is called an infimum of S if for all lower bounds y of S in P, y ≤ a. Similarly, a bound of a subset S of a partially ordered set is an element b of P such that b ≥ x for all x in S. An upper bound b of S is called a supremum of S if for all upper bounds z of S in P, z ≥ b, infima and suprema do not necessarily exist. Existence of an infimum of a subset S of P can fail if S has no lower bound at all, however, if an infimum or supremum does exist, it is unique. Consequently, partially ordered sets for which certain infima are known to exist become especially interesting, more information on the various classes of partially ordered sets that arise from such considerations are found in the article on completeness properties. If the supremum of a subset S exists, it is unique, if S contains a greatest element, then that element is the supremum, otherwise, the supremum does not belong to S. Likewise, if the infimum exists, it is unique. If S contains a least element, then that element is the infimum, otherwise, the infimum of a subset S of a partially ordered set P, assuming it exists, does not necessarily belong to S. If it does, it is a minimal or least element of S. Similarly, if the supremum of S belongs to S, for example, consider the set of negative real numbers. This set has no greatest element, since for every element of the set, there is another, larger, for instance, for any negative real number x, there is another negative real number x 2, which is greater. On the other hand, every real number greater than or equal to zero is certainly an upper bound on this set, hence,0 is the least upper bound of the negative reals, so the supremum is 0
36.
Construction of the real numbers
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In mathematics, there are several ways of defining the real number system as an ordered field. The synthetic approach gives a list of axioms for the numbers as a complete ordered field. Under the usual axioms of set theory, one can show that these axioms are categorical, in the sense there is a model for the axioms. Any one of these models must be constructed, and most of these models are built using the basic properties of the rational number system as an ordered field. The synthetic approach axiomatically defines the number system as a complete ordered field. A model for the number system consists of a set R, two distinct elements 0 and 1 of R, two binary operations + and × on R, and a binary relation ≤ on R. In other words, For all x, y, and z in R, x + = + z and x × = × z, for all x and y in R, x + y = y + x and x × y = y × x. For all x, y, and z in R, x × = +, for all x in R, x +0 = x.0 is not equal to 1, and for all x in R, x ×1 = x. For every x in R, there exists an element −x in R, for every x ≠0 in R, there exists an element x−1 in R, such that x × x−1 =1. In other words, For all x in R, x ≤ x, for all x and y in R, if x ≤ y and y ≤ x, then x = y. For all x, y, and z in R, if x ≤ y and y ≤ z, for all x and y in R, x ≤ y or y ≤ x. The field operations + and × on R are compatible with the order ≤, in other words, For all x, y and z in R, if x ≤ y, then x + z ≤ y + z. In other words, If A is a non-empty subset of R, the rational numbers Q satisfy the first three axioms but Q does not satisfy axiom 4. So axiom 4, which requires the order to be Dedekind-complete, is crucial, axiom 4 implies the Archimedean property. Several models for axioms 1-4 are given below, any two models for axioms 1-4 are isomorphic, and so up to isomorphism, there is only one complete ordered Archimedean field. When we say that any two models of the axioms are isomorphic, we mean that for any two models and, there is a bijection f, R → S preserving both the field operations and the order. Explicitly, f is injective and surjective. For all x and y in R, f = f +S f and f = f ×S f, for all x and y in R, x ≤R y if and only if f ≤S f
37.
Axiomatic set theory
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Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics, the language of set theory can be used in the definitions of nearly all mathematical objects. The modern study of set theory was initiated by Georg Cantor, Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals. Mathematical topics typically emerge and evolve through interactions among many researchers, Set theory, however, was founded by a single paper in 1874 by Georg Cantor, On a Property of the Collection of All Real Algebraic Numbers. Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, especially notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1867–71, with Cantors work on number theory, an 1872 meeting between Cantor and Richard Dedekind influenced Cantors thinking and culminated in Cantors 1874 paper. Cantors work initially polarized the mathematicians of his day, while Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. This utility of set theory led to the article Mengenlehre contributed in 1898 by Arthur Schoenflies to Kleins encyclopedia, in 1899 Cantor had himself posed the question What is the cardinal number of the set of all sets. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics, in 1906 English readers gained the book Theory of Sets of Points by William Henry Young and his wife Grace Chisholm Young, published by Cambridge University Press. The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment, the work of Zermelo in 1908 and Abraham Fraenkel in 1922 resulted in the set of axioms ZFC, which became the most commonly used set of axioms for set theory. The work of such as Henri Lebesgue demonstrated the great mathematical utility of set theory. Set theory is used as a foundational system, although in some areas category theory is thought to be a preferred foundation. Set theory begins with a binary relation between an object o and a set A. If o is a member of A, the notation o ∈ A is used, since sets are objects, the membership relation can relate sets as well. A derived binary relation between two sets is the relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, for example, is a subset of, and so is but is not. As insinuated from this definition, a set is a subset of itself, for cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined
38.
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
39.
Order theory
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Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations. It provides a framework for describing statements such as this is less than that or this precedes that. This article introduces the field and provides basic definitions, a list of order-theoretic terms can be found in the order theory glossary. Orders are everywhere in mathematics and related fields like computer science. The first order often discussed in primary school is the order on the natural numbers e. g.2 is less than 3,10 is greater than 5. This intuitive concept can be extended to orders on sets of numbers, such as the integers. The idea of being greater than or less than another number is one of the basic intuitions of number systems in general, other familiar examples of orderings are the alphabetical order of words in a dictionary and the genealogical property of lineal descent within a group of people. The notion of order is very general, extending beyond contexts that have an immediate, in other contexts orders may capture notions of containment or specialization. Abstractly, this type of order amounts to the relation, e. g. Pediatricians are physicians. However, many other orders do not and those orders like the subset-of relation for which there exist incomparable elements are called partial orders, orders for which every pair of elements is comparable are total orders. Order theory captures the intuition of orders that arises from such examples in a general setting and this is achieved by specifying properties that a relation ≤ must have to be a mathematical order. This more abstract approach makes sense, because one can derive numerous theorems in the general setting. These insights can then be transferred to many less abstract applications. Driven by the wide usage of orders, numerous special kinds of ordered sets have been defined. In addition, order theory does not restrict itself to the classes of ordering relations. A simple example of an order theoretic property for functions comes from analysis where monotone functions are frequently found and this section introduces ordered sets by building upon the concepts of set theory, arithmetic, and binary relations. Suppose that P is a set and that ≤ is a relation on P, a set with a partial order on it is called a partially ordered set, poset, or just an ordered set if the intended meaning is clear. By checking these properties, one sees that the well-known orders on natural numbers, integers, rational numbers
40.
