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.
Transcendental number
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In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a non-zero polynomial equation with integer coefficients. The best-known transcendental numbers are π and e, though only a few classes of transcendental numbers are known, transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the numbers are countable while the sets of real. All real transcendental numbers are irrational, since all numbers are algebraic. Another irrational number that is not transcendental is the ratio, φ or ϕ. The name transcendental comes from the root trans meaning across and length of numbers, euler was probably the first person to define transcendental numbers in the modern sense. Johann Heinrich Lambert conjectured that e and π were both transcendental numbers in his 1768 paper proving the number π is irrational, and proposed a tentative sketch of a proof of πs transcendence. In other words, the nth digit of this number is 1 only if n is one of the numbers 1. Liouville showed that number is what we now call a Liouville number. Liouville showed that all Liouville numbers are transcendental, the first number to be proven transcendental without having been specifically constructed for the purpose was e, by Charles Hermite in 1873. In 1874, Georg Cantor proved that the numbers are countable. He also gave a new method for constructing transcendental numbers, in 1878, Cantor published a construction that proves there are as many transcendental numbers as there are real numbers. Cantors work established the ubiquity of transcendental numbers, in 1882, Ferdinand von Lindemann published a proof that the number π is transcendental. He first showed that ea is transcendental when a is algebraic, then, since eiπ = −1 is algebraic, iπ and therefore π must be transcendental. This approach was generalized by Karl Weierstrass to the Lindemann–Weierstrass theorem, the transcendence of π allowed the proof of the impossibility of several ancient geometric constructions involving compass and straightedge, including the most famous one, squaring the circle. The affirmative answer was provided in 1934 by the Gelfond–Schneider theorem and this work was extended by Alan Baker in the 1960s in his work on lower bounds for linear forms in any number of logarithms. The set of numbers is uncountably infinite. Since the polynomials with rational coefficients are countable, and since each such polynomial has a number of zeroes
3.
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
4.
Mathematical constant
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A mathematical constant is a special number, usually a real number, that is significantly interesting in some way. Constants arise in areas of mathematics, with constants such as e and π occurring in such diverse contexts as geometry, number theory. The more popular constants have been studied throughout the ages and computed to many decimal places, all mathematical constants are definable numbers and usually are also computable numbers. These are constants which one is likely to encounter during pre-college education in many countries, however, its ubiquity is not limited to pure mathematics. It appears in many formulas in physics, and several physical constants are most naturally defined with π or its reciprocal factored out and it is debatable, however, if such appearances are fundamental in any sense. For example, the textbook nonrelativistic ground state wave function of the atom is ψ =11 /2 e − r / a 0. This formula contains a π, but it is unclear if that is fundamental in a physical sense, furthermore, this formula gives only an approximate description of physical reality, as it omits spin, relativity, and the quantal nature of the electromagnetic field itself. The numeric value of π is approximately 3.1415926535, memorizing increasingly precise digits of π is a world record pursuit. The constant e also has applications to probability theory, where it arises in a way not obviously related to exponential growth, suppose a slot machine with a one in n probability of winning is played n times. Then, for large n the probability that nothing will be won is approximately 1/e, another application of e, discovered in part by Jacob Bernoulli along with French mathematician Pierre Raymond de Montmort, is in the problem of derangements, also known as the hat check problem. Here n guests are invited to a party, and at the door each guest checks his hat with the butler who then places them into labelled boxes, the butler does not know the name of the guests, and so must put them into boxes selected at random. The problem of de Montmort is, what is the probability that none of the hats gets put into the right box, the answer is p n =1 −11. + ⋯ + n 1 n. and as n tends to infinity, the numeric value of e is approximately 2.7182818284. The square root of 2, often known as root 2, radical 2, or Pythagorass constant, and written as √2, is the algebraic number that. It is more called the principal square root of 2. Geometrically the square root of 2 is the length of a diagonal across a square sides of one unit of length. It was probably the first number known to be irrational and its numerical value truncated to 65 decimal places is,1.41421356237309504880168872420969807856967187537694807317667973799. The quick approximation 99/70 for the root of two is frequently used
5.
Decimal
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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.
