1.
Irrational number
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In mathematics, the irrational numbers are all the real numbers, which are not rational numbers, the latter being the numbers constructed from ratios of integers. Irrational numbers may also be dealt with via non-terminating continued fractions, for example, the decimal representation of the number π starts with 3.14159265358979, but no finite number of digits can represent π exactly, nor does it repeat. Mathematicians do not generally take terminating or repeating to be the definition of the concept of rational number, as a consequence of Cantors proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational. The first proof of the existence of numbers is usually attributed to a Pythagorean. The then-current Pythagorean method would have claimed that there must be sufficiently small. However, Hippasus, in the 5th century BC, was able to deduce that there was in no common unit of measure. His reasoning is as follows, Start with a right triangle with side lengths of integers a, b. The ratio of the hypotenuse to a leg is represented by c, b, assume a, b, and c are in the smallest possible terms. By the Pythagorean theorem, c2 = a2+b2 = b2+b2 = 2b2, since c2 = 2b2, c2 is divisible by 2, and therefore even. Since c2 is even, c must be even, since c is even, dividing c by 2 yields an integer. Squaring both sides of c = 2y yields c2 =2, or c2 = 4y2, substituting 4y2 for c2 in the first equation gives us 4y2= 2b2. Dividing by 2 yields 2y2 = b2, since y is an integer, and 2y2 = b2, b2 is divisible by 2, and therefore even. Since b2 is even, b must be even and we have just show that both b and c must be even. Hence they have a factor of 2. However this contradicts the assumption that they have no common factors and this contradiction proves that c and b cannot both be integers, and thus the existence of a number that cannot be expressed as a ratio of two integers. Greek mathematicians termed this ratio of incommensurable magnitudes alogos, or inexpressible. ”Another legend states that Hippasus was merely exiled for this revelation, the discovery of incommensurable ratios was indicative of another problem facing the Greeks, the relation of the discrete to the continuous. Brought into light by Zeno of Elea, who questioned the conception that quantities are discrete and composed of a number of units of a given size. ”However Zeno found that in fact “ in general are not discrete collections of units. That in fact, these divisions of quantity must necessarily be infinite, for example, consider a line segment, this segment can be split in half, that half split in half, the half of the half in half, and so on
2.
Square root of 2
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The square root of 2, or the th power of 2, written in mathematics as √2 or 2 1⁄2, is the positive algebraic number that, when multiplied by itself, gives the number 2. Technically, it is called the 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, the rational approximation of the square root of two,665, 857/470,832, derived from the fourth step in the Babylonian algorithm starting with a0 =1, is too large by approx. 1. 6×10−12, its square is 2. 0000000000045… The rational approximation 99/70 is frequently used, despite having a denominator of only 70, it differs from the correct value by less than 1/10,000. The numerical value for the root of two, truncated to 65 decimal places, is,1. 41421356237309504880168872420969807856967187537694807317667973799….41421296 ¯. That is,1 +13 +13 ×4 −13 ×4 ×34 =577408 =1.4142156862745098039 ¯. This approximation is the seventh in a sequence of increasingly accurate approximations based on the sequence of Pell numbers, despite having a smaller denominator, it is only slightly less accurate than the Babylonian approximation. Pythagoreans discovered that the diagonal of a square is incommensurable with its side, or in modern language, little is known with certainty about the time or circumstances of this discovery, but the name of Hippasus of Metapontum is often mentioned. For a while, the Pythagoreans treated as a secret the discovery that the square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it. The square root of two is occasionally called Pythagoras number or Pythagoras constant, for example by Conway & Guy, there are a number of algorithms for approximating √2, which in expressions as a ratio of integers or as a decimal can only be approximated. The most common algorithm for this, one used as a basis in many computers and calculators, is the Babylonian method of computing square roots, which is one of many methods of computing square roots. It goes as follows, First, pick a guess, a0 >0, then, using that guess, iterate through the following recursive computation, a n +1 = a n +2 a n 2 = a n 2 +1 a n. The more iterations through the algorithm, the approximation of the square root of 2 is achieved. Each iteration approximately doubles the number of correct digits, starting with a0 =1 the next approximations are 3/2 =1.5 17/12 =1.416. The value of √2 was calculated to 137,438,953,444 decimal places by Yasumasa Kanadas team in 1997, in February 2006 the record for the calculation of √2 was eclipsed with the use of a home computer. Shigeru Kondo calculated 1 trillion decimal places in 2010, for a development of this record, see the table below. Among mathematical constants with computationally challenging decimal expansions, only π has been calculated more precisely, such computations aim to check empirically whether such numbers are normal
3.
Square root of 3
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The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is more called the principal square root of 3. The square root of 3 is an irrational number and it is also known as Theodorus constant, named after Theodorus of Cyrene, who proved its irrationality. The fraction 97/56 for the root of three can be used as an approximation. Despite having a denominator of only 56, it differs from the value by less than 1/10,000. The rounded value of 1.732 is correct to within 0. 01% of the actual value, archimedes reported 2 >3 >2, accurate to 1/608400 and 2/23409, respectively. It can be expressed as the fraction, expanded on the right. It can also be expressed by generalized continued fractions such as =2 −14 −14 −14 − ⋱ which is evaluated at every second term. The following nested square expressions converge to 3,3 =2 −22 =74 −42. This irrationality proof for the root of 3 uses Fermats method of infinite descent, Suppose that √3 is rational. Therefore, multiplying by 1 will give an expression, m n where q is the largest integer smaller than √3. Note that both the numerator and the denominator have been multiplied by a smaller than 1. An alternate proof of this is, assuming √3 = m/n with m/n being a reduced fraction. Since the left side is divisible by 3, so is the right side, thus, as both n and m are divisible by 3, they have a common factor and m/n is not a fully reduced fraction, contradicting the original premise. The square root of 3 can be found as the leg length of a triangle that encompasses a circle with a diameter of 1. From this the trigonometric function tangent of 60 degrees equals √3, and the sine of 60° and it is the distance between parallel sides of a regular hexagon with sides of length 1. On the complex plane, this distance is expressed as i√3 mentioned below and it is the length of the space diagonal of a unit cube. The vesica piscis has an axis to minor axis ratio equal to 1, √3
4.