Dedekind cut
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Dedekind cuts are one method of construction of the real numbers. The set B may or may not have a smallest element among the rationals, if B has a smallest element among the rationals, the cut corresponds to that rational. Otherwise, that cut defines a unique number which, loosely speaking. In other words, A contains every rational number less than the cut, an irrational cut is equated to an irrational number which is in neither set. Every real number, rational or not, is equated to one, whenever, then, we have to do with a cut produced by no rational number, we create a new irrational number, which we regard as completely defined by this cut. From now on, therefore, to every definite cut there corresponds a definite rational or irrational number. More generally, a Dedekind cut is a partition of an ordered set into two non-empty parts A and B, such that A is closed downwards and B is closed upwards. It is straightforward to show that a Dedekind cut among the numbers is uniquely defined by the corresponding cut among the rational numbers. Similarly, every cut of reals is identical to the cut produced by a real number. In other words, the line where every real number is defined as a Dedekind cut of rationals is a complete continuum without any further gaps. Dedekind used the German word Schnitt in a visual sense rooted in Euclidean geometry and his theorem asserting the completeness of the real number system is nevertheless a theorem about numbers and not geometry. In David Hilberts axiom system, continuity is provided by the Axiom of Archimedes, in mathematical logic, the identification of the real numbers with the real number line is provided by the Cantor–Dedekind axiom. It is more symmetrical to use the notation for Dedekind cuts and it can be a simplification, in terms of notation if nothing more, to concentrate on one half — say, the lower one — and call any downward closed set A without greatest element a Dedekind cut. If the ordered set S is complete, then, for every Dedekind cut of S, the set B must have an element b, hence we must have that A is the interval. In this case, we say that b is represented by the cut, the important purpose of the Dedekind cut is to work with number sets that are not complete. The cut itself can represent a number not in the collection of numbers. The cut can represent a number b, even though the numbers contained in the two sets A and B do not actually include the number b that their cut represents. Even though there is no value for √2, if the rational numbers are partitioned into A and B this way
41.
Cauchy sequence
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In mathematics, a Cauchy sequence, named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a number of elements of the sequence are less than that given distance from each other. It is not sufficient for each term to become close to the preceding term. For instance, in the harmonic series ∑1 n a difference between consecutive terms decreases as 1 n, however the series does not converge, rather, it is required that all terms get arbitrarily close to each other, starting from some point. More formally, for any given ε >0 there exists an N such that for any m, n > N. The notions above are not as unfamiliar as they might at first appear, the customary acceptance of the fact that any real number x has a decimal expansion is an implicit acknowledgment that a particular Cauchy sequence of rational numbers has the real limit x. In some cases it may be difficult to describe x independently of such a process involving rational numbers. Generalizations of Cauchy sequences in more abstract uniform spaces exist in the form of Cauchy filters, in a similar way one can define Cauchy sequences of rational or complex numbers. Cauchy formulated such a condition by requiring x m − x n to be infinitesimal for every pair of infinite m, n, to define Cauchy sequences in any metric space X, the absolute value |xm - xn| is replaced by the distance d between xm and xn. A metric space X in which every Cauchy sequence converges to an element of X is called complete, the real numbers are complete under the metric induced by the usual absolute value, and one of the standard constructions of the real numbers involves Cauchy sequences of rational numbers. A rather different type of example is afforded by a metric space X which has the discrete metric, any Cauchy sequence of elements of X must be constant beyond some fixed point, and converges to the eventually repeating term. The rational numbers Q are not complete, There are sequences of rationals that converge to irrational numbers, if one considers this as a sequence of real numbers, however, it converges to the real number φ = /2, the Golden ratio, which is irrational. Every Cauchy sequence of numbers is bounded. Every Cauchy sequence of numbers is bounded, hence by Bolzano-Weierstrass has a convergent subsequence, hence is itself convergent. It should be noted, though, that proof of the completeness of the real numbers implicitly makes use of the least upper bound axiom. The alternative approach, mentioned above, of constructing the real numbers as the completion of the rational numbers, makes the completeness of the real numbers tautological. Such a series ∑ n =1 ∞ x n is considered to be convergent if and only if the sequence of sums is convergent. It is a matter to determine whether the sequence of partial sums is Cauchy or not
42.
Richard Dedekind
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Julius Wilhelm Richard Dedekind was a German mathematician who made important contributions to abstract algebra, algebraic number theory and the definition of the real numbers. Dedekinds father was Julius Levin Ulrich Dedekind, an administrator of Collegium Carolinum in Braunschweig, as an adult, he never used the names Julius Wilhelm. He was born, lived most of his life, and died in Braunschweig and he first attended the Collegium Carolinum in 1848 before transferring to the University of Göttingen in 1850. There, Dedekind was taught number theory by professor Moritz Stern, Gauss was still teaching, although mostly at an elementary level, and Dedekind became his last student. Dedekind received his doctorate in 1852, for a thesis titled Über die Theorie der Eulerschen Integrale and this thesis did not display the talent evident by Dedekinds subsequent publications. At that time, the University of Berlin, not Göttingen, was the facility for mathematical research in Germany. Thus Dedekind went to Berlin for two years of study, where he and Bernhard Riemann were contemporaries, they were awarded the habilitation in 1854. Dedekind returned to Göttingen to teach as a Privatdozent, giving courses on probability and he studied for a while with Peter Gustav Lejeune Dirichlet, and they became good friends. Because of lingering weaknesses in his knowledge, he studied elliptic. Yet he was also the first at Göttingen to lecture concerning Galois theory, about this time, he became one of the first people to understand the importance of the notion of groups for algebra and arithmetic. In 1858, he began teaching at the Polytechnic school in Zürich, when the Collegium Carolinum was upgraded to a Technische Hochschule in 1862, Dedekind returned to his native Braunschweig, where he spent the rest of his life, teaching at the Institute. He retired in 1894, but did occasional teaching and continued to publish and he never married, instead living with his sister Julia. Dedekind was elected to the Academies of Berlin and Rome, and he received honorary doctorates from the universities of Oslo, Zurich, and Braunschweig. While teaching calculus for the first time at the Polytechnic school, Dedekind developed the now known as a Dedekind cut. The idea of a cut is that an irrational number divides the rational numbers into two classes, with all the numbers of one class being strictly greater than all the numbers of the other class. Every location on the number line continuum contains either a rational or an irrational number, thus there are no empty locations, gaps, or discontinuities. Dedekind published his thoughts on irrational numbers and Dedekind cuts in his pamphlet Stetigkeit und irrationale Zahlen, in modern terminology, Vollständigkeit, Dedekinds theorem states that if there existed a one-to-one correspondence between two sets, then Dedekind said that the two sets were similar. Thus the set N of natural numbers can be shown to be similar to the subset of N whose members are the squares of every member of N, N12345678910
43.