On-Line Encyclopedia of Integer Sequences
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The On-Line Encyclopedia of Integer Sequences, also cited simply as Sloanes, is an online database of integer sequences. It was created and maintained by Neil Sloane while a researcher at AT&T Labs, Sloane continues to be involved in the OEIS in his role as President of the OEIS Foundation. OEIS records information on integer sequences of interest to professional mathematicians and amateurs, and is widely cited. As of 30 December 2016 it contains nearly 280,000 sequences, the database is searchable by keyword and by subsequence. Neil Sloane started collecting integer sequences as a student in 1965 to support his work in combinatorics. The database was at first stored on punched cards and he published selections from the database in book form twice, A Handbook of Integer Sequences, containing 2,372 sequences in lexicographic order and assigned numbers from 1 to 2372. The Encyclopedia of Integer Sequences with Simon Plouffe, containing 5,488 sequences and these books were well received and, especially after the second publication, mathematicians supplied Sloane with a steady flow of new sequences. The collection became unmanageable in book form, and when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service, as a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998. The database continues to grow at a rate of some 10,000 entries a year, Sloane has personally managed his sequences for almost 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the database. In 2004, Sloane celebrated the addition of the 100, 000th sequence to the database, A100000, in 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki at OEIS. org was created to simplify the collaboration of the OEIS editors and contributors, besides integer sequences, the OEIS also catalogs sequences of fractions, the digits of transcendental numbers, complex numbers and so on by transforming them into integer sequences. Sequences of rationals are represented by two sequences, the sequence of numerators and the sequence of denominators, important irrational numbers such as π =3.1415926535897. are catalogued under representative integer sequences such as decimal expansions, binary expansions, or continued fraction expansions. The OEIS was limited to plain ASCII text until 2011, yet it still uses a form of conventional mathematical notation. Greek letters are represented by their full names, e. g. mu for μ. Every sequence is identified by the letter A followed by six digits, sometimes referred to without the leading zeros, individual terms of sequences are separated by commas. Digit groups are not separated by commas, periods, or spaces, a represents the nth term of the sequence. Zero is often used to represent non-existent sequence elements, for example, A104157 enumerates the smallest prime of n² consecutive primes to form an n×n magic square of least magic constant, or 0 if no such magic square exists. The value of a is 2, a is 1480028129, but there is no such 2×2 magic square, so a is 0
7.
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
8.
Floor and ceiling functions
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In mathematics and computer science, the floor and ceiling functions map a real number to the greatest preceding or the least succeeding integer, respectively. More precisely, floor = ⌊ x ⌋ is the greatest integer less than or equal to x, carl Friedrich Gauss introduced the square bracket notation for the floor function in his third proof of quadratic reciprocity. This remained the standard in mathematics until Kenneth E. Iverson introduced the names floor and ceiling, both notations are now used in mathematics, this article follows Iverson. e. The value of x rounded to an integer towards 0, the language APL uses ⌊x, other computer languages commonly use notations like entier, INT, or floor. In mathematics, it can also be written with boldface or double brackets, the ceiling function is usually denoted by ceil or ceiling in non-APL computer languages that have a notation for this function. The J Programming Language, a follow on to APL that is designed to use standard symbols, uses >. for ceiling. In mathematics, there is another notation with reversed boldface or double brackets ] ] x x[\. x[, the fractional part is the sawtooth function, denoted by for real x and defined by the formula = x − ⌊ x ⌋. HTML4.0 uses the names, &lfloor, &rfloor, &lceil. Unicode contains codepoints for these symbols at U+2308–U+230B, ⌈x⌉, ⌊x⌋, in the following formulas, x and y are real numbers, k, m, and n are integers, and Z is the set of integers. Floor and ceiling may be defined by the set equations ⌊ x ⌋ = max, ⌈ x ⌉ = min. Since there is exactly one integer in an interval of length one. Then ⌊ x ⌋ = m and ⌈ x ⌉ = n may also be taken as the definition of floor and these formulas can be used to simplify expressions involving floors and ceilings. In the language of order theory, the function is a residuated mapping. These formulas show how adding integers to the arguments affect the functions, negating the argument complements the fractional part, + = {0 if x ∈ Z1 if x ∉ Z. The floor, ceiling, and fractional part functions are idempotent, the result of nested floor or ceiling functions is the innermost function, ⌊ ⌈ x ⌉ ⌋ = ⌈ x ⌉, ⌈ ⌊ x ⌋ ⌉ = ⌊ x ⌋. If m and n are integers and n ≠0,0 ≤ ≤1 −1 | n |. If n is a positive integer ⌊ x + m n ⌋ = ⌊ ⌊ x ⌋ + m n ⌋, ⌈ x + m n ⌉ = ⌈ ⌈ x ⌉ + m n ⌉. For m =2 these imply n = ⌊ n 2 ⌋ + ⌈ n 2 ⌉
9.