Square root of 5
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The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. It is more called the principal square root of 5. This number appears in the expression for the golden ratio. It can be denoted in surd form as,5 and it is an irrational algebraic number. The first sixty significant digits of its expansion are,2. 23606797749978969640917366873127623544061835961152572427089…. Which can be rounded down to 2.236 to within 99. 99% accuracy, the approximation 161/72 for the square root of five can be used. Despite having a denominator of only 72, it differs from the value by less than 1/10,000. As of December 2013, its value in decimal has been computed to at least ten billion digits. This irrationality proof for the root of 5 uses Fermats method of infinite descent, Suppose that √5 is rational. Then √5 can be expressed in terms as 5n − 2m/m − 2n. This irrationality proof is a proof by contradiction, Suppose that √5 = a/b where a/b is in reduced form, thus 5 = a2/b2 and 5b2 = a2. If b were even, b2, a2, and a would be making the fraction a/b not in reduced form. Thus b is odd, and by following a similar process, now, let a = 2m +1 and b = 2n +1 where m and n are integers. So what this is saying is 5 × even +1 = even, there is no integer thats both even and odd thus weve reached a contradiction and √5 is irrational. Geometrically, √5 corresponds to the diagonal of a rectangle whose sides are of length 1 and 2, such a rectangle can be obtained by halving a square, or by placing two equal squares side by side. The number √5 can be algebraically and geometrically related to √2 and √3, as it is the length of the hypotenuse of a triangle with catheti measuring √2. This follows from the relationships between a cube and the quantities √2, √3 and √5. A root-5 rectangle is particularly notable in that it can be split into a square and it can also be decomposed as the union of two equal golden rectangles whose intersection forms a square
5.
Golden ratio
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In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship, expressed algebraically, for quantities a and b with a > b >0, a + b a = a b = def φ, where the Greek letter phi represents the golden ratio. Its value is, φ =1 +52 =1.6180339887 …, A001622 The golden ratio is also called the golden mean or golden section. Other names include extreme and mean ratio, medial section, divine proportion, divine section, golden proportion, golden cut, the golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts. The golden ratio has also used to analyze the proportions of natural objects as well as man-made systems such as financial markets. Two quantities a and b are said to be in the golden ratio φ if a + b a = a b = φ, one method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ, a + b a =1 + b a =1 +1 φ, multiplying by φ gives φ +1 = φ2 which can be rearranged to φ2 − φ −1 =0. First, the line segment A B ¯ is about doubled and then the semicircle with the radius A S ¯ around the point S is drawn, now the semicircle is drawn with the radius A B ¯ around the point B. The arising intersection point E corresponds 2 φ, next up, the perpendicular on the line segment A E ¯ from the point D will be establish. The subsequent parallel F S ¯ to the line segment C M ¯, produces, as it were and it is well recognizable, this triangle and the triangle M S C are similar to each other. The hypotenuse F S ¯ has due to the cathetuses S D ¯ =1 and D F ¯ =2 according the Pythagorean theorem, finally, the circle arc is drawn with the radius 5 around the point F. The golden ratio has been claimed to have held a fascination for at least 2,400 years. But the fascination with the Golden Ratio is not confined just to mathematicians, biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics. Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry, the division of a line into extreme and mean ratio is important in the geometry of regular pentagrams and pentagons. Euclid explains a construction for cutting a line in extreme and mean ratio, throughout the Elements, several propositions and their proofs employ the golden ratio. The golden ratio is explored in Luca Paciolis book De divina proportione, since the 20th century, the golden ratio has been represented by the Greek letter φ or less commonly by τ. Timeline according to Priya Hemenway, Phidias made the Parthenon statues that seem to embody the golden ratio, plato, in his Timaeus, describes five possible regular solids, some of which are related to the golden ratio
6.
Pi
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The number π is a mathematical constant, the ratio of a circles circumference to its diameter, commonly approximated as 3.14159. It has been represented by the Greek letter π since the mid-18th century, being an irrational number, π cannot be expressed exactly as a fraction. Still, fractions such as 22/7 and other numbers are commonly used to approximate π. The digits appear to be randomly distributed, in particular, the digit sequence of π is conjectured to satisfy a specific kind of statistical randomness, but to date no proof of this has been discovered. Also, π is a number, i. e. a number that is not the root of any non-zero polynomial having rational coefficients. This transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass, ancient civilizations required fairly accurate computed values for π for practical reasons. It was calculated to seven digits, using techniques, in Chinese mathematics. The extensive calculations involved have also used to test supercomputers. Because its definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, and spheres. Because of its role as an eigenvalue, π appears in areas of mathematics. It is also found in cosmology, thermodynamics, mechanics, attempts to memorize the value of π with increasing precision have led to records of over 70,000 digits. In English, π is pronounced as pie, in mathematical use, the lowercase letter π is distinguished from its capitalized and enlarged counterpart ∏, which denotes a product of a sequence, analogous to how ∑ denotes summation. The choice of the symbol π is discussed in the section Adoption of the symbol π, π is commonly defined as the ratio of a circles circumference C to its diameter d, π = C d The ratio C/d is constant, regardless of the circles size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d. This definition of π implicitly makes use of geometry, although the notion of a circle can be extended to any curved geometry. Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be defined independently of geometry using limits. An integral such as this was adopted as the definition of π by Karl Weierstrass, definitions of π such as these that rely on a notion of circumference, and hence implicitly on concepts of the integral calculus, are no longer common in the literature. One such definition, due to Richard Baltzer, and popularized by Edmund Landau, is the following, the cosine can be defined independently of geometry as a power series, or as the solution of a differential equation
7.
Binary number
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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
8.
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
9.