Decimal fraction
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This article aims to be an accessible introduction. For the mathematical definition, see Decimal representation, the decimal numeral system has ten as its base, which, in decimal, is written 10, as is the base in every positional numeral system. It is the base most widely used by modern civilizations. Decimal fractions have terminating decimal representations and other fractions have repeating decimal representations, Decimal notation is the writing of numbers in a base-ten numeral system. Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, Roman numerals have symbols for the decimal powers and secondary symbols for half these values. Brahmi numerals have symbols for the nine numbers 1–9, the nine decades 10–90, plus a symbol for 100, Chinese numerals have symbols for 1–9, and additional symbols for powers of ten, which in modern usage reach 1072. Positional decimal systems include a zero and use symbols for the ten values to represent any number, positional notation uses positions for each power of ten, units, tens, hundreds, thousands, etc. The position of each digit within a number denotes the multiplier multiplied with that position has a value ten times that of the position to its right. There were at least two independent sources of positional decimal systems in ancient civilization, the Chinese counting rod system. Ten is the number which is the count of fingers and thumbs on both hands, the English word digit as well as its translation in many languages is also the anatomical term for fingers and toes. In English, decimal means tenth, decimate means reduce by a tenth, however, the symbols used in different areas are not identical, for instance, Western Arabic numerals differ from the forms used by other Arab cultures. A decimal fraction is a fraction the denominator of which is a power of ten. g, Decimal fractions 8/10, 1489/100, 24/100000, and 58900/10000 are expressed in decimal notation as 0.8,14.89,0.00024,5.8900 respectively. In English-speaking, some Latin American and many Asian countries, a period or raised period is used as the separator, in many other countries, particularly in Europe. The integer part, or integral part of a number is the part to the left of the decimal separator. The part from the separator to the right is the fractional part. It is usual for a number that consists only of a fractional part to have a leading zero in its notation. Any rational number with a denominator whose only prime factors are 2 and/or 5 may be expressed as a decimal fraction and has a finite decimal expansion. 1/2 =0.5 1/20 =0.05 1/5 =0.2 1/50 =0.02 1/4 =0.25 1/40 =0.025 1/25 =0.04 1/8 =0.125 1/125 =0.008 1/10 =0
44.
Eduard Heine
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Heinrich Eduard Heine was a German mathematician. Heine became known for results on special functions and in real analysis, in particular, he authored an important treatise on spherical harmonics and Legendre functions. He also investigated basic hypergeometric series, heinrich Eduard Heine was born on 16 March 1821 in Berlin, as the eighth child of banker Karl Heine and his wife Henriette Märtens. Eduard was initially home schooled, then studied at the Friedrichswerdersche Gymnasium, in 1840 Heine returned to Berlin, where he studied mathematics under Peter Gustav Lejeune Dirichlet, while also attending classes of Jakob Steiner and Johann Franz Encke. In 1842 he was an awarded a Ph. D. by the University of Berlin for a thesis on differential equations submitted with Enno Dirksen, Heine dedicated the doctoral thesis to his professor Gustav Dirichlet. Next he went to the University of Königsberg to participate in the seminar of Carl Gustav Jacobi. In Königsberg Heine got in contact with fellow students Gustav Kirchhoff, in 1844 Heine went for a teaching position at the University of Bonn, passing his habilitation and starting as a privatdozent. He continued his research in mathematics in Bonn and, in 1848, was promoted to extraordinary professor, in 1850 he married Sophie Wolff, the daughter of a Berlin merchant, the couple had five children, four daughters and one son. In 1856 Heine moved as a professor to the University of Halle. From 1864 to 1865, he served as a rector of the university, in 1875 the University of Göttingen offered Heine a mathematics chair but he decided to reject the offer and remain in Halle. In 1877, at the centenary of Gauss birth, he was awarded the Gauss medal for his research, Eduard Heine died on 21 October 1881 in Halle. Robertson, Edmund F. Eduard Heine, MacTutor History of Mathematics archive, Eduard Heine at the Mathematics Genealogy Project
45.
Georg Cantor
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Georg Ferdinand Ludwig Philipp Cantor was a German mathematician. He invented set theory, which has become a theory in mathematics. In fact, Cantors method of proof of this theorem implies the existence of an infinity of infinities and he defined the cardinal and ordinal numbers and their arithmetic. Cantors work is of great philosophical interest, a fact of which he was well aware, E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections. Cantor, a devout Lutheran, believed the theory had been communicated to him by God, Kronecker objected to Cantors proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum. The harsh criticism has been matched by later accolades, in 1904, the Royal Society awarded Cantor its Sylvester Medal, the highest honor it can confer for work in mathematics. David Hilbert defended it from its critics by declaring, From his paradise that Cantor with us unfolded, we hold our breath in awe, knowing, we shall not be expelled. Georg Cantor was born in the merchant colony in Saint Petersburg, Russia. Georg, the oldest of six children, was regarded as an outstanding violinist and his grandfather Franz Böhm was a well-known musician and soloist in a Russian imperial orchestra. In 1860, Cantor graduated with distinction from the Realschule in Darmstadt, his skills in mathematics. In 1862, Cantor entered the Swiss Federal Polytechnic and he spent the summer of 1866 at the University of Göttingen, then and later a center for mathematical research. Cantor submitted his dissertation on number theory at the University of Berlin in 1867, after teaching briefly in a Berlin girls school, Cantor took up a position at the University of Halle, where he spent his entire career. He was awarded the habilitation for his thesis, also on number theory. In 1874, Cantor married Vally Guttmann and they had six children, the last born in 1886. Cantor was able to support a family despite modest academic pay, during his honeymoon in the Harz mountains, Cantor spent much time in mathematical discussions with Richard Dedekind, whom he had met two years earlier while on Swiss holiday. Cantor was promoted to Extraordinary Professor in 1872 and made full Professor in 1879, however, his work encountered too much opposition for that to be possible. Worse yet, Kronecker, a figure within the mathematical community and Cantors former professor. Cantor came to believe that Kroneckers stance would make it impossible for him ever to leave Halle, in 1881, Cantors Halle colleague Eduard Heine died, creating a vacant chair
46.
Secondary schools
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A secondary school is both an organization that delivers level 2 junior secondary education or level 3 secondary education phases of the ISCED scale, and the building where this takes place. Level 2 junior secondary education is considered to be the second, Secondary schools typically follow on from primary schools and lead into vocational and tertiary education. Attendance is compulsory in most countries for students between the ages 11 and 16, the systems and terminology remain unique to each country. School building design does not happen in isolation, schools need to accommodate students, staff, storage, mechanical and electrical systems, storage, support staff, ancillary staff and administration. The number of rooms required can be determined from the roll of the school. A general classroom for 30 students needs to be 55m2, or more generously 62m2, a general art room for 30 students needs to be 83m2, but 104 m2 for 3D textile work. A drama studio or a specialist science laboratory for 30 needs to be 90 m2, examples are given on how this can be configured for a 1,200 place secondary. The building providing the education has to fulfil the needs of, The students, the teachers, the support staff, the adminstrators. It has to should meet health requirements, minimal functional requirements- such as classrooms, toilets and showers, electricity, textbooks, Government accountants having read the advice then publish minimum guidelines on schools. These enable environmental modelling and establish building costs. Future plans are audited to ensure that standards are not exceeded. The UK government published this downwardly revised space formula in 2014 and it said the floor area should be 1050m² +6. 3m²/pupil place for 11- to 16-year-olds + 7m²/pupil place for post-16s. The external finishes were to be downgraded to meet a build cost of £1113/m², a secondary school, locally may be called high school, junior high school, senior high school. Sweden, gymnasium Switzerland, gymnasium, secondary school, collège or lycée Taiwan, Junior High School, Senior High School, Vocational High School, Military School, in Nigeria, secondary school starts from JSS1 until SSS3. Most students start at the age of 10 or 11 and finish at 16 or 17, Students are required to sit for the West African Senior Secondary Certificate Examination. To progress to university students must obtain at least a credit in Maths, English, in Somalia, secondary school starts from 9th grade until 12th. Students start it when they are around 14 to 15 years of age, Students are required to study Somali and Arabic, with the option of either English or Italian depending on the type of school. Religion, chemistry, physics, biology, physical education, textile, art, design, when secondary school has been completed, students are sent to national training camp before going to either college, or military training. In South Africa, high school begins at grade 8, Students study for five years, at the end of which they write a Matriculation examination
47.