Floor function
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In mathematics and computer science, the floor and ceiling functions map a real number to the greatest preceding or the least succeeding integer, respectively. More precisely, floor = ⌊ x ⌋ is the greatest integer less than or equal to x, carl Friedrich Gauss introduced the square bracket notation for the floor function in his third proof of quadratic reciprocity. This remained the standard in mathematics until Kenneth E. Iverson introduced the names floor and ceiling, both notations are now used in mathematics, this article follows Iverson. e. The value of x rounded to an integer towards 0, the language APL uses ⌊x, other computer languages commonly use notations like entier, INT, or floor. In mathematics, it can also be written with boldface or double brackets, the ceiling function is usually denoted by ceil or ceiling in non-APL computer languages that have a notation for this function. The J Programming Language, a follow on to APL that is designed to use standard symbols, uses >. for ceiling. In mathematics, there is another notation with reversed boldface or double brackets ] ] x x[\. x[, the fractional part is the sawtooth function, denoted by for real x and defined by the formula = x − ⌊ x ⌋. HTML4.0 uses the names, &lfloor, &rfloor, &lceil. Unicode contains codepoints for these symbols at U+2308–U+230B, ⌈x⌉, ⌊x⌋, in the following formulas, x and y are real numbers, k, m, and n are integers, and Z is the set of integers. Floor and ceiling may be defined by the set equations ⌊ x ⌋ = max, ⌈ x ⌉ = min. Since there is exactly one integer in an interval of length one. Then ⌊ x ⌋ = m and ⌈ x ⌉ = n may also be taken as the definition of floor and these formulas can be used to simplify expressions involving floors and ceilings. In the language of order theory, the function is a residuated mapping. These formulas show how adding integers to the arguments affect the functions, negating the argument complements the fractional part, + = {0 if x ∈ Z1 if x ∉ Z. The floor, ceiling, and fractional part functions are idempotent, the result of nested floor or ceiling functions is the innermost function, ⌊ ⌈ x ⌉ ⌋ = ⌈ x ⌉, ⌈ ⌊ x ⌋ ⌉ = ⌊ x ⌋. If m and n are integers and n ≠0,0 ≤ ≤1 −1 | n |. If n is a positive integer ⌊ x + m n ⌋ = ⌊ ⌊ x ⌋ + m n ⌋, ⌈ x + m n ⌉ = ⌈ ⌈ x ⌉ + m n ⌉. For m =2 these imply n = ⌊ n 2 ⌋ + ⌈ n 2 ⌉
10.
Champernowne constant
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In mathematics, the Champernowne constant C10 is a transcendental real constant whose decimal expansion has important properties. It is named after economist and mathematician D. G. Champernowne, for base 10, the number is defined by concatenating representations of successive integers, C10 =0. 12345678910111213141516… . Champernowne constants can also be constructed in other bases, similarly, for example, the Champernowne word or Barbier word is the sequence of digits of Ck. X is said to be normal in base b if its digits in base b follow a uniform distribution, if we denote a digit string as, then, in base ten, we would expect strings. To occur 1/10 of the time, strings, to occur 1/100 of the time, and so on, in a normal number. Champernowne proved that C10 is normal in base ten, while Nakai and Shiokawa proved a general theorem. It is an open problem whether C k is normal in bases b ≠ k, the simple continued fraction expansion of Champernownes constant has been studied as well. Kurt Mahler showed that the constant is transcendental, therefore its continued fraction does not terminate and is aperiodic, the terms in the continued fraction expansion exhibit very erratic behaviour, with huge terms appearing between many small ones. For example, in base 10, C10 =, the large number at position 19 has 166 digits, and the next very large term at position 41 of the continued fraction has 2504 digits. It can be understood from infinite series expression of C10, then we ignore the terms for higher n. For example, if we keep lowest order of n, it is equivalent to truncating before the 4th partial quotient,123456790 ¯ which approximates Champernownes constant with an error of about 1 × 10−9. The irrationality measure of C10 is μ =10, copeland–Erdős constant, a similar normal number, defined using the prime numbers Liouvilles constant, another constant defined by its decimal representation Cassaigne, J. Nicolas, F. Encyclopedia of Mathematics and its Applications, the construction of decimals normal in the scale of ten, Journal of the London Mathematical Society,8, 254–260, doi,10. 1112/jlms/s1-8.4.254. Discrepancy estimates for a class of numbers, Acta Arithmetica,62
11.