Hexadecimal
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In mathematics and computing, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, Hexadecimal numerals are widely used by computer system designers and programmers. As each hexadecimal digit represents four binary digits, it allows a more human-friendly representation of binary-coded values, one hexadecimal digit represents a nibble, which is half of an octet or byte. For example, a byte can have values ranging from 00000000 to 11111111 in binary form. In a non-programming context, a subscript is typically used to give the radix, several notations are used to support hexadecimal representation of constants in programming languages, usually involving a prefix or suffix. The prefix 0x is used in C and related languages, where this value might be denoted as 0x2AF3, in contexts where the base is not clear, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously, a numerical subscript can give the base explicitly,15910 is decimal 159,15916 is hexadecimal 159, which is equal to 34510. Some authors prefer a text subscript, such as 159decimal and 159hex, or 159d and 159h. example. com/name%20with%20spaces where %20 is the space character, thus ’, represents the right single quotation mark, Unicode code point number 2019 in hex,8217. In the Unicode standard, a value is represented with U+ followed by the hex value. Color references in HTML, CSS and X Window can be expressed with six hexadecimal digits prefixed with #, white, CSS allows 3-hexdigit abbreviations with one hexdigit per component, #FA3 abbreviates #FFAA33. *nix shells, AT&T assembly language and likewise the C programming language, to output an integer as hexadecimal with the printf function family, the format conversion code %X or %x is used. In Intel-derived assembly languages and Modula-2, hexadecimal is denoted with a suffixed H or h, some assembly languages use the notation HABCD. Ada and VHDL enclose hexadecimal numerals in based numeric quotes, 16#5A3#, for bit vector constants VHDL uses the notation x5A3. Verilog represents hexadecimal constants in the form 8hFF, where 8 is the number of bits in the value, the Smalltalk language uses the prefix 16r, 16r5A3 PostScript and the Bourne shell and its derivatives denote hex with prefix 16#, 16#5A3. For PostScript, binary data can be expressed as unprefixed consecutive hexadecimal pairs, in early systems when a Macintosh crashed, one or two lines of hexadecimal code would be displayed under the Sad Mac to tell the user what went wrong. Common Lisp uses the prefixes #x and #16r, setting the variables *read-base* and *print-base* to 16 can also used to switch the reader and printer of a Common Lisp system to Hexadecimal number representation for reading and printing numbers. Thus Hexadecimal numbers can be represented without the #x or #16r prefix code, MSX BASIC, QuickBASIC, FreeBASIC and Visual Basic prefix hexadecimal numbers with &H, &H5A3 BBC BASIC and Locomotive BASIC use & for hex. TI-89 and 92 series uses a 0h prefix, 0h5A3 ALGOL68 uses the prefix 16r to denote hexadecimal numbers, binary, quaternary and octal numbers can be specified similarly
10.
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
11.
Number theory
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Number theory or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called The Queen of Mathematics because of its place in the discipline. Number theorists study prime numbers as well as the properties of objects out of integers or defined as generalizations of the integers. Integers can be considered either in themselves or as solutions to equations, questions in number theory are often best understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. One may also study real numbers in relation to rational numbers, the older term for number theory is arithmetic. By the early century, it had been superseded by number theory. The use of the arithmetic for number theory regained some ground in the second half of the 20th century. In particular, arithmetical is preferred as an adjective to number-theoretic. The first historical find of a nature is a fragment of a table. The triples are too many and too large to have been obtained by brute force, the heading over the first column reads, The takiltum of the diagonal which has been subtracted such that the width. The tables layout suggests that it was constructed by means of what amounts, in language, to the identity 2 +1 =2. If some other method was used, the triples were first constructed and then reordered by c / a, presumably for use as a table. It is not known what these applications may have been, or whether there could have any, Babylonian astronomy, for example. It has been suggested instead that the table was a source of examples for school problems. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt, Euclid IX 21—34 is very probably Pythagorean, it is very simple material, but it is all that is needed to prove that 2 is irrational. Pythagorean mystics gave great importance to the odd and the even, the discovery that 2 is irrational is credited to the early Pythagoreans. This forced a distinction between numbers, on the one hand, and lengths and proportions, on the other hand, the Pythagorean tradition spoke also of so-called polygonal or figurate numbers
12.
Riemann zeta function
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More general representations of ζ for all s are given below. The Riemann zeta function plays a role in analytic number theory and has applications in physics, probability theory. As a function of a variable, Leonhard Euler first introduced and studied it in the first half of the eighteenth century without using complex analysis. The values of the Riemann zeta function at even positive integers were computed by Euler, the first of them, ζ, provides a solution to the Basel problem. In 1979 Apéry proved the irrationality of ζ, the values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms. Many generalizations of the Riemann zeta function, such as Dirichlet series, the Riemann zeta function ζ is a function of a complex variable s = σ + it. It can also be defined by the integral ζ =1 Γ ∫0 ∞ x s −1 e x −1 d x where Γ is the gamma function. The Riemann zeta function is defined as the continuation of the function defined for σ >1 by the sum of the preceding series. Leonhard Euler considered the series in 1740 for positive integer values of s. The above series is a prototypical Dirichlet series that converges absolutely to a function for s such that σ >1. Riemann showed that the function defined by the series on the half-plane of convergence can be continued analytically to all complex values s ≠1, for s =1 the series is the harmonic series which diverges to +∞, and lim s →1 ζ =1. Thus the Riemann zeta function is a function on the whole complex s-plane. For any positive even integer 2n, ζ = n +1 B2 n 2 n 2, where B2n is the 2nth Bernoulli number. For odd positive integers, no simple expression is known, although these values are thought to be related to the algebraic K-theory of the integers. For nonpositive integers, one has ζ = B n +1 n +1 for n ≥0 In particular, ζ = −12, Similarly to the above, this assigns a finite result to the series 1 +1 +1 +1 + ⋯. ζ ≈ −1.4603545 This is employed in calculating of kinetic boundary layer problems of linear kinetic equations, ζ =1 +12 +13 + ⋯ = ∞, if we approach from numbers larger than 1. Then this is the harmonic series, but its Cauchy principal value lim ε →0 ζ + ζ2 exists which is the Euler–Mascheroni constant γ =0. 5772…. ζ ≈2.612, This is employed in calculating the critical temperature for a Bose–Einstein condensate in a box with periodic boundary conditions, and for spin wave physics in magnetic systems
13.
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
14.