Equivalence relation
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In mathematics, an equivalence relation is a binary relation that is at the same time a reflexive relation, a symmetric relation and a transitive relation. As a consequence of these properties an equivalence relation provides a partition of a set into equivalence classes, a given binary relation ~ on a set X is said to be an equivalence relation if and only if it is reflexive, symmetric and transitive. That is, for all a, b and c in X, a ~ b if and only if b ~ a. if a ~ b and b ~ c then a ~ c. X together with the relation ~ is called a setoid, the equivalence class of a under ~, denoted, is defined as =. Let the set have the equivalence relation, the following sets are equivalence classes of this relation, =, = =. The set of all classes for this relation is. The following are all equivalence relations, Has the same birthday as on the set of all people, is similar to on the set of all triangles. Is congruent to on the set of all triangles, is congruent to, modulo n on the integers. Has the same image under a function on the elements of the domain of the function, has the same absolute value on the set of real numbers Has the same cosine on the set of all angles. The relation ≥ between real numbers is reflexive and transitive, but not symmetric, for example,7 ≥5 does not imply that 5 ≥7. It is, however, a total order, the relation has a common factor greater than 1 with between natural numbers greater than 1, is reflexive and symmetric, but not transitive. The empty relation R on a non-empty set X is vacuously symmetric and transitive, a partial order is a relation that is reflexive, antisymmetric, and transitive. Equality is both a relation and a partial order. Equality is also the relation on a set that is reflexive. In algebraic expressions, equal variables may be substituted for one another, the equivalence classes of an equivalence relation can substitute for one another, but not individuals within a class. A strict partial order is irreflexive, transitive, and asymmetric, a partial equivalence relation is transitive and symmetric. Transitive and symmetric imply reflexive if and only if for all a ∈ X, a reflexive and symmetric relation is a dependency relation, if finite, and a tolerance relation if infinite. A preorder is reflexive and transitive, a congruence relation is an equivalence relation whose domain X is also the underlying set for an algebraic structure, and which respects the additional structure
48.
Base (exponentiation)
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In exponentiation, the base is the number b in an expression of the form bn. The number n is called the exponent and the expression is known formally as exponentiation of b by n or the exponential of n with base b and it is more commonly expressed as the nth power of b, b to the nth power or b to the power n. For example, the power of 10 is 10,000 because 104 =10 ×10 ×10 ×10 =10,000. The term power strictly refers to the expression, but is sometimes used to refer to the exponent. When the nth power of b equals a number a, or a = bn, for example,10 is a fourth root of 10,000. The inverse function to exponentiation with base b is called the logarithm to base b, for example, log1010,000 =4
49.
Binary numeral system
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The base-2 system is a positional notation with a radix of 2. Because of its implementation in digital electronic circuitry using logic gates. Each digit is referred to as a bit, the modern binary number system was devised by Gottfried Leibniz in 1679 and appears in his article Explication de lArithmétique Binaire. Systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, Leibniz was specifically inspired by the Chinese I Ching. The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions, the method used for ancient Egyptian multiplication is also closely related to binary numbers. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus, the I Ching dates from the 9th century BC in China. The binary notation in the I Ching is used to interpret its quaternary divination technique and it is based on taoistic duality of yin and yang. Eight trigrams and a set of 64 hexagrams, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China. The Song Dynasty scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, the Indian scholar Pingala developed a binary system for describing prosody. He used binary numbers in the form of short and long syllables, Pingalas Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter. The binary representations in Pingalas system increases towards the right, the residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Africa, sets of binary combinations similar to the I Ching have also been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy. The base-2 system utilized in geomancy had long been applied in sub-Saharan Africa. Leibnizs system uses 0 and 1, like the modern binary numeral system, Leibniz was first introduced to the I Ching through his contact with the French Jesuit Joachim Bouvet, who visited China in 1685 as a missionary. Leibniz saw the I Ching hexagrams as an affirmation of the universality of his own beliefs as a Christian. Binary numerals were central to Leibnizs theology and he believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing. Is not easy to impart to the pagans, is the ex nihilo through Gods almighty power. In 1854, British mathematician George Boole published a paper detailing an algebraic system of logic that would become known as Boolean algebra
50.
Ternary numeral system
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The ternary numeral system has three as its base. Analogous to a bit, a digit is a trit. One trit is equivalent to bits of information. Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary, for example, decimal 365 corresponds to binary 101101101 and to ternary 111112. However, they are far less compact than the corresponding representations in bases such as decimal – see below for a compact way to codify ternary using nonary. The value of a number with n bits that are all 1 is 2n −1. Then N = M, N = /, and N = bd −1, for a three-digit ternary number, N =33 −1 =26 =2 ×32 +2 ×31 +2 ×30 =18 +6 +2. Nonary or septemvigesimal can be used for representation of ternary. A base-three system is used in Islam to keep track of counting Tasbih to 99 or to 100 on a hand for counting prayers. In certain analog logic, the state of the circuit is often expressed ternary and this is most commonly seen in Transistor–transistor logic using 7406 open collector logic. The output is said to either be low, high, or open, in this configuration the output of the circuit is actually not connected to any voltage reference at all. Where the signal is usually grounded to a reference, or at a certain voltage level. Thus, the voltage level is sometimes unpredictable. A rare ternary point is used to denote fractional parts of an inning in baseball, since each inning consists of three outs, each out is considered one third of an inning and is denoted as.1. For example, if a player pitched all of the 4th, 5th and 6th innings, plus 2 outs of the 7th inning, his Innings pitched column for that game would be listed as 3.2, meaning 3⅔. In this usage, only the part of the number is written in ternary form. Ternary numbers can be used to convey self-similar structures like the Sierpinski triangle or the Cantor set conveniently, additionally, it turns out that the ternary representation is useful for defining the Cantor set and related point sets, because of the way the Cantor set is constructed. The Cantor set consists of the points from 0 to 1 that have an expression that does not contain any instance of the digit 1
51.
Golden ratio base
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Golden ratio base is a non-integer positional numeral system that uses the golden ratio as its base. It is sometimes referred to as base-φ, golden mean base, phi-base, or, colloquially, any non-negative real number can be represented as a base-φ numeral using only the digits 0 and 1, and avoiding the digit sequence 11 – this is called a standard form. A base-φ numeral that includes the digit sequence 11 can always be rewritten in standard form, despite using an irrational number base, when using standard form, all non-negative integers have a unique representation as a terminating base-φ expansion. Other numbers have standard representations in base-φ, with rational numbers having recurring representations and these representations are unique, except that numbers with a terminating expansion also have a non-terminating expansion, as they do in base-10, for example,1 =0. 99999…. In the following example the notation 1 is used to represent −1. 211. 01φ is not a standard base-φ numeral, since it contains a 11 and a 2, which isnt a 0 or 1, and contains a 1 = −1, which isnt a 0 or 1 either. To standardize a numeral, we can use the following substitutions, 011φ = 100φ, 0200φ = 1001φ, 010φ = 101φ and we can apply the substitutions in any order we like, as the result is the same. Below, the applied to the number on the previous line are on the right. Any positive number with a non-standard terminating base-φ representation can be standardized in this manner. If we get to a point where all digits are 0 or 1, except for the first digit being negative and this can be converted to the negative of a base-φ representation by negating every digit, standardizing the result, and then marking it as negative. For example, use a sign, or some other significance to denote negative numbers. If the arithmetic is being performed on a computer, a message may be returned. We can either consider our integer to be the digit of a nonstandard base-φ numeral, therefore, we can compute + =, − = and × =. So, using integer values only, we can add, subtract and multiply numbers of the form, > if and only if 2 − > × √5. If one side is negative, the positive, the comparison is trivial. Otherwise, square sides, to get an integer comparison, reversing the comparison direction if both sides were negative. On squaring both sides, the √5 is replaced with the integer 5, so, using integer values only, we can also compare numbers of the form. To convert an integer x to a number, note that x =
52.