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
12.
Logarithmic scale
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A logarithmic scale is a nonlinear scale used when there is a large range of quantities. Common uses include the strength, sound loudness, light intensity. It is based on orders of magnitude, rather than a linear scale. In particular our sense of hearing perceives equal ratios of frequencies as equal differences in pitch, the top left graph is linear in the X and Y axis, and the Y-axis ranges from 0 to 10. A base-10 log scale is used for the Y axis of the left graph. The top right graph uses a scale for just the X axis. A slide rule has logarithmic scales, and nomograms often employ logarithmic scales, the geometric mean of two numbers is midway between the numbers. Before the advent of graphics, logarithmic graph paper was a commonly used scientific tool. If both the vertical and horizontal axes of a plot are scaled logarithmically, the plot is referred to as a log–log plot, if only the ordinate or abscissa is scaled logarithmically, the plot is referred to as a semi-logarithmic plot. Bit Byte Decade John Napier Level Logarithm Logarithmic mean Preferred number Dehaene, Stanislas, Izard, Véronique, Spelke, Elizabeth, Pica, distinct intuitions of the number scale in Western and Amazonian indigene cultures. American Association for the Advancement of Science, why using logarithmic scale to display share prices
13.
Kurt Mahler
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Kurt Mahler FRS was a mathematician. Mahler was a student at the universities in Frankfurt and Göttingen, graduating with a Ph. D. from Johann Wolfgang Goethe University of Frankfurt am Main in 1927 and he left Germany with the rise of Hitler and accepted an invitation by Louis Mordell to go to Manchester. However, at the start of World War II he was interned as an alien in Central Camp in Douglas, Isle of Man. He became a British citizen in 1946 and he was elected a member of the Royal Society in 1948 and a member of the Australian Academy of Science in 1965. He was awarded the London Mathematical Societys Senior Berwick Prize in 1950, the De Morgan Medal,1971, and he spoke fluent Chinese and was an expert photographer. Mahler proved that the Prouhet–Thue–Morse constant and the Champernowne constant 0.1234567891011121314151617181920. are transcendental numbers, Mahlers inequality Mahler measure Mahler polynomial Mahler volume Mahlers theorem Mahlers compactness theorem Skolem–Mahler–Lech theorem
14.
Rational number
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In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number, the decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number and these statements hold true not just for base 10, but also for any other integer base. A real number that is not rational is called irrational, irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating, since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. Rational numbers can be defined as equivalence classes of pairs of integers such that q ≠0, for the equivalence relation defined by ~ if. The rational numbers together with addition and multiplication form field which contains the integers and is contained in any field containing the integers, finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the numbers form a dense subset of the real numbers. The real numbers can be constructed from the numbers by completion, using Cauchy sequences, Dedekind cuts. The term rational in reference to the set Q refers to the fact that a number represents a ratio of two integers. In mathematics, rational is often used as a noun abbreviating rational number, the adjective rational sometimes means that the coefficients are rational numbers. However, a curve is not a curve defined over the rationals. Any integer n can be expressed as the rational number n/1, a b = c d if and only if a d = b c. Where both denominators are positive, a b < c d if and only if a d < b c. If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations, − a − b = a b, two fractions are added as follows, a b + c d = a d + b c b d. A b − c d = a d − b c b d, the rule for multiplication is, a b ⋅ c d = a c b d. Where c ≠0, a b ÷ c d = a d b c, note that division is equivalent to multiplying by the reciprocal of the divisor fraction, a d b c = a b × d c. Additive and multiplicative inverses exist in the numbers, − = − a b = a − b and −1 = b a if a ≠0
15.
Prime number
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n, algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits, there are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n
16.