Gyromagnetic ratio
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In physics, the gyromagnetic ratio of a particle or system is the ratio of its magnetic moment to its angular momentum, and it is often denoted by the symbol γ, gamma. Its SI unit is the radian per second per tesla or, equivalently, the term gyromagnetic ratio is often used as a synonym for a different but closely related quantity, the g-factor. The g-factor, unlike the gyromagnetic ratio, is dimensionless, for more on the g-factor, see below, or see the article g-factor. For this reason, values of γ/, in units of hertz per tesla, are often quoted instead of γ. The derivation of this relation is as follows, First we must prove that the torque resulting from subjecting a magnetic moment m ¯ to a magnetic field B ¯ is T ¯ = m ¯ × B ¯. By classical mechanics the torque on this needle is T ¯ = l ¯ × B ¯ ⋅ q m = q m ⋅ l ¯ × B ¯. But as previously stated q m ⋅ l ¯ = I π r 2 = m ¯, the model of the spinning electron we use in the derivation has an evident analogy with a gyroscope. For any rotating body the rate of change of the angular momentum J ¯ equals the applied torque T ¯, d J ¯ d t = T ¯, note as an example the precession of a gyroscope. Replace the gravity with a flux density B. D J ¯ d t represents the velocity of the pike of the arrow J ¯ along a circle whose radius is J ⋅ sin ϕ where ϕ is the angle between J ¯ and the vertical. Consequently, f = γ2 π B q. e. d, the angular precession frequency has an important physical meaning, It is the angular cyclotron frequency. The resonance frequency of an ionized plasma being under the influence of a static magnetic field. Consider a charged body rotating about an axis of symmetry, according to the laws of classical physics, it has both a magnetic dipole moment and an angular momentum due to its rotation. It can be shown that as long as its charge and mass are distributed identically, its ratio is γ = q 2 m where q is its charge. The derivation of this relation is as follows, It suffices to demonstrate this for a narrow circular ring within the body. Suppose the ring has radius r, area A = πr2, mass m, charge q, and angular momentum L = mvr. Then the magnitude of the dipole moment is μ = I A = q v 2 π r × π r 2 = q 2 m × m v r = q 2 m L. An isolated electron has a momentum and a magnetic moment resulting from its spin
15.
Quantum electrodynamics
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In particle physics, quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved, in technical terms, QED can be described as a perturbation theory of the electromagnetic quantum vacuum. Dirac described the quantization of the field as an ensemble of harmonic oscillators with the introduction of the concept of creation and annihilation operators of particles. At higher orders in the series emerged, making such computations meaningless. With no solution for this problem known at the time, it appeared that a fundamental incompatibility existed between special relativity and quantum mechanics, difficulties with the theory increased through the end of 1940. Improvements in microwave technology made it possible to more precise measurements of the shift of the levels of a hydrogen atom, now known as the Lamb shift. These experiments exposed discrepancies which the theory was unable to explain, a first indication of a possible way out was given by Hans Bethe in 1947, after attending the Shelter Island Conference. While he was traveling by train from the conference to Schenectady he made the first non-relativistic computation of the shift of the lines of the atom as measured by Lamb. Despite the limitations of the computation, agreement was excellent, the idea was simply to attach infinities to corrections of mass and charge that were actually fixed to a finite value by experiments. In this way, the infinities get absorbed in those constants, sin-Itiro Tomonaga, Julian Schwinger and Richard Feynman were jointly awarded with a Nobel prize in physics in 1965 for their work in this area. Even though renormalization works very well in practice, Feynman was never comfortable with its mathematical validity, even referring to renormalization as a shell game. QED has served as the model and template for all subsequent quantum field theories. One such subsequent theory is quantum chromodynamics, which began in the early 1960s and attained its present form in the 1975 work by H. David Politzer, Sidney Coleman, David Gross and Frank Wilczek. Near the end of his life, Richard P. Feynman gave a series of lectures on QED intended for the lay public. These lectures were transcribed and published as Feynman, QED, The strange theory of light and matter, the key components of Feynmans presentation of QED are three basic actions. A photon goes from one place and time to another place, an electron goes from one place and time to another place and time. An electron emits or absorbs a photon at a certain place and these can all be seen in the adjacent diagram. It is important not to over-interpret these diagrams, nothing is implied about how a particle gets from one point to another
16.
Gamma function
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In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers. That is, if n is an integer, Γ =. The gamma function is defined for all numbers except the non-positive integers. The gamma function can be seen as a solution to the interpolation problem. The simple formula for the factorial, x. =1 ×2 × … × x, a good solution to this is the gamma function. There are infinitely many continuous extensions of the factorial to non-integers, the gamma function is the most useful solution in practice, being analytic, and it can be characterized in several ways. The Bohr–Mollerup theorem proves that these properties, together with the assumption that f be logarithmically convex, uniquely determine f for positive, from there, the gamma function can be extended to all real and complex values by using the unique analytic continuation of f. Also see Eulers infinite product definition below where the properties f =1 and f = x f together with the requirement that limn→+∞. nx / f =1 uniquely define the same function. The notation Γ is due to Legendre, if the real part of the complex number z is positive, then the integral Γ = ∫0 ∞ x z −1 e − x d x converges absolutely, and is known as the Euler integral of the second kind. The identity Γ = Γ z can be used to extend the integral formulation for Γ to a meromorphic function defined for all complex numbers z. It is this version that is commonly referred to as the gamma function. When seeking to approximate z. for a number z, it turns out that it is effective to first compute n. for some large integer n. And then use the relation m. = m. backwards n times. Furthermore, this approximation is exact in the limit as n goes to infinity, specifically, for a fixed integer m, it is the case that lim n → + ∞ n. m. =1, and we can ask that the formula is obeyed when the arbitrary integer m is replaced by an arbitrary complex number z lim n → + ∞ n. z. =1. Multiplying both sides by z. gives z. = lim n → + ∞ n. z, Z = lim n → + ∞1 ⋯ n ⋯ z = ∏ n =1 + ∞. Similarly for the function, the definition as an infinite product due to Euler is valid for all complex numbers z except the non-positive integers. By this construction, the function is the unique function that simultaneously satisfies Γ =1, Γ = z Γ for all complex numbers z except the non-positive integers
17.