Almost all
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In mathematics, the phrase almost all has a number of specialised uses which extend its intuitive meaning. Almost all is used synonymously with all but finitely many or all but a countable set. A simple example is that almost all numbers are odd. Perversely, if we allow almost all to all but a countable set, then it follows that almost all prime numbers are even. When speaking about the reals, sometimes it means all reals, in this sense almost all reals are not a member of the Cantor set even though the Cantor set is uncountable. More generally, almost all is used in the sense of almost everywhere in measure theory. Thus, almost all positive integers are composite, however there are still a number of primes. Generic property Sufficiently large Weisstein, Eric W
53.
Non-standard positional numeral systems
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Each numeral represents one of the values 0,1,2, etc. up to b −1, but the value also depends on the position of the digit in a number. The value of a string like pqrs in base b is given by the polynomial form p × b 3 + q × b 2 + r × b + s. The numbers written in superscript represent the powers of the base used, and a minus sign −, all real numbers can be represented. This article summarizes facts on some non-standard positional numeral systems, in most cases, the polynomial form in the description of standard systems still applies. Some historical numeral systems may be described as non-standard positional numeral systems, however, most of the non-standard systems listed below have never been intended for general use, but are deviced by mathematicians or engineers for special academic or technical use. A bijective numeral system with base b uses b different numerals to represent all non-negative integers, however, the numerals have values 1,2,3, etc. up to and including b, whereas zero is represented by an empty digit string. For example, it is possible to have decimal without a zero, unary is the bijective numeral system with base b =1. In unary, one numeral is used to represent all positive integers, the value of the digit string pqrs given by the polynomial form can be simplified into p + q + r + s since bn =1 for all n. Non-standard features of this include, The value of a digit does not depend on its position. Thus, one can argue that unary is not a positional system at all. Introducing a radix point in this system will not enable representation of non-integer values, the single numeral represents the value 1, not the value 0 = b −1. The value 0 cannot be represented, in some systems, while the base is a positive integer, negative digits are allowed. Non-adjacent form is a system where the base is b =2. In the balanced system, the base is b =3. The reflected binary code, also known as the Gray code, is related to binary numbers. A few positional systems have been suggested in which the base b is not a positive integer, negative-base systems include negabinary, negaternary and negadecimal, in base −b the number of different numerals used is b. All integers, positive and negative, can be represented without a sign, in purely imaginary base bi the b2 numbers from 0 to b2 −1 are used as digits. It can be generalized to other bases, Complex-base system
54.
Balanced ternary
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Balanced ternary is a non-standard positional numeral system, useful for comparison logic. While it is a number system, in the standard ternary system. The digits in the balanced ternary system have values −1,0, different sources use different glyphs used to represent the three digits in balanced ternary. In this article, T represents −1, while 0 and 1 represent themselves, other conventions include using − and + to represent −1 and 1 respectively, or using Greek letter theta, which resembles a minus sign in a circle, to represent −1. In Setun printings, −1 is represented as overturned 1,1, the notation has a number of computational advantages over regular binary. Particularly, the plus–minus consistency cuts down the rate in multi-digit multiplication. Balanced ternary also has a number of advantages over traditional ternary. Particularly, the multiplication table has no carries in balanced ternary. A possible use of balanced ternary is to represent if a list of values in a list is less than, equal to or greater than the corresponding value in a second list. Balanced ternary can also represent all integers without using a separate minus sign, in the balanced ternary system the value of a digit n places left of the radix point is the product of the digit and 3n. This is useful when converting between decimal and balanced ternary, in the following the strings denoting balanced ternary carry the suffix, bal3. For instance, −2/3dec = −1 + 1/3 = −1×30 + 1×3−1 = T. 1bal3, an integer is divisible by three if and only if the digit in the units place is zero. We may check the parity of a balanced ternary integer by checking the parity of the sum of all trits and this sum has the same parity as the integer itself. Balanced ternary can also be extended to fractional numbers similar to how decimal numbers are written to the right of the radix point, in decimal or binary, integer values and terminating fractions have multiple representations. For example,110 =0.1 =0.10 =0.09, and,12 =0. 1bin =0. 10bin =0. 01bin. Some balanced ternary fractions have multiple representations too, for example,16 =0. 1Tbal3 =0. 01bal3. Certainly, in the decimal and binary, we may omit the rightmost trailing infinite 0s after the radix point, but, in balanced ternary, we cant omit the rightmost trailing infinite –1s after the radix point in order to gain a representations of integer or terminating fraction. Donald Knuth has pointed out that truncation and rounding are the operation in balanced ternary — they produce exactly the same result
55.
Factorial number system
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In combinatorics, the factorial number system, also called factoradic, is a mixed radix numeral system adapted to numbering permutations. It is also called factorial base, although factorials do not function as base, by converting a number less than n. General mixed radix systems were studied by Georg Cantor, the term factorial number system is used by Knuth, while the French equivalent numération factorielle was first used in 1888. The term factoradic, which is a portmanteau of factorial and mixed radix, appears to be of more recent date. The factorial number system is a mixed radix numeral system, the i-th digit from the right has base i, which means that the digit must be less than i. From this it follows that the rightmost digit is always 0, the second can be 0 or 1, the third 0,1 or 2, the factorial number system is sometimes defined with the 0. Place omitted because it is always zero, in this article, a factorial number representation will be flagged by a subscript. Stands for 354413021100, whose value is = 3×5, general properties of mixed radix number systems also apply to the factorial number system. Reading the remainders backward gives 341010, in principle, this system may be extended to represent fractional numbers, though rather than the natural extension of place values. Etc. which are undefined, the choice of radix values n =0,1,2,3,4. Again, the 0 and 1 places may be omitted as these are always zero, the corresponding place values are therefore 1/1, 1/1, 1/2, 1/6, 1/24. The following sortable table shows the 24 permutations of four elements with different inversion related vectors, the left and right inversion counts l and r are particularly eligible to be interpreted as factorial numbers. L gives the position in reverse colexicographic order, and the latter the position in lexicographic order. Sorting by a column that has the omissible 0 on the right makes the numbers in that column correspond to the index numbers in the immovable column on the left. The small columns are reflections of the next to them. The rightmost column shows the digit sums of the factorial numbers, for another example, the greatest number that could be represented with six digits would be 543210. Which equals 719 in decimal, 5×5, clearly the next factorial number representation after 543210. is 1000000. =72010, the value for the radix-7 digit
56.