Jeffrey Shallit
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Jeffrey Outlaw Shallit is a computer scientist, number theorist, a noted advocate for civil liberties on the Internet, and a noted critic of intelligent design. He is married to Anna Lubiw, also a computer scientist, Shallit was born in Philadelphia, Pennsylvania in 1957. His father was Joseph Shallit, a journalist and author, and his mother was Louise Lee Outlaw Shallit, a writer. He has one brother, Jonathan Shallit, a music professor and he earned a Bachelors degree in mathematics from Princeton University in June 1979. He received a Ph. D. also in mathematics, from the University of California and his doctoral thesis was entitled Metric Theory of Pierce Expansions and his advisor was Manuel Blum. Since 1996, Shallit has held the position of Vice-President of Electronic Frontier Canada and this triggered a public exchange of letters between him and Irving. Shallit has been a critic of the work of William Dembski promoting intelligent design, Shallit is currently a Professor in the School of Computer Science at the University of Waterloo and the editor-in-chief of the Journal of Integer Sequences. His primary academic interests are combinatorics on words, formal languages, automata theory and he has been recognized by the Association for Computing Machinery as a Distinguished Scientist. He has an Erdős number of 1, from a joint publication with Paul Erdős in 1991 concerning Engel expansion, home page of Jeffrey O. Shallit Shallits blog, Recursivity Electronic Frontier Canada Holocaust Revised, Lies of our Times
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Cambridge University Press
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Cambridge University Press is the publishing business of the University of Cambridge. Granted letters patent by Henry VIII in 1534, it is the worlds oldest publishing house and it also holds letters patent as the Queens Printer. The Presss mission is To further the Universitys mission by disseminating knowledge in the pursuit of education, learning, Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. With a global presence, publishing hubs, and offices in more than 40 countries. Its publishing includes journals, monographs, reference works, textbooks. Cambridge University Press is an enterprise that transfers part of its annual surplus back to the university. Cambridge University Press is both the oldest publishing house in the world and the oldest university press and it originated from Letters Patent granted to the University of Cambridge by Henry VIII in 1534, and has been producing books continuously since the first University Press book was printed. Cambridge is one of the two privileged presses, authors published by Cambridge have included John Milton, William Harvey, Isaac Newton, Bertrand Russell, and Stephen Hawking. In 1591, Thomass successor, John Legate, printed the first Cambridge Bible, the London Stationers objected strenuously, claiming that they had the monopoly on Bible printing. The universitys response was to point out the provision in its charter to print all manner of books. In July 1697 the Duke of Somerset made a loan of £200 to the university towards the house and presse and James Halman, Registrary of the University. It was in Bentleys time, in 1698, that a body of scholars was appointed to be responsible to the university for the Presss affairs. The Press Syndicates publishing committee still meets regularly, and its role still includes the review, John Baskerville became University Printer in the mid-eighteenth century. Baskervilles concern was the production of the finest possible books using his own type-design, a technological breakthrough was badly needed, and it came when Lord Stanhope perfected the making of stereotype plates. This involved making a mould of the surface of a page of type. The Press was the first to use this technique, and in 1805 produced the technically successful, under the stewardship of C. J. Clay, who was University Printer from 1854 to 1882, the Press increased the size and scale of its academic and educational publishing operation. An important factor in this increase was the inauguration of its list of schoolbooks, during Clays administration, the Press also undertook a sizable co-publishing venture with Oxford, the Revised Version of the Bible, which was begun in 1870 and completed in 1885. It was Wright who devised the plan for one of the most distinctive Cambridge contributions to publishing—the Cambridge Histories, the Cambridge Modern History was published between 1902 and 1912
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International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
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Journal of Number Theory
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The Journal of Number Theory is a mathematics journal that publishes a broad spectrum of original research in number theory. The journal was established in 1969 by R. P. Bambah, P. Roquette, A. Ross, A. Woods and it is currently published monthly by Elsevier, with 6 volumes per year
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London Mathematical Society
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The London Mathematical Society is one of the United Kingdoms learned societies for mathematics. The Society was established on 16 January 1865, the first president being Augustus De Morgan, the earliest meetings were held in University College, but the Society soon moved into Burlington House, Piccadilly. The initial activities of the Society included talks and publication of a journal, the LMS was used as a model for the establishment of the American Mathematical Society in 1888. The Society was granted a charter in 1965, a century after its foundation. In 1998 the Society moved from rooms in Burlington House into De Morgan House, at 57–58 Russell Square, Bloomsbury, the Society is also a member of the UK Science Council. On 4 July 2008, the Joint Planning Group for the LMS, the proposal was the result of eight years of consultations and the councils of both societies commended the report to their members. Those in favour of the merger argued a single society would give mathematics in the UK a coherent voice when dealing with Research Councils, while accepted by the IMA membership, the proposal was rejected by the LMS membership on 29 May 2009 by 591 to 458. It also publishes the journal Compositio Mathematica on behalf of its owning foundation, in addition, the Society jointly with the Institute of Mathematics and its Applications awards the David Crighton Medal every three years. London Mathematical Society website A History of the London Mathematical Society MacTutor, The London Mathematical Society