Debye model
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In thermodynamics and solid state physics, the Debye model is a method developed by Peter Debye in 1912 for estimating the phonon contribution to the specific heat in a solid. It treats the vibrations of the lattice as phonons in a box, in contrast to the Einstein model. The Debye model correctly predicts the low temperature dependence of the heat capacity, just like the Einstein model, it also recovers the Dulong–Petit law at high temperatures. But due to simplifying assumptions, its accuracy suffers at intermediate temperatures, see M. Shubin and T. Sunada for a rigorous treatment of the Debye model. The Debye model is an equivalent of Plancks law of black body radiation. The Debye model treats atomic vibrations as phonons in a box, most of the calculation steps are identical. Consider a cube of side L, from the particle in a box article, the resonating modes of the sonic disturbances inside the box have wavelengths given by λ n =2 L n, where n is an integer. The energy of a phonon is E n = h ν n, in three dimensions we will use, E n 2 = p n 2 c s 2 =2, in which p n is the magnitude of the three-dimensional momentum of the phonon. Lets now compute the energy in the box, E = ∑ n E n N ¯. In other words, the energy is equal to the sum of energy multiplied by the number of phonons with that energy. In 3 dimensions we have, U = ∑ n x ∑ n y ∑ n z E n N ¯, now, this is where Debye model and Plancks law of black body radiation differ. Unlike electromagnetic radiation in a box, there is a number of phonon energy states because a phonon cannot have infinite frequency. Its frequency is bound by the medium of its propagation—the atomic lattice of the solid, consider an illustration of a transverse phonon below. It is reasonable to assume that the wavelength of a phonon is twice the atom separation. There are N atoms in a solid and our solid is a cube, which means there are N3 atoms per edge. Atom separation is then given by L / N3, and this is the upper limit of the triple energy sum U = ∑ n x N3 ∑ n y N3 ∑ n z N3 E n N ¯. For slowly varying, well-behaved functions, a sum can be replaced with an integral U ≈ ∫0 N3 ∫0 N3 ∫0 N3 E N ¯ d n x d n y d n z. So far, there has no mention of N ¯
18.
France
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France, officially the French Republic, is a country with territory in western Europe and several overseas regions and territories. The European, or metropolitan, area of France extends from the Mediterranean Sea to the English Channel and the North Sea, Overseas France include French Guiana on the South American continent and several island territories in the Atlantic, Pacific and Indian oceans. France spans 643,801 square kilometres and had a population of almost 67 million people as of January 2017. It is a unitary republic with the capital in Paris. Other major urban centres include Marseille, Lyon, Lille, Nice, Toulouse, during the Iron Age, what is now metropolitan France was inhabited by the Gauls, a Celtic people. The area was annexed in 51 BC by Rome, which held Gaul until 486, France emerged as a major European power in the Late Middle Ages, with its victory in the Hundred Years War strengthening state-building and political centralisation. During the Renaissance, French culture flourished and a colonial empire was established. The 16th century was dominated by civil wars between Catholics and Protestants. France became Europes dominant cultural, political, and military power under Louis XIV, in the 19th century Napoleon took power and established the First French Empire, whose subsequent Napoleonic Wars shaped the course of continental Europe. Following the collapse of the Empire, France endured a succession of governments culminating with the establishment of the French Third Republic in 1870. Following liberation in 1944, a Fourth Republic was established and later dissolved in the course of the Algerian War, the Fifth Republic, led by Charles de Gaulle, was formed in 1958 and remains to this day. Algeria and nearly all the colonies became independent in the 1960s with minimal controversy and typically retained close economic. France has long been a centre of art, science. It hosts Europes fourth-largest number of cultural UNESCO World Heritage Sites and receives around 83 million foreign tourists annually, France is a developed country with the worlds sixth-largest economy by nominal GDP and ninth-largest by purchasing power parity. In terms of household wealth, it ranks fourth in the world. France performs well in international rankings of education, health care, life expectancy, France remains a great power in the world, being one of the five permanent members of the United Nations Security Council with the power to veto and an official nuclear-weapon state. It is a member state of the European Union and the Eurozone. It is also a member of the Group of 7, North Atlantic Treaty Organization, Organisation for Economic Co-operation and Development, the World Trade Organization, originally applied to the whole Frankish Empire, the name France comes from the Latin Francia, or country of the Franks
19.
Mathematician
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A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems. Mathematics is concerned with numbers, data, quantity, structure, space, models, one of the earliest known mathematicians was Thales of Miletus, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, the number of known mathematicians grew when Pythagoras of Samos established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was All is number. It was the Pythagoreans who coined the term mathematics, and with whom the study of mathematics for its own sake begins, the first woman mathematician recorded by history was Hypatia of Alexandria. She succeeded her father as Librarian at the Great Library and wrote works on applied mathematics. Because of a dispute, the Christian community in Alexandria punished her, presuming she was involved, by stripping her naked. Science and mathematics in the Islamic world during the Middle Ages followed various models and it was extensive patronage and strong intellectual policies implemented by specific rulers that allowed scientific knowledge to develop in many areas. As these sciences received wider attention from the elite, more scholars were invited and funded to study particular sciences, an example of a translator and mathematician who benefited from this type of support was al-Khawarizmi. A notable feature of many working under Muslim rule in medieval times is that they were often polymaths. Examples include the work on optics, maths and astronomy of Ibn al-Haytham, the Renaissance brought an increased emphasis on mathematics and science to Europe. As time passed, many gravitated towards universities. Moving into the 19th century, the objective of universities all across Europe evolved from teaching the “regurgitation of knowledge” to “encourag productive thinking. ”Thus, seminars, overall, science became the focus of universities in the 19th and 20th centuries. Students could conduct research in seminars or laboratories and began to produce doctoral theses with more scientific content. According to Humboldt, the mission of the University of Berlin was to pursue scientific knowledge. ”Mathematicians usually cover a breadth of topics within mathematics in their undergraduate education, and then proceed to specialize in topics of their own choice at the graduate level. In some universities, a qualifying exam serves to test both the breadth and depth of an understanding of mathematics, the students, who pass, are permitted to work on a doctoral dissertation. Mathematicians involved with solving problems with applications in life are called applied mathematicians. Applied mathematicians are mathematical scientists who, with their knowledge and professional methodology. With professional focus on a variety of problems, theoretical systems
20.