Lexicographic ordering
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In mathematics, the lexicographic or lexicographical order is a generalization of the way the alphabetical order of words is based on the alphabetical order of their component letters. This generalization consists primarily in defining a total order over the sequences of elements of a totally ordered set. There are several variants and generalizations of the lexicographical ordering, another generalization defines an order on a Cartesian product of partially ordered sets, this order is a total order if and only if the factors of the Cartesian product are totally ordered. The word lexicographic is derived from lexicon, the set of words that are used in language and appear in dictionaries. The lexicographic order has thus been introduced for sorting the entries of dictionaries and this has been formalized in the following way. Consider a finite set A, often called alphabet, which is totally ordered, in dictionaries, this is the common alphabet, ordered by the alphabetical order. In book indexes, the alphabet is generally extended to all alphanumeric characters, the lexicographic order is a total order on the sequences of elements of A, often called words on A, which is defined as follows. Given two different sequences of the length, a1a2. ak and b1b2. bk, the first one is smaller than the second one for the lexicographical order, if ai<bi, for the first i where ai. To compare sequences of different lengths, the sequence is usually padded at the end with enough blanks. This way of comparing sequences of different lengths is always used in dictionaries, however, in combinatorics, an other convention is frequently used, whereby a shorter sequence is always smaller than a longer sequence. This variant of the order is sometimes called shortlex order. In dictionary order, the word Thomas appears before Thompson because the letter a comes before the letter p in the alphabet, the 5th letter is the first that is different in the two words, the first four letters are Thom in both. Because it is the first difference, the 5th letter is the most significant difference for the alphabetical ordering. An important property of the order on words of a fixed length on a finite alphabet is that it is a well-order. The lexicographical order is used not only in dictionaries, but also commonly for numbers, one of the drawbacks of the Roman numeral system is that it is not always immediately obvious which of two numbers is the smaller. When negative numbers are considered, one has to reverse the order for comparing negative numbers. This is not usually a problem for humans, but it may be for computers and this is one of the reasons for adopting twos complement representation for representing signed integers in computers. Another example of a use of lexicographical ordering appears in the ISO8601 standard for dates
57.
Partition of a set
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In mathematics, a partition of a set is a grouping of the sets elements into non-empty subsets, in such a way that every element is included in one and only one of the subsets. A partition of a set X is a set of nonempty subsets of X such that every element x in X is in one of these subsets. Equivalently, a family of sets P is a partition of X if and only if all of the following conditions hold, the union of the sets in P is equal to X. The sets in P are said to cover X, the intersection of any two distinct sets in P is empty. The elements of P are said to be pairwise disjoint, the sets in P are called the blocks, parts or cells of the partition. The rank of P is |X| − |P|, if X is finite, every singleton set has exactly one partition, namely. The empty set ∅ has exactly one partition, namely ∅, for any nonempty set X, P = is a partition of X, called the trivial partition. For any non-empty proper subset A of a set U, the set A together with its complement form a partition of U, the set has these five partitions, sometimes written 1|2|3. The following are not partitions of, is not a partition because one of its elements is the empty set, is not a partition because the element 2 is contained in more than one block. Is not a partition of because none of its blocks contains 3, however, thus the notions of equivalence relation and partition are essentially equivalent. The axiom of choice guarantees for any partition of a set X the existence of a subset of X containing exactly one element from each part of the partition and this implies that given an equivalence relation on a set one can select a canonical representative element from every equivalence class. Informally, this means that α is a fragmentation of ρ. In that case, it is written that α ≤ ρ and this finer-than relation on the set of partitions of X is a partial order. Each set of elements has a least upper bound and a greatest lower bound, so that it forms a lattice, the partition lattice of a 4-element set has 15 elements and is depicted in the Hasse diagram on the left. These atomic partitions correspond one-for-one with the edges of a complete graph, in this way, the lattice of partitions corresponds to the lattice of flats of the graphic matroid of the complete graph. Another example illustrates the refining of partitions from the perspective of equivalence relations, if D is the set of cards in a standard 52-card deck, the same-color-as relation on D – which can be denoted ~C – has two equivalence classes, the sets and. The 2-part partition corresponding to ~C has a refinement that yields the same-suit-as relation ~S, which has the four equivalence classes, and. In other words, given distinct numbers a, b, c in N, with a < b < c, if a ~ c, it follows that also a ~ b and b ~ c, that is b is also in C
58.
String (computer science)
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In computer programming, a string is traditionally a sequence of characters, either as a literal constant or as some kind of variable. The latter may allow its elements to be mutated and the length changed, a string is generally understood as a data type and is often implemented as an array of bytes that stores a sequence of elements, typically characters, using some character encoding. A string may also more general arrays or other sequence data types and structures. When a string appears literally in source code, it is known as a literal or an anonymous string. In formal languages, which are used in logic and theoretical computer science. Let Σ be a non-empty finite set of symbols, called the alphabet, no assumption is made about the nature of the symbols. A string over Σ is any sequence of symbols from Σ. For example, if Σ =, then 01011 is a string over Σ, the length of a string s is the number of symbols in s and can be any non-negative integer, it is often denoted as |s|. The empty string is the string over Σ of length 0. The set of all strings over Σ of length n is denoted Σn, for example, if Σ =, then Σ2 =. Note that Σ0 = for any alphabet Σ, the set of all strings over Σ of any length is the Kleene closure of Σ and is denoted Σ*. In terms of Σn, Σ ∗ = ⋃ n ∈ N ∪ Σ n For example, if Σ =, although the set Σ* itself is countably infinite, each element of Σ* is a string of finite length. A set of strings over Σ is called a language over Σ. For example, if Σ =, the set of strings with an number of zeros, is a formal language over Σ. Concatenation is an important binary operation on Σ*, for any two strings s and t in Σ*, their concatenation is defined as the sequence of symbols in s followed by the sequence of characters in t, and is denoted st. For example, if Σ =, s = bear, and t = hug, then st = bearhug, String concatenation is an associative, but non-commutative operation. The empty string ε serves as the identity element, for any string s, therefore, the set Σ* and the concatenation operation form a monoid, the free monoid generated by Σ. In addition, the length function defines a monoid homomorphism from Σ* to the non-negative integers, a string s is said to be a substring or factor of t if there exist strings u and v such that t = usv
59.