Legendre polynomials
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In mathematics, Legendre functions are solutions to Legendres differential equation, They are named after Adrien-Marie Legendre. This ordinary differential equation is frequently encountered in physics and other technical fields, in particular, it occurs when solving Laplaces equation in spherical coordinates. The Legendre differential equation may be solved using the power series method. The equation has regular singular points at x = ±1 so, in general, when n is an integer, the solution Pn that is regular at x =1 is also regular at x = −1, and the series for this solution terminates. These solutions for n =0,1,2, form a polynomial sequence of orthogonal polynomials called the Legendre polynomials. Each Legendre polynomial Pn is an nth-degree polynomial and it may be expressed using Rodrigues formula, P n =12 n n. The Pn can also be defined as the coefficients in a Taylor series expansion, In physics, expanding the Taylor series in Equation for the first two terms gives P0 =1, P1 = x for the first two Legendre Polynomials. Replacing the quotient of the root with its definition in. This relation, along with the first two polynomials P0 and P1, allows the Legendre Polynomials to be generated recursively, in fact, an alternative derivation of the Legendre polynomials is by carrying out the Gram–Schmidt process on the polynomials with respect to this inner product. The series converges when r > r ′, the expression gives the gravitational potential associated to a point mass or the Coulomb potential associated to a point charge. The expansion using Legendre polynomials might be useful, for instance, when integrating this expression over a continuous mass or charge distribution. Where z ^ is the axis of symmetry and θ is the angle between the position of the observer and the z ^ axis, the solution for the potential will be Φ = ∑ ℓ =0 ∞ P ℓ. A ℓ and B ℓ are to be determined according to the condition of each problem. They also appear when solving Schrödinger equation in three dimensions for a central force. Legendre polynomials are also useful in expanding functions of the form,11 + η2 −2 η x = ∑ k =0 ∞ η k P k which arise naturally in multipole expansions. The left-hand side of the equation is the function for the Legendre polynomials. This expansion is used to develop the normal multipole expansion, conversely, if the radius r of the observation point P is smaller than a, the potential may still be expanded in the Legendre polynomials as above, but with a and r exchanged. This expansion is the basis of interior multipole expansion, the trigonometric functions cos n θ, also denoted as the Chebyshev polynomials T n ≡ cos n θ, can also be multipole expanded by the Legendre polynomials P n
21.
Almost integer
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In recreational mathematics, an almost integer is any number that is very close to an integer. Almost integers are considered interesting when they arise in some context in which they are unexpected, the constant e π163 is sometimes referred to as Ramanujans constant. Almost integers that involve the mathematical constants π and e have often puzzled mathematicians. An example is, e π − π =19.999099979189 … To date, no explanation has been given for why Gelfonds constant is identical to π +20. Markovitch Coincidence, data compression, and Machs concept of economy of thought
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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
23.
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
24.
Srinivasa Ramanujan
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Srinivasa Iyengar Ramanujan FRS was an Indian mathematician and autodidact who lived during the British Raj. Though he had almost no training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series. Ramanujan initially developed his own research in isolation, it was quickly recognized by Indian mathematicians. When his skills became obvious and known to the mathematical community, centred in Europe at the time. The Cambridge professor realized that Srinivasa Ramanujan had produced new theorems in addition to rediscovering previously known ones, during his short life, Ramanujan independently compiled nearly 3,900 results. Nearly all his claims have now been proven correct and his original and highly unconventional results, such as the Ramanujan prime and the Ramanujan theta function, have inspired a vast amount of further research. The Ramanujan Journal, a scientific journal, was established to publish work in all areas of mathematics influenced by Ramanujan. Deeply religious, Ramanujan credited his substantial mathematical capacities to divinity, An equation for me has no meaning, he once said, the name Ramanujan means younger brother of the god Rama. Iyengar is a caste of Hindu Brahmins of Tamil origin whose members follow the Visishtadvaita philosophy propounded by Ramanuja, Ramanujan was born on 22 December 1887 into a Tamil Brahmin Iyengar family in Erode, Madras Presidency, at the residence of his maternal grandparents. His father, K. Srinivasa Iyengar, worked as a clerk in a sari shop and his mother, Komalatammal, was a housewife and also sang at a local temple. They lived in a traditional home on Sarangapani Sannidhi Street in the town of Kumbakonam. The family home is now a museum, when Ramanujan was a year and a half old, his mother gave birth to a son, Sadagopan, who died less than three months later. In December 1889, Ramanujan contracted smallpox, but unlike the thousands in the Thanjavur district who died of the disease that year and he moved with his mother to her parents house in Kanchipuram, near Madras. His mother gave birth to two children, in 1891 and 1894, but both died in infancy. On 1 October 1892, Ramanujan was enrolled at the local school, after his maternal grandfather lost his job as a court official in Kanchipuram, Ramanujan and his mother moved back to Kumbakonam and he was enrolled in the Kangayan Primary School. When his paternal grandfather died, he was sent back to his maternal grandparents and he did not like school in Madras, and tried to avoid attending. His family enlisted a local constable to make sure the boy attended school, within six months, Ramanujan was back in Kumbakonam. Since Ramanujans father was at work most of the day, his mother took care of the boy as a child and he had a close relationship with her
25.
Johan Jensen (mathematician)
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Johan Ludwig William Valdemar Jensen, mostly known as Johan Jensen, was a Danish mathematician and engineer. He was the president of the Danish Mathematical Society from 1892 to 1903, Jensen was born in Nakskov, Denmark, but spent much of his childhood in northern Sweden, because his father obtained a job there as the manager of an estate. Their family returned to Denmark before 1876, when Jensen enrolled to the Copenhagen College of Technology, instead, he was a successful engineer for the Copenhagen Telephone Company between 1881 and 1924, and became head of the technical department in 1890. All his mathematics research was carried out in his spare time, Jensen is mostly renowned for his famous inequality, Jensens inequality. In 1915, Jensen also proved Jensens formula in complex analysis
26.
Polygamma function
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Thus ψ = ψ = Γ ′ Γ holds where ψ is the digamma function and Γ is the gamma function. They are holomorphic on ℂ \ −ℕ0, at all the nonpositive integers these polygamma functions have a pole of order m +1. The function ψ is sometimes called the trigamma function, for m =0 see the digamma function definition. It satisfies the recurrence relation ψ = ψ + m m, Z m +1 which – considered for positive integer argument – leads to a presentation of the sum of reciprocals of the powers of the natural numbers, ψ m +1 m. This is a consequence of the Bohr–Mollerup theorem for the gamma function where strictly logarithmic convexity on ℝ+ is demanded additionally. The case m =0 must be treated differently because ψ is not normalizable at infinity and they obey the recursion equation P0 = x P m +1 = −. The multiplication theorem gives k m +1 ψ = ∑ n =0 k −1 ψ m ≥1 and k ψ = k log + ∑ n =0 k −1 ψ for the digamma function. The polygamma function has the series representation ψ = m +1 m, ∑ k =0 ∞1 m +1 which holds for m >0 and any complex z not equal to a negative integer. This representation can be written compactly in terms of the Hurwitz zeta function as ψ = m +1 m. Alternately, the Hurwitz zeta can be understood to generalize the polygamma to arbitrary, one more series may be permitted for the polygamma functions. As given by Schlömilch,1 Γ = z e γ z ∏ n =1 ∞ e − z n and this is a result of the Weierstrass factorization theorem. Thus, the function may now be defined as, Γ = e − γ z z ∏ n =1 ∞ −1 e z n. Now, the logarithm of the gamma function is easily representable. Finally, we arrive at a summation representation for the polygamma function, Z n +1 + ∑ k =1 ∞ Where δn0 is the Kronecker delta. Also the Lerch transcendent Φ = ∑ k =0 ∞ k m +1 can be denoted in terms of polygamma function Φ =1 m +1 m, the Taylor series at z =1 is ψ = ∑ k =0 ∞ m + k +1. ζ z k m ≥1 and ψ = − γ + ∑ k =1 ∞ k +1 ζ z k which converges for | z | <1, here, ζ is the Riemann zeta function. This series is derived from the corresponding Taylor series for the Hurwitz zeta function. This series may be used to derive a number of rational zeta series and these non-converging series can be used to get quickly an approximation value with a certain numeric at-least-precision for large arguments, ψ ∼ m +1 ∑ k =0 ∞
27.
Adrien-Marie Legendre
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Adrien-Marie Legendre was a French mathematician. Legendre made numerous contributions to mathematics, well-known and important concepts such as the Legendre polynomials and Legendre transformation are named after him. Adrien-Marie Legendre was born in Paris on 18 September 1752 to a wealthy family and he received his education at the Collège Mazarin in Paris, and defended his thesis in physics and mathematics in 1770. He taught at the École Militaire in Paris from 1775 to 1780, at the same time, he was associated with the Bureau des Longitudes. In 1782, the Berlin Academy awarded Legendre a prize for his treatise on projectiles in resistant media and this treatise also brought him to the attention of Lagrange. The Académie des Sciences made Legendre an adjoint member in 1783, in 1789 he was elected a Fellow of the Royal Society. He assisted with the Anglo-French Survey to calculate the distance between the Paris Observatory and the Royal Greenwich Observatory by means of trigonometry. To this end in 1787 he visited Dover and London together with Dominique, comte de Cassini, the three also visited William Herschel, the discoverer of the planet Uranus. Legendre lost his fortune in 1793 during the French Revolution. That year, he also married Marguerite-Claudine Couhin, who helped him put his affairs in order, in 1795 Legendre became one of six members of the mathematics section of the reconstituted Académie des Sciences, renamed the Institut National des Sciences et des Arts. Later, in 1803, Napoleon reorganized the Institut National, and his pension was partially reinstated with the change in government in 1828. In 1831 he was made an officer of the Légion dHonneur, Legendre died in Paris on 10 January 1833, after a long and painful illness, and Legendres widow carefully preserved his belongings to memorialize him. Upon her death in 1856, she was buried next to her husband in the village of Auteuil, where the couple had lived, Legendres name is one of the 72 names inscribed on the Eiffel Tower. Today, the term least squares method is used as a translation from the French méthode des moindres carrés. Around 1811 he named the gamma function and introduced the symbol Γ normalizing it to Γ = n, in 1830 he gave a proof of Fermats last theorem for exponent n =5, which was also proven by Lejeune Dirichlet in 1828. In number theory, he conjectured the quadratic reciprocity law, subsequently proved by Gauss, in connection to this and he also did pioneering work on the distribution of primes, and on the application of analysis to number theory. His 1798 conjecture of the prime number theorem was proved by Hadamard. He is known for the Legendre transformation, which is used to go from the Lagrangian to the Hamiltonian formulation of classical mechanics, in thermodynamics it is also used to obtain the enthalpy and the Helmholtz and Gibbs energies from the internal energy
28.
Thomas Joannes Stieltjes
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Thomas Joannes Stieltjes was a Dutch mathematician. He was born in Zwolle and died in Toulouse, France and he was a pioneer in the field of moment problems and contributed to the study of continued fractions. The Thomas Stieltjes Institute for Mathematics at the University of Leiden, dissolved in 2011, was named after him, Stieltjes was born in Zwolle on 29 December 1856. His father was a engineer and politician. Stieltjes Sr. was responsible for the construction of various harbours around Rotterdam, Stieltjes Jr. went to university at the Polytechnical School in Delft in 1873. Instead of attending lectures, he spent his student years reading the works of Gauss, there were 2 further failures, and his father despaired. His father was friends with H. G. van de Sande Bakhuyzen, soon afterwards, Stieltjes began a correspondence with Charles Hermite which lasted for the rest of his life. Stieltjes originally wrote to Hermite concerning celestial mechanics, but the subject turned to mathematics. The director of Leiden Observatory, van de Sande-Bakhuyzen, responded quickly to Stieltjes request on 1 January 1883 to stop his work to allow him to work more on mathematical topics. In 1883, he also married Elizabeth Intveld in May and she also encouraged him to move from astronomy to mathematics. And in September, Stieltjes was asked to substitute at University of Delft for F J van den Berg, from then until December of that year, he lectured on analytical geometry and on descriptive geometry. He resigned his post at the observatory at the end of that year, in 1884, Stieltjes applied for a chair in Groningen. He was initially accepted, but in the end turned down by the Department of Education, in 1884, Hermite and professor David Bierens de Haan arranged for an honorary doctorate to be granted to Stieltjes by Leiden University, enabling him to become a professor. In 1885, he was appointed as member of the Royal Dutch Academy of Sciences, in 1889, he was appointed professor of differential and integral calculus at Toulouse University. Stieltjes worked on almost all branches of analysis, continued fractions and number theory and his work is also seen as important as a first step towards the theory of Hilbert spaces. Other important contributions to mathematics that he made involved discontinuous functions and divergent series, differential equations, interpolation, Stieltjes work on continued fractions earned him the Ormoy Prize of the Académie des Sciences. Robertson, Edmund F. Thomas Joannes Stieltjes, MacTutor History of Mathematics archive, Thomas Joannes Stieltjes at the Mathematics Genealogy Project Œuvres complètes de Thomas Jan Stieltjes, pub. par les soins de la Société mathématique dAmsterdam
29.