Supremum
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In mathematics, the infimum of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists. Consequently, the term greatest lower bound is also commonly used, the supremum of a subset S of a partially ordered set T is the least element in T that is greater than or equal to all elements of S, if such an element exists. Consequently, the supremum is also referred to as the least upper bound, the infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are special cases that are important in analysis. However, the general definitions remain valid in the abstract setting of order theory where arbitrary partially ordered sets are considered. The concepts of infimum and supremum are similar to minimum and maximum, for instance, the positive real numbers ℝ+* does not have a minimum, because any given element of ℝ+* could simply be divided in half resulting in a smaller number that is still in ℝ+*. There is, however, exactly one infimum of the real numbers,0. A lower bound of a subset S of an ordered set is an element a of P such that a ≤ x for all x in S. A lower bound a of S is called an infimum of S if for all lower bounds y of S in P, y ≤ a. Similarly, a bound of a subset S of a partially ordered set is an element b of P such that b ≥ x for all x in S. An upper bound b of S is called a supremum of S if for all upper bounds z of S in P, z ≥ b, infima and suprema do not necessarily exist. Existence of an infimum of a subset S of P can fail if S has no lower bound at all, however, if an infimum or supremum does exist, it is unique. Consequently, partially ordered sets for which certain infima are known to exist become especially interesting, more information on the various classes of partially ordered sets that arise from such considerations are found in the article on completeness properties. If the supremum of a subset S exists, it is unique, if S contains a greatest element, then that element is the supremum, otherwise, the supremum does not belong to S. Likewise, if the infimum exists, it is unique. If S contains a least element, then that element is the infimum, otherwise, the infimum of a subset S of a partially ordered set P, assuming it exists, does not necessarily belong to S. If it does, it is a minimal or least element of S. Similarly, if the supremum of S belongs to S, for example, consider the set of negative real numbers. This set has no greatest element, since for every element of the set, there is another, larger, for instance, for any negative real number x, there is another negative real number x 2, which is greater. On the other hand, every real number greater than or equal to zero is certainly an upper bound on this set, hence,0 is the least upper bound of the negative reals, so the supremum is 0
60.
Infimum
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In mathematics, the infimum of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists. Consequently, the term greatest lower bound is also commonly used, the supremum of a subset S of a partially ordered set T is the least element in T that is greater than or equal to all elements of S, if such an element exists. Consequently, the supremum is also referred to as the least upper bound, the infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are special cases that are important in analysis. However, the general definitions remain valid in the abstract setting of order theory where arbitrary partially ordered sets are considered. The concepts of infimum and supremum are similar to minimum and maximum, for instance, the positive real numbers ℝ+* does not have a minimum, because any given element of ℝ+* could simply be divided in half resulting in a smaller number that is still in ℝ+*. There is, however, exactly one infimum of the real numbers,0. A lower bound of a subset S of an ordered set is an element a of P such that a ≤ x for all x in S. A lower bound a of S is called an infimum of S if for all lower bounds y of S in P, y ≤ a. Similarly, a bound of a subset S of a partially ordered set is an element b of P such that b ≥ x for all x in S. An upper bound b of S is called a supremum of S if for all upper bounds z of S in P, z ≥ b, infima and suprema do not necessarily exist. Existence of an infimum of a subset S of P can fail if S has no lower bound at all, however, if an infimum or supremum does exist, it is unique. Consequently, partially ordered sets for which certain infima are known to exist become especially interesting, more information on the various classes of partially ordered sets that arise from such considerations are found in the article on completeness properties. If the supremum of a subset S exists, it is unique, if S contains a greatest element, then that element is the supremum, otherwise, the supremum does not belong to S. Likewise, if the infimum exists, it is unique. If S contains a least element, then that element is the infimum, otherwise, the infimum of a subset S of a partially ordered set P, assuming it exists, does not necessarily belong to S. If it does, it is a minimal or least element of S. Similarly, if the supremum of S belongs to S, for example, consider the set of negative real numbers. This set has no greatest element, since for every element of the set, there is another, larger, for instance, for any negative real number x, there is another negative real number x 2, which is greater. On the other hand, every real number greater than or equal to zero is certainly an upper bound on this set, hence,0 is the least upper bound of the negative reals, so the supremum is 0
61.
Disjoint sets
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In mathematics, two sets are said to be disjoint if they have no element in common. Equivalently, disjoint sets are sets whose intersection is the empty set, for example, and are disjoint sets, while and are not. This definition of disjoint sets can be extended to any family of sets, a family of sets is pairwise disjoint or mutually disjoint if every two different sets in the family are disjoint. For example, the collection of sets is pairwise disjoint, two sets are said to be almost disjoint sets if their intersection is small in some sense. For instance, two sets whose intersection is a finite set may be said to be almost disjoint. In topology, there are notions of separated sets with more strict conditions than disjointness. For instance, two sets may be considered to be separated when they have disjoint closures or disjoint neighborhoods, similarly, in a metric space, positively separated sets are sets separated by a nonzero distance. Disjointness of two sets, or of a family of sets, may be expressed in terms of their intersections, two sets A and B are disjoint if and only if their intersection A ∩ B is the empty set. It follows from definition that every set is disjoint from the empty set. A family F of sets is pairwise disjoint if, for two sets in the family, their intersection is empty. If the family more than one set, this implies that the intersection of the whole family is also empty. However, a family of one set is pairwise disjoint, regardless of whether that set is empty. Additionally, a family of sets may have an empty intersection without being pairwise disjoint, for instance, the three sets have an empty intersection but are not pairwise disjoint. In fact, there are no two disjoint sets in this collection, also the empty family of sets is pairwise disjoint. A Helly family is a system of sets within which the only subfamilies with empty intersections are the ones that are pairwise disjoint. For instance, the intervals of the real numbers form a Helly family, if a family of closed intervals has an empty intersection and is minimal. A partition of a set X is any collection of mutually disjoint non-empty sets whose union is X, every partition can equivalently be described by an equivalence relation, a binary relation that describes whether two elements belong to the same set in the partition. A disjoint union may mean one of two things, most simply, it may mean the union of sets that are disjoint
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Mixed radix
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Mixed radix numeral systems are non-standard positional numeral systems in which the numerical base varies from position to position. Such numerical representation applies when a quantity is expressed using a sequence of units that are each a multiple of the smaller one. 32,5,7,45,15,500. ∞,7,24,60,60,1000 or as 32∞577244560.15605001000 In the tabular format, the digits are written above their base, and a semicolon indicates the radix point. In numeral format, each digit has its base attached as a subscript. The base for each digit is the number of corresponding units that make up the larger unit. As a consequence there is no base for the first digit, the most familiar example of mixed radix systems is in timekeeping and calendars. Western time radices include decimal centuries, decades and years as well as duodecimal months, trigesimal days, overlapped with base 52 weeks, one variant uses tridecimal months, quaternary weeks, and septenary days. Time is further divided by quadrivigesimal hours, sexagesimal minutes and seconds, a mixed radix numeral system can often benefit from a tabular summary. m. On Wednesday, and 070201202602460 would be 12,02,24 a. m. on Sunday, ad hoc notations for mixed radix numeral systems are commonplace. The Maya calendar consists of several overlapping cycles of different radices, a short count tzolkin overlaps vigesimal named days with tridecimal numbered days. A haab consists of vigesimal days, octodecimal months, and base-52 years forming a round, in addition, a long count of vigesimal days, octodecimal winal, then vigesimal tun, katun, baktun, etc. tracks historical dates. So, for example, in the UK, banknotes are printed for £50, £20, £10 and £5, mixed-radix numbers of the same base can be manipulated using a generalization of manual arithmetic algorithms. APL and J include operators to convert to and from mixed-radix systems, another proposal is the so-called factorial number system, For example, the biggest number that could be represented with six digits would be 543210 which equals 719 in decimal, 5×5. It might not be clear at first sight but the factorial based numbering system is unambiguous and complete. Every number can be represented in one and only one way because the sum of respective factorials multiplied by the index is always the next factorial minus one, −1 There is a natural mapping between the integers 0. N. −1 and permutations of n elements in lexicographic order, the above equation is a particular case of the following general rule for any radix base representation which expresses the fact that any radix base representation is unambiguous and complete. The Art of Computer Programming, Volume 2, Seminumerical Algorithms, Über einfache Zahlensysteme, Zeitschrift für Math. Mixed Radix Calculator — Mixed Radix Calculator in C#
63.