Multiplicative inverse
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In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity,1. The multiplicative inverse of a fraction a/b is b/a, for the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth, the reciprocal function, the function f that maps x to 1/x, is one of the simplest examples of a function which is its own inverse. In the phrase multiplicative inverse, the qualifier multiplicative is often omitted, multiplicative inverses can be defined over many mathematical domains as well as numbers. In these cases it can happen that ab ≠ ba, then inverse typically implies that an element is both a left and right inverse. The notation f −1 is sometimes used for the inverse function of the function f. For example, the multiplicative inverse 1/ = −1 is the cosecant of x, only for linear maps are they strongly related. The terminology difference reciprocal versus inverse is not sufficient to make this distinction, since many authors prefer the opposite naming convention, in the real numbers, zero does not have a reciprocal because no real number multiplied by 0 produces 1. With the exception of zero, reciprocals of every real number are real, reciprocals of every rational number are rational, the property that every element other than zero has a multiplicative inverse is part of the definition of a field, of which these are all examples. On the other hand, no other than 1 and −1 has an integer reciprocal. In modular arithmetic, the multiplicative inverse of a is also defined. This multiplicative inverse exists if and only if a and n are coprime, for example, the inverse of 3 modulo 11 is 4 because 4 ·3 ≡1. The extended Euclidean algorithm may be used to compute it, the sedenions are an algebra in which every nonzero element has a multiplicative inverse, but which nonetheless has divisors of zero, i. e. nonzero elements x, y such that xy =0. A square matrix has an inverse if and only if its determinant has an inverse in the coefficient ring, the linear map that has the matrix A−1 with respect to some base is then the reciprocal function of the map having A as matrix in the same base. Thus, the two notions of the inverse of a function are strongly related in this case, while they must be carefully distinguished in the general case. A ring in which every element has a multiplicative inverse is a division ring. As mentioned above, the reciprocal of every complex number z = a + bi is complex. In particular, if ||z||=1, then 1 / z = z ¯, consequently, the imaginary units, ±i, have additive inverse equal to multiplicative inverse, and are the only complex numbers with this property
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Probability
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Probability is the measure of the likelihood that an event will occur. Probability is quantified as a number between 0 and 1, the higher the probability of an event, the more certain that the event will occur. A simple example is the tossing of a fair coin, since the coin is unbiased, the two outcomes are both equally probable, the probability of head equals the probability of tail. Since no other outcomes are possible, the probability is 1/2 and this type of probability is also called a priori probability. Probability theory is used to describe the underlying mechanics and regularities of complex systems. For example, tossing a coin twice will yield head-head, head-tail, tail-head. The probability of getting an outcome of head-head is 1 out of 4 outcomes or 1/4 or 0.25 and this interpretation considers probability to be the relative frequency in the long run of outcomes. A modification of this is propensity probability, which interprets probability as the tendency of some experiment to yield a certain outcome, subjectivists assign numbers per subjective probability, i. e. as a degree of belief. The degree of belief has been interpreted as, the price at which you would buy or sell a bet that pays 1 unit of utility if E,0 if not E. The most popular version of subjective probability is Bayesian probability, which includes expert knowledge as well as data to produce probabilities. The expert knowledge is represented by some prior probability distribution and these data are incorporated in a likelihood function. The product of the prior and the likelihood, normalized, results in a probability distribution that incorporates all the information known to date. The scientific study of probability is a development of mathematics. Gambling shows that there has been an interest in quantifying the ideas of probability for millennia, there are reasons of course, for the slow development of the mathematics of probability. Whereas games of chance provided the impetus for the study of probability. According to Richard Jeffrey, Before the middle of the century, the term probable meant approvable. A probable action or opinion was one such as people would undertake or hold. However, in legal contexts especially, probable could also apply to propositions for which there was good evidence, the sixteenth century Italian polymath Gerolamo Cardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes
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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
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Coprime integers
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In number theory, two integers a and b are said to be relatively prime, mutually prime, or coprime if the only positive integer that divides both of them is 1. That is, the common positive factor of the two numbers is 1. This is equivalent to their greatest common divisor being 1, the numerator and denominator of a reduced fraction are coprime. In addition to gcd =1 and =1, the notation a ⊥ b is used to indicate that a and b are relatively prime. For example,14 and 15 are coprime, being divisible by only 1. The numbers 1 and −1 are the only integers coprime to every integer, a fast way to determine whether two numbers are coprime is given by the Euclidean algorithm. The number of integers coprime to an integer n, between 1 and n, is given by Eulers totient function φ. A set of integers can also be called if its elements share no common positive factor except 1. A set of integers is said to be pairwise coprime if a and b are coprime for every pair of different integers in it, a number of conditions are individually equivalent to a and b being coprime, No prime number divides both a and b. There exist integers x and y such that ax + by =1, the integer b has a multiplicative inverse modulo a, there exists an integer y such that by ≡1. In other words, b is a unit in the ring Z/aZ of integers modulo a, the least common multiple of a and b is equal to their product ab, i. e. LCM = ab. As a consequence of the point, if a and b are coprime and br ≡ bs. That is, we may divide by b when working modulo a, as a consequence of the first point, if a and b are coprime, then so are any powers ak and bl. If a and b are coprime and a divides the product bc and this can be viewed as a generalization of Euclids lemma. In a sense that can be made precise, the probability that two randomly chosen integers are coprime is 6/π2, which is about 61%, two natural numbers a and b are coprime if and only if the numbers 2a −1 and 2b −1 are coprime. As a generalization of this, following easily from the Euclidean algorithm in base n >1, a set of integers S = can also be called coprime or setwise coprime if the greatest common divisor of all the elements of the set is 1. For example, the integers 6,10,15 are coprime because 1 is the positive integer that divides all of them. If every pair in a set of integers is coprime, then the set is said to be pairwise coprime, pairwise coprimality is a stronger condition than setwise coprimality, every pairwise coprime finite set is also setwise coprime, but the reverse is not true
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Wadim Zudilin
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Wadim Zudilin is a Russian number theorist who is active in studying hypergeometric functions and zeta constants. He studied under Yuri V. Nesterenko and worked at Moscow State University, the Steklov Institute of Mathematics and he now works at the University of Newcastle, Australia. He has reproved Apérys theorem and expanded it and he proved that at least one of the four numbers ζ, ζ, ζ, or ζ is irrational. For that accomplishment he won the Distinguished Award of the Hardy-Ramanujan Society in 2001, Wadim Zudilins homepage Wadim Zudilins research profile Wadim Zudilins list of published works Wadim Zudilin at the Mathematics Genealogy Project