Monotonic
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In mathematics, a monotonic function is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was generalized to the more abstract setting of order theory. In calculus, a function f defined on a subset of the numbers with real values is called monotonic if. That is, as per Fig.1, a function that increases monotonically does not exclusively have to increase, a function is called monotonically increasing, if for all x and y such that x ≤ y one has f ≤ f, so f preserves the order. Likewise, a function is called monotonically decreasing if, whenever x ≤ y, then f ≥ f, if the order ≤ in the definition of monotonicity is replaced by the strict order <, then one obtains a stronger requirement. A function with this property is called strictly increasing, again, by inverting the order symbol, one finds a corresponding concept called strictly decreasing. The terms non-decreasing and non-increasing should not be confused with the negative qualifications not decreasing, for example, the function of figure 3 first falls, then rises, then falls again. It is therefore not decreasing and not increasing, but it is neither non-decreasing nor non-increasing, the term monotonic transformation can also possibly cause some confusion because it refers to a transformation by a strictly increasing function. Notably, this is the case in economics with respect to the properties of a utility function being preserved across a monotonic transform. A function f is said to be absolutely monotonic over an interval if the derivatives of all orders of f are nonnegative or all nonpositive at all points on the interval, F can only have jump discontinuities, f can only have countably many discontinuities in its domain. The discontinuities, however, do not necessarily consist of isolated points and these properties are the reason why monotonic functions are useful in technical work in analysis. In addition, this result cannot be improved to countable, see Cantor function, if f is a monotonic function defined on an interval, then f is Riemann integrable. An important application of functions is in probability theory. If X is a variable, its cumulative distribution function F X = Prob is a monotonically increasing function. A function is unimodal if it is monotonically increasing up to some point, when f is a strictly monotonic function, then f is injective on its domain, and if T is the range of f, then there is an inverse function on T for f. A map f, X → Y is said to be if each of its fibers is connected i. e. for each element y in Y the set f−1 is connected. A subset G of X × X∗ is said to be a set if for every pair. G is said to be monotone if it is maximal among all monotone sets in the sense of set inclusion
64.
Bijection
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In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments and images are related or mapped to each other. A function maps elements from its domain to elements in its codomain, given a function f, X → Y The function is injective if every element of the codomain is mapped to by at most one element of the domain. An injective function is an injection, notationally, ∀ x, x ′ ∈ X, f = f ⇒ x = x ′. Or, equivalently, ∀ x, x ′ ∈ X, x ≠ x ′ ⇒ f ≠ f, the function is surjective if every element of the codomain is mapped to by at least one element of the domain. A surjective function is a surjection, notationally, ∀ y ∈ Y, ∃ x ∈ X such that y = f. The function is bijective if every element of the codomain is mapped to by one element of the domain. A bijective function is a bijection, an injective function need not be surjective, and a surjective function need not be injective. The four possible combinations of injective and surjective features are illustrated in the diagrams to the right, a function is injective if every possible element of the codomain is mapped to by at most one argument. Equivalently, a function is injective if it maps distinct arguments to distinct images, an injective function is an injection. The formal definition is the following, the function f, X → Y is injective iff for all x, x ′ ∈ X, we have f = f ⇒ x = x ′. A function f, X → Y is injective if and only if X is empty or f is left-invertible, here f is the image of f. Since every function is surjective when its codomain is restricted to its image, more precisely, every injection f, X → Y can be factored as a bijection followed by an inclusion as follows. Let fR, X → f be f with codomain restricted to its image, a dual factorisation is given for surjections below. The composition of two injections is again an injection, but if g o f is injective, then it can only be concluded that f is injective, a function is surjective if every possible image is mapped to by at least one argument. In other words, every element in the codomain has non-empty preimage, equivalently, a function is surjective if its image is equal to its codomain. A surjective function is a surjection, the formal definition is the following. The function f, X → Y is surjective iff for all y ∈ Y, there is x ∈ X such that f = y. A function f, X → Y is surjective if and only if it is right-invertible, by collapsing all arguments mapping to a given fixed image, every surjection induces a bijection defined on a quotient of its domain
65.
Point-set topology
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In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, another name for general topology is point-set topology. The fundamental concepts in topology are continuity, compactness, and connectedness, Continuous functions, intuitively. Compact sets are those that can be covered by finitely many sets of small size. Connected sets are sets that cannot be divided into two pieces that are far apart, the words nearby, arbitrarily small, and far apart can all be made precise by using open sets. If we change the definition of open set, we change what continuous functions, compact sets, each choice of definition for open set is called a topology. A set with a topology is called a topological space, metric spaces are an important class of topological spaces where a real, non-negative distance, also called a metric, can be defined on pairs of points in the set. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces, General topology assumed its present form around 1940. It captures, one might say, almost everything in the intuition of continuity, let X be a set and let τ be a family of subsets of X. The notation Xτ may be used to denote a set X endowed with the particular topology τ, the members of τ are called open sets in X. A subset of X is said to be closed if its complement is in τ, a subset of X may be open, closed, both, or neither. The empty set and X itself are both closed and open. A base B for a topological space X with topology T is a collection of sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T, every subset of a topological space can be given the subspace topology in which the open sets are the intersections of the open sets of the larger space with the subset. For any indexed family of spaces, the product can be given the product topology. For example, in products, a basis for the product topology consists of all products of open sets. For infinite products, there is the requirement that in a basic open set, all. In other words, the quotient topology is the finest topology on Y for which f is continuous, a common example of a quotient topology is when an equivalence relation is defined on the topological space X
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Continuous function (topology)
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In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a discontinuous function, a continuous function with a continuous inverse function is called a homeomorphism. Continuity of functions is one of the concepts of topology. The introductory portion of this focuses on the special case where the inputs and outputs of functions are real numbers. In addition, this article discusses the definition for the general case of functions between two metric spaces. In order theory, especially in theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article, as an example, consider the function h, which describes the height of a growing flower at time t. By contrast, if M denotes the amount of money in an account at time t, then the function jumps at each point in time when money is deposited or withdrawn. A form of the definition of continuity was first given by Bernard Bolzano in 1817. Cauchy defined infinitely small quantities in terms of quantities. The formal definition and the distinction between pointwise continuity and uniform continuity were first given by Bolzano in the 1830s but the work wasnt published until the 1930s, all three of those nonequivalent definitions of pointwise continuity are still in use. Eduard Heine provided the first published definition of continuity in 1872. This is not a definition of continuity since the function f =1 x is continuous on its whole domain of R ∖ A function is continuous at a point if it does not have a hole or jump. A “hole” or “jump” in the graph of a function if the value of the function at a point c differs from its limiting value along points that are nearby. Such a point is called a discontinuity, a function is then continuous if it has no holes or jumps, that is, if it is continuous at every point of its domain. Otherwise, a function is discontinuous, at the points where the value of the function differs from its limiting value, there are several ways to make this definition mathematically rigorous. These definitions are equivalent to one another, so the most convenient definition can be used to determine whether a function is continuous or not. In the definitions below, f, I → R. is a function defined on a subset I of the set R of real numbers and this subset I is referred to as the domain of f